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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 146137 10 pageshttpdxdoiorg1011552013146137
Research ArticleExact Asymptotic Stability Analysis and Region-of-AttractionEstimation for Nonlinear Systems
Min Wu1 Zhengfeng Yang1 and Wang Lin2
1 Shanghai Key Laboratory of Trustworthy Computing East China Normal University Shanghai 200062 China2 College of Mathematics and Information Science Wenzhou University Wenzhou Zhejiang 325035 China
Correspondence should be addressed to Wang Lin linwangwzueducn
Received 14 November 2012 Revised 11 February 2013 Accepted 20 February 2013
Academic Editor Fabio M Camilli
Copyright copy 2013 Min Wu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We address the problem of asymptotic stability and region-of-attraction analysis of nonlinear dynamical systems A hybridsymbolic-numeric method is presented to compute exact Lyapunov functions and exact estimates of regions of attraction ofnonlinear systems efficiently A numerical Lyapunov function and an estimate of region of attraction can be obtained by solvingan (bilinear) SOS programming via BMI solver then the modified Newton refinement and rational vector recovery techniques areapplied to obtain exact Lyapunov functions and verified estimates of regions of attraction with rational coefficients Experimentson some benchmarks are given to illustrate the efficiency of our algorithm
1 Introduction
Lyapunovrsquos stability theory which concern is with the behav-ior of trajectories of differential systems plays an impor-tant role in analysis and synthesis of nonlinear continuoussystems Lyapunov functions can be used to verify theasymptotic stability including locally asymptotic stability andglobally asymptotic stability In the literature there has beenlots of work on constructing Lyapunov functions [1ndash12]For example [4 5] used the linear matrix inequality (LMI)method to compute quadratic Lyapunov functions In [8 13]based on sum of squares (SOS) decomposition a methodis proposed to construct high-degree numerical polynomialLyapunov functions Reference [9] proposed a new methodfor computing polynomials Lyapunov functions by solvingsemialgebraic constraints via the tool DISCOVERER [14]Reference [15] constructed Lyapunov functions beyond poly-nomials by using radial basis functions In [11] the Grobnerbased method is used to choose the parameters in Lyapunovfunctions in an optimal way
Since the analysis of asymptotic stability is not sufficientfor safety critical systems the analysis of the region ofattraction (ROA) of an asymptotically stable equilibrium
point is a topic of significant importance The ROA is theset of initial states from which the system converges tothe equilibrium point Computing exact regions of attrac-tion (ROAs) for nonlinear dynamical systems is very hardif not impossible therefore researchers have focused onfinding estimates of the actual ROAs There are many well-established techniques for computation of ROAs [5 16ndash22]Among all methods those based on Lyapunov functionsare dominant in the literature These methods computeboth a Lyapunov function as a stability certificate and thesublevel sets of this Lyapunov function which give riseto estimates of the ROA In [17] the authors computedquadratic Lyapunov functions to optimize the volume of anellipsoid contained inROAby using nonlinear programmingReference [5] employed LMI based method to computethe optimal quadratic Lyapunov function for maximizingthe volume of an ellipsoid contained in the ROA of oddpolynomial systems In [22] based on SOS decompositiona method is presented to search for polynomial Lyapunovfunctions that enlarge a provable ROA of nonlinear polyno-mial system Reference [18] used discretization (in time) toflow invariant sets backwards along the flow of the vectorfield obtaining larger and larger estimates for the ROA
2 Abstract and Applied Analysis
Reference [21] employed quantifier elimination (QE)methodvia QEPCAD to find Lyapunov functions for estimating theROA
Taking advantage of the efficiency of numerical compu-tation and the error-free property of symbolic computationin this paper we present a hybrid symbolic-numeric algo-rithm to compute exact Lyapunov functions for asymptoticstability and verified estimates of the ROAs of continuoussystems The algorithm is based on SOS relaxation [23] ofa parametric polynomial optimization problem with LMIor bilinear matrix inequality (BMI) constraints which canbe solved directly using LMI or BMI solver in MAT-LAB such as SOSTOOLS [24] YALMIP [25] SeDuMi[26] PENBMI [27] and exact SOS representation recoverytechniques presented in [28 29] Unlike the numericalapproaches our method can yield exact Lyapunov functionsand verified estimates of ROAs which can overcome theunsoundness in analysis of nonlinear systems caused bynumerical errors [30] In comparison with some symbolicapproaches based on qualifier elimination technique ourapproach is more efficient and practical because parametricpolynomial optimization problem based on SOS relaxationwith fixed size can be solved in polynomial time theoreti-cally
The rest of the paper is organized as follows In Section 2we introduce some definitions and notions about Lyapunovstability and ROA Section 3 is devoted to transform theproblem of computing Lyapunov functions and estimatesof ROAs into a parametric program with LMI or BMIconstraints In Section 4 a symbolic-numeric approach viaSOS relaxation and exact rational vector recovery is proposedto compute Lyapunov functions and estimates of ROAsand an algorithm is described In Section 5 experiments onsome benchmarks are shown to illustrate our algorithm onasymptotic stability and ROA analysis At last we concludeour results in Section 6
2 Lyapunov Stability and Region of Attraction
In this section we will present the notion of Lyapunov stabil-ity and regions of attraction (ROAs) of dynamical systems
Consider the autonomous system
x = f (x) (1)
where f R119899 rarr R119899 is continuous and f satisfies theLipschitz condition Denote by 120601(119905 x
0) the solution of (1)
corresponding to the initial state x0= x(0) evaluated at time
119905 gt 0A vector x isin R119899 is an equilibrium point of the system (1)
if f(x) = 0 Since any equilibrium point can be shifted to theorigin 0 via a change of variables without loss of generalitywe may always assume that the equilibrium point of interestoccurs at the origin
Lyapunov theory is concerned with the behavior of thesolution 120601(119905 x
0) where the initial state x
0is not at the
equilibrium 0 but is ldquocloserdquo to it
Definition 1 (Lyapunov stability) The equilibrium point 0 of(1) is
(i) stable if for any 120598 gt 0 there exists 120575 = 120575(120598) such that1003817
1003817
1003817
1003817
x0
1003817
1003817
1003817
1003817
lt 120575 997904rArr
1003817
1003817
1003817
1003817
120601 (119905 x0)
1003817
1003817
1003817
1003817
lt 120598 forall119905 gt 0 (2)here sdot denotes any norm defined on R119899
(ii) unstable if it is not stable(iii) asymptotically stable if it is stable and 120575 can be chosen
such that1003817
1003817
1003817
1003817
x0
1003817
1003817
1003817
1003817
lt 120575 997904rArr lim119905rarrinfin
120601 (119905 x0) = 0 (3)
(iv) globally asymptotically stable if it is stable and for allx0isin R119899
lim119905rarrinfin
120601 (119905 x0) = 0 (4)
Intuitively the equilibrium point 0 is stable if all solutionsstarting near the origin (meaning that the initial conditionsare in a neighborhood of the origin) remain near theorigin for all time The equilibrium point 0 is asymptoticallystable if all solutions starting at nearby points not only staynearby but also converge to the equilibrium point as timeapproaches infinity And the equilibrium point 0 is globallyasymptotically stable if it is asymptotically stable for all initialconditions x
0isin R119899 The stability is an important property
in practice because it means arbitrarily small perturbationsof the initial state about the equilibrium point 0 result inarbitrarily small perturbations in the corresponding solutiontrajectories of (1)
A sufficient condition for stability of the zero equilibriumis the existence of a Lyapunov function as shown in thefollowing theorem
Theorem 2 ([31 Theorem 41]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R such that
119881 (0) = 0 119881 (x) gt 0 in 119863 0 (5)
119881 (x) = 120597119881
120597xsdot f (x) le 0 in 119863
(6)
then the origin is stable Moreover if
119881 (x) lt 0 in 119863 0 (7)then the origin is asymptotically stable
A function 119881(x) satisfying the conditions (5) and (6) inTheorem 2 is commonly known as a Lyapunov function Andwe can verify globally asymptotic stability of system (1) byusing Lyapunov functions stated as follows
Theorem 3 ([31 Theorem 42]) Let the origin be an equilib-rium point for (1) If there exists a continuously differentiablefunction 119881 R119899 rarr R such that
119881 (0) = 0 119881 (x) gt 0 forallx = 0 (8)x 997888rarr infin 997904rArr 119881(x) 997888rarr infin (9)
119881 (x) lt 0 forallx = 0 (10)then the origin is globally asymptotically stable
Abstract and Applied Analysis 3
Remark that a function satisfying condition (9) is said tobe radially unbounded
It is known that globally asymptotic stability is verydesirable but in many applications it is difficult to achieveWhen the equilibrium point is asymptotically stable we areinterested in determining how far from the origin the trajec-tory can be and still converge to the origin as 119905 approachesinfinThis gives rise to the following definition
Definition 4 (region of attraction) The region of attraction(ROA) Ω of the equilibrium point 0 is defined as Ω = x isin
R119899 | lim119905rarrinfin
120601(119905 x) = 0
The ROA of the equilibrium point 0 is a collection ofall points x such that any trajectory starting at initial state xat time 0 will be attracted to the equilibrium point In theliterature the terms ldquodomain of attractionrdquo and ldquoattractionbasinrdquo are also used instead of ldquoregion of attractionrdquo
Definition 5 (positively invariant set) A set119872 sub R119899 is calleda positively invariant set of the system (1) if x
0isin 119872 implies
120601(119905 x0) isin 119872 for all 119905 ge 0 Namely if a solution belongs to a
positively invariant set119872 at some time instant then it belongsto119872 for all future time
In general finding the exact ROA analytically might bedifficult or even impossible Usually Lyapunov functions areused to find underestimates of the ROA that is sets containedin the region of attraction as stated in the following theorem
Theorem 6 ([20 Theorem 10]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R satisfying(5) and (7) then any region Ω
119888= x isin R119899 | 119881(x) le 119888 with
119888 ge 0 such that Ω119888sube 119863 is a positively invariant set contained
in the ROA of the equilibrium 0
Theorem 6 describes an approach to compute estimatesof the ROA through Lyapunov functionsMore specifically inthe case of asymptotic stability ifΩ
119888= x isin R119899 | 119881(x) le 119888 is
bounded and contained in119863 then every trajectory starting inΩ
119888remains inΩ
119888and approaches the origin as 119905 rarr infinThus
Ω
119888is an estimate of the ROA Remark that when the origin is
globally asymptotically stable the region of attraction is thewhole space R119899
3 Problem Reformulation
In this section we will transform the problem of the asymp-totic stability and ROA analysis of system (1) to a parametricprogram with LMI or BMI constraints In the sequel wesuppose that system (1) is a polynomial dynamical systemwith 119891
119894(x) isin R[x] for 1 le 119894 le 119899 where R[x] denotes the
ring of real polynomials in the variables x
31 Asymptotic Stability Firstly we consider the asymptoticstability of system (1) As shown in Theorem 2 the existenceof a Lyapunov function 119881(x) which satisfies the conditions(5) and (7) is a certificate for asymptotical stability of the
equilibrium point 0 and the problem of computing such a119881(x) can be transformed into the following problem
find 119881 (x) isin R [x]
st 119881 (0) = 0
119881 (x) gt 0 forallx isin 119863 0
119881 (x) = 120597119881
120597xf (x) lt 0 forallx isin 119863 0
(11)
In general 119863 can be an arbitrary neighborhood of theequilibrium point 0 However to simplify calculation inpractice we can assume 119863 = x isin R119899 119892(x) le 0 where119892(x) isin R[x] is chosen to be 119892(x) = sum
119899
119894=1119909
2
119894minus 119903
2 for 1 le 119894 le 119899with a given constant 119903 isin R
+ for instance 119903 = 10
minus2Let us first predetermine a template of Lyapunov func-
tions with the given degree 119889 We assume that
119881 (x) = sum
120572
119888
120572x120572 (12)
where x120572 = 119909
1205721
1 119909
120572119899
119899 120572 = (120572
1 120572
119899) isin Z119899
ge0with
sum
119899
119894=1120572
119894le 119889 and 119888
120572isin R are parameters We can rewrite
119881(x) = c119879 sdot 119879(x) where 119879(x) is the (column) vector of allterms in 119909
1 119909
119899with total degree le 119889 and c isin R] with
] = (
119899+119889
119899) is the coefficient vector of 119881(x) In the sequel we
write 119881(x) as 119881(x c) for clarityFor a polynomial 120593(x) isin R[x] we say that 120593(x) is positive
definite (resp positive semidefinite) if 120593(x) gt 0 for all x isin
R119899 0 (resp 120593(x) ge 0 for all x isin R119899) Observe that if 120593(x)is a sum of squares (SOS) then 120593(x) is globally nonnegativeAnd to ensure positive definiteness of 120593(x) we can use apolynomial 119897(x) of the form
119897 (x) =119899
sum
119894=1
120598
119894119909
119896
119894 (13)
where 120598
119894isin R+and 119896 is assumed to be even Clearly the
condition that 120593(x) minus 119897(x) is an SOS polynomial guaranteesthe positive definiteness of 120593(x) Therefore to ensure positivedefiniteness of the second and third constraints in (11)we can use two polynomials 119897
1(x) 1198972(x) of the form of
(13) It is notable that SOS programming can be applied todetermine the nonnegativity of a multivariate polynomialover a semialgebraic set Consider the problem of verifyingwhether the implication
119898
⋀
119894=1
(119901
119894(x) ge 0) 997904rArr 119902 (x) ge 0 (14)
holds where 119901
119894(x) isin R[x] for 1 le 119894 le 119898 and 119902(x) isin
R[x] According to Stenglersquos Positivstellensatz SchmudgenrsquosPositivstellensatz or Putinarrsquos Positivstellensatz [32] if thereexist SOS polynomials 120590
119894isin R[x] for 119894 = 0 119898 such that
119902 (x) = 120590
0(x) +
119898
sum
119894=1
120590
119894(x) 119901119894(x) (15)
4 Abstract and Applied Analysis
then the assertion (14) holds Therefore the existence of SOSrepresentations provides a sufficient condition for determin-ing the nonnegativity of 119902(x) over x isin R119899 and119898
119894=1119901
119894(x) ge 0
Based on the above observation the problem (11) can befurther transformed into the following SOS programming
find c isin R]
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x) minus 120590
1(x) 119892 (x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x) minus 120590
2(x) 119892 (x)
(16)
where 120590119895(x) and 120588
119895(x) are SOS inR[x] for 119895 = 1 2 Moreover
the degree bound of those unknown SOS polynomials 120590119895 120588119895
is exponential in 119899 deg(119881) deg(f) and deg(119892) In practiceto avoid high computational complexity we simply set upa truncated SOS programming by fixing a priori (muchsmaller) degree bound 2119890 with 119890 isin Z
+ of the unknown
SOS polynomials Consequently the existence of a solutionc of (16) can guarantee the asymptotic stability of the givensystem
Similarly the problem of computing the Lyapunov func-tion 119881(x) for globally asymptotic stability of system (1) thatsatisfies the conditions in Theorem 3 can be rewritten as thefollowing problem
find 119881 (x) isin R [x]
st 119881 (0) = 0
119881 (x) gt 0 forallx = 0
x 997888rarr infin 997904rArr 119881(x) 997888rarr infin
119881 (x) lt 0 forallx = 0
(17)
By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form
(13) the condition that 119881(x c) minus 119897
1(x) is SOS guarantees
the radially unboundedness of 119881(x c) Consequently theproblem (17) can be further transformed into the followingSOS programming
find c isin R]
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x)
(18)
where 120588119895(x) are SOS polynomials in R[x] for 119895 = 1 2 The
decision variables are the coefficients of all the polynomialsappearing in (18) such as 119881(x c) 119897
119894(x) and 120588
119895(x)
32 Region of Attraction Suppose that the equilibrium point0 is asymptotically stable In this section we will considerhow to find a large enough underestimate of the ROA In thecase where the equilibrium point 0 is globally asymptoticallystable the ROA becomes the whole space R119899
Our idea of computing the estimate of ROA is similar tothat of the algorithm in [20 Section 422] Suppose that119863 isa semialgebraic set
x isin R119899
| 119881 (x) le 1 (19)
given by a Lyapunov function 119881(x) In order to enlarge thecomputed positively invariant set contained in the ROA wedefine a variable sized region
119875
120573= x isin R
119899
| 119901 (x) le 120573 (20)
where119901(x) isin R[x] is a fixed and positive definite polynomialand maximize 120573 subject to the constraint 119875
120573sube 119863
Fix a template of 119881 of the form (12) The problem offinding an estimate 119875
120573of the ROA can be transformed into
the following polynomial optimization problem
maxcisinR]
120573
st 119881 (0 c) = 0
119881 (x c) gt 0 forallx isin 119863 0
119881 (x c) = 120597119881
120597xf (x) lt 0 forallx isin 119863 0
119901 (x) le 120573 ⊨ 119881 (x c) le 1
(21)
By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form (13)
and based on SOS relaxation the problem (21) can be furthertransformed into the following SOS programming
maxcisinR]
120573
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x) minus 120590
1(x) (119881 (x c) minus 1)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x) minus 120590
2(x) (119881 (x c) minus 1)
1 minus 119881 (x c) = 120588
3(x) minus 120590
3(x) (119901 (x) minus 120573)
(22)
where 120590
119894(x) and 120588
119894(x) are SOS in R[x] for 1 le 119894 le 3
In (22) the decision variables are 120573 and the coefficients ofall the polynomials appearing in (22) such as 119881(x c) 120590
119894(x)
and 120588
119894(x) Since 120573 and the coefficients of 119881(x c) and 120590
119894(x)
are unknown some nonlinear terms that are products ofthese coefficients will occur in the second third and fourthconstraints of (22) which yields a nonconvex bilinear matrixinequalities (BMI) problemWe will discuss in Section 4 howto handle the SOS programming (22) directly using the BMIsolver or iterative method
4 Exact Certificate of Sum ofSquares Decomposition
According to Theorems 2 3 and 6 the key of asymptoticstability analysis and ROA estimation lies in finding a real
Abstract and Applied Analysis 5
function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573
41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method
Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that
120595 (x) = 119885(x)119879119876119885 (x) (23)
where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem
inf119882[119895]119876[119895]
2
sum
119895=1
Trace (119882[119895] + 119876
[119895]
)
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus 119901
1(x)119879 sdot 119876[1] sdot 119901
1(x) 119892 (x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
minus 119901
2(x)119879 sdot 119876[2] sdot 119901
2(x) 119892 (x)
(24)
where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2
119895=1Trace(119882[119895] + 119876
[119895]
) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function
Similarly the SOS programming (18) can be rewritten asthe following SDP problem
inf119882[1]119882[2]
Trace (119882[1]) + Trace (119882[2])
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
(25)
where matrices 119882
[1] 119882
[2] are symmetric and positivesemidefinite
Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently
Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem
Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and
minus119909
2
+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat
2119909
119889120593
119889119909
= 120601
0(119909) + 120601
1(119909) (1 minus 119909
2
) + 120601
2(119909) 120593 (119909)
(26)
where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1
deg(1206010) = 2 and deg(120601
1) = deg(120601
2) = 0 and that 120593(119909) = 119906
0+
119906
1119909 1206011= 119906
2and 1206012= V1 with 119906
0 119906
1 119906
2 V1isin R parameters
From (1) we have
120601
0(119909) = 119906
2119909
2
+ (2119906
1minus 119906
1V1) 119909 minus 119906
2minus 119906
0V1 (27)
whose square matrix representation (SMR) [33] is 1206010(119909) =
119885
119879
119876119885 where
119876 =
[
[
[
minus119906
2minus 119906
0V1
119906
1minus
1
2
119906
1V1
119906
1minus
1
2
119906
1V1
119906
2
]
]
]
119885 = [
1
119909
] (28)
Since all the 120601119894(119909) are SOSes we have 119906
2ge 0 V
1ge 0 and
119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint
B (119906
0 119906
1 119906
2 V1) = 119906
1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 1
0 0 1 0
]
]
]
]
+ 119906
2
[
[
[
[
1 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 1
]
]
]
]
+ V1
[
[
[
[
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
]
]
]
]
+ 119906
0V1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 0
]
]
]
]
+ 119906
1V1
[
[
[
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 minus
1
2
0 0 minus
1
2
0
]
]
]
]
]
]
]
⪰ 0
(29)
Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form
inf minus120573
st B (u k) = 119860
0+
119898
sum
119894=1
119906
119894119860
119894+
119896
sum
119895=1
V119895119860
119898+119895
+ sum
1le119894le119898
sum
1le119895le119896
119906
119894V119895119861
119894119895⪰ 0
(30)
where 119860
119894 119861
119894119895are constant symmetric matrices u =
(119906
1 119906
119898) k = (V
1 V
119896) are parameter coefficients of the
SOSpolynomials occurring in the original SOSprogramming(22)
Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI
6 Abstract and Applied Analysis
solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ
119901that penalizes the inequality constraints This
function satisfies a number of properties [35] such that forany 119901 gt 0 we have
B (u k) ⪰ 0 lArrrArr Φ
119901(B (u k)) ⪰ 0 (31)
This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem
inf minus120573
st Φ
119901(B (u k)) ⪰ 0
(32)
The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)
119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ
119901(B (u k))) (33)
where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]
Alternatively observing (30) B(u k) involves no cross-ing products like 119906
119894119906
119895and V
119894V119895 Taking this special form
into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice
42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints
Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem
find c isin R]
st 119881
1(x c) = m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) = m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
119882
[119894]
⪰ 0 119894 = 1 2 3
(34)
where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3
are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the
numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is
119881
1(x c) asymp m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) asymp m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
(35)
as illustrated by the following example
Example 8 Consider the following nonlinear system
[
1
2
] =
[
[
[
[
[
[
[
[
[
minus05119909
1+ 119909
2+ 024999119909
2
1
+ 0083125119909
3
1+ 00205295119909
4
1
+ 00046875119909
5
1+ 00015191119909
6
1
minus119909
2+ 119909
1119909
2minus 049991119909
3
1+ 0040947119909
5
1
]
]
]
]
]
]
]
]
]
(36)
We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2 When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 3112937368119909
2
1+ 3112937368119909
2
2
120573 = 032124
(37)
However119881(1199091 119909
2) and120573 cannot satisfy the conditions in (21)
exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore
Ω
119881= (119909
1 119909
2) isin R2
| 119881 (119909
1 119909
2) le 1 (38)
is not an estimate of the ROA of this system
In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows
We first convert 119882[3] to a nearby rational positive semi-definite matrix
119882
[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882
[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix
119882
[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882
[2]
simultaneously with respect to a given tolerance 120591
Abstract and Applied Analysis 7
Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and
119882
[2] that satisfy exactly
119881
1(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(39)
Since the equations in (39) are affine in entries of c and
119882
1
119882
2 one can define an affine linear hyperplane
L = c119882[1]119882[2] | 1198811(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(40)
Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882
[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices
119882
[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le
119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]
43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)
Algorithm 9 Verified Parametric Optimization SolverInput
(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z
gt0 the bound of the common denominator
(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to
construct the SOS programming(iv) 120591 isin R
gt0 the given tolerance
Output
(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3
positive semidefinite
(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882
[119894]
⪸ 0 1 le 119894 le 3
(2) (Compute the verified solution c)
(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix
119882
[3] by non-negativetruncated PLDLTPT-decomposition
(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]
(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices
119882
[1]
119882
[2] can beobtained by orthogonal projection if 119882[1]119882
[2] are of full rank or by rational vectorrecovery method otherwise
(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and
119882
[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo
5 Experiments
Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems
Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system
x = f (x) =
[
[
[
[
[
[
[
[
1
2
3
4
5
6
]
]
]
]
]
]
]
]
=
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119909
3
1+ 4119909
3
2minus 6119909
3119909
4
minus119909
1minus 119909
2+ 119909
3
5
119909
1119909
4minus 119909
3+ 119909
4119909
6
119909
1119909
3+ 119909
3119909
6minus 119909
3
4
minus2119909
3
2minus 119909
5+ 119909
6
minus3119909
3119909
4minus 119909
3
5minus 119909
6
]
]
]
]
]
]
]
]
]
]
]
]
]
(41)
Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909
2
1+119909
2
2+119909
2
3minus00001 When 119889 = 4 we obtain a feasible
solution of the associated SDP system Here we just list oneapproximate polynomial
119881 (x) = 081569119909
2
1+ 018066119909
2
2+ 16479119909
2
3+ 21429119909
2
4
+ 12996119909
2
6+ sdot sdot sdot + 067105119909
4
4+ 052855119909
4
5
(42)
Let the tolerance 120591 = 10
minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and
8 Abstract and Applied Analysis
the corresponding SOS polynomials Here we only list theLyapunov function
119881 (x) = 41
50
119909
2
1+
9
50
119909
2
2+
33
20
119909
2
3+
107
50
119909
2
4
+
69
100
119909
2
5+
13
10
119909
2
6+ sdot sdot sdot +
67
100
119909
4
4+
53
100
119909
4
5
(43)
Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic
stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain
119881 (x) = 078997119909
2
1+ 019119909
2
2+ 22721119909
2
3
+ 14213119909
2
6+ sdot sdot sdot + 14213119909
4
2+ 071066119909
4
5
(44)
By use of the rational SOS recovery technique described inSection 42 we then obtain
119881 (x) = 79
100
119909
2
1+
19
50
119909
1119909
2+
9
100
119909
1119909
5+
19
100
119909
2
2+
227
100
119909
2
3
+
199
100
119909
2
4+
9
100
119909
2119909
5+
71
100
119909
2
5
+
71
50
119909
2
6+
71
50
119909
4
2+
71
100
119909
4
5
(45)
which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied
Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10
minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds
Example 11 ([41 Example A]) Consider a nonlinear contin-uous system
x = f (x) = [
1
2
] = [
minus119909
2
119909
1+ (119909
2
1minus 1) 119909
2
] (46)
Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909
1 119909
2)
with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909
2
1+ 119909
2
2minus 00001 When 119889 = 2 we obtain
119881 (x) = 14957119909
2
1minus 07279119909
1119909
2+ 11418119909
2
2 (47)
Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)
Example 119899 Num deg(119881) Time(s)
1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830
Table 2 Algorithm performance on benchmarks (ROA)
Example119899 Num deg(119881)
120573 120573
Time(s)
1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915
Let the tolerance 120591 = 10
minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function
119881 (x) = minus
62
85
119909
1119909
2+
127
85
119909
2
1+
97
85
119909
2
2 (48)
Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified
estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2
When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 06174455119909
2
1minus 040292119909
1119909
2+ 043078119909
2
2
120573 = 13402225
(49)
Let the tolerance 120591 = 10
minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10
5By use of the rational SOS recovery technique described inSection 42 we obtain
119881 (119909
1 119909
2) =
6732
10903
119909
2
1minus
4393
10903
119909
1119909
2+
42271
10903
119909
2
2
120573 =
6701
5000
(50)
which exactly satisfy the conditions inTheorem 6Therefore
Ω
= (119909
1 119909
2) isin R2
|
119881 (119909
1 119909
2) le 1 (51)
is a certified estimate of the ROA of the given system
Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite
Abstract and Applied Analysis 9
polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =
10
minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and
120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds
6 Conclusion
In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system
Acknowledgments
This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)
References
[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997
[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000
[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000
[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006
[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005
[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012
[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003
[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002
[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009
[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009
[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991
[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011
[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004
[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007
[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008
[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989
[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971
[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001
[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985
[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003
[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002
[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006
[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
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MathematicsJournal of
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
Reference [21] employed quantifier elimination (QE)methodvia QEPCAD to find Lyapunov functions for estimating theROA
Taking advantage of the efficiency of numerical compu-tation and the error-free property of symbolic computationin this paper we present a hybrid symbolic-numeric algo-rithm to compute exact Lyapunov functions for asymptoticstability and verified estimates of the ROAs of continuoussystems The algorithm is based on SOS relaxation [23] ofa parametric polynomial optimization problem with LMIor bilinear matrix inequality (BMI) constraints which canbe solved directly using LMI or BMI solver in MAT-LAB such as SOSTOOLS [24] YALMIP [25] SeDuMi[26] PENBMI [27] and exact SOS representation recoverytechniques presented in [28 29] Unlike the numericalapproaches our method can yield exact Lyapunov functionsand verified estimates of ROAs which can overcome theunsoundness in analysis of nonlinear systems caused bynumerical errors [30] In comparison with some symbolicapproaches based on qualifier elimination technique ourapproach is more efficient and practical because parametricpolynomial optimization problem based on SOS relaxationwith fixed size can be solved in polynomial time theoreti-cally
The rest of the paper is organized as follows In Section 2we introduce some definitions and notions about Lyapunovstability and ROA Section 3 is devoted to transform theproblem of computing Lyapunov functions and estimatesof ROAs into a parametric program with LMI or BMIconstraints In Section 4 a symbolic-numeric approach viaSOS relaxation and exact rational vector recovery is proposedto compute Lyapunov functions and estimates of ROAsand an algorithm is described In Section 5 experiments onsome benchmarks are shown to illustrate our algorithm onasymptotic stability and ROA analysis At last we concludeour results in Section 6
2 Lyapunov Stability and Region of Attraction
In this section we will present the notion of Lyapunov stabil-ity and regions of attraction (ROAs) of dynamical systems
Consider the autonomous system
x = f (x) (1)
where f R119899 rarr R119899 is continuous and f satisfies theLipschitz condition Denote by 120601(119905 x
0) the solution of (1)
corresponding to the initial state x0= x(0) evaluated at time
119905 gt 0A vector x isin R119899 is an equilibrium point of the system (1)
if f(x) = 0 Since any equilibrium point can be shifted to theorigin 0 via a change of variables without loss of generalitywe may always assume that the equilibrium point of interestoccurs at the origin
Lyapunov theory is concerned with the behavior of thesolution 120601(119905 x
0) where the initial state x
0is not at the
equilibrium 0 but is ldquocloserdquo to it
Definition 1 (Lyapunov stability) The equilibrium point 0 of(1) is
(i) stable if for any 120598 gt 0 there exists 120575 = 120575(120598) such that1003817
1003817
1003817
1003817
x0
1003817
1003817
1003817
1003817
lt 120575 997904rArr
1003817
1003817
1003817
1003817
120601 (119905 x0)
1003817
1003817
1003817
1003817
lt 120598 forall119905 gt 0 (2)here sdot denotes any norm defined on R119899
(ii) unstable if it is not stable(iii) asymptotically stable if it is stable and 120575 can be chosen
such that1003817
1003817
1003817
1003817
x0
1003817
1003817
1003817
1003817
lt 120575 997904rArr lim119905rarrinfin
120601 (119905 x0) = 0 (3)
(iv) globally asymptotically stable if it is stable and for allx0isin R119899
lim119905rarrinfin
120601 (119905 x0) = 0 (4)
Intuitively the equilibrium point 0 is stable if all solutionsstarting near the origin (meaning that the initial conditionsare in a neighborhood of the origin) remain near theorigin for all time The equilibrium point 0 is asymptoticallystable if all solutions starting at nearby points not only staynearby but also converge to the equilibrium point as timeapproaches infinity And the equilibrium point 0 is globallyasymptotically stable if it is asymptotically stable for all initialconditions x
0isin R119899 The stability is an important property
in practice because it means arbitrarily small perturbationsof the initial state about the equilibrium point 0 result inarbitrarily small perturbations in the corresponding solutiontrajectories of (1)
A sufficient condition for stability of the zero equilibriumis the existence of a Lyapunov function as shown in thefollowing theorem
Theorem 2 ([31 Theorem 41]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R such that
119881 (0) = 0 119881 (x) gt 0 in 119863 0 (5)
119881 (x) = 120597119881
120597xsdot f (x) le 0 in 119863
(6)
then the origin is stable Moreover if
119881 (x) lt 0 in 119863 0 (7)then the origin is asymptotically stable
A function 119881(x) satisfying the conditions (5) and (6) inTheorem 2 is commonly known as a Lyapunov function Andwe can verify globally asymptotic stability of system (1) byusing Lyapunov functions stated as follows
Theorem 3 ([31 Theorem 42]) Let the origin be an equilib-rium point for (1) If there exists a continuously differentiablefunction 119881 R119899 rarr R such that
119881 (0) = 0 119881 (x) gt 0 forallx = 0 (8)x 997888rarr infin 997904rArr 119881(x) 997888rarr infin (9)
119881 (x) lt 0 forallx = 0 (10)then the origin is globally asymptotically stable
Abstract and Applied Analysis 3
Remark that a function satisfying condition (9) is said tobe radially unbounded
It is known that globally asymptotic stability is verydesirable but in many applications it is difficult to achieveWhen the equilibrium point is asymptotically stable we areinterested in determining how far from the origin the trajec-tory can be and still converge to the origin as 119905 approachesinfinThis gives rise to the following definition
Definition 4 (region of attraction) The region of attraction(ROA) Ω of the equilibrium point 0 is defined as Ω = x isin
R119899 | lim119905rarrinfin
120601(119905 x) = 0
The ROA of the equilibrium point 0 is a collection ofall points x such that any trajectory starting at initial state xat time 0 will be attracted to the equilibrium point In theliterature the terms ldquodomain of attractionrdquo and ldquoattractionbasinrdquo are also used instead of ldquoregion of attractionrdquo
Definition 5 (positively invariant set) A set119872 sub R119899 is calleda positively invariant set of the system (1) if x
0isin 119872 implies
120601(119905 x0) isin 119872 for all 119905 ge 0 Namely if a solution belongs to a
positively invariant set119872 at some time instant then it belongsto119872 for all future time
In general finding the exact ROA analytically might bedifficult or even impossible Usually Lyapunov functions areused to find underestimates of the ROA that is sets containedin the region of attraction as stated in the following theorem
Theorem 6 ([20 Theorem 10]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R satisfying(5) and (7) then any region Ω
119888= x isin R119899 | 119881(x) le 119888 with
119888 ge 0 such that Ω119888sube 119863 is a positively invariant set contained
in the ROA of the equilibrium 0
Theorem 6 describes an approach to compute estimatesof the ROA through Lyapunov functionsMore specifically inthe case of asymptotic stability ifΩ
119888= x isin R119899 | 119881(x) le 119888 is
bounded and contained in119863 then every trajectory starting inΩ
119888remains inΩ
119888and approaches the origin as 119905 rarr infinThus
Ω
119888is an estimate of the ROA Remark that when the origin is
globally asymptotically stable the region of attraction is thewhole space R119899
3 Problem Reformulation
In this section we will transform the problem of the asymp-totic stability and ROA analysis of system (1) to a parametricprogram with LMI or BMI constraints In the sequel wesuppose that system (1) is a polynomial dynamical systemwith 119891
119894(x) isin R[x] for 1 le 119894 le 119899 where R[x] denotes the
ring of real polynomials in the variables x
31 Asymptotic Stability Firstly we consider the asymptoticstability of system (1) As shown in Theorem 2 the existenceof a Lyapunov function 119881(x) which satisfies the conditions(5) and (7) is a certificate for asymptotical stability of the
equilibrium point 0 and the problem of computing such a119881(x) can be transformed into the following problem
find 119881 (x) isin R [x]
st 119881 (0) = 0
119881 (x) gt 0 forallx isin 119863 0
119881 (x) = 120597119881
120597xf (x) lt 0 forallx isin 119863 0
(11)
In general 119863 can be an arbitrary neighborhood of theequilibrium point 0 However to simplify calculation inpractice we can assume 119863 = x isin R119899 119892(x) le 0 where119892(x) isin R[x] is chosen to be 119892(x) = sum
119899
119894=1119909
2
119894minus 119903
2 for 1 le 119894 le 119899with a given constant 119903 isin R
+ for instance 119903 = 10
minus2Let us first predetermine a template of Lyapunov func-
tions with the given degree 119889 We assume that
119881 (x) = sum
120572
119888
120572x120572 (12)
where x120572 = 119909
1205721
1 119909
120572119899
119899 120572 = (120572
1 120572
119899) isin Z119899
ge0with
sum
119899
119894=1120572
119894le 119889 and 119888
120572isin R are parameters We can rewrite
119881(x) = c119879 sdot 119879(x) where 119879(x) is the (column) vector of allterms in 119909
1 119909
119899with total degree le 119889 and c isin R] with
] = (
119899+119889
119899) is the coefficient vector of 119881(x) In the sequel we
write 119881(x) as 119881(x c) for clarityFor a polynomial 120593(x) isin R[x] we say that 120593(x) is positive
definite (resp positive semidefinite) if 120593(x) gt 0 for all x isin
R119899 0 (resp 120593(x) ge 0 for all x isin R119899) Observe that if 120593(x)is a sum of squares (SOS) then 120593(x) is globally nonnegativeAnd to ensure positive definiteness of 120593(x) we can use apolynomial 119897(x) of the form
119897 (x) =119899
sum
119894=1
120598
119894119909
119896
119894 (13)
where 120598
119894isin R+and 119896 is assumed to be even Clearly the
condition that 120593(x) minus 119897(x) is an SOS polynomial guaranteesthe positive definiteness of 120593(x) Therefore to ensure positivedefiniteness of the second and third constraints in (11)we can use two polynomials 119897
1(x) 1198972(x) of the form of
(13) It is notable that SOS programming can be applied todetermine the nonnegativity of a multivariate polynomialover a semialgebraic set Consider the problem of verifyingwhether the implication
119898
⋀
119894=1
(119901
119894(x) ge 0) 997904rArr 119902 (x) ge 0 (14)
holds where 119901
119894(x) isin R[x] for 1 le 119894 le 119898 and 119902(x) isin
R[x] According to Stenglersquos Positivstellensatz SchmudgenrsquosPositivstellensatz or Putinarrsquos Positivstellensatz [32] if thereexist SOS polynomials 120590
119894isin R[x] for 119894 = 0 119898 such that
119902 (x) = 120590
0(x) +
119898
sum
119894=1
120590
119894(x) 119901119894(x) (15)
4 Abstract and Applied Analysis
then the assertion (14) holds Therefore the existence of SOSrepresentations provides a sufficient condition for determin-ing the nonnegativity of 119902(x) over x isin R119899 and119898
119894=1119901
119894(x) ge 0
Based on the above observation the problem (11) can befurther transformed into the following SOS programming
find c isin R]
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x) minus 120590
1(x) 119892 (x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x) minus 120590
2(x) 119892 (x)
(16)
where 120590119895(x) and 120588
119895(x) are SOS inR[x] for 119895 = 1 2 Moreover
the degree bound of those unknown SOS polynomials 120590119895 120588119895
is exponential in 119899 deg(119881) deg(f) and deg(119892) In practiceto avoid high computational complexity we simply set upa truncated SOS programming by fixing a priori (muchsmaller) degree bound 2119890 with 119890 isin Z
+ of the unknown
SOS polynomials Consequently the existence of a solutionc of (16) can guarantee the asymptotic stability of the givensystem
Similarly the problem of computing the Lyapunov func-tion 119881(x) for globally asymptotic stability of system (1) thatsatisfies the conditions in Theorem 3 can be rewritten as thefollowing problem
find 119881 (x) isin R [x]
st 119881 (0) = 0
119881 (x) gt 0 forallx = 0
x 997888rarr infin 997904rArr 119881(x) 997888rarr infin
119881 (x) lt 0 forallx = 0
(17)
By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form
(13) the condition that 119881(x c) minus 119897
1(x) is SOS guarantees
the radially unboundedness of 119881(x c) Consequently theproblem (17) can be further transformed into the followingSOS programming
find c isin R]
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x)
(18)
where 120588119895(x) are SOS polynomials in R[x] for 119895 = 1 2 The
decision variables are the coefficients of all the polynomialsappearing in (18) such as 119881(x c) 119897
119894(x) and 120588
119895(x)
32 Region of Attraction Suppose that the equilibrium point0 is asymptotically stable In this section we will considerhow to find a large enough underestimate of the ROA In thecase where the equilibrium point 0 is globally asymptoticallystable the ROA becomes the whole space R119899
Our idea of computing the estimate of ROA is similar tothat of the algorithm in [20 Section 422] Suppose that119863 isa semialgebraic set
x isin R119899
| 119881 (x) le 1 (19)
given by a Lyapunov function 119881(x) In order to enlarge thecomputed positively invariant set contained in the ROA wedefine a variable sized region
119875
120573= x isin R
119899
| 119901 (x) le 120573 (20)
where119901(x) isin R[x] is a fixed and positive definite polynomialand maximize 120573 subject to the constraint 119875
120573sube 119863
Fix a template of 119881 of the form (12) The problem offinding an estimate 119875
120573of the ROA can be transformed into
the following polynomial optimization problem
maxcisinR]
120573
st 119881 (0 c) = 0
119881 (x c) gt 0 forallx isin 119863 0
119881 (x c) = 120597119881
120597xf (x) lt 0 forallx isin 119863 0
119901 (x) le 120573 ⊨ 119881 (x c) le 1
(21)
By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form (13)
and based on SOS relaxation the problem (21) can be furthertransformed into the following SOS programming
maxcisinR]
120573
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x) minus 120590
1(x) (119881 (x c) minus 1)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x) minus 120590
2(x) (119881 (x c) minus 1)
1 minus 119881 (x c) = 120588
3(x) minus 120590
3(x) (119901 (x) minus 120573)
(22)
where 120590
119894(x) and 120588
119894(x) are SOS in R[x] for 1 le 119894 le 3
In (22) the decision variables are 120573 and the coefficients ofall the polynomials appearing in (22) such as 119881(x c) 120590
119894(x)
and 120588
119894(x) Since 120573 and the coefficients of 119881(x c) and 120590
119894(x)
are unknown some nonlinear terms that are products ofthese coefficients will occur in the second third and fourthconstraints of (22) which yields a nonconvex bilinear matrixinequalities (BMI) problemWe will discuss in Section 4 howto handle the SOS programming (22) directly using the BMIsolver or iterative method
4 Exact Certificate of Sum ofSquares Decomposition
According to Theorems 2 3 and 6 the key of asymptoticstability analysis and ROA estimation lies in finding a real
Abstract and Applied Analysis 5
function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573
41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method
Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that
120595 (x) = 119885(x)119879119876119885 (x) (23)
where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem
inf119882[119895]119876[119895]
2
sum
119895=1
Trace (119882[119895] + 119876
[119895]
)
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus 119901
1(x)119879 sdot 119876[1] sdot 119901
1(x) 119892 (x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
minus 119901
2(x)119879 sdot 119876[2] sdot 119901
2(x) 119892 (x)
(24)
where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2
119895=1Trace(119882[119895] + 119876
[119895]
) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function
Similarly the SOS programming (18) can be rewritten asthe following SDP problem
inf119882[1]119882[2]
Trace (119882[1]) + Trace (119882[2])
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
(25)
where matrices 119882
[1] 119882
[2] are symmetric and positivesemidefinite
Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently
Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem
Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and
minus119909
2
+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat
2119909
119889120593
119889119909
= 120601
0(119909) + 120601
1(119909) (1 minus 119909
2
) + 120601
2(119909) 120593 (119909)
(26)
where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1
deg(1206010) = 2 and deg(120601
1) = deg(120601
2) = 0 and that 120593(119909) = 119906
0+
119906
1119909 1206011= 119906
2and 1206012= V1 with 119906
0 119906
1 119906
2 V1isin R parameters
From (1) we have
120601
0(119909) = 119906
2119909
2
+ (2119906
1minus 119906
1V1) 119909 minus 119906
2minus 119906
0V1 (27)
whose square matrix representation (SMR) [33] is 1206010(119909) =
119885
119879
119876119885 where
119876 =
[
[
[
minus119906
2minus 119906
0V1
119906
1minus
1
2
119906
1V1
119906
1minus
1
2
119906
1V1
119906
2
]
]
]
119885 = [
1
119909
] (28)
Since all the 120601119894(119909) are SOSes we have 119906
2ge 0 V
1ge 0 and
119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint
B (119906
0 119906
1 119906
2 V1) = 119906
1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 1
0 0 1 0
]
]
]
]
+ 119906
2
[
[
[
[
1 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 1
]
]
]
]
+ V1
[
[
[
[
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
]
]
]
]
+ 119906
0V1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 0
]
]
]
]
+ 119906
1V1
[
[
[
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 minus
1
2
0 0 minus
1
2
0
]
]
]
]
]
]
]
⪰ 0
(29)
Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form
inf minus120573
st B (u k) = 119860
0+
119898
sum
119894=1
119906
119894119860
119894+
119896
sum
119895=1
V119895119860
119898+119895
+ sum
1le119894le119898
sum
1le119895le119896
119906
119894V119895119861
119894119895⪰ 0
(30)
where 119860
119894 119861
119894119895are constant symmetric matrices u =
(119906
1 119906
119898) k = (V
1 V
119896) are parameter coefficients of the
SOSpolynomials occurring in the original SOSprogramming(22)
Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI
6 Abstract and Applied Analysis
solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ
119901that penalizes the inequality constraints This
function satisfies a number of properties [35] such that forany 119901 gt 0 we have
B (u k) ⪰ 0 lArrrArr Φ
119901(B (u k)) ⪰ 0 (31)
This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem
inf minus120573
st Φ
119901(B (u k)) ⪰ 0
(32)
The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)
119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ
119901(B (u k))) (33)
where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]
Alternatively observing (30) B(u k) involves no cross-ing products like 119906
119894119906
119895and V
119894V119895 Taking this special form
into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice
42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints
Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem
find c isin R]
st 119881
1(x c) = m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) = m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
119882
[119894]
⪰ 0 119894 = 1 2 3
(34)
where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3
are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the
numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is
119881
1(x c) asymp m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) asymp m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
(35)
as illustrated by the following example
Example 8 Consider the following nonlinear system
[
1
2
] =
[
[
[
[
[
[
[
[
[
minus05119909
1+ 119909
2+ 024999119909
2
1
+ 0083125119909
3
1+ 00205295119909
4
1
+ 00046875119909
5
1+ 00015191119909
6
1
minus119909
2+ 119909
1119909
2minus 049991119909
3
1+ 0040947119909
5
1
]
]
]
]
]
]
]
]
]
(36)
We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2 When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 3112937368119909
2
1+ 3112937368119909
2
2
120573 = 032124
(37)
However119881(1199091 119909
2) and120573 cannot satisfy the conditions in (21)
exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore
Ω
119881= (119909
1 119909
2) isin R2
| 119881 (119909
1 119909
2) le 1 (38)
is not an estimate of the ROA of this system
In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows
We first convert 119882[3] to a nearby rational positive semi-definite matrix
119882
[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882
[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix
119882
[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882
[2]
simultaneously with respect to a given tolerance 120591
Abstract and Applied Analysis 7
Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and
119882
[2] that satisfy exactly
119881
1(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(39)
Since the equations in (39) are affine in entries of c and
119882
1
119882
2 one can define an affine linear hyperplane
L = c119882[1]119882[2] | 1198811(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(40)
Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882
[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices
119882
[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le
119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]
43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)
Algorithm 9 Verified Parametric Optimization SolverInput
(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z
gt0 the bound of the common denominator
(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to
construct the SOS programming(iv) 120591 isin R
gt0 the given tolerance
Output
(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3
positive semidefinite
(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882
[119894]
⪸ 0 1 le 119894 le 3
(2) (Compute the verified solution c)
(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix
119882
[3] by non-negativetruncated PLDLTPT-decomposition
(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]
(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices
119882
[1]
119882
[2] can beobtained by orthogonal projection if 119882[1]119882
[2] are of full rank or by rational vectorrecovery method otherwise
(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and
119882
[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo
5 Experiments
Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems
Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system
x = f (x) =
[
[
[
[
[
[
[
[
1
2
3
4
5
6
]
]
]
]
]
]
]
]
=
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119909
3
1+ 4119909
3
2minus 6119909
3119909
4
minus119909
1minus 119909
2+ 119909
3
5
119909
1119909
4minus 119909
3+ 119909
4119909
6
119909
1119909
3+ 119909
3119909
6minus 119909
3
4
minus2119909
3
2minus 119909
5+ 119909
6
minus3119909
3119909
4minus 119909
3
5minus 119909
6
]
]
]
]
]
]
]
]
]
]
]
]
]
(41)
Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909
2
1+119909
2
2+119909
2
3minus00001 When 119889 = 4 we obtain a feasible
solution of the associated SDP system Here we just list oneapproximate polynomial
119881 (x) = 081569119909
2
1+ 018066119909
2
2+ 16479119909
2
3+ 21429119909
2
4
+ 12996119909
2
6+ sdot sdot sdot + 067105119909
4
4+ 052855119909
4
5
(42)
Let the tolerance 120591 = 10
minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and
8 Abstract and Applied Analysis
the corresponding SOS polynomials Here we only list theLyapunov function
119881 (x) = 41
50
119909
2
1+
9
50
119909
2
2+
33
20
119909
2
3+
107
50
119909
2
4
+
69
100
119909
2
5+
13
10
119909
2
6+ sdot sdot sdot +
67
100
119909
4
4+
53
100
119909
4
5
(43)
Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic
stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain
119881 (x) = 078997119909
2
1+ 019119909
2
2+ 22721119909
2
3
+ 14213119909
2
6+ sdot sdot sdot + 14213119909
4
2+ 071066119909
4
5
(44)
By use of the rational SOS recovery technique described inSection 42 we then obtain
119881 (x) = 79
100
119909
2
1+
19
50
119909
1119909
2+
9
100
119909
1119909
5+
19
100
119909
2
2+
227
100
119909
2
3
+
199
100
119909
2
4+
9
100
119909
2119909
5+
71
100
119909
2
5
+
71
50
119909
2
6+
71
50
119909
4
2+
71
100
119909
4
5
(45)
which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied
Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10
minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds
Example 11 ([41 Example A]) Consider a nonlinear contin-uous system
x = f (x) = [
1
2
] = [
minus119909
2
119909
1+ (119909
2
1minus 1) 119909
2
] (46)
Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909
1 119909
2)
with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909
2
1+ 119909
2
2minus 00001 When 119889 = 2 we obtain
119881 (x) = 14957119909
2
1minus 07279119909
1119909
2+ 11418119909
2
2 (47)
Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)
Example 119899 Num deg(119881) Time(s)
1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830
Table 2 Algorithm performance on benchmarks (ROA)
Example119899 Num deg(119881)
120573 120573
Time(s)
1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915
Let the tolerance 120591 = 10
minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function
119881 (x) = minus
62
85
119909
1119909
2+
127
85
119909
2
1+
97
85
119909
2
2 (48)
Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified
estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2
When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 06174455119909
2
1minus 040292119909
1119909
2+ 043078119909
2
2
120573 = 13402225
(49)
Let the tolerance 120591 = 10
minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10
5By use of the rational SOS recovery technique described inSection 42 we obtain
119881 (119909
1 119909
2) =
6732
10903
119909
2
1minus
4393
10903
119909
1119909
2+
42271
10903
119909
2
2
120573 =
6701
5000
(50)
which exactly satisfy the conditions inTheorem 6Therefore
Ω
= (119909
1 119909
2) isin R2
|
119881 (119909
1 119909
2) le 1 (51)
is a certified estimate of the ROA of the given system
Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite
Abstract and Applied Analysis 9
polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =
10
minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and
120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds
6 Conclusion
In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system
Acknowledgments
This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)
References
[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997
[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000
[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000
[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006
[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005
[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012
[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003
[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002
[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009
[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009
[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991
[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011
[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004
[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007
[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008
[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989
[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971
[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001
[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985
[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003
[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002
[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006
[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Remark that a function satisfying condition (9) is said tobe radially unbounded
It is known that globally asymptotic stability is verydesirable but in many applications it is difficult to achieveWhen the equilibrium point is asymptotically stable we areinterested in determining how far from the origin the trajec-tory can be and still converge to the origin as 119905 approachesinfinThis gives rise to the following definition
Definition 4 (region of attraction) The region of attraction(ROA) Ω of the equilibrium point 0 is defined as Ω = x isin
R119899 | lim119905rarrinfin
120601(119905 x) = 0
The ROA of the equilibrium point 0 is a collection ofall points x such that any trajectory starting at initial state xat time 0 will be attracted to the equilibrium point In theliterature the terms ldquodomain of attractionrdquo and ldquoattractionbasinrdquo are also used instead of ldquoregion of attractionrdquo
Definition 5 (positively invariant set) A set119872 sub R119899 is calleda positively invariant set of the system (1) if x
0isin 119872 implies
120601(119905 x0) isin 119872 for all 119905 ge 0 Namely if a solution belongs to a
positively invariant set119872 at some time instant then it belongsto119872 for all future time
In general finding the exact ROA analytically might bedifficult or even impossible Usually Lyapunov functions areused to find underestimates of the ROA that is sets containedin the region of attraction as stated in the following theorem
Theorem 6 ([20 Theorem 10]) Let 119863 sub R119899 be a domaincontaining the equilibrium point 0 of (1) If there exists acontinuously differentiable function 119881 119863 rarr R satisfying(5) and (7) then any region Ω
119888= x isin R119899 | 119881(x) le 119888 with
119888 ge 0 such that Ω119888sube 119863 is a positively invariant set contained
in the ROA of the equilibrium 0
Theorem 6 describes an approach to compute estimatesof the ROA through Lyapunov functionsMore specifically inthe case of asymptotic stability ifΩ
119888= x isin R119899 | 119881(x) le 119888 is
bounded and contained in119863 then every trajectory starting inΩ
119888remains inΩ
119888and approaches the origin as 119905 rarr infinThus
Ω
119888is an estimate of the ROA Remark that when the origin is
globally asymptotically stable the region of attraction is thewhole space R119899
3 Problem Reformulation
In this section we will transform the problem of the asymp-totic stability and ROA analysis of system (1) to a parametricprogram with LMI or BMI constraints In the sequel wesuppose that system (1) is a polynomial dynamical systemwith 119891
119894(x) isin R[x] for 1 le 119894 le 119899 where R[x] denotes the
ring of real polynomials in the variables x
31 Asymptotic Stability Firstly we consider the asymptoticstability of system (1) As shown in Theorem 2 the existenceof a Lyapunov function 119881(x) which satisfies the conditions(5) and (7) is a certificate for asymptotical stability of the
equilibrium point 0 and the problem of computing such a119881(x) can be transformed into the following problem
find 119881 (x) isin R [x]
st 119881 (0) = 0
119881 (x) gt 0 forallx isin 119863 0
119881 (x) = 120597119881
120597xf (x) lt 0 forallx isin 119863 0
(11)
In general 119863 can be an arbitrary neighborhood of theequilibrium point 0 However to simplify calculation inpractice we can assume 119863 = x isin R119899 119892(x) le 0 where119892(x) isin R[x] is chosen to be 119892(x) = sum
119899
119894=1119909
2
119894minus 119903
2 for 1 le 119894 le 119899with a given constant 119903 isin R
+ for instance 119903 = 10
minus2Let us first predetermine a template of Lyapunov func-
tions with the given degree 119889 We assume that
119881 (x) = sum
120572
119888
120572x120572 (12)
where x120572 = 119909
1205721
1 119909
120572119899
119899 120572 = (120572
1 120572
119899) isin Z119899
ge0with
sum
119899
119894=1120572
119894le 119889 and 119888
120572isin R are parameters We can rewrite
119881(x) = c119879 sdot 119879(x) where 119879(x) is the (column) vector of allterms in 119909
1 119909
119899with total degree le 119889 and c isin R] with
] = (
119899+119889
119899) is the coefficient vector of 119881(x) In the sequel we
write 119881(x) as 119881(x c) for clarityFor a polynomial 120593(x) isin R[x] we say that 120593(x) is positive
definite (resp positive semidefinite) if 120593(x) gt 0 for all x isin
R119899 0 (resp 120593(x) ge 0 for all x isin R119899) Observe that if 120593(x)is a sum of squares (SOS) then 120593(x) is globally nonnegativeAnd to ensure positive definiteness of 120593(x) we can use apolynomial 119897(x) of the form
119897 (x) =119899
sum
119894=1
120598
119894119909
119896
119894 (13)
where 120598
119894isin R+and 119896 is assumed to be even Clearly the
condition that 120593(x) minus 119897(x) is an SOS polynomial guaranteesthe positive definiteness of 120593(x) Therefore to ensure positivedefiniteness of the second and third constraints in (11)we can use two polynomials 119897
1(x) 1198972(x) of the form of
(13) It is notable that SOS programming can be applied todetermine the nonnegativity of a multivariate polynomialover a semialgebraic set Consider the problem of verifyingwhether the implication
119898
⋀
119894=1
(119901
119894(x) ge 0) 997904rArr 119902 (x) ge 0 (14)
holds where 119901
119894(x) isin R[x] for 1 le 119894 le 119898 and 119902(x) isin
R[x] According to Stenglersquos Positivstellensatz SchmudgenrsquosPositivstellensatz or Putinarrsquos Positivstellensatz [32] if thereexist SOS polynomials 120590
119894isin R[x] for 119894 = 0 119898 such that
119902 (x) = 120590
0(x) +
119898
sum
119894=1
120590
119894(x) 119901119894(x) (15)
4 Abstract and Applied Analysis
then the assertion (14) holds Therefore the existence of SOSrepresentations provides a sufficient condition for determin-ing the nonnegativity of 119902(x) over x isin R119899 and119898
119894=1119901
119894(x) ge 0
Based on the above observation the problem (11) can befurther transformed into the following SOS programming
find c isin R]
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x) minus 120590
1(x) 119892 (x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x) minus 120590
2(x) 119892 (x)
(16)
where 120590119895(x) and 120588
119895(x) are SOS inR[x] for 119895 = 1 2 Moreover
the degree bound of those unknown SOS polynomials 120590119895 120588119895
is exponential in 119899 deg(119881) deg(f) and deg(119892) In practiceto avoid high computational complexity we simply set upa truncated SOS programming by fixing a priori (muchsmaller) degree bound 2119890 with 119890 isin Z
+ of the unknown
SOS polynomials Consequently the existence of a solutionc of (16) can guarantee the asymptotic stability of the givensystem
Similarly the problem of computing the Lyapunov func-tion 119881(x) for globally asymptotic stability of system (1) thatsatisfies the conditions in Theorem 3 can be rewritten as thefollowing problem
find 119881 (x) isin R [x]
st 119881 (0) = 0
119881 (x) gt 0 forallx = 0
x 997888rarr infin 997904rArr 119881(x) 997888rarr infin
119881 (x) lt 0 forallx = 0
(17)
By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form
(13) the condition that 119881(x c) minus 119897
1(x) is SOS guarantees
the radially unboundedness of 119881(x c) Consequently theproblem (17) can be further transformed into the followingSOS programming
find c isin R]
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x)
(18)
where 120588119895(x) are SOS polynomials in R[x] for 119895 = 1 2 The
decision variables are the coefficients of all the polynomialsappearing in (18) such as 119881(x c) 119897
119894(x) and 120588
119895(x)
32 Region of Attraction Suppose that the equilibrium point0 is asymptotically stable In this section we will considerhow to find a large enough underestimate of the ROA In thecase where the equilibrium point 0 is globally asymptoticallystable the ROA becomes the whole space R119899
Our idea of computing the estimate of ROA is similar tothat of the algorithm in [20 Section 422] Suppose that119863 isa semialgebraic set
x isin R119899
| 119881 (x) le 1 (19)
given by a Lyapunov function 119881(x) In order to enlarge thecomputed positively invariant set contained in the ROA wedefine a variable sized region
119875
120573= x isin R
119899
| 119901 (x) le 120573 (20)
where119901(x) isin R[x] is a fixed and positive definite polynomialand maximize 120573 subject to the constraint 119875
120573sube 119863
Fix a template of 119881 of the form (12) The problem offinding an estimate 119875
120573of the ROA can be transformed into
the following polynomial optimization problem
maxcisinR]
120573
st 119881 (0 c) = 0
119881 (x c) gt 0 forallx isin 119863 0
119881 (x c) = 120597119881
120597xf (x) lt 0 forallx isin 119863 0
119901 (x) le 120573 ⊨ 119881 (x c) le 1
(21)
By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form (13)
and based on SOS relaxation the problem (21) can be furthertransformed into the following SOS programming
maxcisinR]
120573
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x) minus 120590
1(x) (119881 (x c) minus 1)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x) minus 120590
2(x) (119881 (x c) minus 1)
1 minus 119881 (x c) = 120588
3(x) minus 120590
3(x) (119901 (x) minus 120573)
(22)
where 120590
119894(x) and 120588
119894(x) are SOS in R[x] for 1 le 119894 le 3
In (22) the decision variables are 120573 and the coefficients ofall the polynomials appearing in (22) such as 119881(x c) 120590
119894(x)
and 120588
119894(x) Since 120573 and the coefficients of 119881(x c) and 120590
119894(x)
are unknown some nonlinear terms that are products ofthese coefficients will occur in the second third and fourthconstraints of (22) which yields a nonconvex bilinear matrixinequalities (BMI) problemWe will discuss in Section 4 howto handle the SOS programming (22) directly using the BMIsolver or iterative method
4 Exact Certificate of Sum ofSquares Decomposition
According to Theorems 2 3 and 6 the key of asymptoticstability analysis and ROA estimation lies in finding a real
Abstract and Applied Analysis 5
function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573
41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method
Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that
120595 (x) = 119885(x)119879119876119885 (x) (23)
where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem
inf119882[119895]119876[119895]
2
sum
119895=1
Trace (119882[119895] + 119876
[119895]
)
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus 119901
1(x)119879 sdot 119876[1] sdot 119901
1(x) 119892 (x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
minus 119901
2(x)119879 sdot 119876[2] sdot 119901
2(x) 119892 (x)
(24)
where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2
119895=1Trace(119882[119895] + 119876
[119895]
) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function
Similarly the SOS programming (18) can be rewritten asthe following SDP problem
inf119882[1]119882[2]
Trace (119882[1]) + Trace (119882[2])
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
(25)
where matrices 119882
[1] 119882
[2] are symmetric and positivesemidefinite
Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently
Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem
Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and
minus119909
2
+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat
2119909
119889120593
119889119909
= 120601
0(119909) + 120601
1(119909) (1 minus 119909
2
) + 120601
2(119909) 120593 (119909)
(26)
where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1
deg(1206010) = 2 and deg(120601
1) = deg(120601
2) = 0 and that 120593(119909) = 119906
0+
119906
1119909 1206011= 119906
2and 1206012= V1 with 119906
0 119906
1 119906
2 V1isin R parameters
From (1) we have
120601
0(119909) = 119906
2119909
2
+ (2119906
1minus 119906
1V1) 119909 minus 119906
2minus 119906
0V1 (27)
whose square matrix representation (SMR) [33] is 1206010(119909) =
119885
119879
119876119885 where
119876 =
[
[
[
minus119906
2minus 119906
0V1
119906
1minus
1
2
119906
1V1
119906
1minus
1
2
119906
1V1
119906
2
]
]
]
119885 = [
1
119909
] (28)
Since all the 120601119894(119909) are SOSes we have 119906
2ge 0 V
1ge 0 and
119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint
B (119906
0 119906
1 119906
2 V1) = 119906
1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 1
0 0 1 0
]
]
]
]
+ 119906
2
[
[
[
[
1 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 1
]
]
]
]
+ V1
[
[
[
[
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
]
]
]
]
+ 119906
0V1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 0
]
]
]
]
+ 119906
1V1
[
[
[
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 minus
1
2
0 0 minus
1
2
0
]
]
]
]
]
]
]
⪰ 0
(29)
Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form
inf minus120573
st B (u k) = 119860
0+
119898
sum
119894=1
119906
119894119860
119894+
119896
sum
119895=1
V119895119860
119898+119895
+ sum
1le119894le119898
sum
1le119895le119896
119906
119894V119895119861
119894119895⪰ 0
(30)
where 119860
119894 119861
119894119895are constant symmetric matrices u =
(119906
1 119906
119898) k = (V
1 V
119896) are parameter coefficients of the
SOSpolynomials occurring in the original SOSprogramming(22)
Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI
6 Abstract and Applied Analysis
solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ
119901that penalizes the inequality constraints This
function satisfies a number of properties [35] such that forany 119901 gt 0 we have
B (u k) ⪰ 0 lArrrArr Φ
119901(B (u k)) ⪰ 0 (31)
This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem
inf minus120573
st Φ
119901(B (u k)) ⪰ 0
(32)
The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)
119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ
119901(B (u k))) (33)
where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]
Alternatively observing (30) B(u k) involves no cross-ing products like 119906
119894119906
119895and V
119894V119895 Taking this special form
into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice
42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints
Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem
find c isin R]
st 119881
1(x c) = m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) = m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
119882
[119894]
⪰ 0 119894 = 1 2 3
(34)
where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3
are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the
numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is
119881
1(x c) asymp m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) asymp m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
(35)
as illustrated by the following example
Example 8 Consider the following nonlinear system
[
1
2
] =
[
[
[
[
[
[
[
[
[
minus05119909
1+ 119909
2+ 024999119909
2
1
+ 0083125119909
3
1+ 00205295119909
4
1
+ 00046875119909
5
1+ 00015191119909
6
1
minus119909
2+ 119909
1119909
2minus 049991119909
3
1+ 0040947119909
5
1
]
]
]
]
]
]
]
]
]
(36)
We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2 When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 3112937368119909
2
1+ 3112937368119909
2
2
120573 = 032124
(37)
However119881(1199091 119909
2) and120573 cannot satisfy the conditions in (21)
exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore
Ω
119881= (119909
1 119909
2) isin R2
| 119881 (119909
1 119909
2) le 1 (38)
is not an estimate of the ROA of this system
In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows
We first convert 119882[3] to a nearby rational positive semi-definite matrix
119882
[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882
[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix
119882
[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882
[2]
simultaneously with respect to a given tolerance 120591
Abstract and Applied Analysis 7
Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and
119882
[2] that satisfy exactly
119881
1(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(39)
Since the equations in (39) are affine in entries of c and
119882
1
119882
2 one can define an affine linear hyperplane
L = c119882[1]119882[2] | 1198811(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(40)
Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882
[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices
119882
[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le
119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]
43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)
Algorithm 9 Verified Parametric Optimization SolverInput
(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z
gt0 the bound of the common denominator
(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to
construct the SOS programming(iv) 120591 isin R
gt0 the given tolerance
Output
(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3
positive semidefinite
(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882
[119894]
⪸ 0 1 le 119894 le 3
(2) (Compute the verified solution c)
(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix
119882
[3] by non-negativetruncated PLDLTPT-decomposition
(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]
(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices
119882
[1]
119882
[2] can beobtained by orthogonal projection if 119882[1]119882
[2] are of full rank or by rational vectorrecovery method otherwise
(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and
119882
[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo
5 Experiments
Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems
Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system
x = f (x) =
[
[
[
[
[
[
[
[
1
2
3
4
5
6
]
]
]
]
]
]
]
]
=
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119909
3
1+ 4119909
3
2minus 6119909
3119909
4
minus119909
1minus 119909
2+ 119909
3
5
119909
1119909
4minus 119909
3+ 119909
4119909
6
119909
1119909
3+ 119909
3119909
6minus 119909
3
4
minus2119909
3
2minus 119909
5+ 119909
6
minus3119909
3119909
4minus 119909
3
5minus 119909
6
]
]
]
]
]
]
]
]
]
]
]
]
]
(41)
Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909
2
1+119909
2
2+119909
2
3minus00001 When 119889 = 4 we obtain a feasible
solution of the associated SDP system Here we just list oneapproximate polynomial
119881 (x) = 081569119909
2
1+ 018066119909
2
2+ 16479119909
2
3+ 21429119909
2
4
+ 12996119909
2
6+ sdot sdot sdot + 067105119909
4
4+ 052855119909
4
5
(42)
Let the tolerance 120591 = 10
minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and
8 Abstract and Applied Analysis
the corresponding SOS polynomials Here we only list theLyapunov function
119881 (x) = 41
50
119909
2
1+
9
50
119909
2
2+
33
20
119909
2
3+
107
50
119909
2
4
+
69
100
119909
2
5+
13
10
119909
2
6+ sdot sdot sdot +
67
100
119909
4
4+
53
100
119909
4
5
(43)
Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic
stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain
119881 (x) = 078997119909
2
1+ 019119909
2
2+ 22721119909
2
3
+ 14213119909
2
6+ sdot sdot sdot + 14213119909
4
2+ 071066119909
4
5
(44)
By use of the rational SOS recovery technique described inSection 42 we then obtain
119881 (x) = 79
100
119909
2
1+
19
50
119909
1119909
2+
9
100
119909
1119909
5+
19
100
119909
2
2+
227
100
119909
2
3
+
199
100
119909
2
4+
9
100
119909
2119909
5+
71
100
119909
2
5
+
71
50
119909
2
6+
71
50
119909
4
2+
71
100
119909
4
5
(45)
which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied
Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10
minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds
Example 11 ([41 Example A]) Consider a nonlinear contin-uous system
x = f (x) = [
1
2
] = [
minus119909
2
119909
1+ (119909
2
1minus 1) 119909
2
] (46)
Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909
1 119909
2)
with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909
2
1+ 119909
2
2minus 00001 When 119889 = 2 we obtain
119881 (x) = 14957119909
2
1minus 07279119909
1119909
2+ 11418119909
2
2 (47)
Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)
Example 119899 Num deg(119881) Time(s)
1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830
Table 2 Algorithm performance on benchmarks (ROA)
Example119899 Num deg(119881)
120573 120573
Time(s)
1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915
Let the tolerance 120591 = 10
minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function
119881 (x) = minus
62
85
119909
1119909
2+
127
85
119909
2
1+
97
85
119909
2
2 (48)
Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified
estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2
When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 06174455119909
2
1minus 040292119909
1119909
2+ 043078119909
2
2
120573 = 13402225
(49)
Let the tolerance 120591 = 10
minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10
5By use of the rational SOS recovery technique described inSection 42 we obtain
119881 (119909
1 119909
2) =
6732
10903
119909
2
1minus
4393
10903
119909
1119909
2+
42271
10903
119909
2
2
120573 =
6701
5000
(50)
which exactly satisfy the conditions inTheorem 6Therefore
Ω
= (119909
1 119909
2) isin R2
|
119881 (119909
1 119909
2) le 1 (51)
is a certified estimate of the ROA of the given system
Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite
Abstract and Applied Analysis 9
polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =
10
minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and
120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds
6 Conclusion
In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system
Acknowledgments
This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)
References
[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997
[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000
[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000
[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006
[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005
[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012
[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003
[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002
[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009
[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009
[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991
[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011
[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004
[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007
[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008
[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989
[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971
[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001
[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985
[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003
[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002
[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006
[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
then the assertion (14) holds Therefore the existence of SOSrepresentations provides a sufficient condition for determin-ing the nonnegativity of 119902(x) over x isin R119899 and119898
119894=1119901
119894(x) ge 0
Based on the above observation the problem (11) can befurther transformed into the following SOS programming
find c isin R]
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x) minus 120590
1(x) 119892 (x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x) minus 120590
2(x) 119892 (x)
(16)
where 120590119895(x) and 120588
119895(x) are SOS inR[x] for 119895 = 1 2 Moreover
the degree bound of those unknown SOS polynomials 120590119895 120588119895
is exponential in 119899 deg(119881) deg(f) and deg(119892) In practiceto avoid high computational complexity we simply set upa truncated SOS programming by fixing a priori (muchsmaller) degree bound 2119890 with 119890 isin Z
+ of the unknown
SOS polynomials Consequently the existence of a solutionc of (16) can guarantee the asymptotic stability of the givensystem
Similarly the problem of computing the Lyapunov func-tion 119881(x) for globally asymptotic stability of system (1) thatsatisfies the conditions in Theorem 3 can be rewritten as thefollowing problem
find 119881 (x) isin R [x]
st 119881 (0) = 0
119881 (x) gt 0 forallx = 0
x 997888rarr infin 997904rArr 119881(x) 997888rarr infin
119881 (x) lt 0 forallx = 0
(17)
By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form
(13) the condition that 119881(x c) minus 119897
1(x) is SOS guarantees
the radially unboundedness of 119881(x c) Consequently theproblem (17) can be further transformed into the followingSOS programming
find c isin R]
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x)
(18)
where 120588119895(x) are SOS polynomials in R[x] for 119895 = 1 2 The
decision variables are the coefficients of all the polynomialsappearing in (18) such as 119881(x c) 119897
119894(x) and 120588
119895(x)
32 Region of Attraction Suppose that the equilibrium point0 is asymptotically stable In this section we will considerhow to find a large enough underestimate of the ROA In thecase where the equilibrium point 0 is globally asymptoticallystable the ROA becomes the whole space R119899
Our idea of computing the estimate of ROA is similar tothat of the algorithm in [20 Section 422] Suppose that119863 isa semialgebraic set
x isin R119899
| 119881 (x) le 1 (19)
given by a Lyapunov function 119881(x) In order to enlarge thecomputed positively invariant set contained in the ROA wedefine a variable sized region
119875
120573= x isin R
119899
| 119901 (x) le 120573 (20)
where119901(x) isin R[x] is a fixed and positive definite polynomialand maximize 120573 subject to the constraint 119875
120573sube 119863
Fix a template of 119881 of the form (12) The problem offinding an estimate 119875
120573of the ROA can be transformed into
the following polynomial optimization problem
maxcisinR]
120573
st 119881 (0 c) = 0
119881 (x c) gt 0 forallx isin 119863 0
119881 (x c) = 120597119881
120597xf (x) lt 0 forallx isin 119863 0
119901 (x) le 120573 ⊨ 119881 (x c) le 1
(21)
By introducing two polynomials 119897119894(x) 119894 = 1 2 of the form (13)
and based on SOS relaxation the problem (21) can be furthertransformed into the following SOS programming
maxcisinR]
120573
st 119881 (0 c) = 0
119881 (x c) minus 119897
1(x) = 120588
1(x) minus 120590
1(x) (119881 (x c) minus 1)
minus
120597119881
120597xf (x) minus 119897
2(x) = 120588
2(x) minus 120590
2(x) (119881 (x c) minus 1)
1 minus 119881 (x c) = 120588
3(x) minus 120590
3(x) (119901 (x) minus 120573)
(22)
where 120590
119894(x) and 120588
119894(x) are SOS in R[x] for 1 le 119894 le 3
In (22) the decision variables are 120573 and the coefficients ofall the polynomials appearing in (22) such as 119881(x c) 120590
119894(x)
and 120588
119894(x) Since 120573 and the coefficients of 119881(x c) and 120590
119894(x)
are unknown some nonlinear terms that are products ofthese coefficients will occur in the second third and fourthconstraints of (22) which yields a nonconvex bilinear matrixinequalities (BMI) problemWe will discuss in Section 4 howto handle the SOS programming (22) directly using the BMIsolver or iterative method
4 Exact Certificate of Sum ofSquares Decomposition
According to Theorems 2 3 and 6 the key of asymptoticstability analysis and ROA estimation lies in finding a real
Abstract and Applied Analysis 5
function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573
41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method
Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that
120595 (x) = 119885(x)119879119876119885 (x) (23)
where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem
inf119882[119895]119876[119895]
2
sum
119895=1
Trace (119882[119895] + 119876
[119895]
)
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus 119901
1(x)119879 sdot 119876[1] sdot 119901
1(x) 119892 (x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
minus 119901
2(x)119879 sdot 119876[2] sdot 119901
2(x) 119892 (x)
(24)
where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2
119895=1Trace(119882[119895] + 119876
[119895]
) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function
Similarly the SOS programming (18) can be rewritten asthe following SDP problem
inf119882[1]119882[2]
Trace (119882[1]) + Trace (119882[2])
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
(25)
where matrices 119882
[1] 119882
[2] are symmetric and positivesemidefinite
Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently
Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem
Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and
minus119909
2
+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat
2119909
119889120593
119889119909
= 120601
0(119909) + 120601
1(119909) (1 minus 119909
2
) + 120601
2(119909) 120593 (119909)
(26)
where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1
deg(1206010) = 2 and deg(120601
1) = deg(120601
2) = 0 and that 120593(119909) = 119906
0+
119906
1119909 1206011= 119906
2and 1206012= V1 with 119906
0 119906
1 119906
2 V1isin R parameters
From (1) we have
120601
0(119909) = 119906
2119909
2
+ (2119906
1minus 119906
1V1) 119909 minus 119906
2minus 119906
0V1 (27)
whose square matrix representation (SMR) [33] is 1206010(119909) =
119885
119879
119876119885 where
119876 =
[
[
[
minus119906
2minus 119906
0V1
119906
1minus
1
2
119906
1V1
119906
1minus
1
2
119906
1V1
119906
2
]
]
]
119885 = [
1
119909
] (28)
Since all the 120601119894(119909) are SOSes we have 119906
2ge 0 V
1ge 0 and
119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint
B (119906
0 119906
1 119906
2 V1) = 119906
1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 1
0 0 1 0
]
]
]
]
+ 119906
2
[
[
[
[
1 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 1
]
]
]
]
+ V1
[
[
[
[
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
]
]
]
]
+ 119906
0V1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 0
]
]
]
]
+ 119906
1V1
[
[
[
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 minus
1
2
0 0 minus
1
2
0
]
]
]
]
]
]
]
⪰ 0
(29)
Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form
inf minus120573
st B (u k) = 119860
0+
119898
sum
119894=1
119906
119894119860
119894+
119896
sum
119895=1
V119895119860
119898+119895
+ sum
1le119894le119898
sum
1le119895le119896
119906
119894V119895119861
119894119895⪰ 0
(30)
where 119860
119894 119861
119894119895are constant symmetric matrices u =
(119906
1 119906
119898) k = (V
1 V
119896) are parameter coefficients of the
SOSpolynomials occurring in the original SOSprogramming(22)
Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI
6 Abstract and Applied Analysis
solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ
119901that penalizes the inequality constraints This
function satisfies a number of properties [35] such that forany 119901 gt 0 we have
B (u k) ⪰ 0 lArrrArr Φ
119901(B (u k)) ⪰ 0 (31)
This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem
inf minus120573
st Φ
119901(B (u k)) ⪰ 0
(32)
The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)
119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ
119901(B (u k))) (33)
where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]
Alternatively observing (30) B(u k) involves no cross-ing products like 119906
119894119906
119895and V
119894V119895 Taking this special form
into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice
42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints
Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem
find c isin R]
st 119881
1(x c) = m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) = m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
119882
[119894]
⪰ 0 119894 = 1 2 3
(34)
where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3
are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the
numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is
119881
1(x c) asymp m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) asymp m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
(35)
as illustrated by the following example
Example 8 Consider the following nonlinear system
[
1
2
] =
[
[
[
[
[
[
[
[
[
minus05119909
1+ 119909
2+ 024999119909
2
1
+ 0083125119909
3
1+ 00205295119909
4
1
+ 00046875119909
5
1+ 00015191119909
6
1
minus119909
2+ 119909
1119909
2minus 049991119909
3
1+ 0040947119909
5
1
]
]
]
]
]
]
]
]
]
(36)
We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2 When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 3112937368119909
2
1+ 3112937368119909
2
2
120573 = 032124
(37)
However119881(1199091 119909
2) and120573 cannot satisfy the conditions in (21)
exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore
Ω
119881= (119909
1 119909
2) isin R2
| 119881 (119909
1 119909
2) le 1 (38)
is not an estimate of the ROA of this system
In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows
We first convert 119882[3] to a nearby rational positive semi-definite matrix
119882
[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882
[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix
119882
[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882
[2]
simultaneously with respect to a given tolerance 120591
Abstract and Applied Analysis 7
Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and
119882
[2] that satisfy exactly
119881
1(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(39)
Since the equations in (39) are affine in entries of c and
119882
1
119882
2 one can define an affine linear hyperplane
L = c119882[1]119882[2] | 1198811(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(40)
Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882
[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices
119882
[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le
119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]
43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)
Algorithm 9 Verified Parametric Optimization SolverInput
(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z
gt0 the bound of the common denominator
(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to
construct the SOS programming(iv) 120591 isin R
gt0 the given tolerance
Output
(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3
positive semidefinite
(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882
[119894]
⪸ 0 1 le 119894 le 3
(2) (Compute the verified solution c)
(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix
119882
[3] by non-negativetruncated PLDLTPT-decomposition
(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]
(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices
119882
[1]
119882
[2] can beobtained by orthogonal projection if 119882[1]119882
[2] are of full rank or by rational vectorrecovery method otherwise
(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and
119882
[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo
5 Experiments
Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems
Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system
x = f (x) =
[
[
[
[
[
[
[
[
1
2
3
4
5
6
]
]
]
]
]
]
]
]
=
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119909
3
1+ 4119909
3
2minus 6119909
3119909
4
minus119909
1minus 119909
2+ 119909
3
5
119909
1119909
4minus 119909
3+ 119909
4119909
6
119909
1119909
3+ 119909
3119909
6minus 119909
3
4
minus2119909
3
2minus 119909
5+ 119909
6
minus3119909
3119909
4minus 119909
3
5minus 119909
6
]
]
]
]
]
]
]
]
]
]
]
]
]
(41)
Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909
2
1+119909
2
2+119909
2
3minus00001 When 119889 = 4 we obtain a feasible
solution of the associated SDP system Here we just list oneapproximate polynomial
119881 (x) = 081569119909
2
1+ 018066119909
2
2+ 16479119909
2
3+ 21429119909
2
4
+ 12996119909
2
6+ sdot sdot sdot + 067105119909
4
4+ 052855119909
4
5
(42)
Let the tolerance 120591 = 10
minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and
8 Abstract and Applied Analysis
the corresponding SOS polynomials Here we only list theLyapunov function
119881 (x) = 41
50
119909
2
1+
9
50
119909
2
2+
33
20
119909
2
3+
107
50
119909
2
4
+
69
100
119909
2
5+
13
10
119909
2
6+ sdot sdot sdot +
67
100
119909
4
4+
53
100
119909
4
5
(43)
Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic
stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain
119881 (x) = 078997119909
2
1+ 019119909
2
2+ 22721119909
2
3
+ 14213119909
2
6+ sdot sdot sdot + 14213119909
4
2+ 071066119909
4
5
(44)
By use of the rational SOS recovery technique described inSection 42 we then obtain
119881 (x) = 79
100
119909
2
1+
19
50
119909
1119909
2+
9
100
119909
1119909
5+
19
100
119909
2
2+
227
100
119909
2
3
+
199
100
119909
2
4+
9
100
119909
2119909
5+
71
100
119909
2
5
+
71
50
119909
2
6+
71
50
119909
4
2+
71
100
119909
4
5
(45)
which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied
Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10
minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds
Example 11 ([41 Example A]) Consider a nonlinear contin-uous system
x = f (x) = [
1
2
] = [
minus119909
2
119909
1+ (119909
2
1minus 1) 119909
2
] (46)
Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909
1 119909
2)
with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909
2
1+ 119909
2
2minus 00001 When 119889 = 2 we obtain
119881 (x) = 14957119909
2
1minus 07279119909
1119909
2+ 11418119909
2
2 (47)
Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)
Example 119899 Num deg(119881) Time(s)
1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830
Table 2 Algorithm performance on benchmarks (ROA)
Example119899 Num deg(119881)
120573 120573
Time(s)
1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915
Let the tolerance 120591 = 10
minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function
119881 (x) = minus
62
85
119909
1119909
2+
127
85
119909
2
1+
97
85
119909
2
2 (48)
Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified
estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2
When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 06174455119909
2
1minus 040292119909
1119909
2+ 043078119909
2
2
120573 = 13402225
(49)
Let the tolerance 120591 = 10
minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10
5By use of the rational SOS recovery technique described inSection 42 we obtain
119881 (119909
1 119909
2) =
6732
10903
119909
2
1minus
4393
10903
119909
1119909
2+
42271
10903
119909
2
2
120573 =
6701
5000
(50)
which exactly satisfy the conditions inTheorem 6Therefore
Ω
= (119909
1 119909
2) isin R2
|
119881 (119909
1 119909
2) le 1 (51)
is a certified estimate of the ROA of the given system
Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite
Abstract and Applied Analysis 9
polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =
10
minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and
120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds
6 Conclusion
In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system
Acknowledgments
This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)
References
[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997
[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000
[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000
[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006
[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005
[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012
[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003
[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002
[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009
[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009
[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991
[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011
[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004
[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007
[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008
[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989
[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971
[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001
[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985
[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003
[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002
[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006
[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
function 119881(x) and a constant 120573 satisfying the desired con-ditions We will present a symbolic-numeric hybrid methodbased on SOS relaxation and rational vector recovery tocompute the exact polynomial 119881(x) and constant 120573
41 Approximate Solution from LMI or BMI Solver In thissection wewill discuss how to handle the SOS programmings(16) (18) and (22) directly using the LMI and BMI solver oriterative method
Using Gram matrix representation method [23] (alsoknown as square matrix representation (SMR) [33]) a poly-nomial 120595(x) of degree 2119898 is SOS if and only if there exists apositive semidefinite matrix 119876 such that
120595 (x) = 119885(x)119879119876119885 (x) (23)
where 119885(x) is a monomial vector in x of degree 119898 Thusthe SOS programming (16) is equivalent to the followingsemidefinite program (SDP) problem
inf119882[119895]119876[119895]
2
sum
119895=1
Trace (119882[119895] + 119876
[119895]
)
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus 119901
1(x)119879 sdot 119876[1] sdot 119901
1(x) 119892 (x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
minus 119901
2(x)119879 sdot 119876[2] sdot 119901
2(x) 119892 (x)
(24)
where all the matrices 119882[119895] 119876[119895] are symmetric and positivesemidefinite andsum2
119895=1Trace(119882[119895] + 119876
[119895]
) denotes the sum oftraces of all these matrices which acts as a dummy objectivefunction commonly used in SDP for optimization problemwith no objective function
Similarly the SOS programming (18) can be rewritten asthe following SDP problem
inf119882[1]119882[2]
Trace (119882[1]) + Trace (119882[2])
st 119881 (0) = 0
119881 (x c) minus 119897
1(x) = 119898
1(x)119879 sdot 119882[1] sdot 119898
1(x)
minus
120597119881
120597xf (x) minus 119897
2(x) = 119898
2(x)119879 sdot 119882[2] sdot 119898
2(x)
(25)
where matrices 119882
[1] 119882
[2] are symmetric and positivesemidefinite
Many Matlab packages of SDP solvers such as SOS-TOOLS [24] YALMIP [25] and SeDuMi [26] are availableto solve the problem (24) and (25) efficiently
Now let us consider the problem (22) The followingexample shows how to transform nonlinear parametric poly-nomial constraints into a BMI problem
Example 7 To find a polynomial 120593(119909) satisfying 120593(119909) ge 0 and
minus119909
2
+ 1 ge 0 ⊨ 2119909(119889120593119889119909) ge 0 it suffices to find 120593(119909) suchthat
2119909
119889120593
119889119909
= 120601
0(119909) + 120601
1(119909) (1 minus 119909
2
) + 120601
2(119909) 120593 (119909)
(26)
where1206010(119909)1206011(119909)1206012(119909) are SOSes Suppose that deg(120593) = 1
deg(1206010) = 2 and deg(120601
1) = deg(120601
2) = 0 and that 120593(119909) = 119906
0+
119906
1119909 1206011= 119906
2and 1206012= V1 with 119906
0 119906
1 119906
2 V1isin R parameters
From (1) we have
120601
0(119909) = 119906
2119909
2
+ (2119906
1minus 119906
1V1) 119909 minus 119906
2minus 119906
0V1 (27)
whose square matrix representation (SMR) [33] is 1206010(119909) =
119885
119879
119876119885 where
119876 =
[
[
[
minus119906
2minus 119906
0V1
119906
1minus
1
2
119906
1V1
119906
1minus
1
2
119906
1V1
119906
2
]
]
]
119885 = [
1
119909
] (28)
Since all the 120601119894(119909) are SOSes we have 119906
2ge 0 V
1ge 0 and
119876 ⪰ 0 Therefore the original constraint is translated into aBMI constraint
B (119906
0 119906
1 119906
2 V1) = 119906
1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 1
0 0 1 0
]
]
]
]
+ 119906
2
[
[
[
[
1 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 1
]
]
]
]
+ V1
[
[
[
[
0 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
]
]
]
]
+ 119906
0V1
[
[
[
[
0 0 0 0
0 0 0 0
0 0 minus1 0
0 0 0 0
]
]
]
]
+ 119906
1V1
[
[
[
[
[
[
[
0 0 0 0
0 0 0 0
0 0 0 minus
1
2
0 0 minus
1
2
0
]
]
]
]
]
]
]
⪰ 0
(29)
Similar to Example 7 the SOS programming (22) can betransformed into a BMI problem of the form
inf minus120573
st B (u k) = 119860
0+
119898
sum
119894=1
119906
119894119860
119894+
119896
sum
119895=1
V119895119860
119898+119895
+ sum
1le119894le119898
sum
1le119895le119896
119906
119894V119895119861
119894119895⪰ 0
(30)
where 119860
119894 119861
119894119895are constant symmetric matrices u =
(119906
1 119906
119898) k = (V
1 V
119896) are parameter coefficients of the
SOSpolynomials occurring in the original SOSprogramming(22)
Manymethods can be used to solve the BMI problem (30)directly such as interior-point constrained trust regionmeth-ods [34] an augmented Lagrangian strategy [35] PENBMI
6 Abstract and Applied Analysis
solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ
119901that penalizes the inequality constraints This
function satisfies a number of properties [35] such that forany 119901 gt 0 we have
B (u k) ⪰ 0 lArrrArr Φ
119901(B (u k)) ⪰ 0 (31)
This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem
inf minus120573
st Φ
119901(B (u k)) ⪰ 0
(32)
The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)
119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ
119901(B (u k))) (33)
where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]
Alternatively observing (30) B(u k) involves no cross-ing products like 119906
119894119906
119895and V
119894V119895 Taking this special form
into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice
42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints
Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem
find c isin R]
st 119881
1(x c) = m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) = m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
119882
[119894]
⪰ 0 119894 = 1 2 3
(34)
where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3
are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the
numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is
119881
1(x c) asymp m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) asymp m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
(35)
as illustrated by the following example
Example 8 Consider the following nonlinear system
[
1
2
] =
[
[
[
[
[
[
[
[
[
minus05119909
1+ 119909
2+ 024999119909
2
1
+ 0083125119909
3
1+ 00205295119909
4
1
+ 00046875119909
5
1+ 00015191119909
6
1
minus119909
2+ 119909
1119909
2minus 049991119909
3
1+ 0040947119909
5
1
]
]
]
]
]
]
]
]
]
(36)
We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2 When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 3112937368119909
2
1+ 3112937368119909
2
2
120573 = 032124
(37)
However119881(1199091 119909
2) and120573 cannot satisfy the conditions in (21)
exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore
Ω
119881= (119909
1 119909
2) isin R2
| 119881 (119909
1 119909
2) le 1 (38)
is not an estimate of the ROA of this system
In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows
We first convert 119882[3] to a nearby rational positive semi-definite matrix
119882
[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882
[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix
119882
[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882
[2]
simultaneously with respect to a given tolerance 120591
Abstract and Applied Analysis 7
Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and
119882
[2] that satisfy exactly
119881
1(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(39)
Since the equations in (39) are affine in entries of c and
119882
1
119882
2 one can define an affine linear hyperplane
L = c119882[1]119882[2] | 1198811(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(40)
Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882
[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices
119882
[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le
119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]
43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)
Algorithm 9 Verified Parametric Optimization SolverInput
(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z
gt0 the bound of the common denominator
(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to
construct the SOS programming(iv) 120591 isin R
gt0 the given tolerance
Output
(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3
positive semidefinite
(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882
[119894]
⪸ 0 1 le 119894 le 3
(2) (Compute the verified solution c)
(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix
119882
[3] by non-negativetruncated PLDLTPT-decomposition
(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]
(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices
119882
[1]
119882
[2] can beobtained by orthogonal projection if 119882[1]119882
[2] are of full rank or by rational vectorrecovery method otherwise
(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and
119882
[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo
5 Experiments
Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems
Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system
x = f (x) =
[
[
[
[
[
[
[
[
1
2
3
4
5
6
]
]
]
]
]
]
]
]
=
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119909
3
1+ 4119909
3
2minus 6119909
3119909
4
minus119909
1minus 119909
2+ 119909
3
5
119909
1119909
4minus 119909
3+ 119909
4119909
6
119909
1119909
3+ 119909
3119909
6minus 119909
3
4
minus2119909
3
2minus 119909
5+ 119909
6
minus3119909
3119909
4minus 119909
3
5minus 119909
6
]
]
]
]
]
]
]
]
]
]
]
]
]
(41)
Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909
2
1+119909
2
2+119909
2
3minus00001 When 119889 = 4 we obtain a feasible
solution of the associated SDP system Here we just list oneapproximate polynomial
119881 (x) = 081569119909
2
1+ 018066119909
2
2+ 16479119909
2
3+ 21429119909
2
4
+ 12996119909
2
6+ sdot sdot sdot + 067105119909
4
4+ 052855119909
4
5
(42)
Let the tolerance 120591 = 10
minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and
8 Abstract and Applied Analysis
the corresponding SOS polynomials Here we only list theLyapunov function
119881 (x) = 41
50
119909
2
1+
9
50
119909
2
2+
33
20
119909
2
3+
107
50
119909
2
4
+
69
100
119909
2
5+
13
10
119909
2
6+ sdot sdot sdot +
67
100
119909
4
4+
53
100
119909
4
5
(43)
Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic
stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain
119881 (x) = 078997119909
2
1+ 019119909
2
2+ 22721119909
2
3
+ 14213119909
2
6+ sdot sdot sdot + 14213119909
4
2+ 071066119909
4
5
(44)
By use of the rational SOS recovery technique described inSection 42 we then obtain
119881 (x) = 79
100
119909
2
1+
19
50
119909
1119909
2+
9
100
119909
1119909
5+
19
100
119909
2
2+
227
100
119909
2
3
+
199
100
119909
2
4+
9
100
119909
2119909
5+
71
100
119909
2
5
+
71
50
119909
2
6+
71
50
119909
4
2+
71
100
119909
4
5
(45)
which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied
Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10
minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds
Example 11 ([41 Example A]) Consider a nonlinear contin-uous system
x = f (x) = [
1
2
] = [
minus119909
2
119909
1+ (119909
2
1minus 1) 119909
2
] (46)
Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909
1 119909
2)
with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909
2
1+ 119909
2
2minus 00001 When 119889 = 2 we obtain
119881 (x) = 14957119909
2
1minus 07279119909
1119909
2+ 11418119909
2
2 (47)
Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)
Example 119899 Num deg(119881) Time(s)
1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830
Table 2 Algorithm performance on benchmarks (ROA)
Example119899 Num deg(119881)
120573 120573
Time(s)
1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915
Let the tolerance 120591 = 10
minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function
119881 (x) = minus
62
85
119909
1119909
2+
127
85
119909
2
1+
97
85
119909
2
2 (48)
Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified
estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2
When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 06174455119909
2
1minus 040292119909
1119909
2+ 043078119909
2
2
120573 = 13402225
(49)
Let the tolerance 120591 = 10
minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10
5By use of the rational SOS recovery technique described inSection 42 we obtain
119881 (119909
1 119909
2) =
6732
10903
119909
2
1minus
4393
10903
119909
1119909
2+
42271
10903
119909
2
2
120573 =
6701
5000
(50)
which exactly satisfy the conditions inTheorem 6Therefore
Ω
= (119909
1 119909
2) isin R2
|
119881 (119909
1 119909
2) le 1 (51)
is a certified estimate of the ROA of the given system
Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite
Abstract and Applied Analysis 9
polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =
10
minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and
120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds
6 Conclusion
In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system
Acknowledgments
This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)
References
[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997
[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000
[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000
[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006
[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005
[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012
[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003
[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002
[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009
[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009
[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991
[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011
[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004
[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007
[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008
[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989
[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971
[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001
[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985
[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003
[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002
[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006
[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
solver [27] is a Matlab package to solve the general BMIproblem whose idea is based on a choice of penaltybarrierfunction Φ
119901that penalizes the inequality constraints This
function satisfies a number of properties [35] such that forany 119901 gt 0 we have
B (u k) ⪰ 0 lArrrArr Φ
119901(B (u k)) ⪰ 0 (31)
This means that for any 119901 gt 0 the problem (30) is equivalentto the following augmented problem
inf minus120573
st Φ
119901(B (u k)) ⪰ 0
(32)
The associated Lagrangian of (32) can be viewed as a(generalized) augmented Lagrangian of (30)
119865 (u k 119880 119901) = minus120573 + 119879119903 (119880Φ
119901(B (u k))) (33)
where 119880 is the Lagrangian multiplier associated with theinequality constraint Remark that 119880 is a real symmetricmatrix of the same order as the matrix operatorB For moredetails please refer to [35]
Alternatively observing (30) B(u k) involves no cross-ing products like 119906
119894119906
119895and V
119894V119895 Taking this special form
into account an iterative method can be applied by fixing uand k alternatively which leads to a sequential convex LMIproblem [20] Remark that although the convergence of theiterative method cannot be guaranteed this method is easierto implement than PENBMI solver and can yield a feasiblesolution (u k) efficiently in practice
42 Exact SOS Recovery Since the SDP (LMI) or BMIsolvers in Matlab are running in fixed precision applyingthe techniques in Section 41 only yields numerical solutionsof (16) (18) and (22) In the sequel we will propose animproved algorithm based on a modified Newton refinementand rational vector recovery technique to compute exactsolutions of polynomial optimization problems with LMI orBMI constraints
Without loss of generality we can reduce the problems(16) (18) and (22) to the following problem
find c isin R]
st 119881
1(x c) = m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) = m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
119882
[119894]
⪰ 0 119894 = 1 2 3
(34)
where the coefficients of the polynomials 119881119894(x c) 1 le 119894 le 3
are affine in c Note that (34) involves both LMI and BMIconstraints After solving the SDP system (34) by applyingthe techniques in Section 41 the numerical vector c and the
numerical positive semidefinite matrices119882[119894] 119894 = 1 2 3maynot satisfy the conditions in (34) exactly that is
119881
1(x c) asymp m
1(x)119879 sdot 119882[1] sdotm
1(x)
119881
2(x c) asymp m
2(x)119879 sdot 119882[2] sdotm
2(x)
+ (m3(x)119879 sdot 119882[3] sdotm
3(x)) sdot 119881
3(x c)
(35)
as illustrated by the following example
Example 8 Consider the following nonlinear system
[
1
2
] =
[
[
[
[
[
[
[
[
[
minus05119909
1+ 119909
2+ 024999119909
2
1
+ 0083125119909
3
1+ 00205295119909
4
1
+ 00046875119909
5
1+ 00015191119909
6
1
minus119909
2+ 119909
1119909
2minus 049991119909
3
1+ 0040947119909
5
1
]
]
]
]
]
]
]
]
]
(36)
We want to find a certified estimate of the ROA In theassociated SOS programming (22) with BMI constraints wesuppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2 When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 3112937368119909
2
1+ 3112937368119909
2
2
120573 = 032124
(37)
However119881(1199091 119909
2) and120573 cannot satisfy the conditions in (21)
exactly because there exists a sample point (1532 85256)such that the third condition in (21) cannot be satisfiedTherefore
Ω
119881= (119909
1 119909
2) isin R2
| 119881 (119909
1 119909
2) le 1 (38)
is not an estimate of the ROA of this system
In our former papers [36 37] we applied Gauss-Newtoniteration and rational vector recovery to obtain exact solu-tions that satisfy the constraints in (34) exactly Howeverthese techniques may fail in some cases as shown in [38Example 2] The reason may lie in that we recover thevector c and the associated positive semidefinite matricesseparately Here we will compute exact solutions of (34)by using a modified Newton refinement and rational vectorrecovery technique [38] which applies on the vector c andthe associated positive semidefinite matrices simultaneouslyThe main ideal is as follows
We first convert 119882[3] to a nearby rational positive semi-definite matrix
119882
[3] by nonnegative truncated PLDLTPT-decomposition In practice 119882
[3] is numerical diagonalmatrix in other words the off-diagonal entries are verytiny and the diagonal entries are nonnegative Therefore bysetting the small entries of 119882[3] to zeros we easily get thenearby rational positive semidefinite matrix
119882
[3] We thenapply Gauss-Newton iteration to refine c 119882[1] and 119882
[2]
simultaneously with respect to a given tolerance 120591
Abstract and Applied Analysis 7
Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and
119882
[2] that satisfy exactly
119881
1(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(39)
Since the equations in (39) are affine in entries of c and
119882
1
119882
2 one can define an affine linear hyperplane
L = c119882[1]119882[2] | 1198811(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(40)
Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882
[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices
119882
[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le
119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]
43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)
Algorithm 9 Verified Parametric Optimization SolverInput
(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z
gt0 the bound of the common denominator
(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to
construct the SOS programming(iv) 120591 isin R
gt0 the given tolerance
Output
(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3
positive semidefinite
(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882
[119894]
⪸ 0 1 le 119894 le 3
(2) (Compute the verified solution c)
(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix
119882
[3] by non-negativetruncated PLDLTPT-decomposition
(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]
(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices
119882
[1]
119882
[2] can beobtained by orthogonal projection if 119882[1]119882
[2] are of full rank or by rational vectorrecovery method otherwise
(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and
119882
[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo
5 Experiments
Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems
Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system
x = f (x) =
[
[
[
[
[
[
[
[
1
2
3
4
5
6
]
]
]
]
]
]
]
]
=
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119909
3
1+ 4119909
3
2minus 6119909
3119909
4
minus119909
1minus 119909
2+ 119909
3
5
119909
1119909
4minus 119909
3+ 119909
4119909
6
119909
1119909
3+ 119909
3119909
6minus 119909
3
4
minus2119909
3
2minus 119909
5+ 119909
6
minus3119909
3119909
4minus 119909
3
5minus 119909
6
]
]
]
]
]
]
]
]
]
]
]
]
]
(41)
Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909
2
1+119909
2
2+119909
2
3minus00001 When 119889 = 4 we obtain a feasible
solution of the associated SDP system Here we just list oneapproximate polynomial
119881 (x) = 081569119909
2
1+ 018066119909
2
2+ 16479119909
2
3+ 21429119909
2
4
+ 12996119909
2
6+ sdot sdot sdot + 067105119909
4
4+ 052855119909
4
5
(42)
Let the tolerance 120591 = 10
minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and
8 Abstract and Applied Analysis
the corresponding SOS polynomials Here we only list theLyapunov function
119881 (x) = 41
50
119909
2
1+
9
50
119909
2
2+
33
20
119909
2
3+
107
50
119909
2
4
+
69
100
119909
2
5+
13
10
119909
2
6+ sdot sdot sdot +
67
100
119909
4
4+
53
100
119909
4
5
(43)
Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic
stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain
119881 (x) = 078997119909
2
1+ 019119909
2
2+ 22721119909
2
3
+ 14213119909
2
6+ sdot sdot sdot + 14213119909
4
2+ 071066119909
4
5
(44)
By use of the rational SOS recovery technique described inSection 42 we then obtain
119881 (x) = 79
100
119909
2
1+
19
50
119909
1119909
2+
9
100
119909
1119909
5+
19
100
119909
2
2+
227
100
119909
2
3
+
199
100
119909
2
4+
9
100
119909
2119909
5+
71
100
119909
2
5
+
71
50
119909
2
6+
71
50
119909
4
2+
71
100
119909
4
5
(45)
which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied
Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10
minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds
Example 11 ([41 Example A]) Consider a nonlinear contin-uous system
x = f (x) = [
1
2
] = [
minus119909
2
119909
1+ (119909
2
1minus 1) 119909
2
] (46)
Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909
1 119909
2)
with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909
2
1+ 119909
2
2minus 00001 When 119889 = 2 we obtain
119881 (x) = 14957119909
2
1minus 07279119909
1119909
2+ 11418119909
2
2 (47)
Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)
Example 119899 Num deg(119881) Time(s)
1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830
Table 2 Algorithm performance on benchmarks (ROA)
Example119899 Num deg(119881)
120573 120573
Time(s)
1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915
Let the tolerance 120591 = 10
minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function
119881 (x) = minus
62
85
119909
1119909
2+
127
85
119909
2
1+
97
85
119909
2
2 (48)
Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified
estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2
When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 06174455119909
2
1minus 040292119909
1119909
2+ 043078119909
2
2
120573 = 13402225
(49)
Let the tolerance 120591 = 10
minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10
5By use of the rational SOS recovery technique described inSection 42 we obtain
119881 (119909
1 119909
2) =
6732
10903
119909
2
1minus
4393
10903
119909
1119909
2+
42271
10903
119909
2
2
120573 =
6701
5000
(50)
which exactly satisfy the conditions inTheorem 6Therefore
Ω
= (119909
1 119909
2) isin R2
|
119881 (119909
1 119909
2) le 1 (51)
is a certified estimate of the ROA of the given system
Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite
Abstract and Applied Analysis 9
polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =
10
minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and
120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds
6 Conclusion
In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system
Acknowledgments
This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)
References
[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997
[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000
[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000
[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006
[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005
[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012
[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003
[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002
[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009
[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009
[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991
[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011
[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004
[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007
[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008
[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989
[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971
[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001
[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985
[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003
[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002
[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006
[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
Now we will discuss how to recover from the refinedc 119882[1] 119882[2] the rational vector c and rational positivesemidefinite matrices 119882[1] and
119882
[2] that satisfy exactly
119881
1(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(39)
Since the equations in (39) are affine in entries of c and
119882
1
119882
2 one can define an affine linear hyperplane
L = c119882[1]119882[2] | 1198811(x c) minusm
1(x)119879 sdot 119882[1] sdotm
1(x) = 0
119881
2(x c) minusm
2(x)119879 sdot 119882[2] sdotm
2(x) minus 119881
3(x c)m
3(x)119879
sdot
119882
[3]
sdotm3(x) = 0
(40)
Note that the hyperplane (40) can be constructed from alinear system 119860y = b where y consists of the entries of c119882
[1] and119882[2] If 119860 has full row rank such a hyperplane isguaranteed to existThen for a given bound119863 of the commondenominator the rationalized SOS solutions of (34) can becomputed by orthogonal projection if thematrices119882[1] 119882[2]have full ranks with respect to 120591 or by the rational vectorrecovery otherwise Finally we check whether the matrices
119882
[119894] are positive semidefinite for 119894 = 1 2 If so the rationalvector c and rational positive semidefinite matrices 119882[119894] 1 le
119894 le 3 can satisfy the conditions of problem (34) exactly Formore details the reader can refer to [38]
43 Algorithm The results in Sections 41 and 42 yield analgorithm to find exact solutions to the problem (34)
Algorithm 9 Verified Parametric Optimization SolverInput
(i) A polynomial optimization problem of the form (34)(ii) 119863 isin Z
gt0 the bound of the common denominator
(iii) 119890 isin Zge0 the degree bound 2119890 of the SOSes used to
construct the SOS programming(iv) 120591 isin R
gt0 the given tolerance
Output
(i) The verified solution c of (34) with the 119882[119894] 1 le 119894 le 3
positive semidefinite
(1) (Compute the numerical solutions) Apply LMIor BMI solver to compute numerical solutionsof the associated polynomial optimization prob-lem (34) If this problem has no feasible solu-tions return ldquowe canrsquot find solutions of (34)withthe given degree bound 2119890rdquo Otherwise obtain cand119882
[119894]
⪸ 0 1 le 119894 le 3
(2) (Compute the verified solution c)
(21) Convert 119882[3] to a nearby rational positivesemidefinite matrix
119882
[3] by non-negativetruncated PLDLTPT-decomposition
(22) For the tolerance 120591 apply the modifiedNewton iteration to refine c and119882[1]119882[2]
(23) Determine the singularity of119882[1] and119882[2]with respect to 120591 Then for a given com-mon denominator the rational vector cand rational matrices
119882
[1]
119882
[2] can beobtained by orthogonal projection if 119882[1]119882
[2] are of full rank or by rational vectorrecovery method otherwise
(24) Check whether the matrices 119882[1] 119882[2] arepositive semidefinite If so return c and
119882
[119894] 1 le 119894 le 3 Otherwise return ldquowe canrsquotfind the solutions of (34) with the givendegree boundrdquo
5 Experiments
Let us present some examples of asymptotic stability andROAs analysis of nonlinear systems
Example 10 ([8 Example 1]) Consider a nonlinear continu-ous system
x = f (x) =
[
[
[
[
[
[
[
[
1
2
3
4
5
6
]
]
]
]
]
]
]
]
=
[
[
[
[
[
[
[
[
[
[
[
[
[
minus119909
3
1+ 4119909
3
2minus 6119909
3119909
4
minus119909
1minus 119909
2+ 119909
3
5
119909
1119909
4minus 119909
3+ 119909
4119909
6
119909
1119909
3+ 119909
3119909
6minus 119909
3
4
minus2119909
3
2minus 119909
5+ 119909
6
minus3119909
3119909
4minus 119909
3
5minus 119909
6
]
]
]
]
]
]
]
]
]
]
]
]
]
(41)
Firstly we will certify the locally asymptotic stability of thissystem According toTheorem 2 we need to find a Lyapunovfunction119881(x) which satisfies all the conditions inTheorem 2We can set up an associated SOS programming (16) with119892(x) = 119909
2
1+119909
2
2+119909
2
3minus00001 When 119889 = 4 we obtain a feasible
solution of the associated SDP system Here we just list oneapproximate polynomial
119881 (x) = 081569119909
2
1+ 018066119909
2
2+ 16479119909
2
3+ 21429119909
2
4
+ 12996119909
2
6+ sdot sdot sdot + 067105119909
4
4+ 052855119909
4
5
(42)
Let the tolerance 120591 = 10
minus2 and let the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we can obtain an exact Lyapunov function and
8 Abstract and Applied Analysis
the corresponding SOS polynomials Here we only list theLyapunov function
119881 (x) = 41
50
119909
2
1+
9
50
119909
2
2+
33
20
119909
2
3+
107
50
119909
2
4
+
69
100
119909
2
5+
13
10
119909
2
6+ sdot sdot sdot +
67
100
119909
4
4+
53
100
119909
4
5
(43)
Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic
stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain
119881 (x) = 078997119909
2
1+ 019119909
2
2+ 22721119909
2
3
+ 14213119909
2
6+ sdot sdot sdot + 14213119909
4
2+ 071066119909
4
5
(44)
By use of the rational SOS recovery technique described inSection 42 we then obtain
119881 (x) = 79
100
119909
2
1+
19
50
119909
1119909
2+
9
100
119909
1119909
5+
19
100
119909
2
2+
227
100
119909
2
3
+
199
100
119909
2
4+
9
100
119909
2119909
5+
71
100
119909
2
5
+
71
50
119909
2
6+
71
50
119909
4
2+
71
100
119909
4
5
(45)
which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied
Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10
minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds
Example 11 ([41 Example A]) Consider a nonlinear contin-uous system
x = f (x) = [
1
2
] = [
minus119909
2
119909
1+ (119909
2
1minus 1) 119909
2
] (46)
Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909
1 119909
2)
with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909
2
1+ 119909
2
2minus 00001 When 119889 = 2 we obtain
119881 (x) = 14957119909
2
1minus 07279119909
1119909
2+ 11418119909
2
2 (47)
Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)
Example 119899 Num deg(119881) Time(s)
1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830
Table 2 Algorithm performance on benchmarks (ROA)
Example119899 Num deg(119881)
120573 120573
Time(s)
1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915
Let the tolerance 120591 = 10
minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function
119881 (x) = minus
62
85
119909
1119909
2+
127
85
119909
2
1+
97
85
119909
2
2 (48)
Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified
estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2
When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 06174455119909
2
1minus 040292119909
1119909
2+ 043078119909
2
2
120573 = 13402225
(49)
Let the tolerance 120591 = 10
minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10
5By use of the rational SOS recovery technique described inSection 42 we obtain
119881 (119909
1 119909
2) =
6732
10903
119909
2
1minus
4393
10903
119909
1119909
2+
42271
10903
119909
2
2
120573 =
6701
5000
(50)
which exactly satisfy the conditions inTheorem 6Therefore
Ω
= (119909
1 119909
2) isin R2
|
119881 (119909
1 119909
2) le 1 (51)
is a certified estimate of the ROA of the given system
Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite
Abstract and Applied Analysis 9
polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =
10
minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and
120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds
6 Conclusion
In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system
Acknowledgments
This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)
References
[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997
[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000
[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000
[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006
[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005
[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012
[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003
[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002
[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009
[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009
[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991
[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011
[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004
[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007
[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008
[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989
[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971
[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001
[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985
[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003
[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002
[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006
[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
the corresponding SOS polynomials Here we only list theLyapunov function
119881 (x) = 41
50
119909
2
1+
9
50
119909
2
2+
33
20
119909
2
3+
107
50
119909
2
4
+
69
100
119909
2
5+
13
10
119909
2
6+ sdot sdot sdot +
67
100
119909
4
4+
53
100
119909
4
5
(43)
Therefore the locally asymptotic stability is certifiedFurthermore we will consider the globally asymptotic
stability It suffices to find a Lyapunov function 119881(x) withrational coefficients which satisfies all the conditions inTheorem 3 By solving the associated SOS programming (18)we obtain
119881 (x) = 078997119909
2
1+ 019119909
2
2+ 22721119909
2
3
+ 14213119909
2
6+ sdot sdot sdot + 14213119909
4
2+ 071066119909
4
5
(44)
By use of the rational SOS recovery technique described inSection 42 we then obtain
119881 (x) = 79
100
119909
2
1+
19
50
119909
1119909
2+
9
100
119909
1119909
5+
19
100
119909
2
2+
227
100
119909
2
3
+
199
100
119909
2
4+
9
100
119909
2119909
5+
71
100
119909
2
5
+
71
50
119909
2
6+
71
50
119909
4
2+
71
100
119909
4
5
(45)
which exactly satisfies the conditions in Theorem 3 There-fore the globally asymptotic stability of this system is certi-fied
Table 1 shows the performance of Algorithm 9 on anothersix examples for globally asymptotic stability analysis ofdynamical systems in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 1 Examples 1ndash3 correspond to [9 Examples 2 3 9] and Examples 4ndash6correspond to [39 Example 7] [6 Example 22] and [40page 1341] For all these examples let the degree bound of theSOSes be 4 and set 120591 = 10
minus2 119863 = 100 Here 119899 and Numdenote the number of system variables and the number ofdecision variables in the LMI problem respectively deg(119881)denotes the degree of 119881(x) obtained fromAlgorithm 9 Timeis that for the entire computation runAlgorithm 9 in seconds
Example 11 ([41 Example A]) Consider a nonlinear contin-uous system
x = f (x) = [
1
2
] = [
minus119909
2
119909
1+ (119909
2
1minus 1) 119909
2
] (46)
Firstly we will certify the locally asymptotic stability of thissystem It suffices to find a Lyapunov function 119881(119909
1 119909
2)
with rational coefficients which satisfies all the conditions inTheorem 2 We can set up an associated SOS programming(16) with 119892(x) = 119909
2
1+ 119909
2
2minus 00001 When 119889 = 2 we obtain
119881 (x) = 14957119909
2
1minus 07279119909
1119909
2+ 11418119909
2
2 (47)
Table 1 Algorithm performance on benchmarks (globally asymp-totic stability)
Example 119899 Num deg(119881) Time(s)
1 2 11 2 13912 2 11 2 14183 3 18 2 16744 4 81 4 29515 2 24 4 18326 2 24 4 1830
Table 2 Algorithm performance on benchmarks (ROA)
Example119899 Num deg(119881)
120573 120573
Time(s)
1 2 28 2 54159277 0593 194172 3 46 2 26509902 276 266353 2 28 2 9941 205 9915
Let the tolerance 120591 = 10
minus2 and the bound of the commondenominator of the polynomial coefficients vector be 100By use of the rational SOS recovery technique described inSection 42 we then obtain a Lyapunov function
119881 (x) = minus
62
85
119909
1119909
2+
127
85
119909
2
1+
97
85
119909
2
2 (48)
Therefore the locally asymptotic stability is certifiedWe now construct Lyapunov functions to find certified
estimates of the ROA In the associated SOS programming(22) with BMI constraints we suppose 119901(119909
1 119909
2) = 119909
2
1+ 119909
2
2
When 119889 = 2 we obtain
119881 (119909
1 119909
2) = 06174455119909
2
1minus 040292119909
1119909
2+ 043078119909
2
2
120573 = 13402225
(49)
Let the tolerance 120591 = 10
minus5 and the bound of the commondenominator of the polynomial coefficients vector be 10
5By use of the rational SOS recovery technique described inSection 42 we obtain
119881 (119909
1 119909
2) =
6732
10903
119909
2
1minus
4393
10903
119909
1119909
2+
42271
10903
119909
2
2
120573 =
6701
5000
(50)
which exactly satisfy the conditions inTheorem 6Therefore
Ω
= (119909
1 119909
2) isin R2
|
119881 (119909
1 119909
2) le 1 (51)
is a certified estimate of the ROA of the given system
Table 2 shows the performance ofAlgorithm 9on anotherthree examples for computing verified estimates of ROAsof dynamical systems with the same fixed positive definite
Abstract and Applied Analysis 9
polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =
10
minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and
120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds
6 Conclusion
In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system
Acknowledgments
This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)
References
[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997
[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000
[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000
[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006
[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005
[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012
[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003
[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002
[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009
[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009
[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991
[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011
[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004
[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007
[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008
[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989
[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971
[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001
[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985
[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003
[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002
[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006
[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
polynomial 119901(x) given in the literature All the computationshave been performed on an Intel Core 2 Duo 20GHzprocessor with 2GB of memory In Table 2 Examples 1ndash3correspond to [22 Examples 1 2] and [20 page 75] For allthese examples let the degree bound of the SOSes be 4 120591 =
10
minus4 and 119863 = 10000 Here 119899 and Num denote the numberof system variables and the number of decision variablesin the BMI problem respectively deg(119881) and
120573 denoterespectively the degree of 119881(x) and value of 120573 obtained fromAlgorithm 9 whereas 120573 is reported in the literature time isthat for the entire computation run Algorithm 9 in seconds
6 Conclusion
In this paper we present a symbolic-numeric method onasymptotic stability andROAanalysis of nonlinear dynamicalsystems A numerical Lyapunov function and an estimateof ROA can be obtained by solving an (bilinear) SOS pro-gramming via BMI solverThen a method based on modifiedNewton iteration and rational vector recovery techniquesis deployed to obtain exact polynomial Lyapunov functionsand verified estimates of ROAs with rational coefficientsSome experimental results are given to show the efficiencyof our algorithm For future work we will consider theproblemof stability region analysis of nonpolynomial systemsby applying a rigorous polynomial approximate techniqueto compute an uncertain polynomial system whose set oftrajectories contains that of the given nonpolynomial system
Acknowledgments
This material is supported in part by the National NaturalScience Foundation of China under Grants 91118007 and61021004 (Wu Yang) the Fundamental Research Funds forthe Central Universities under Grant 78210043 (Wu Yang)and the Education Department of Zhejiang Province Projectunder Grant Y201120383 (Lin)
References
[1] A Polanski ldquoLyapunov function construction by linear pro-grammingrdquo Transactions on Automatic Control vol 42 no 7pp 1013ndash1016 1997
[2] T A Johansen ldquoComputation of Lyapunov functions forsmooth nonlinear systems using convex optimizationrdquo Auto-matica vol 36 no 11 pp 1617ndash1626 2000
[3] L Grune and FWirth ldquoComputing control Lyapunov functionsvia a Zubov type algorithmrdquo in Proceedings of the 39th IEEEConfernce on Decision and Control vol 3 pp 2129ndash2134 IEEEDecember 2000
[4] R C L F Oliveira and P L D Peres ldquoLMI conditions for robuststability analysis based on polynomially parameter-dependentLyapunov functionsrdquo Systems amp Control Letters vol 55 no 1pp 52ndash61 2006
[5] G Chesi A Garulli A Tesi and A Vicino ldquoLMI-based com-putation of optimal quadratic Lyapunov functions for odd poly-nomial systemsrdquo International Journal of Robust and NonlinearControl vol 15 no 1 pp 35ndash49 2005
[6] J Liu N Zhan and H Zhao ldquoAutomatically discoveringrelaxed Lyapunov functions for polynomial dynamical systemsrdquoMathematics in Computer Science vol 6 no 4 pp 395ndash4082012
[7] T Nguyen Y Mori T Mori and Y Kuroe ldquoQE approach tocommon Lyapunov function problemrdquo Journal of Japan Societyfor Symbolic and Algebraic Computation vol 10 no 1 pp 52ndash622003
[8] A Papachristodoulou and S Prajna ldquoOn the construction ofLyapunov functions using the sum of squares decompositionrdquoin Proceedings of the 41st IEEE Conference on Decision andControl vol 3 pp 3482ndash3487 IEEE December 2002
[9] Z She B Xia R Xiao and Z Zheng ldquoA semi-algebraicapproach for asymptotic stability analysisrdquo Nonlinear AnalysisHybrid Systems vol 3 no 4 pp 588ndash596 2009
[10] BGrosman andDR Lewin ldquoLyapunov-based stability analysisautomated by genetic programmingrdquo Automatica vol 45 no 1pp 252ndash256 2009
[11] K Forsman ldquoConstruction of Lyapunov functions using Grob-ner basesrdquo in Proceedings of the 30th IEEE Conference onDecision and Control pp 798ndash799 IEEE December 1991
[12] Z She B Xue and Z Zheng ldquoAlgebraic analysis on asymptoticstability of continuous dynamical systemsrdquo in Proceedings ofthe 36th International Symposium on Symbolic and AlgebraicComputation pp 313ndash320 ACM 2011
[13] S Prajna P A Parrilo and A Rantzer ldquoNonlinear controlsynthesis by convex optimizationrdquo Transactions on AutomaticControl vol 49 no 2 pp 310ndash314 2004
[14] B Xia ldquoDISCOVERER a tool for solving semi-algebraic sys-temsrdquo ACM Communications in Computer Algebra vol 41 no3 pp 102ndash103 2007
[15] P Giesl ldquoConstruction of a local and global Lyapunov functionusing radial basis functionsrdquo IMA Journal of AppliedMathemat-ics vol 73 no 5 pp 782ndash802 2008
[16] H-D Chiang and J S Thorp ldquoStability regions of nonlineardynamical systems a constructive methodologyrdquo Transactionson Automatic Control vol 34 no 12 pp 1229ndash1241 1989
[17] E J Davison and E M Kurak ldquoA computational methodfor determining quadratic Lyapunov functions for non-linearsystemsrdquo vol 7 no 5 pp 627ndash636 1971
[18] E Cruck R Moitie and N Seube ldquoEstimation of basinsof attraction for uncertain systems with affine and Lipschitzdynamicsrdquo Dynamics and Control vol 11 no 3 pp 211ndash2272001
[19] R Genesio M Tartaglia and A Vicino ldquoOn the estimation ofasymptotic stability regions state of the art and new proposalsrdquoTransactions on Automatic Control vol 30 no 8 pp 747ndash7551985
[20] Z Jarvis-Wloszek Lyapunov based analysis and controller syn-thesis for polynomial systems using sum-of-squares optimization[PhD thesis] University of California Berkeley Calif USA2003
[21] S Prakash J Vanualailai and T Soma ldquoObtaining approximateregion of asymptotic stability by computer algebra a case studyrdquoTheSouth Pacific Journal of Natural andApplied Sciences vol 20no 1 pp 56ndash61 2002
[22] T Weehong and A Packard ldquoStability region analysis usingsum of squares programmingrdquo in Proceedings of the AmericanControl Conference (ACC rsquo06) pp 2297ndash2302 IEEE June 2006
[23] P A Parrilo ldquoSemidefinite programming relaxations for semi-algebraic problemsrdquoMathematical Programming B vol 96 no2 pp 293ndash320 2003
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
[24] S Prajna A Papachristodoulou and P Parrilo 2002 SOS-TOOLS Sum of squares optimization toolbox for MATLABhttpwwwcdscaltechedusostools
[25] J Lofberg ldquoYALMIP a toolbox for modeling and optimizationin MATLABrdquo in Proceedings of the 2004 IEEE InternationalSymposium on Computer Aided Control SystemDesign pp 284ndash289 September 2004
[26] J F Sturm ldquoUsing SeDuMi 102 a MATLAB toolbox foroptimization over symmetric conesrdquoOptimizationMethods andSoftware vol 11 no 1ndash4 pp 625ndash653 1999
[27] M Kocvara and M Stingl PENBMI Userrsquos Guide (version 2 0)httpwwwpenoptcom 2005
[28] E Kaltofen B Li Z Yang and L Zhi ldquoExact certification ofglobal optimality of approximate factorizations via rationalizingsums-of-squares with floating point scalarsrdquo in Proceedings ofthe 21th International Symposium on Symbolic and AlgebraicComputation (ISSAC rsquo08) pp 155ndash164 ACM Hagenberg Aus-tria July 2008
[29] E L Kaltofen B Li Z Yang and L Zhi ldquoExact certification inglobal polynomial optimization via sums-of-squares of rationalfunctions with rational coefficientsrdquo Journal of Symbolic Com-putation vol 47 no 1 pp 1ndash15 2012
[30] A Platzer and E M Clarke ldquoThe image computation problemin hybrid systems model checkingrdquo in Proceedings of the 10thInternational Conference on Hybrid Systems Computation andControl (HSCC rsquo07) vol 4416 pp 473ndash486 Springer April 2007
[31] H Khalil Nonlinear Systems Prentice hall Upper Saddle RiverNJ USA 3rd edition 2002
[32] J Bochnak M Coste and M Roy Real Algebraic GeometrySpringer Berlin Germany 1998
[33] G Chesi ldquoLMI techniques for optimization over polynomialsin control a surveyrdquo Transactions on Automatic Control vol 55no 11 pp 2500ndash2510 2010
[34] F Leibfritz and E M E Mostafa ldquoAn interior point constrainedtrust regionmethod for a special class of nonlinear semidefiniteprogramming problemsrdquo SIAM Journal onOptimization vol 12no 4 pp 1048ndash1074 2002
[35] M Kocvara and M Stingl ldquoPennon a code for convex nonlin-ear and semidefinite programmingrdquo Optimization Methods ampSoftware vol 18 no 3 pp 317ndash333 2003
[36] M Wu and Z Yang ldquoGenerating invariants of hybrid systemsvia sums-of-squares of polynomials with rational coefficientsrdquoin Proceedings of the International Workshop on Symbolic-Numeric Computation pp 104ndash111 ACM Press 2011
[37] W Lin M Wu Z Yang and Z Zeng ldquoExact safety verificationof hybrid systems using sums-of-squares representationrdquo Sci-ence China Information Sciences 15 pages 2012
[38] Z Yang M Wu and W Lin ldquoExact verification of hybridsystems based on bilinear SOS representationrdquo 19 pages 2012httparxivorgabs12014219
[39] A Papachristodoulou and S Prajna ldquoA tutorial on sum ofsquares techniques for systems analysisrdquo in Proceedings of theAmerican Control Conference (ACC rsquo05) pp 2686ndash2700 IEEEJune 2005
[40] J Lofberg ldquoModeling and solving uncertain optimization prob-lems in YALMIPrdquo in Proceedings of the 17th World CongressInternational Federation of Automatic Control (IFAC rsquo08) pp1337ndash1341 July 2008
[41] U Topcu A Packard and P Seiler ldquoLocal stability analysisusing simulations and sum-of-squares programmingrdquo Auto-matica vol 44 no 10 pp 2669ndash2675 2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of