IMPROVED QUANTITATIVE MEASURES OF ROBUSTNESS FOR MULTIVARIABLE SYSTEMS

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 4, APRIL 1994 807

Improved Quantitative Measures

of

Robustness for Multivariable Systems Homg-Giou Chen and Kuang-Wei Han

Abs*uct- Asymptotically stable linear systems subject to unstruc- tured time varying perturbations are considered. Allowable perturbation bounds are obtained such that the perturbed systems remain stable. These bounds are derived iteratively by means of adjusting a sequence of Lya- punov matrices. In comparison with existing methods, less conservative quantitative measures of robustness are obtained.

I. NOMENCLATURE R" P , A' g m a x (A) g m i n (A) A-l

llvll

P > O P > O P > Q P > Q

Real vector space of dimension n.

QRBM (Quantitative Robustness-Bound Measure). Transposed matrix of A.

Maximum singular value of matrix A. Minimum singular value of matrix A. Inverse matrix of an invertible matrix A. Square root of positive-definite matrix A. Euclidean norm of vector U.

Square symmetric matrix P being positive-definite. Square symmetric matrix P being

positive-semidefinite.

Square symmetric matrices P and Q that satisfy Square symmetric matrices P and Q that satisfy P - Q > O .

P - Q > O .

11. INTRODUCTION

Recently, the aspect of developing explicit upper bound on the perturbation of linear systems, such that the perturbed systems remain stable, has received much attention. Starting with Patel et al. [l], Patel and Toda [2], and Lee [3], considerable effort has been given on the reduction of conservatism in quantitative measures of robustness with increasingly complicated ways of defining the structure of perturbations [4]-[7]. In these literatures, perturbations are broadly categorized as being unstructured or structured. For unstructurally perturbed systems, while it is assumed that only the norm bound of the perturbation is available, robustness measures can be derived by use of the Lyapunov theory [I], [2]. Since the robustness measures derived for the unstructured cases are generally used to propose robustness bounds for the structured cases, it is desirable to always derive less conservative robustness measures for unstructurally perturbed systems.

In this correspondence, new robustness measures for unstruc- turally perturbed systems are derived. By way of integrating matrices specified within a pair of Lyapunov equations, less conservative Quantitative Robustness-Bound Measure (QRBM) can be achieved.

Manuscript received March 21, 1992; revised November 17, 1992 and April H.-G. Chen is with the Institute of Electronics, National Chiao-Tung K.-W. Han is with the Center of System Development, Chung-Shan Institute IEEE Log Number 9216419.

9, 1993.

University, 1001 Ta-Hsueh Road, Hsinchu, Taiwan, ROC. of Science and Technology, Hsinchu, Taiwan, ROC.

111. PROBLEM FORMULATION We consider systems described by the differential equation

z ( t ) / d t = A z ( t )

+

f ( z ( t ) , t ) (1) where A E R"'" is the stable nominal matrix, and f ( z ( t ) , t) is an unstructured perturbing function with the property of f ( 0 , t) = 0 for all time t. It is assumed that, while an exact expression of the perturbation cannot be written explicitly, an estimate of some bound on the perturbation is available. The problem which we investigate in this correspondence is to derive a less conservative quantitative bound on the perturbing function f ( z ( t ) , t) such that the perturbed system described by (1) remains stable.

For unstructurally perturbed systems described by ( l ) , the Lyapunov-based method of deriving QRBM' s has been considered as well established for a long time [l], [2]. Existing results are the following two theorems where sufficient conditions for the stability of the system described by (1) were expressed as upper norm-bounds of the perturbing function f ( z ( t ) , t ) .

Theorem I: (Theorem 1 of [2]) The perturbed system described by (1) is stable if

Ilf(t,

t)lllll~II

5

PQ E gmin(Q)/gmax(P) (2) for all ( 2 , t) E R"+lwith nonzero t, where Q E R"'" is some symmetric positive-definite matrix, and P E R"

'

"

is the symmetric positive-definite matrix that fulfils the Lyapunov equation

(3) Theorem2: (Lemma 2 of [2]) The bound PQ defined in (2) is maximum (i.e., the least conservative result of PQ denoted as PI) when the matrix Q = I is assigned in the Lyapunov equation (3), where 1 is the n x n identity matrix.

It can be shown that these theorems generally produce QRBM's that are very conservative. In the following, another Lyapunov-based method of deriving QRBM's is proposed. The basic idea comes from the generally accepted fact that, more often than not, several Lyapunov functions are better than one. Since the results of the existing two theorems are not flexible enough for mingling Lyapunov functions, it is necessary for us to rederive the problem specified in Theorem 1.

Theorem 3: The perturbed system described by (1) is stable if A ' P

+

P A = -2Q.

for all (z, t) E R"+l with nonzero z , where matrices

P

and Q fulfil the Lyapunov equation (3).

Proof: Followed from the Lyapunov equation (3), a Lyapunov function of the stable nominal system matrix A is given by

V(z) = z ' P z . ( 5 )

Employing the quadratic function V(z) on the perturbed system described by (l), a sufficient condition for justifying the stability of the perturbed system is given by

0

>

d V ( z ) / d t = ( d z / d t ) ' P z

+

z ' P ( d z / d t )

= z'(A' P

+

P A ) a

+

2 f ' P z . (6) Using Lyapunov equation (3), the sufficient condition for stability becomes

f ' P z

5

z'Qx. (7)

808

Making a slight modification, condition (7) is equivalently given by

and a sufficient condition for stability can be given as

(9) Knowing that

and

condition (9) is sufficiently justified by

o m a x

(Q-

P)

II f

I1

i

o m i n

(Q"'

II

5

I1

(12)

which complete the proof. 0

We note that Theorem 1 was derived without the modification stage from (7) to (8) given in the proof of Theorem 3. Thus, it is implicitly assumed in Theorem 1 that matrices P and Q of the Lyapunov equation (3) are structurally unrelated. This is certainly not realistic, and Theorem 3 can be looked at as the rectified version of Theorem 1 in this particular point of view. Comparing the proposed bound given in (4) with the existing bound given in (2), since

o m i n (&'/')/omax (Q-'/'P) = o m i n ( Q 1 l 2 ) o m i n (P-l Q1l2)

2

o m i n ( Q ' / 2 ) o m i n ( Q 1 / 2 ) ~ m i n (P-l)

= o m i n ( Q ) / o m a x ( P ) (13)

the proposed bound ( p c ) is always better than the existing bound

( p ~ ) for all possible choices of matrix Q in the Lyapunov equation (3). In this correspondence, the following two facts are used:

i) d m i n (A

+

B)

2

o m i n (A)

+

o m i n (B), where A

2

0, B

2

0;

ii) c m i n ( A B )

2

o m i n ( A ) U m i n ( B ) .

More discussion of singular values and their properties can be found in various texts [8].

Heuristically, there is a particular choice of matrix Q which will bring forth possibly the least conservative bound. The structural relation between matrices P and Q of the Lyapunov equation (3) can not be resolved, however, without first having the Lyapunov equation solved. This is a common difficulty confronting the robustness analyses by ways of the Lyapunov-based methods. Thus, we are obliged to create an iterative process such that a proper choice of matrix Q for the Lyapunov equation (3) may be acquired with less conservative QRBM ( p c ) .

IV. REDUCING CONSERVATISM IN QRBM

Let A be a stable system matrix, so will A' be stable. The following Lyapunov equations specify two symmetric positive-definite matrices Pi and P 2 :

A'Pi + P I A

+

2 Q i = 0 ₍₁₄₎

PT'A'

+

APF'

+

2 Q 2 = 0 ₍₁₅₎

where matrices Q1 and QZ are symmetric positivs-definite. Simul-

taneously, we have the following alternative expressions of (14) and (15):

PTIA'

+

APT'

+

2PF1Q1PT1 = 0 (16)

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 4. APRIL 1994

Employing Theorem 3, the Lyapunov equations (14) and (17) pro- vide us with two QRBM's ( p c ) for the perturbed system described by (1). These are

112 = Umin(Q:/z)omin(PzQ:/2). (19)

Without loss of generality, it is assumed in the followings that matrices Q1 and QZ are always normalized such that

(20)

o m i n ( Q 1 ) = c m i n ( Q 2 ) = 1.

Thus, in matrix sense, relations given in (20) are expressed by

Qi

2 1

and Q 2 2 1 (21)

Pc'Qil'c'

2

p : I (22) and relations given in (18) and (19) are expressed by

We note that (14) and (17) can be interpolated to make new Lyapunov equations, so can (15) and (16) in dual manner.

Lemma 1: Given Lyapunov equations (14) and (15), the following interpolated Lyapunov equations are fulfilled for all interpolating parameters U E (0, 1/&) and b E ( 0 , l/p?):

A'Xi

+

XiA

+

2 Y 1 = 0 ₍₂₄₎

where

xi

= (1

-

u p ; ) Pl

+

U P ! , (26)

Y2 = (1

-

bp:)Q2

+

bPl-lQiPF1. (29) Pro@ Equation (24) is the direct interpolated result of (14) and

(17). Equation (25) is the direct interpolated result of (15) and (16). 0

Employing Theorem 3, the interpolated Lyapunov equations (24) and (25) provide us with two QRBM's ( p c ) for the perturbed system described by (1). These are

The following lemmas provide some qualitative properties that are useful for producing iterative interpolating procedures from which a proper choice of matrix Q for the Lyapunov equation (3) can be acquired with less conservative QRBM ( p c ) .

Lemma 2: Given Lyapunov equations (14) and (15) with their QRBM's p1 and p2 given in (18) and (19), the interpolated Lyapunov equations (24) and (25) defined in Lemma 1 produce new QRBM's

vi and v2 given in (30) and (31) such that, for all interpolating parameters a E (0, 1/&) and b E (0, l/p?)

(32) min(v1, v 2 )

2

min{pi, p 2 ) .

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 4, A P W 1994 809

Pruufi We shall prove that VI

2

min(p1, p z } , and the other relation showing that V P

2

min {PI, p ~ } can be proved in a similar way.

k m m a 3: Given Lyapunov equations (14) and (15) with their

QRBM's p 1 and p2 given in (18) and (19), the interpolated Lyapunov equations (24) and (25) defined in Lemma 1 produce new QRBM's

V I and v2 given in (30) and (31). There are interpolating parameters By use of relations given in (27). (20), and (19), we obtain

a E [amin, l/pz) and b E [bmin, l/p:) such that cmin(Y1) = c m i n ( ( 1 - ap;)Q1

+

a 9 Q 2 5 )

2

(1 - a&)cmin(Ql)

+

aUmin(PZQ2Pz) max(v1, V Z }

2

max{pl, pa} (45)

-

- 1

-

up;

+

up; = 1. (334

Thus, the interpolated Lyapunov equation (24) produce the QRBM

if the following conditions are fulfilled:

p z ( Q 2

-

1)s

+

(Pz

-

p i P ~ ) ~

>

p i ( p ; p :

-

Q i ) (46)

V I given in (30) such that

which is rewritten, in matrix sense, by x;lYlx,l

2

EZI

and bmin E [0, l/p:) is the smallest value that fulfils

bmin{p~l(Ql-1)Pr1+(PF1 - ~ L L : P F ~ ) ~ }

2

p:PL2-Qz. (49) (36)

Proofi Lemma 2 implies the result for the case of p1 = pz. We shall prove that, given pz

>

p1, conditions (46) and (47) lead to an

improved QRBM with v1

2

pz. In a similar way, the other relation, showing that uz

2

p1, can be proved for the case of p1

>

pz.

Given P z

>

P I , by Use Of the matrix relation given in (23), it can be shown that

or

(37) By definitions of matrices

x1

and yl given in (26) and (271, condition (37) becomes

Yl

2

E2X,2.

(1

-

&)Q1

+

aPzQ2Pz

2

E2[(1 - ap;)Pi

+

UP^]'.

(38) p:pFz

-

Qz

<

p i p . '

-

Q2

5

0. (50) Thus, condition (47) is automatically fulfilled and is redundant. (amin, l/p;) such that

R e - and post-multiplied by the matrix P;', we have (1 - ap;)PF1&1Pr1

+

aPF1PzQzPzP;l

Let condition (48) be fulfilled; there are values of a

E

(51)

2

E Z [ ( l

-

a & ) 1 + aP,-'P2][(1

-

ap;)1+ aPzP,-']. (39) By use of the matrix relation given in (21) and (22), condition (39) is sufficiently justified if

(1 -

+

aPclP2PzP;l

+

a2piPz(Qz -I)&,

a{Pz(Qz

-

1)pz

+

(p2 - p;Pl)'}

>

- p;p:

-

Q1.

Given matrix relations (21), since QZ

2

I, we have

0

5

(1

-

a&){a{Pz(Qz - I ) &

+

(Pz

- P ; S ) ~ } + & I - p i p ? }

2

t2[(l

-

ad)1

+

aPc1P21[(1 - a & ) I + apzp;'] (40) _{= (1}

_-

a p ; ) { a ( p z

-

p ; p l ) z

+

Q1

-

+

&(Q2

-

I ) p 2 ,

= -a2p;P,2

-

apg(1- a p ; ) ( p 2 ~ 1

+

~ 1 ~

-

2p i ( 1 ) - a p ; ) ' ~ ?

which is rearranged to be

( 1 - ap;)[pT - ( 1 - a p ; ) E 2 ] 1

+

a(1

-

ap)P;'P2PZP;1

+

(1 - a&)Q1

+

u P z Q ~ P z ,

= (1

-

ap;)Ql+ UPZQZPZ - &[(I

-

ap;)P1

+

aP212. (52)

2

a ( l -

ap;)EZ[Pr1P2

+

P2PF1]. (41)

By definitions of matrices X1 and Y1 given in (26) and (27), relation (52) becomes

Since it is known that (1 - a t Z )

2

(1

-

u p ; ) , condition (41) is sufficiently justified if Yl

2

pix?

(53) umin(XTIYiXF1)

2

pi. (54) [p:

-

(1 - a p ; ) t 2 ] 1

-

aEZ(P;lP2

+

P2P;') or +aPl-lPzPzP;l

2

0 (42) which is equivalently expressed by

On the other hand, it has been shown in the proof of Lemma 2 that

+[p?

-

(1

-

a p i ) ~ ~ - a t 4 ] 1

2

0. (43a) cmin(Y1)

2

1. (33b) Thus, followed from the definition of the interpolated QRBM given in (30), relations (33) and (54) are combined to provide

a(P;'P, - E21)(PzP;l

-

( 2 1 )

Thus, V I

2

is sufficiently justified if

V; = ami,(X;'YlX,')emin(Yl)

2

pi- (55)

0 Utilizing these lemmas, the following procedure is devised to interpolate Lyapunov equations and to extract the improved QRBM

p: - (1 - - a t 4

2

0 (43b) which is always fulfilled, for all a E (0, l/pz), since

p? - (1 -

.&)E2

-

at4

= (& -

t 2 )

+

-

t 2 )

2

0. (4) 0 (clcl.

810 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 4, APRIL 1994

Algorithm I:

1) Assigning Q1 = QZ = I.

2) Equate Lyapunov equations (14) and (15), and acquire the QRBMSs p1 and pz from (18) and (19).

3) Compute a m i n and bmin from relations (48) and (49) and make

the following adjustments: if amin

<

0, we make a m i n = 0;

and if a m i n

>

l/pz, we make a m i n = 1/&; if bmin

<

0, we

make bmin = 0; and if bmin

>

l / p f , we make b ~ n = l / p f .

4) Selecting U = 0.5(1/&

+

amin) and b = 0.5(1/p:

+

bmin),

the interpolated matrices YI and YZ are obtained from (27) and (29).

5) Normalize matrices YI and

YZ

to form new matrices QI and

Q2

that fulfil (20), and repeat from Step 2 until a convergent condition is detected.

We note that, in each iteration, two robustness measures (p1 and 112) are obtained in Step 2 of the proposed algorithm. To make appropriate use of Lemmas 2 and 3, the qualitative property of interpolating Lyapunov equations is investigated in Step 3. There are only two cases:

amin E [0, l/p;), then both Lemma 2 and Lemma 3 are applied in Step 4 such that the magnitudes of both max(p1, p2} and min(p1, pz} are improved in the next iteration.

ii) If a m i n = 0 and bmin = l/p:, or if bmin = 0 and a m i n = l/p;,

then only Lemma 2 is applied in Step 4 such that the magnitude of min (p1, p z } is improved in the next iteration. Since Lemma 3 is not applicable in this case, the process attempts to preserve the magnitude of max {pl

,

p z } . Nevertheless, it is still possible that the magnitude of max {PI, p2} is improved in the next iteration.

Obviously, the iterative process persistently improves the mag- nitude of max(p1, p2) until it is found that the magnitude of max{pl, p 2 } is trivially affected by further interpolations.

Example 1: Consider the system given in [2], the nominal matrix is given by

i) If a m i n = 0 and bmin E [0, l / p f ) , oc if bmin = 0 and

-3 -2

A =

[

1 01.

The bound pr in Theorem 2 is known to be 0.3820. By use of the proposed procedure, the result is

p c = 0.4495

where matrices in the Lyapunov equation (3) are given by

5.2361 2.6180 2.1817 1.3090

= [2.6180 2.61801 and = [1.3090 3.0544]. An improvement of 18% in QRBM is observed.

the nominal matrix is

Example 2: Consider the stabilized STOL aircraft given in [2],

1

1

-0.201 0.755 0.351 -0.075 0.033 -0.149 -0.696 -0.160 0.110 -0.048 A = 0.081 0.004 -0.189 -0.003 0.001

.

-0.173 0.802 0.251 -0.804 0.056 0.092 -0.467 -0.127 0.075 -1.162 p c = 0.0929

i

The bound

proposed procedure, the result is

in Theorem 2 is known to be 0.0774. By use of the

where matrices in the Lyapunov equation (3) are given by

2.2744 -1.7412 -1.7071 1.0771 -1.0905 -1.7412 23.9420 7.2540 -11.8002 10.3020 -1.7071 7.2540 4.7158 -3.1945 2.5491 1.0771 -11.8002 -3.1945 12.3753 -0.8221 -1.0905 10.3020 2.5491 -0.8221 27.1638 and 9.2069 2.1312 -0.6771 0.0540 -0.4574 2.1312 26.6291 8.3456 -6.0982 4.6249 P = -0.6771 8.3456 14.9825 -0.8588 0.7625 0.0540 -6.0982 -0.8588 14.6391 0.8907

I

-0.4574 4.6249 0.7625 0.8907 23.2163 An improvement of 20% in QRBM is observed.

v.

CONCLUDING &MARKS

The main theme of this correspondence is to derive a less conser- vative QRBM for the unstructurally perturbed systems. In this regard, interpolated Lyapunov equations have been examined, and an iterative computational procedure has been proposed. The two examples show that, with a handful of iterations, our method achieves a considerable improvement over old results.

REFERENCES

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[2] R. V. Patel and M. Toda, “Quantitative measures of robustness for multi- variable systems,” in Proc. Joint Automat. Conrr. Con&. San hancisco, [3] W. H. Lee, “Robustness analysis for state space models,” Alphatech

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[4] R. K. Yedavalli, “Improved measures of stability robustness for linear state space models,” ZEEE Trans. Automat. Contr., vol. AC-30, pp. 577-579, Jun 1985.

[5] R. K. Yedavalli and Z. Liang, “Reduced conservation in stability robustness bounds by state transformation,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 863-866, Sep 1986.

[6] K. Zhou and P. P. Khargonekar, “Stability robustness bounds for linear state space models with structured uncertainty,” IEEE Trans. Aufomat. Contr., vol. AC-32, pp. 621-623, Jul 1987.

[7] R. K. Yedavalli, “Stability robustness measures under dependent uncer- tainty” in Proc. American Contr. Con$, Atlanta, GA, pp. 820-823, Jun

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[8] G. W. Stewart, Infroduction to Matrix Computations. New York Aca- demic, 1973.