06811177

192 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 1, JANUARY 2015

Haar-Based Stability Analysis of LPV SystemsLeonardo O. de Araújo, Paulo C. Pellanda, Member, IEEE,

Juraci F. Galdino, and Alberto M. Simões

Abstract—A new gridding-based algorithm for stability analysisof Linear Parameter-Varying (LPV) systems is presented. Thealgorithm inherits the main features of classical gridding tech-niques: it can handle a vast class of parametric dependencies aswell as non-convex parametric domains. Novelty of the proposedapproach lies in the use of Haar wavelet transform theory to guar-antee constraint satisfaction over the entire parametric domain,even for an arbitrarily sparse grid. It represents a major improve-ment over traditional gridding approaches, which fail to providesuch a certificate without requiring a posteriori verification tests.The resulting algorithms rely on semidefinite programing and arerelated to sufficient stability conditions whose degree of conser-vatism decreases as the grid density and the Haar truncation levelincrease. Two numerical examples corroborate the validity of theproposed algorithms.

Index Terms—Haar wavelet, linear parameter-varying (LPV)system, parameter-dependent Lyapunov function, stability anal-ysis.

I. INTRODUCTION

Stability analysis of Linear Parameter-Varying (LPV) systems re-mains a challenge, despite the remarkable recent progress in dynamicalsystem theory. Most of the analysis methods for uncertain or time-varying linear systems based on Lyapunov stability theory turn out tobe inadequate in the particular case of LPV systems. First, parametersare generally assumed to take value in some convex polytope, usually ahyperrectangle [1], [2] or a simplex [3]–[5]. Unfortunately, such an as-sumption is invalid for the large class of LPV systems in which the setof allowable parameter trajectories defines more irregular domains. Tocircumvent the eventual non-convexity of the parametric domain, thosemethods have to recur to some sort of convex covering technique, seee.g. [6]. However, such a scheme is likely to introduce conservatismsince non-realistic trajectories are taken into account. Second, existingapproaches can generally handle only a limited class of parametric de-pendencies of system matrices, basically linear [3]–[5], [7], affine [1],[2], [8] or rational [9]–[11]. Consequently, those methods are unableto directly address more general dependencies encountered in someapplications of great practical interest, like, for example, quasi-LPVsystems arising in aerospace applications where some of the endoge-nous parameters enter system matrices via trigonometric functions[12]. To apply the above methods to such problems, one has to eitherresort to some kind of linearization scheme or to embed the systeminto a polytopic one, which is recognizably a conservative procedure.

Manuscript received May 28, 2013; revised December 30, 2013; acceptedApril 24, 2014. Date of publication May 7, 2014; date of current versionDecember 22, 2014. This work was supported in part by the Brazilian AgenciesCNPq under Grants 309846/2011-0 and 308046/2009-9 and FAPERJ underGrant E-26/102.269/2009. Recommended by Associate Editor F. Blanchini.

The authors are with the Electrical and Defense Engineering GraduatePrograms, Military Institute of Engineering, Rio de Janeiro, RJ, Brazil (e-mail:[email protected]; [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TAC.2014.2322436

A well-known strategy to overcome the above drawbacks is thegridding approach [13], [14]. One of its most appealing features is themuch more general class of parameter-dependent Lyapunov Functions(LF) or systems that can be handled, including non-convex parametricdomains. On the other hand, a major flaw of traditional gridding-based techniques is that computed solutions are guaranteed to satisfyconstraints only at those points in the grid, not necessarily over theentire parametric domain. Consequently, an over-optimistic estimateof the stability domain may follow. In practice, one must select thegrid as dense as possible and then hope that constraints will be satisfiedover the entire domain. Obviously, the denser the grid, the higher theassociated computational burden.

In this work, new gridding-based algorithms for stability analysisof LPV systems are presented. The proposed algorithms build on thewavelet theory [15], [16] to overcome the above limitations of theclassical gridding schemes. Novelty of the proposed approach lies inthe use of wavelet theory to guarantee constraint satisfaction over theentire analysis domain, even when a sparse parameter grid is con-sidered. It represents a major improvement over traditional griddingapproaches, which fail to provide such a certificate without requiringadditional verification tests. To that end, the parameter-dependent statematrix is initially replaced in matrix inequalities by an arbitrarilyprecise approximation obtained by a truncated Haar series expansion.Then a quadratic-in-the-state Haar-based parameter-dependent LF issought, while somehow taking into account the neglected terms of theHaar expansion of the state matrix. The resulting computational testinvolves solving a semidefinite program (SDP) and corresponds to asufficient stability condition whose degree of conservatism decreasesas the grid density and/or the resolution level of the Haar decomposi-tion increases. The algorithm also inherits the main benefits of griddingtechniques: it can directly address a large class of systems as well asnon-convex parametric domains.

Notation: For a given real matrix M ∈ Rn×m, MT denotes its

transpose, ‖M‖ its induced 2-norm, and Mpq its (p, q)th entry.

Notation S(M)Δ= M +MT is used in large matrix inequalities. Sn

stands for the set of real symmetric matrices of size n. For two matricesM,N ∈ S

n, notation M � N means that M −N is positive definiteand M � N that M −N is positive semi-definite. Moreover, notation±M � N means that both inequalities M � N and −M � N aresatisfied. L2(R) denotes the space of square-integrable real-valuedfunctions. For a matrix-valued mapping M : R �→ R

n×m, M(·) ∈L2(R) means that Mpq(·) ∈ L2(R), for all entry (p, q). For a compactdomain C ⊂ R

n, IC(·) stands for the indicator function of C

IC(θ)Δ={1, if θ ∈ C,0, if θ ∈ C.

For a mapping f : C → R, f(·) ∈ LC2 (R) denotes that f(·)IC(·) ∈

L2(R). The inner product of two functions f, g ∈ L2(R) is denoted

〈f(θ), g(θ)〉 Δ=

∞∫−∞

f(θ)g(θ)dx.

0018-9286 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 60, NO. 1, JANUARY 2015 193

For two mappings F : R → Rn×m and g : R → R, 〈〈F (θ), g(θ)〉〉

stands for a matrix whose (p, q)th entry is given by 〈F pq(θ), g(θ)〉.⊕ stands for the direct sum of disjunct subspaces. In denotes theidentity matrix of dimension n× n. Dn denotes the n-dimensionalunit hypercube with vertexes whose coordinates are either 0 or 1. N≥0

(R≥0) and N+ (R+) denote, respectively, nonnegative and strictlypositive natural (real) numbers.

II. PRELIMINARIES

A. Problem Formulation

Consider an LPV state-space representation

x(t) =A (θ(t))x(t) +B (θ(t))u(t)

y(t) =C (θ(t))x(t) +D (θ(t))u(t) (1)

where x ∈ Rn, u ∈ R

m, y ∈ Rp and θ ∈ R

r denote state, input,output and time-varying parameter vectors respectively. A large classof nonlinear dynamical systems, including many systems of interest inaerospace and mechatronics fields, are known to admit such a state-space representation.

Only a very few assumptions on LPV model (1) are made here.Firstly, parameter trajectories θ(t) are assumed to be contained ina given compact domain Θ, whereas the rate of variation θ(t) issupposed to be valued in a given hyperrectangle Θd. It is also assumedthat mapping θ �→ A(θ) ∈ LΘ

2 (R). If, on the one hand, such a generalparameter dependence renders representation (1) flexible enough toencompass a wide range of practical applications, on the other handobtaining stability analysis tools for such a large class of systemsconsists in a particularly challenging task.

A well-known sufficient condition to the stability of the system (1)is the existence of a quadratic parameter-dependent LF [17]

V (x, θ) = xTP (θ)x (2)

with mapping P : Θ → Sn piecewise differentiable. Time dependence

has been omitted in (2) for notational simplicity. It can be easily shownthat V (·, ·) in (2) represents an LF for (1) if and only if P (θ) is afeasible solution to the following set of Parameterized Linear MatrixInequalities (PLMI):

P (θ) � 0, ∀θ ∈ Θ, (3)

∂P (θ)

∂θθ + S (P (θ)A(θ)) ≺ 0, ∀θ ∈ Θ, ∀θ ∈ Θd. (4)

The feasibility problem defined by PLMI (3), (4) is recognizablyhard to solve by virtue of the infinitely many constraints involved.Here the originally infinite-dimensional feasibility program (3), (4)is solved by dint of a gridding scheme in which the PLMI problemis reduced to a finite-dimensional program. As stressed before, mostof the techniques proposed in the literature to solve this problemrequire stronger assumptions on the parametric dependence of stateand Lyapunov matrices, or also on the shape of domain Θ, see, e.g.,[18], [19].

B. The Haar Transform

The Haar transform [15], [16] is a discrete wavelet transformproviding an infinite-dimensional representation of square-integrablefunctions. To begin with, consider the so-called father and mother Haarfunctions, φ : R → R and ψ : R → R respectively, given by

φ(θ)Δ=

{1, if 0 ≤ θ < 1,0, otherwise,

ψ(θ)Δ=

{1, if 0 ≤ θ < 0.5,−1, if 0.5 ≤ θ < 1,0, otherwise.

(5)

Next, consider the following sets of basis functions:{φj(θ)

Δ= φ(θ − j), j ∈ N

},{

ψij(θ)Δ= 2

i2ψ(2iθ − j), i ∈ N, 0 ≤ j < 2i

}. (6)

Also, consider functional subspaces V0Δ= span(φj : j ∈ N), Wi

Δ=

span(ψij : 0 ≤ j < 2i) and let Vi+1Δ= Vi ⊕Wi. An important prop-

erty concerning the above subspaces is that V0 ⊂ · · ·Vi ⊂ Vi+j ⊂· · · ⊂ L2, with Vi → L2 as i → ∞.

With the above definitions in mind, the Haar transform fΣ∞ ofa given square-integrable function f : D → R ∈ L2(R) is such thatfΣ∞(θ) = f(θ), ∀θ, and can be obtained as

fΣ∞(θ)Δ= v0φ0(θ) +

∞∑i=0

2i−1∑j=0

wijψij(θ) (7)

where coefficients wij and v0 of the above Haar series expansion aregiven by the inner products wij = 〈f, ψij〉 and v0 = 〈f, φ0〉. The Haartransform can be seen as a function which is piecewise constant ininfinitely many intervals.

As the resolution index i increases in (7), each new projectionof the original function f(θ) onto the subspace Wi, added to itsprojection onto the subspace Vi, provides further information for itsHaar representation. If the outer sum in the second term on the right-hand side of (7) is truncated, then one has only an approximation tof(·) instead of an exact representation. In the sequel, notation fΣ∞(·)is used to denote the precise representation of a function f(·) given byits Haar transform (7), whereas fΣI

(·) stands for the approximation tof obtained by truncating the expansion (7) as follows:

fΣI(θ)

Δ= v0φ0(θ) +

I∑i=0

2i−1∑j=0

wijψij(θ). (8)

Obviously, fΣI(·) → fΣ∞(·) = f(·) as I → ∞.

The Haar transform presents an energy conservation property simi-lar to that in Parseval theorem. The energy in Haar coefficients tends todecrease as the resolution i increases [20]. At resolution level I , there

exist scalars {τj}2I−1

j=0∈ R+ such that

|wIj | ≤ 2−I(s+ 12 )τj (9)

where s ∈ R≥0 represents the Hölder exponent of f(·).Now, let Γi,j ⊂ R denote the sub-domain where ψij(·) is different

from zero, as illustrated in Fig. 1 for the particular case of ψ21(·). Forhigher decomposition levels (i > I), it can be shown that there existscalars κij such that

|wij | ≤ 2−i(s+ 12 )κij (10)

where κij ≤ τm, for m ∈ {0, 1, . . . , 2I − 1} representing the indexfor which the inclusion Γi,j ⊂ ΓI,m holds. Hence, if f represented asignal, then its energy would be mostly concentrated in low-resolutioncoefficients.

On the basis of the scaling property of the Haar father function, itcan be shown that for any decomposition level i > I

2i−1∑j=0

φj(2iθ) =

2I−1∑j=0

φj(2Iθ) = φ0(θ), ∀θ ∈ D. (11)


Fig. 1. Haar basis functions.

For a matrix-valued function M : D → Rn×m ∈ L2(R

n×m), itsHaar transform is defined as

MΣ∞(θ)Δ= M0φ0(θ) +

∞∑i=0

2i−1∑j=0

Mijψij(θ), (12)

with M0 = 〈〈M,φ0〉〉, Mij = 〈〈M,ψij〉〉 and MΣ∞(θ) = M(θ). Al-though the Haar transform formulation presented here involves a uni-dimensional domain, it can be easily extended to the multidimensionalcase [15].

Consider momentarily the affine mapping fa : θ �→ aθ + b, witha, b ∈ R, and θ ∈ D. The Haar transform of such an affine mappingpresents some interesting properties that play a central role in thiswork. First, the coefficients of the Haar transform of fa(θ) can beshown to be given by

wij = 〈fa(θ), ψij(θ)〉 = −2−i(1+ 12 )

a

4. (13)

Thus, for a given resolution level i the Haar coefficients of the affinemapping fa(θ) are the same for all indexes j. Another particularproperty for the affine mapping fa that follows from (13) is that

w(i+m)j = 2−3m2 wij . (14)

Let DIΔ= {θk}2

I+1−1k=0 ⊂ D denote a set of 2I+1 linearly spaced

points, where each point is located at the center of intervals definedby subspace VI+1, i.e.

θk =(k +

1

2

)1

2I+1=

2k + 1

2I+2. (15)

Finally, let fΣIrepresent the truncated Haar series expansion of fa.

It can be verified that for any resolution level I ∈ N+ it follows thatfΣI

(θk) = fa(θk), ∀θk ∈ DI . In other words, at every point in DI

the truncated Haar series expansion of such an affine mapping alwaysprovides the exact value of the function.

III. A NEW HAAR-BASED SUFFICIENT CONDITION

FOR QUADRATIC STABILITY

In this section, a new sufficient condition for quadratic stabilityof LPV system (1) on the basis of a parameter-independent LF isobtained. For the sake of clarity, the scalar case θ ∈ D is discussed.

Following the notation in Section II-B, let AΣ∞(θ) represent theHaar transform of state matrix A(θ)

AΣ∞(θ)Δ= AΣI

(θ) +AE(θ) (16)

where AΣI(θ) corresponds to a truncated series expansion similar to

(8), and AE(θ) represents the residual matrix given by

AE(θ) =

∞∑i=I+1

2i−1∑j=0

Aijψij(θ). (17)

As previously mentioned, a fundamental property of the Haartransform is that the coefficients of highest resolution gathered inAE(θ) are those of lowest energy level. Therefore, for a sufficienthigh decomposition level I in (16) the residue AE(θ) carries poor orirrelevant information about the parametric dependence of the statematrix. In such a case, the finite dimensional series expansion AΣI

(θ)becomes a very precise representation of A(θ). Notice that AΣI

(θ)is piecewise constant, and hence can be interpreted as some sort ofdiscretization of A(θ).

A central idea here consists in considering in PLMI (4) the truncatedHaar series expansion AΣI

(θ) instead of the state matrix A(θ), whiletaking into account some bound on the norm of residue AE(θ). Tobegin with, consider such a bound provided in the following lemma.

Lemma 1: Let AE(θ) represent the residual matrix (17) appearingin the Haar transform (16) of A(θ). Then there exist real-valued

piecewise constant functions {αj(θ)}2I−1

j=0such that, for all θ ∈ D

‖AE(θ)‖ ≤2I−1∑j=0

αj(θ). (18)

Proof: According to (6), AE(θ) can be rewritten as

AE(θ) =

∞∑i=I+1

2i−1∑j=0

2i2Aijψ(2

iθ − j).

For each entry (p, q) of matrices Aij , consider the upper bounds givenby (9), (10): |Apq

ij | ≤ κpqij 2

−i(s+(1/2)), where s ∈ N+ and κpqij ∈ R+.

Next, the following sets of parameter-dependent matrices are defined:{Mij ∈R

n×n, i≥I, 0≤j<2i : Mpqij

Δ=∣∣∣2 i

2Apqij ψ(2

iθ−j)∣∣∣} ,{

Mij ∈Rn×n, i≥I, 0≤j<2i :Mpq

ij

Δ=κpq

ij 2−is

∣∣ψ(2iθ−j)∣∣}.

Notice that Mpqij (θ) ≤ Mpq

ij (θ), ∀θ. Moreover

‖AE(θ)‖ ≤

∥∥∥∥∥∥∞∑

i=I+1

2i−1∑j=0

Mij(θ)

∥∥∥∥∥∥ ≤

∥∥∥∥∥∥∞∑

i=I+1

2i−1∑j=0

Mij(θ)

∥∥∥∥∥∥ . (19)

The (p, q)th entry of the matrix on the right-hand side of (19) can berewritten as follows:

∞∑i=I+1

2i−1∑j=0

κpqij 2

−is∣∣ψ(2iθ − j)

∣∣=

∞∑i=0

2(I+i+1)−1∑j=0

κpq(I+i+1)j2

−(I+i+1)sφj(2I+i+1θ) (20)

= 2−(I+1)s

∞∑i=0

2−is

2(I+i+1)−1∑j=0

κpq(I+i+1)jφj(2

I+i+1θ)


≤

(2−(I+1)s

∞∑i=0

2−is

)2I−1∑m=0

τpqm φm(2Iθ) (21)

=2−(I+1)s

1− 2−s

2I−1∑m=0

τpqm φm(2Iθ). (22)

Expression (20) comes from the equivalence |ψ(θ − j)| = φj(θ),whereas (21) follows from property (11) and the definitions in (9),(10). Expression (22) is easily obtained after identifying the firstsummation in (21) as the sum of a geometric series. Consider nowthe following set of matrices:{M j(θ) ∈ R

n×n, 0 ≤ j < 2I : Mpq

j (θ)

Δ=

2−(I+1)sτpqj

1− 2−sφj(2

Iθ)

}. (23)

Inequality (18) finally follows from (19), (22) and the definition

αj(θ)Δ= ‖M j(θ)‖. �

For a prescribed resolution level I , let DIΔ= {θk}2

I+1−1k=0 ⊂ D

denote as in Section II-B the grid consisting of 2I+1 linearly spacedpoints in (15). The next theorem states the main result of this section.

Thmeorem 1: Consider the LPV system (1) and the upper bound(18) on the residue of the state matrix. Then the LPV system isquadratically asymptotically stable if there exist γ ∈ R+ and P ∈Sn � 0 such that the following set of LMI holds:

P − γIn � 0, (24)

S (PAΣI(θk))+2γ

2I−1∑j=0

αj(θk)In≺0, ∀θk ∈ DI . (25)

Proof: Since P ∈ Sn � 0, constraint (24) implies ‖P‖ ≤ γ.

Therefore, for all θ ∈ D

S (PAE(θ)) � ‖S (PAE(θ))‖ In � 2‖P‖ ‖AE(θ)‖ In

� 2‖P‖2I−1∑j=0

αj(θ)In � 2γ

2I−1∑j=0

αj(θ)In.

(26)

Next, since AΣI(·) is by construction piecewise constant, then for all

θ ∈ D there exist at least one discrete value θk ∈ DI such that

S (PAΣI(θ)) + 2γ

2I−1∑j=0

αj(θ)In

= S (PAΣI(θk)) + 2γ

2I−1∑j=0

αj(θk)In ≺ 0. (27)

Hence, for all θ ∈ D, it follows that:

S (PA(θ)) =S (PAΣ∞(θ)) = S (PAΣI(θ) + PAE(θ))

�S (PAΣI(θ)) + 2γ

2I−1∑j=0

αj(θ)In ≺ 0.

Thus, PLMI (3), (4) are satisfied for all θ ∈ D, which implies thequadratic stability of LPV system (1). �

In summary, feasibility of the finite-dimensional program on Theo-rem 1 implies feasibility of the infinite-dimensional problem given byPLMI (3), (4), which in turn allows to infer stability of the LPV system

(1). Notice that the conservatism of the condition of Theorem 1 tendsto decrease with increasing decomposition level I of the state matrix,since AΣI

(θ) → AΣ∞(θ) and AE(θ) → 0 as I → ∞. The number ofdecision variables on program (24), (25) depends only on the numbern of states and is given by 1 + (n(n+ 1)/2). On the other hand, anincreasing decomposition level I implies an increasing grid densityand an exponentially increasing number of n× n matrix inequalitiesconstraints: 2I+2.

IV. HAAR-BASED PARAMETER-DEPENDENT LFFOR QUADRATIC STABILITY ANALYSIS

Conservatism in Theorem 1 can be reduced by adopting aparameter-dependent LF. To that end, consider initially the followingpiecewise constant function constructed by Haar basis functions:

Q(θ)Δ= Q0φ0(θ) +

G∑g=0

2g−1∑h=0

Qghψgh(θ) (28)

with Q0, {Qgh} ∈ Sn. The candidate Lyapunov matrix P (θ) is then

selected as

P (θ)Δ=

∫Q(θ)∂θ = Q0φ0(θ) +

G∑g=0

2g−1∑h=0

Qghψgh(θ) + Q (29)

where Q ∈ Sn and

φ0(θ)Δ=

∫φ0(θ)∂θ,

{ψgh(θ)

Δ=

∫ψgh(θ)∂θ, g ∈ N, 0 ≤ h < 2g

}.

(30)

Alternatively to (29), P (θ) can also be precisely represented by theinfinite series expansion corresponding to its Haar transform

PΣ∞(θ)Δ= PΣI

(θ) + PE(θ) (31)

where PΣI(θ) represents, as before, the truncated series expansion

PΣI(θ) = P0φ0 +

I∑i=0

2i−1∑j=0

Pijψij(θ)

and PE(θ) corresponds to the residue

PE(θ) =

∞∑i=I+1

2i−1∑j=0

Pijψij(θ) (32)

with P0 = 〈〈P (θ), φ0(θ)〉〉 and Pij = 〈〈P (θ), ψij(θ)〉〉.Notice that P (θ) in (29) is piecewise affine. Hence, it follows

by construction that, for any resolution level I ≥ G, it presents thefollowing key property:

PΣI(θk) = P (θk), ∀θk ∈ DI . (33)

Analogously to Lemma 1, the next lemma provides a bound on theresidual matrix PE(θ). First, let

λj(θ)Δ=⟨φ0(θ), ψIj(θ)

⟩, χghj(θ)

Δ=⟨ψgh(θ), ψIj(θ)

⟩(34)

represent the Haar coefficients at level I of expansion set (30).Lemma 2: Consider P (θ) in (29) and its Haar transform (31). For

any decomposition level I≥G∈N+, there exist scalars {πh}2G−1

h=0 ∈


R+ such that the following statements are equivalent:

1) There exist matrices Q0, {Qgh} ∈ Sn such that

±2I2

2I−1∑j=0

(Q0λj +

G∑g=0

2g−1∑h=0

Qghχghj

)φj(2

Iθk)

�2G−1∑h=0

πhφh(2Gθk)In, ∀θk ∈ DI ; (35)

2) The following PLMI holds:

PE(θ) �2G−1∑h=0

πhφh(2Gθ)In, ∀θ ∈ D. (36)

Proof: Only implication (1) =⇒ (2) is proved. Consider P (θ) in(29). The coefficients at level I of PΣ∞(θ) are given by

PIj(θ) = 〈〈P (θ), ψIj(θ)〉〉 = Q0λj +

G∑g=0

2g−1∑h=0

Qghχghj . (37)

In (37), the equality 〈〈Q, ψIj(θ)〉〉 = 0 was used. Inequality (35) canthen be rewritten as

±2I2

2I−1∑j=0

PIjφj(2Iθk) �

2G−1∑h=0

πhφh(2Gθk)In. (38)

Next, since {φj} are piecewise constant, then for all θ ∈ D there existat least one discrete value θk ∈ DI such that

2I2

2I−1∑j=0

PIjφj(2Iθk) = 2

I2

2I−1∑j=0

PIjφj(2Iθ). (39)

The right-hand side of (39) can be rewritten as follows:

2

(2

I2

2

) 2I−1∑j=0

PIjφj(2Iθ) =

∞∑i=0

1

2i

(2

I2

2

) 2I−1∑j=0

PIjφj(2Iθ)

=

∞∑i=0

2I−1∑j=0

2I2 PIj

2i+1φj(2

Iθ)

=

∞∑i=0

2I−1∑j=0

2I+i+1

2 P(I+i+1)jφj(2Iθ) (40)

=

∞∑i=0

2I+i+1−1∑j=0

2I+i+1

2 P(I+i+1)jφj(2I+i+1θ) (41)

=

∞∑i=0

2I+i+1−1∑j=0

P(I+i+1)j

∣∣ψ(I+i+1)j(θ)∣∣ (42)

=

∞∑i=I+1

2i−1∑j=0

Pij |ψij(θ)|. (43)

Step (40) follows from property (14) of affine mapping P (·) in (29),whereas (41) follows from property (11). Step (42) follows from theequivalence φj(2

I+mθ)=2−((I+m)/2)|ψ(I+m)j(θ)|. Now, notice that

Pijψij(θ)=

{Pij |ψij(θ)| , if j 1

2i+1 ≤ θ < (j + 1) 12i+1 ,

−Pij |ψij(θ)| , if (j + 1) 12i+1 ≤ θ < (j + 2) 1

2i+1 ,0, otherwise

which leads to the conclusion that

±∞∑

i=I+1

2i−1∑j=0

Pij |ψij(θ)| �∞∑

i=I+1

2i−1∑j=0

Pijψij(θ) = PE(θ). (44)

Finally, (36) follows from (44), considering (38), (39) and (43). �The bound on PE(θ) obtained on Lemma 2 will be used in the next

theorem, in which the main result of this section is presented.Theorem 2: Consider the LPV system (1) and upper bound (18) on

the residue of the state matrix. Suppose that θ(t) ∈ D, |θ(t)| ≤ ρ ∈R, ∀t. Also, consider the candidate matrix P (θ) defined in (29) andits partial derivative Q(θ) in (28). Then the LPV system is quadrati-cally asymptotically stable if there exist matrices Q0, {Qgh}, Q ∈ S

n,

and scalars {πh}2G−1

h=0 , {γj}2I−1

j=0∈ R+, such that LMI (35) plus the

following set of LMI hold for all θk ∈ DI and given resolution levelsG ≤ I ∈ N+:

P (θk) �2G−1∑h=0

πhφh(2Gθk)In, (45)

P (θk) +

2G−1∑h=0

πhφh(2Gθk)In �

2I−1∑j=0

γjφj(2Iθk)In, (46)

± ρQ(θk) + S (P (θk)AΣI(θk)) + ξ(θk)In ≺ 0 (47)

where

ξ(θk)Δ=2

2G−1∑h=0

‖AΣI(θk)‖πhφh(2

Gθk)+2

2I−1∑j=0

γjαj(θk)φj(2Iθk).

Proof: First, notice that since P (·) is piecewise affine, thenPΣI

(θk) = P (θk), ∀θk ∈ DI . Consequently, (45) can be rewritten inthe equivalent form

PΣI(θk) �

2G−1∑h=0

πhφh(2Gθk)In � 0.

Since PΣI(·) is piecewise constant, then for all θ ∈ D there exist one

discrete value θk ∈ DI such that PΣI(θ) = PΣI

(θk), and hence

PΣI(θ) �

2G−1∑h=0

πhφh(2Gθ)In � 0, ∀θ ∈ D. (48)

Now, let PE(θ) represent the unknown residue (32) associated to P (θ).Thus, from (48) it follows that:

PΣI(θ) + PE(θ) = P (θ) �

2G−1∑h=0

πhφh(2Gθ)In + PE(θ). (49)

Since (35) is verified by hypothesis, then (36) holds according to

Lemma 2. Consequently, since {πh}2G−1

h=0 ∈ R+, it follows from (49)and (36) that P (θ) � 0, ∀θ ∈ D, which is exactly PLMI (3).

Next, following the same arguments leading to (48), it follows from(46) that

PΣI(θ)+

2G−1∑h=0

πhφh(2Gθ)In�

2I−1∑j=0

γjφj(2Iθ)In, ∀θ∈D. (50)

Thus, from (36) one can conclude that, ∀θ ∈ D

P (θ) � PΣI(θ) +

2G−1∑h=0

πhφh(2Gθ)In �

2I−1∑j=0

γjφj(2Iθ)In. (51)


Finally, once again by the arguments leading to (48) and (50), itfollows that inequality (47) also holds for all θ ∈ D. Consequently,from (36) and (51) it follows that:

0 � ± ρ∂P

∂θ(θ) + S (PΣI

(θ)AΣI(θ)) + ξ(θ)In

� ± ρ∂P

∂θ(θ) + S (PΣI

(θ)AΣI(θ))

+ 2 ‖PE(θ)‖ ‖AΣI(θ)‖ In + 2 ‖P (θ)‖ ‖AE(θ)‖ In

� ± ρ∂P

∂θ(θ)

+ S ((PΣI(θ) + PE(θ))AΣI

(θ) + P (θ)AE(θ)) ,

= ± ρ∂P

∂θ(θ) + S (P (θ)A(θ)) . (52)

Now, since, by assumption, |θ(t)| ≤ ρ ∈ R, ∀t, then it follows from(52) that (4) is also satisfied. The quadratic stability of LPV system (1)finally comes from the fact that both PLMI (3), (4) are satisfied. �

The quantity ξ(θk) appearing in LMI (47) provides a measure of theconservatism resulting from the Haar truncation process. Notice thatξ(θ), and hence the conservatism, tends to decrease with increasingresolution levels I and G. Indeed, increasing resolution I correspondsto taking more information about the system into account, whereasincreasing resolution G corresponds to providing more degree offreedom to the Lyapunov candidate with respect to its parametricdependence. On the other hand, increasing level I leads to an increasein the number of LMI constraints, whereas increasing G impliesmore decision variables. The total number of decision variables is2I + 2G(1 + n(n+ 1)) + n(n+ 1)/2, whereas the number of n× nmatrix constraints is given by 6(2I+1).

For resolution levels I and G tending to infinity, ξ(θ) vanishesasymptotically, and hence the only source of conservatism remainingin Theorem 2 comes from the use of a quadratic-in-the-state LF. If, onthe one hand, it can be argued that the quadratic state dependence istoo restrictive, on the other hand, the parametric dependence of theconsidered LF is much more general. It encompasses as particularinstances the great majority of dependencies in the literature, e.g.,the popular homogeneous polynomial case. In the limit, the stabilitycondition on Theorem 2 involves a quadratic-in-the-state LΘ

2 -in-the-parameter LF.

Although the discussion has been limited to the scalar case θ ∈ D,the above results can also be extended to the multi-dimensional case[15]. In such a case, a different truncation level can be consideredfor each one of the parameters, hence allowing for a better balancingbetween the required computational effort and the conservatism of thestability analysis test.

V. NUMERICAL EXPERIMENTS

A. Example 1

Consider the following model introduced in [11]:

A (θ(t)) =

[0 1

−1+θ(t)−θ2(t)1+θ(t)

−1

](53)

where the parameter θ(t) ∈ [0, ζ], ∀t. The objective in this exampleis to determine the maximum ζ, denoted ζ∗, such that the origin isasymptotically stable, even for unbounded parameter variation rate.

In [11], a necessary and sufficient condition for the stability of aclass of LPV systems including (53) has been obtained on the basisof a homogeneous polynomial LF. When applied to (53) considering aquadratic LF, the resulting LMI condition determines ζ∗ = 4.256.

Fig. 2. (a) Estimates of ζ∗ in Example 1 for different resolution levels Igiven by Theorem 1 and the necessary condition (54); the value given by thenecessary and sufficient condition in [11] is also shown; (b) Estimates of ρ∗ inExample 2 for different resolution levels {I, G} given by Theorem 2.

Fig. 2(a) depicts estimates of ζ∗ provided by Theorem 1 for differentresolution levels I . As expected, the estimates approach ζ∗ from belowas I increases, which represents a key property of the proposed ap-proach. For I ≥ 8, the conservatism in Theorem 1 becomes negligible,and hence the estimated ζ∗ virtually coincide with the true boundζ∗ = 4.256 for quadratic stability. For each truncation level I , theestimated ζ∗ is found by a bisection algorithm on the normalizationfactor ζ defining D.

Also in Fig. 2(a) are depicted estimates of ζ∗ obtained by checkinginequalities (3), (4) solely at those points in grid DI . A necessarycondition to the quadratic stability of (1) is that there exists P ∈ S

n �0 such that the following set of LMI is satisfied:

PA(θk) +AT (θk)P ≺ 0, ∀θk ∈ DI . (54)

For a given resolution level I , the estimate of ζ∗ obtained via (54)consists in an upper bound on the one provided by Theorem 1, so theformer serves to assess the conservatism of the latter.

B. Example 2

Consider now the following LPV model taken from [8]:

A (θ(t)) =

[8− 108θ(t) −9 + 9θ(t)

120− 120θ(t) −18 + 17θ(t)

]

where θ(t) ∈ [0, 1] and |θ(t)| ≤ ρ. The problem here is to determinethe maximum ρ, denoted ρ∗, such that the origin is asymptotic stable.In [8], the value ρ∗ = 66.81 has been determined.

Fig. 2(b) depicts estimates of ρ∗ obtained by the LMI condition onTheorem 2 for different pairs of resolution levels {I, G}. As expected,increasing resolution level I leads to a less conservative estimate ofρ∗, as it implies increasing the grid density. Additionally, increasingresolution level G also leads to better results by providing more degreeof freedom to Lyapunov matrix P (θ).

For I = 12 and G = 10, the value ρ∗ = 252.49 is determined. Itconsists in a significant improvement over the result obtained in [8] onthe basis of quadratic-in-the-parameter quadratic-in-the-state LF.

Remark: By comparing (13) with (10), it follows that in the affinecase considered in Example 2, there is no conservatism in upper bound

(10) for s = 1 and κij = a/4, ∀i, j, and hence scalars {αj(θ)}2I−1

j=0

in Lemma 1 can be obtained straightforwardly. In Example 1, since thedenominator is always positive and the Lyapunov matrix is parameter-independent, the analysis problem can be simplified to involve poly-nomial dependence only, so computing the scalars αj(θ) is onceagain straightforward. A systematic procedure for computing thesescalars with as little conservatism as possible for a general parameterdependence remains an area for research.


VI. CONCLUSION

New algorithms for stability analysis of LPV systems have beenpresented. The proposed approach relies on Haar transform theory tosolve the original infinite-dimensional feasibility problem by meansof finite dimensional semidefinite programing. One appealing featureof the proposed approach is that it can handle a very large class ofparametric dependencies as well as non-convex parametric domains.Moreover, realistic parameter variation rates can be easily taken intoaccount. It is worth pointing out that all these features are rarely jointlypresent in existing methods. Extensions of the proposed method forperformance analysis and for LF with more general state dependenciesare currently under investigation.

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