Time-domain robustness and performanceoptimisation off linear quadratic control problems

R.I. Badr, MSc, PhDM.F. Hassan, MSc, DScJ. Bernussou, MSc, DScProf. A.Y. Bilal, MSc, PhD

Indexing terms: Optimisation, Feedback, Control systems, Stability, Algorithms, Robustness

Abstract: To find a single constant state feedbackcontrol law that conserves both concepts of stabil-ity and performance robustness is the purpose ofthe paper. The uncertain linear systems consideredare described by state equations which depend onparameters subject to large variations. The devel-oped procedure results in designing a regulatorhaving optimal performance for each of severaloperating modes of the system, as well as deter-mining the largest set of parameter variations. Thealgorithm for designing such a regulator is illus-trated and applied to two examples, showing theefficiency of the proposed technique.

1 Introduction

In many physical systems, the accurate knowledge ofsome parameters is very difficult or too expensive toobtain. Owing to this fact, robust design of feedback con-trollers has been of major interest for many years.Recently published literature concerning robustness canbe classified into two perspectives: stability and per-formance.

Stability robustness analysis is treated using either thefrequency or time-domain analysis. Singular values haveproven to be of great importance in frequency-domainanalysis [1-4] in spite of their inherent conservatism.Bounds are obtained using eigenvalues [4] and M-matrixanalysis [5]. The time-domain analysis is presented usingLyapunov stability analysis developed by Horisberger [6,7], Davison [8], Wong and Athans [9]; as well as eigen-value analysis and pole placement [10].

On the other hand, performance robustness analysisconsists of finding a single control law, for the uncertainsystem, which satisfies one of the following objectives:

(a) to optimise the performance criterion when theparameters attain their worst values (worst-case design)[11,12]

(b) to guarantee an upper bound on the value of theperformance criterion for admissible parameter varia-tions [13, 14]

Paper 607ID (C8, C9), first received 8th January and in revised form30th October 1987Dr. Badr, Dr. Hassan and Prof. Bilal are with the Electronics & Com-munication Department, Faculty of Engineering, Cairo University,Giza, EgyptMr. Bernussou is Directeur de Recherche, LAAS du CNRS, 7 Avenuedu Colonel Roche, 31077 Toulouse Cedex, France

(c) to consider a finite number of models representingthe possible configurations, the system may attain andoptimise the performance criterion chosen to representthese configurations [15].

Despite the availability of considerable research interestin robust control design, the two perspectives of robust-ness (stability and performance) have always been studiedindependently.

In this paper, a new approach is presented which com-bines both perspectives of robustness. The region ofallowable parameter variations is represented by aconvex polyhedron. Our aim is to design a suitable con-troller stabilising the system for the largest region ofparameter variations, while optimising the performancecriterion for a finite number of operating modes. Thesepoints may lie on, or within, the polyhedron, and providea good representation of the region of operating condi-tions.

This paper is divided into three parts: first, the basicmathematical concepts for stability and performancerobustness are established, secondly, the problem formu-lation is described and the algorithm combining the twoperspectives is developed. Finally, two illustrating exam-ples showing the efficiency of the proposed technique arepresented, and some numerical results are discussed.Detailed mathematical proofs of important theoreticalconcepts used in the first part are given in the Appendixes.

2 Mathematical concept

Stability and performance robustness will be consideredfrom the point of view of time-domain analysis.

2.1 Stability robustnessUsing the Lyapunov stability theory, and the convexpolyhedron idea which was first used by Rosenbrock[16], a sufficient condition for a family of linear auton-omous systems to be asymptotically stable is given asfollows:

Let the family of linear time-invarient autonomoussystems be defined by

p A {S: x = Fx, F € 3F} (1)

where J5" is a closed bounded subset of the set of real(n x n) matrices defined by

^ = {F-Lj=Mv), veV) (2)

where fi}{v) are convex polynomial functions of theparameter vector v which are reduced to a straight linewhen regarded as a function of one component of v

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 4, JULY 1988 223

alone. The set of parameter variations V is assumed to bea convex polyhedron of the form

V ^{v.li^Vi^Ui, VERV} (3)

/ and u are given constant real vectors.The vectors of the polyhedron = N = 2V, and are

denoted by,,(1) ,,(2) ,,(N)

The corresponding state matrices are

F(l> = F(i/°), i = 1, 2, . . . ,N (4)

According to Horisberger and Belanger [6], the followingtheorems and corollary are established:

Theorem I: The family of linear time-invariant auton-omous systems defined by conditions 1 to 4 is asymp-totically stable if there exists a real symmetricpositive-definite matrix P (written as P = PT > 0), suchthat

d) + fd)Tp < o, i = 1, 2, . . . , N

As a result of theorem I, we have

Corollary I :

^maAPFiy) + FT(v)P) ̂ maxi = l , 2 N

x A m a ; c ( P F ( l ) + F ( i ) T P \ v e V a n d P = P T

where Xmax{.) denotes the largest eigenvalue of (.).

To be able to determine whether there exists aP = PT > 0, such that the system is stable for all its ver-tices, define

g{P) = maxi = l , 2 JV

(5)

which is continuous and convex on the whole space ofreal symmetric matrices.

Theorem II: Let g(P) be defined by eqn. 5 and F(l) byeqn. 4. Assume that at least one of the F(l) is asymp-totically stable, then there exists a P = PT > 0, such that

PF(i) + F{i)TP < 0, i = 1, 2, . . . , N

if and only if

min {g(P)} < 0

To solve the above minimax problem, it is thereforeessential to obtain its directional derivative (see theAppendix), which is given by

VD0(P) = max max (tr (wyvTF(i)T + F( 'Wr)D) (6)i 6 J{P) w e WW(P)

where

J{P) £ {i: J(P) = X {PF(i) + F{i)TP) =fl{P)} (7)max

and

W(\P) A {w: wT(PFli) + F(i)TP)w =ft{P), w = 1} (8)

W(l\P) is the collection of normalised eigenvectors corre-sponding to the largest eigenvalue of (PF(i) + F{i)TP).

An algorithm, to minimise g(P) over a closed boundedand convex set Q of real symmetric matrices, is based onthe following lemma [6]:

Lemma I: Let Po e Q, i0 e J{P0) and w0 e W(io\ whereJ(P0) and W{lo) are defined by eqns. 7 and 8, respectively.

Then, VP = PT such that

tr {(M>0 M>O"F(0T + F(I)M>OM>J)(P - Po)} ^ 0

we have

g{P) ^ g{P0)

An efficient algorithm for the search of a minimisingpolicy P* is briefly stated as follows:

First, assume that the original collection of feasiblematrices is defined by Q representing a set of linearinequalities:

Q 4 (p = PT. |pfj.| ^ CoPi( ^ o, P > 0} # ^ (9)

Step 1: Define the set Co = Q and find POECO, let

Step 2; Find an ik e J(Pk) and wk e W(it). Calculate

G t = H>k M

If Gk = 0, an optimal policy is P* = Pk

If Gk # 0, define Ck+1 = Ck n {P: tr (Gk(P - Pk)) ^0} ^ 0 which is also a convex set defined by linearinequalities.

Step 3: Find Pk, the solution of the following linear prog-ramming :

tr (Gk(Pk-Pk))= min {tr (Gk(P - Pk))}

If tr (Gk(Pj, - Pk)) = 0, then P* = Pk, hence stopIf tr (Gk(Pk - Pk)) # 0, define Pk + 1 = Pk + i(Pk - Pk)

replace k by k + 1 and go to step 2.

2.2 Performance robustnessIt is required to optimise a finite number N of possibleconfigurations representing the system. The state equa-tions in the presence of parameter uncertainties may begiven by

i = /*<'V> + B<«V>, i/(i) = Kx«\ x(l)(0) = x0

xeRn, ueRm, i = 1, 2, . . . , N (10)

It is required to design a single controller K such that

0, i=\,2,...,N} (11)

where A(.) denotes the eigenvalue of (.); moreover, it isrequired that K minimises the following quadratic cri-terion :

N N C c

= 1^=1i = l i = l JO

£ = <2r>0, R = R r > 0 (12)

Eqns. 10 and 12 can be reformulated in an augmentedform as

x = Ax + 6, u = Jfjc, Jc(O) = x0

whereC. _ /Vd)7" v(2)T • • . lrW'\T e- tfNnJk- — I «v A- A. ^ t J\

,,(2)1" . . . ,AN)T\T c- nNm

A e RNi"xn) = block diag. (A(i))

B e RN(tt x m) = block diag. (£(0)

K E RNim x n) = block diag. (X), Vi = 1,2,..., N

and

= I (iTQx + UTRu) dtJo

224 IEE PROCEEDINGS, Vol. 135, Pt. D, No. 4, JULY 1988

where

Q = QT>0, Qe RN<n X n) = block diag. (Q)

R = RT>0, Re RNim X m) = block diag. (R)

Assuming a stable response, let

(13)

r(K) * x{t) dt,

where

X(t) = x(t)xT(t)

Thus, the matricial gradient of J(K) is

dJ N

with the stationary conditions

(A + M)r + T(A + BK)1

Q + KTRK + (A + BK)TA + A(A

given by

+ M) = of

(14)

(15)

(16)

Detailed proof of the above equations are found in theAppendix.

Using the gradient technique, we may obtain anoptimal controller K, which minimises the performancecriterion 14 for the system of eqns. 13, through the fol-lowing algorithm:

Step 1: For the iteration index q = 1, determine an initialfeasible gain Kq = block diag. (Kq), Kq e K defined incondition 11.

Step 2: Determine the gradient matrix dJq/dK using eqns.15 and 16.

Step 3: Convergent test; i.e. if \dJq/dK | -> 0, then K* =Kq and stop; if not go to step 4.

Step 4: Update K, i.e. Kq+i = Kq - aq(dJ/dK)q where thestep size aq ^ 0 must be selected such that J(Kq+1)<J(Kq), go to step 2.

Note that:(i) At each iteration the stability of the overall system

must be checked, i.e. A(Kq+1) > 0; otherwise the step sizeaq is decreased according to the following rule.

(ii) The step size aq is modified as follows: aq+1 = naq,if J(Kq+1) < JiK"), and A(Kq+1) > 0, aq + i = £aq, other-wise with 7 t > l , 0 < £ < l and an initial arbitrary valuefor a.

(iii) The convergence of the above algorithm is clari-fied through the following lemma, which proves theexistence of a step size a (required in step 4 above)yielding J(Kq+1) < J(Kq). Moreover, theorem III isintroduced, thus giving a necessary and sufficient condi-tion for the existence of a minimising gain for the case ofuncertain systems under consideration. (The correspond-ing mathematical proofs are developed in Appendixes 8.3and 8.4.

Lemma II: Let J{K) be defined by eqn. 13, which is differ-entiable in K e K of eqn. 11; then there exists a step a > 0such that

J(K -aH)< J(K), H ^dK '

and

K = block diag.

Theorem III: Let J(K) be defined as in eqn. 13, thenthere exists a minimising K* for J(K), if and only if there

exists an initial gain Ko e K of eqn. (11), such that (A+ BK0) is asymptotically stable.

It is obvious that the main difficulty when applying thisalgorithm is to find an initial Koe K which stabilises thesystem at all its configurations. This problem will beovercome in the following Section.

3 Problem formulation

Let us state the problem precisely:(i) The given system is defined by

x = A(vo)x + B(vo)u

A{vo)eRnXn, B(vo)eRn (17)

v0 e Rv ^ nominal parameter vector(ii) A constant feedback gain is to be designed of the

form

u = Kx, KeRnXm (18)

(iii) Some elements of the nominal matrices A(v0),B(v0) are subject to independent variations. We thusdefine a convex polyhedron of parameter perturbationsby

V(e)^{v: -el v0 ^ v eu} (19)

where the vector inequalities are understood to be com-ponentwise, and / and « e Rv are given vectors with non-negative components.

(iv) Combining eqns. 17 and 18, we obtain the closed-loop system in the form

x = F(v) = (A(v) + B(v)K)x (20)

The problem is to find a single constant gain K* yieldingthe largest set of parameter perturbations of the form 19,for which we still can guarantee asymptotic stability ofthe overall system, i.e. which yields the largest e* suchthat F(v) = (A(v) + B(v)K) has its eigenvalues in the openleft half of the complex plane V» e V(e). Moreover, it isrequired that K* e K given by condition 11, optimises thequadratic criterion

J(K*) = min £ |Xeic i = l Jo

+ uii)TRu(i)) dt (21)

where i = 1, 2, . . . , N, represents the possible configu-rations which the system may attain and may lie on (orwithin) the resulting vertices of the complex polyhedronV(8).

As for some e ^ 0, PF(v) + FT(v)P < 0, Vv e V(e), ifand only if the largest eigenvalue

KJLPH') + FTiv)P) < 0, Vr G V(e)

we introduce the following function:

Ke) ± max X^PFWe)) + FT(v{e))P) (22)

where

P = PT > 0

It can be easily shown that the real-valued function h(e) iscontinuous in e e [0, oo[, monotone increasing, and thatthere exists a unique e* > 0 such that h(e*) = 0, and

Ve<e*(23)

We are now able to propose the following algorithm toobtain the optimal solution of eqn. 21, i.e. find a constant

1EE PROCEEDINGS, Vol. 135, Pt. D, No. 4, JULY 1988 225

matrix K* e K satisfying eqns. 15 and 16, as well asfinding the largest allowable parameter variations e*,satisfying inequalities 23:

Step 1: Find an initial Ko e RmX" such that F(v0) = A(v0)+ B(vo)Ko is asymptotically stable. Set IT = 1, qx = 1,

q2 = \.

Step 2: Determine a Pqi-i = i*J, _x e O, given by condi-tion 9, which yields

(A(v0)

Step 3: Define

and determine the first zero crossing eqi, i.e. find eqi suchthat hqi(eqi) = 0 and, for all 0 ^ e ^ eqi, hqi(e) < 0.

Step 4: Define

^ max

and find Pfll which minimises gqi(.) over Q given by con-dition 9, i.e. determine Pqi such that

If 0*t = 0, then e/T = £91 and go to step 5, else replace qxby qr + 1 and go to step 3.

Step 5: From elT, find the resulting vertices and formu-late the new system,

Using the stationary conditions given by eqns. 16, deter-mine the matricial gradient

dJ

using eqn. 15.

Step 6: If

dJdK

= 0,9 2 - 1

then KIT = Kq2_ x and go to step 7, else update, K, i.e.

dJ_dK q2_x

where the step size aq2 _ y $s 0 is selected such thatJ(Kq2) < J(Kq2_x), replace q2 by q2 + 1 and go to step 5.

Step 7: If \\KIT - KIT_X\\ < e, then K* = KIT and e* =e/T, else set qx = I, q2= I and replace / T by IT + 1 andgo to step 2.

Note that the difficulty of finding an initial Ko e K whichstabilises the system at all its N configurations has beenovercome by this algorithm, owing to the fact that Kq2 _ xwill always stabilise the system at the vertices of theresulting convex polyhedron, as well as within the regionincluded between these vertices.

4 Weighted quadratic criterion

In the preceding Section, we should note that the valuesof the nominal parameters of the system must surely

affect the system's performance; especially that the prob-ability of their occurrence should be greater than (or atleast equal to) the probability of occurrence of the systemparameters at the N configurations. Therefore, the per-formance can be reformulated as

J(K)

that is

J(K)

subject

* < • • > =

and

«(') =

— co Jo

•rJO ito

-. A^x^

: KXi()

+

N

£= 0

+

and c0 (24)

, 1,..., N}

(25)

Following the same procedure indicated in Section 3, thematricial gradient of J(K) can be easily found to be

(26)«=o

together with the stationary conditions given byi KTRK + Q=0

B(i)K)T + Xt(0) = 0

Note that, if cx = c2 = • • = cN, then

c,. = ( l - c o ) / N i = l , 2 , . . . , N

(27)

(28)

In the following Section, the proposed technique isapplied on two examples: a 2nd-order and a 5th-ordersystem each having two uncertain parameters [14].

5 Examples

(i) Consider the 2nd-order system of the form of eqn.20 with

[-2 + q, l+q2]where

and -

In each case, the weighting matrices Q and R were chosento be

•• e g R = 10

It is required to find K* e K defined in condition 11, suchthat the quadratic criterion given by eqns. 25 is mini-mised Vq = (qi, q2)

T- The number N of possible configu-rations at the vertices of the required polyhedron isN = 2V = 4, where v = number of varying components.

For the nominal system qy = q2 = 0, an initialKo which stabilises the nominal closed-loop system Fo =(Ao + BK0) is chosen to be the LQR given by

Xo = -R~1BTG = -(0.0248 2.072)

where G is the solution of the Ricatti equation:

AlG + GA0 - GBRlBTG + Q = 0

226 IEE PROCEEDINGS, Vol. 135, PL D, No. 4, JULY 1988

Let »(0(e) be the vector representing parameter variationsof configuration (i), therefore

Q =

vy ' =

»<3> =

+ 1.5e»«> =

Vy ' =1 + 1.5e

and the nominal parameter vector

To fulfill the requirements of our problem, it is requiredthat max £ = e* ^ 1.0. Three different cases of weightedquadratic performance are studied to elaborate theireffect.

Table 1 gives a comparative study between the resultsobtained. Define AJ (in per cent) as the relative measureof the performance criterion, which indicates how farfrom the LQR design we are by applying the resultingcontroller.

Therefore, define

=J0(K*) - J0(KLOR)

x 100 (29)

K

= maximum region stabilised by K*

= required gain which stabilises the closed-loopsystem up to e*ax and minimises the quadraticcriterion at e = 1.00

J0(K) = the performance criterion of the nominal systemwhen the gain K is applied, i.e. for Fo = Ao

+ B0KFrom Table 1, we see that the relative measure of the

performance criterion is improved as the weight c0 on thenominal system increases. This increase also affects themaximum allowable region of parameter uncertaintiese^,x. Thus, an appropriate choice of c0 and c{ is useful toachieve the required region of parameter uncertainties,while keeping the relative measure of performance cri-terion as small as possible.

(ii) The second example considers a 5th-order systemwith uncertain parameters. The data were taken fromReference 14. The system parameters are

-0.85 25.47-0.339 -8.7890.021 -0.5470 10 0

-979.5 32.141.765 0

-1.407 00.0256 00 0

059.89

6.4770

- 2 0

0 0 0 20)r

0 0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0

The weighting matrices Q and R of the quadratic cri-terion are

R

0.130.66

-17.270.17

-2.21

= 2500

0.6610.07

-251.051.26

-35.9

-17.27-251.059633.7-313.41

10.64

0.171.26

-313.4123.7269.29

-2.21-35.9

10.6469.29

359.1

In Reference 14, q was assumed to be scalar and, there-fore, the system was considered for two possible configu-rations only. To elaborate the efficiency of the proposedtechnique, we shall assume q e R2 and it is required tofind the upper and lower values that q may attain (i.e. qu

and qt) and the corresponding K* which stabilises thesystem V# e [_qt, qu~] and which optimises the per-formance index at the vertices, as well as for the nominalvalue, where

qi^q^qu and qt= -3qu

To adapt the above requirements to our algorithm, thevertices of the polyhedron representing parameter varia-tions are given by:

»(1) = (59.89 + 1.71e 6.477 + 3.22e)r

i>(2) = (59.89 + 1.71e 6.477 - 3(3.22)e)r

»(3) = (59.89 - 1.71(3)e 6.477 + 3.22e)r

»(4) = (59.89 - 1.71(3)e 6.477 - 3(3.22)e)T

vi0) = (59.89 6.477)r

An initial Ko which stabilises the nominal closed-loopsystem is

= (0.003 0.084 -1.806 -0.013 -0.344)

Applying the proposed algorithm yields the followingresults:

The closed-loop system with

K* = (0.013 0.0337 -1.066 -0 .04 -0.763)

is asymptotically stable Vf, such that

— 1.291 10.43

i.e. eZax = 0.43, and K* optimises the quadratic per-formance at the vertices of the resulting polyhedron.Note that the relative performance criterion measuregiven by eqn. 29 is found to be +18%.

6 Conclusion

A new technique for designing constant gain feedbackcontrollers for systems with large parameter variations isdeveloped. Despite the fact that recent researches [20, 21,22] have dealt with robustness analysis, no explicit tech-nique for finding a large region of parameter uncer-tainties, while maintaining stability and performancerobustness, has been reported. The proposed techniquegives an efficient method for the synthesis of a constantfeedback gain which allows the largest region of param-eter variations, as well as optimising the system, at afinite number of models, representing the possible systemconfigurations. Throughout the procedure, the problemof finding an initial gain which stabilises the system at itsvarious configurations has been overcome by the use ofthe Lyapunov stability theory. The use of the weighted

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 4, JULY 1988 227

TableCase

»- C

M

CO

1 : Comparative results of example (

Co

0.20.60.9

c,, / = 1, 2, 3,4

0.20.10.025

-K*

(0.2912(0.307(0.2827

4.265)3.811)3.212)

i)

e

111

*max

.47

.28

.01

J0(K0)62.9362.9362.93

82.376.8470.46

kJ, %

30%22%12%

quadratic performance allowed the possibility of a com-promise between increasing the allowable region ofparameter uncertainties, while decreasing the relativemeasure of performance criterion.

Finally, the examples show the effectiveness of the pro-posed technique compared with other methods.

7 References

1 DOYLE, J.C., and STEIN, G.: 'Multivariable feedback design: con-cepts for a classical/modern synthesis', IEEE Trans., 1981, AC-26,pp. 4-16

2 LEHTOMAKI, N.A., SANDELL, N.R., and ATHANS, M.:'Robustness results in linear quadratic Gaussian based multivariablecontrol design', ibid., 1981, AC-26, pp. 75-93

3 LEHTOMAKI, N.A., SANDELL, N.R., and ATHANS, M.:'Robustness tests utilising the structure of modelling errors', 20thCDC, San Diego, December 16th-18th, 1981

4 SOLHEIM, O.A.: 'Robustness analysis of a class of optimal controlsystems', Modelling, Identification & Control, 1983,4, pp. 223-235

5 KANTOR, J.C., and ANDRES, R.P.: 'Characterization of allowableperturbations for robust stability', IEEE Trans., 1983, AC-28,pp. 107-111

6 HORISBERGER, H.P., and BELANGER, P.R.: 'Regulators forlinear, time-invariant plants with uncertain parameters', ibid., 1976,AC-21, pp. 705-708

7 HORISBERGER, H.P.: 'Control of linear systems with largeparameter variations'. PhD dissertation, 1973, McGill University,Montreal, PQ, Canada

8 DAVISON, E.J.: 'The robust control of a servomechanism problemfor linear time-invariant multivariable systems', IEEE Trans., 1976,AC-21, pp. 25-34

9 WONG, P.K., and ATHANS, M.: 'Closed-loop structural stabilityfor linear quadratic optimal systems', ibid., 1977, AC-22, pp. 94-99

10 ACKERMANN, J.: 'Parameter space design of robust controlsystems', ibid., 1980, AC-25, pp. 1058-1071

11 SALMON, D.M.: 'Minimax controller design', ibid., 1968, AC-13,pp. 369-376

12 SCHMITENDORF, W.E.: 'A sufficient condition for minimaxcontrol of systems with uncertainty in the state equations', ibid.,1976, AC-21, pp. 512-515

13 CHANG, S.S.L., and PENG, T.K.C.: 'Adaptive guaranteed costcontrol of systems with uncertain parameters', ibid., 1972, AC-17,pp. 447-483

14 VINKLER, A., and WOOD, L.J.: 'Multistep guaranteed costcontrol of linear systems with uncertain parameters', J. Guid. &Control, 1979, 2, pp. 449-456

15 KOSMIDOU, O.: 'Sur 1'analyse et la commande des systemes incer-tains'. These de Docteur 3eme. cycle, 1984, Universite de Paris-Sud,Centre D'Orsay

16 ROSENBROCK, H.H.: 'A method of investigating stability'. Proc.2nd. Congress of IFAC, Basel, 1963

17 LEVINE, W.S., and ATHANS, M.: 'On the determination of theoptimal constant output feedback gains for linear multivariablesystems', IEEE Trans., 1970, AC-15, pp. 44-48

18 GEROMEL, J.C., and BERNUSSOU, J.: 'An algorithm for optimaldecentralized regulator of linear quadratic interconnected systems',Automatica, 1979,15, pp. 489-491

19 GEROMEL, J.C., and BERNUSSOU, J.: 'Optimal decentralizedcontrol of dynamic systems', ibid., 1982, 18, pp. 545-557

20 YEDAVALLI, R.K.: 'Improved measures of stability robustness forlinear state space models', IEEE Trans., 1985, AC-30, (6)

21 YEDAVALLI, R.K.: 'Perturbation bounds for robust stability inlinear state space models', Int. J. Control, 1985,42, (6)

22 OWENS, D.H., and CHOTAI, A.: 'Robust controller design forlinear dynamic systems using approximate models', IEE Proc. D,Control Theory & Appl., 1983,130, (2), pp. 45-56

23 KLEINMAN, D.L.: 'On the linear regulator problem and thematrix Ricatti equation'. MIT Electronics System Lab., Cambridge,MA, Tech. Rept. ESL-R-271, 1966

24 MARCUS, M., and MINC, H.: 'A survey of matrix theory andmatrix inequalities' (Prindle, Weber and Schmidt, 1964)

8 Appendixes

8.1 Directional derivative ofg(P)From eqn. 5;

g(P)= max ftp)i = l , 2 N

where

fin =

and

= max T(i\w, P)IMI=l

, P) = wT{PF(i) + F(i)TP)w (30)

ftP) is continuous in (H>, P) and has a continuous partialgradient matrix with respect to P. Denoting by W{l\P),the collection of normalised eigenvectors correspondingto the largest eigenvalue of

(ppd) + f

thus:

W(i\P) 4 {w: wT(PF(i) + F(i)TP)w =ft{P); \\w\\ = 1} (31)

and the directional derivative along D = DT, \\D\\ = 1 is

(32)

weWV)(P)

max {w e Wd)(P)

^T + F('WT) • D)

As g(P) is the maximum of a sequence of directionallyderivative functions ftP), therefore the directional deriv-ative of g(P) is given by

(33)

i e J(P)

where;

J(P) = {i: ftF) = g(P)}

8.2 Matricial gradient ofJ(K)According to eqns. 13 and 14, the quadratic performancecan be reformulated as

J(K) = [°° Tr ((Q + KTRK)X) dt (34)Jo

Subject to

X = {A + M)X + X(A + BK)T, X{0) = Xo (35)

the Hamiltonian function is therefore

H{X, A, K) = Tr («5 + KTM)X)

+ Tr (Ar((/4 + BK)X + X(A + BK)T)) (36)

Using the gradient technique (Reference 17 and theorem1, and References 18 and 19), which states that for ageneral dynamic model having the form

= [Jo

, K) dt + g(X(T))

228 IEE PROCEEDINGS, Vol. 135, Pt. D, No. 4, JULY 1988

subject to

X = F(X(t), K); X(0) = Xo

X(t) e R" x" = state variable matrix

K e Rm x" = matrix of parameters

suppose that the functions/, g and F are differentiable,then the matricial gradient of J(K) is given by

where

H(X, A, K) =f(X, K) + Tr (ATF(X, K))

X and A being the solution of the stationary conditions:

^ - X = 0, = Xo

Thus, according to the above theorem, the stationaryconditions for eqns. 36 are

(37)Q + KTRK + (A + M) r A(A + BK)X + X(A + BK)T - X = 0

For the matricial gradient of J:

Note that:

(i) Tr (ATBKX) = Tr (XATM)

therefore,

Following the same steps, we can easily show that

dJ CT N

—(K) = 2 X {(RK - B^AJXM dt" A Jo i = i

For the infinite horizon case, define

T(K) = dt

The matricial gradient of J(K) is therefore given by

dJ N

Together with the stationary conditions:

Q + KTRK + (A + BK)TA + A(A + BK) = 0(A + BK)T + T(A + BK)T + Xo = 0

(38)

(39)

(40)

(41)

8.3 Proof of lemma IIAccording to eqn. 13, the cost function J(K) may be

rewritten as

<t>T(t)(Q + KTRK)4(t) dt • Xo) (42)

o /IEE PROCEEDINGS, Vol. 135, Pt. D, No. 4, JULY 1988

where <£(t) is the fundamental transition matrix

<p(t) = exp (A + BK)t (43)

By developing J(K — aH) in Taylor series around a = 0,we have

dJJ(K - aH) = J(K) + a — {K-aH)

da(44)

a = 0

As J(K) is a trace function, therefore, using Kleinman'slemma [23], we obtain

J(K - aH) = J(K) - a Tr (j^ (K) • H\

Thus

Substituting in eqn. 44

(45)

(46)

dJ1

J(K - aH) - J(K) = -a Tr I — (K) • H

= -alT(HT • H)= -a\\H2\\

Thus, for H # 0, there exists a step a > 0 such that thelemma is true.

8.4 Proof of theorem IIIAccording to eqns. 42 and 43, and assuming the initialstate x(0) is a random variable, uniformly distributed onthe surface of the n-dimensional unit sphere, therefore

X o = E ( x o x l ) = ( l / n ) I (47)

and the expected value J(K) of the performance criterionJ(K) is

J(K) = (l/«) Tr dt

Define

A(t, K) A [Jo

XQ + KTRK)e{A+Bti)* dx

(48)

(49)

8.4.1 Necessity: It shall be shown that for all Ks suchthat (A + BK) has at least one eigenvalue in the closedright half plane, the performance index is infinite, i.e. thecost is really only^ defined for stabilising Ks. Therefore,we assume that (A + BK) has at least one eigenvalue inthe closed right half plane. Let it be in the configuration j ,i.e.

Re IX^Aj + Bj K)-] > 0 (50)

where Xtj is the /th eigenvalue of the configuration pair(Aj, Bj). Recalling from matrix theory [24] the fact that anon-negative definite symmetric matrix has real non-negative eigenvalues, and that the trace is equal to thesum of the eigenvalues; furthermore, from eqn. 49 it isobvious that A(t,, K) is block diagonal, therefore we have

Tr(A(f,/£))=

Tr

r = l

ujA/t,K)Mj, Vt > 0, = 1

229

where lrj is the rth eigenvalue of A; corresponding to the7th configuration, where j e {1, 2, . . . , N}. Taking Uj =uu, such that

\utj\\ =

i.e. utj is the normalised eigenvector corresponding toof eqn. 50, and noting that [24]

(51)

and

Aft,

Therefore

(52).ft, K) = ( V ' + W ^ Q + KTRK)e(A'+B^ dxJo

Tr (A(t, K)) > ujAft, K)Uj

^ ufje^iQ + KTRKle^Uy • dxJo

= uJfQ + KTRK)Ulj \e2RtiX»* • dxJo

where Xu is the complex conjugate of XtJ, and Re {Xtj) ^ 0by assumption, and

KTRK)uu

hence

KM) > 0

for Re (Ay) = 0

for Re (A,,) > 0_ 1

Finally, noting that, from eqn. 48 and eqn. 49,

J{K) = (1/n) lim Tr (A(t, K))f-*OD

we see that J(K) -*• 00 for any K, such that (X + BK) isnot asymptotically stable for all i = 1, 2 , . . . , N.

8.4.2 Sufficiency: Assuming that there exists a Ko e Ksuch that (A + BK0) is asymptotically stable, we firstshow that the set of all K = block diag. [K], such thatJ(K) ^ J(K0), must be bounded. It is shown in Reference7 that, for every stabilising K for a system x = Ax + Bu,J(K) exists and is equal to the unique solution P(K) ofthe following function:

P(K)(A + BK) + (A + BK)TP(K) = -(Q + KTRK) (53)

furthermore, for Q > 0, P(K) is positive-definite. Applyingthis to our case, the augmented form of eqn. 53 for Nconfigurations is

A(K)(A + BK) + (A + BK)TA(K) = -(Q + KT&K)

(54)

where, for Q > 0, A(K) is positive-definite. It is obviousthat A(K) is block diagonal and, therefore, A{(K) is alsopositive-definite, therefore eqn. 54 may be rewritten as

B.tK) + £ {A,1 = 1

+ NQ = -NKTRK (55)

Taking the Euclidean norms, we obtain

N\\Q\\>N\\KTRK\\ (56)

by assumption, we have

n2J\K0) > n2J\K) = (Tr (A(K)))2

( N n \2

= I ZArl<A,.)\i=l r=

Because

XTri = 1 i = 1

therefore, for A,- symmetric,

i= l r=l

r=l \ i = l

N \ \ 2 n / AT

r=l \ i=l

ZAJ

Also

= ZTr(A,-)=

therefore

N

\ > y0/ ^ / •

Now, denote

where

II If II — 1

|| , / v j | | ^— 1

and

a= min NWK^RK^l > 0because i? = RT > 0. Furthermore, denote

b ± c 4

(57)

(58)

(59)

(60)

Using inequality 57 and substituting by conditions 58-60,inequality 56 can be written as

c + pb ^ p2a

hence

(61)

bV c\\K\\ = p ^ — ± /( — J + - = fe, fe is constant

2a \J \2aJ a

As a second step, we demonstrate that J(K) ^ J(K0)implies that

Re (X^Ai + Bt K))^a<0 for i = 1, 2, . . . , N1=1,2,...,n

where

a =

230

-KJQ)2J(K0)

IEE PROCEEDINGS, Vol. 135, Pt. D, No. 4, JULY 1988

Proceeding as in the necessary part of the proof, weobtain

We conclude that all Ks such that J(K) < J(K0) must liein the closed bounded region

J(K0) ^ J(K) = lim Tr (A(t, K))r-»oo

e2 Re (Xri)t _ j

KJQ) V r = l , 2 , . . . , n- 2 Re(ArI) i = 1, 2, . . . , N

where kri is an rth eigenvalue of the ith configuration (At

+ B, K), which is asymptotically stable, hence

^ {K:K = block diag. \\K\\

and Re (Xri{A( + Bt K)) ^ 0, Vi = 1, 2, ..., N and r = 1, 2,..., n}. Over Q, J(X) is continuous because the solutionof eqn. 54 is continuous in K and Tt (A) is a continuousfunction in the elements of A.

Finally, the continuous function J(K) attains aminimum over the closed bounded set Q, i.e. there existsa K* such that J(K*) ^ J(K) for all K.

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