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Robust absolutely stable Lurie systemsLJUBOMIR T. GRUJIĆ a & DJORDJIJA B. PETKOVSKI b

a Faculty of Mechanical Engineering , University of Belgrade , P.O. Box 174, Belgrade ,11001 , Yugoslaviab Faculty of Engineering Sciences , University of Novi Sad , Veljka Vlahovića 3, Novi Sad ,YugoslaviaPublished online: 27 Apr 2007.

To cite this article: LJUBOMIR T. GRUJIĆ & DJORDJIJA B. PETKOVSKI (1987) Robust absolutely stable Lurie systems,International Journal of Control, 46:1, 357-368, DOI: 10.1080/00207178708933903

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INT. J. CONTROL, 1987, VOL. 46, NO. 1,357-368

Robust absolutely stable Lurie systems

LJUBOMIR T. GRUJlCt and DJORDJlJA B. PETKOVSKlt

This paper considers the robustness of Lurie systems with multiple non-linearitiessubject to modelling uncertainties and large parameter variations in the systemdynamics. New robustness results on Lurie systems are presented which lead to areduction of conservatism in robustness tests of absolute stability. It is shown that ifthe perturbations lie along given directions, it is possible to calculate a sector suchthat the perturbations belonging to the given direction do not disturb systemstability. The proposed approaches for robustness analysis are based on the Popovfrequency domain criterion as well as on the algebraic criterion for the absolutestability.The robustness resultsare illustrated by a second-order numericalexample.

I. IntroductionAn important theme in system theory is the preservation of various system­

theoretic properties in the face of perturbations in the system model. In this light,robustness of multivariable systems has received considerable attention in the last fewyears in control theory (Proc. Insrn elect. Engrs 1982, Petkovski 1983).

In this paper we focus on the robustness of Lurie-type systems subject to modelvariations. Robustness is used in the sense of being able to quantify the size ofperturbations which do not disturb system stability. This subject is of special interestsince stability is the most basic system-theoretic issue and since practical feedbacksystems must remain stable in the face of large variations in the system model.

With few exceptions, recently proposed methods to assess stability robustness ofmulti variable control systems are well suited for weakly structured uncertaintiesbecause they require only a matrix norm to bound the uncertainty. In other words, inthese approaches the class of 'allowable perturbations' is completely unstructuredexcept for the definition of the applicable norm. Such robustness criteria tend to bevery conservative for highly structured uncertainties because they fail to exploit theadditional structures. As known, in many practical situations, it may happen thatthere exists some a priori knowledge of the perturbations which may be encountered.In this case there is a need for a more detailed description of the 'allowableperturbations'.

Therefore, in the area of robustness analysis for multivariable control systems, oneof the basic needs is for more refined tests and measures of robustness. To overcomethese difficulties new approaches have been proposed in the literature (see for exampleLehtomaki et al. 1981, Postlethwaite et al. 1981, Yaman et al. 1984). However, thesetechniques have been proposed in the frequency domain. In this paper it is shown thatthe usefulness of the robustness test for physical systems can be greatly enhanced by a

Received 8 September 1986.t Faculty of Mechanical Engineering, University of Belgrade, P.O. Box 174, 11001

Belgrade, Yugoslavia.t Faculty of Engineering Sciences, University of Novi Sad, VeljkaVlahovica 3, Novi Sad,

Yugoslavia.

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358 L. T. Grujic and D. B. Petkovski

judicious choice or robustness criteria combined with physical knowledge or specificmodelling errors, using the state-space model of the system.

In the sequel we consider the robustness of Lurie-type systems with multiple non­linearitics subject to modelling uncertainties and large parameter variations in thesystem dynamics. New robustness results for Lurie systems are presented which leadto reduction of conservatism in a robustness test of absolute stability.

It should be pointed out that the concept or absolute stability, as proposed byLurie and Postnikov (1944), involves the robustness of the asymptotic stabilityproperty of the zero state with respect to any deviation of a non-linearity in a (so­called Lurie) sector. Lurie and Postnikov introduced the absolute stability concept inorder to cope with uncertainty about the non-linear structure. However, we are nowaware that the uncertainty of a mathematical description of a real system is alsorelated to its parameters; and that in turn raises the problem or what is meant by therobustness of a system property, such as the asymptotic stability which is consideredhere. It appears important for effective engineering applications, and for computer­aided design control systems, to study and resolve the problem of robustness of theabsolute stability of Lurie systems with respect to parameter variations.

2. System descriptionA general form of a Lurie system is described by (I):

dxdl = Ax + Bf(I:)

I: = Cx + Of(I:)

(I a)

(I b)

together with t E R denoting time, A E R"X", BE R"xm, C E RmX" and 0 E R">"representing matrices with nominal values or their entries, I: E R" and f: R" -+ R" iscomposed of};,f = (f, ,f2, ... ,fmjT, and belongs to thefamily No(L) on a matrix Luriesector L = [0, K] for K = diag {k" k2 , ••• ,km) E R,,:-xm. More precisely, we have thefollowing.

Definition I

A vector non-linearity f belongs to N o(L) iff:

(i) it is continuous in I: :f(I:) E C(Rm),

(ii) it vanishes at the origin: f(O) = 0,

(iii) it is in the Lurie sector L: };(I:)Ui-1E [0, k,], VI: E R", a, # 0, Vi = 1,2, ... , m,

and

(iv) it guarantees existence on R + and uniqueness of the solution of the system (1)for every initial state Xo E R".

For absolute stability of the system (I) on No(L) its state x = 0 must be its uniqueequilibrium state for every fEN o(L). From Definition 1 and Proposition 7 in Chapter1 of Grujic et al. (1984) the following result is straightforward.

Theorem 1

Positive semi-definiteness or the matrix UN:

U N = [CA-l B - 0 - oT + BT(AT)-lCT]

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Robust absolutely stable Lurie systems 359

is necessary and sufficient for x = 0 to be the unique equilibrium state of the system (I)for every j ENo(L).

Hence, positive semi-definiteness of the matrix UN is required herein as well as thatfor every L = diag {II' 12 , ... , 1m } E L the matrix E(L) = (I - OL) is non-singular. Thelatter condition is needed for explicit solution of 1: = Cx + OJ(1:) in 1: as shown inwhat follows.

The lack of knowledge of physical laws, characteristics or properties of a physicalsystem has unavoidable consequences for the approximations and simplificationsused in modelling the system mathematically. This explains why the accepted valuesof the parameters and/or non-linearities differfrom their true values. It means that thecorrect description of a physical Lurie system should be governed by

di A ...7­dt = i + DJ (1:)

f= Ci+ Ol(f)

(2 a)

(2 b)

Unfortunately we are not able to determine A, aand 1 exactly. Therefore theirdeviations,

tJ.A = A- A, tJ.B = a- Band tJ.j(f) = j(f) - /(f) (2 c)

are not known. Instead of that, the classes A and B of all possible or permissibledeviations tJ.A and tJ.B, respectively, and the class N o(L) of all possible or permissibleJ, are known. Naturally, tJ.j is such that both j and 1 belong to N o(L).

Let

and

A= {A: A= A + tJ.A, tJ.A E A}

B= {B: a= B + tJ.B, tJ.B E B}

(3 a)

(3 b)

3. The statement of a Lurie system robustness problemRobustness of (Lurie) systems can be related to any of their dynamic properties

such as (absolute) stability, controllability, observability and optimality, or to more ofthem. To be specific we shall refer to the following definitions.

Definition 2The system (I) is absolutely stable on No(L) iff its equilibrium state x = 0 is

asymptotically stable on the whole for every j ENo(L).

This is the definition of absolute stability in the classical sense.From the robustness point of view we can restate Definition 2 for the system (2) as

follows.

Definition 3

The system (I) is robust over A x B wit II respect to absolute stability onN o(L) iff its equilibrium state x = 0 is asymptotically stable on the whole for every(A, B,f) E A x Bx No(L).

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360 L. T. Grujic and D. B. Petkovski

Hence, we can pose the following problem.

Problem

Under what conditions absolute stability of the system (1) on No(L) guaranteesrobustness of the system (2) over A x fj with respect to absolute stability over N o(L)?

4. Popov-like approach to the rohustness of Lurie systemsA great advantage of the famous Popov frequency-domain criterion (Popov 1961,

1973) is the possibility to solve the absolute stability problem without solving thematrix Lurie equations (4):

ATH+HA+XXT=-Q, XER"xm

HB-XY= -S, YERmxm, SER"xm

yTy= r

(4 a)

(4 b)

(4 c)

for arbitrarily chosen positive definite Q = QT E R" X", a non-negative diagonalo ER~ xm, positive diagonal M E R'"xm and

S=·t(ATCT0+C™)

f= MK-' -t(D™ + MD + BTCT0 + 0CB)

(5 a)

(5 b)

Let j = p, I be the identity matrix of the appropriate order and He ( . ) denotethe hermitian part of a matrix ( . ).

If A is stable and (A, B)-controllable, then it is well known (Popov 1961, 1973,Narendra and Taylor 1973) that the solution (H, X, V), H = HT> 0, exists if T(jw),

T( -jw,jw) = He {M[K -I + C(A - jwl)-' B] + jw0[C(A - jwl)-' B + D]}

- BT(AT+ jwl)-'Q(A- jwl)-I B (6)

is positive semi-definite for every w ~ o.

Theorem 2If A is stable, (A, B) is controllable, H satisfies (4),and the positive diagonal matrix

M and the non-negative diagonal matrix 0 are such that T( - jw,jw) (6) is positivesemi-definite for every w ~ 0, then positive definiteness of R(6A, 6B, L) (7),

R(6A, 6B, L)=Q- {[6A +6BL(I- DL)-IC]T[H +tCT0L(I- DL)-'C]

+ [H + tCT0L(1 - DL)-' C]T[6A + 6BL(I- DL)-' C]} (7)

for every (6A, 6B, L) E A x B x L is sufficient for robustness of the system (2) onA x fj with respect to the absolute stability on N o(L).

All proofs are presented in the Appendix.

5. An algebraic approach to the robustness of Lurie systemsIn order to avoid solving the matrix Lurie equations (4), which are needed for the

robustness analysis of Lurie systems via the Popov method, purely algebraic methodswill be developed. They are based on the algebraic approach to the absolute stabilityanalysis proposed by Grujic (1978, 1980, 1981 a, b).

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Robust absolutely stable Lurie systems

Introducing the matrix function N: R m..... R" x m by

/li(O) = 0 and /li(r) = j,(r)Ui- 1 iffUi~ 0

then evidently

and

r = E(N)x

where

E(N) = (1- DN)-l C

}

361

(8 a)

(8 b)

(8 c)

(8 d)

(9 a)

is well defined because (I - DL) is non-singular for every L E L, hence [I - DN(r)] isnon-singular for every r E R" because N(r) E L, V r E R":

For fCE) = Lr the system (1) is reduced to the linear system (9),

dxdl = F(L)x

F(L) = A + B LE(L) (9 b)

Stability of the matrix F(L) for every L E L is necessary for absolute stability of thesystem (1). This condition and negative definiteness of the matrix [FT(L)P(L) +pT(L)F(L)] for every L E L, where

pel) = H + !ET (L)L0 C (10)

are sufficient for absolute stability of the system (1) on No(L) (Grujic 1978, 1980,1981 a, b). The matrix H is symmetric and positive definite, and 0 E Rm

xm is adiagonal matrix. The determination of Hand 0 was considered in Grujic (1978, 1980,1981 a, b), which was based on solving the Liapunov matrix equation by completelyavoiding the matrix Lurie equations (4). Knowledge of Hand 0, i.e. of pel), isessential for effective resolution of the problem.

Theorem 3If F(L) (9 b) is stable for every L E L, or equivalently [pel) + pT(L)] (10) is positive

definite for every L E L, and if [0 + FT(L)P(L) + PT(L)F(L)] is negative definite forevery L E L, then positive semi-definiteness of R(~A, ~B, L) (7) for every(~A, ~B, L) E A x B x L is sufficient for robustness of the system (2) on A x fj withrespect to the absolute stability on N o(L).

Theorems 2 and 3 present general sufficient conditions for robustness of the Luriesystem (2) on A x fj with respect to the absolute stability on N o(L). Their effectiveconceptual applications are established below.

6. Perturbation characterizationWhile in some cases the deviation of the perturbed system from the nominal

model, represented by the matrices ~A and ~B, is unknown, there is usually someknowledge of its size. In what follows the results of Theorem 2, i.e. Theorem 3, are

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362 L. T. Grujic and D. B. Petkouski

expressed alternatively in terms of specific bounds on AA and AB, so that the resultsbecome easier to appreciate.

Theorem 4

Let A be stable, (A, B) be controllable, H satisfy (4), and the positive diagonalmatrix M and the non-negative diagonal matrix 8 be such that T( - jw,jw), (6), ispositive semi-definite for every w ~ 0. Then

IIAAII + Ilell -11(1- Dl)-lll -IILII - IIABII

< Amin(0)(2I1HII + IICII2 11(1- DL)-lll . IILII . 11811)-1 (I I)

for every (AA, AB, L) E A x B x L, where II - II denotes the spectral norm, is sufficientfor robustness of the system (2) on A x jj with respect to the absolute stability onNo(L).

In a similar way, following the algebraic approach to the robustness of Luriesystems we give the following theorem.

Theorem 5

If F(L) (9 b) is stable for every L E L, and if [0 + FT(L)P(L) + PT(L)F(L)] is negativedefinite for every L E L, then the condition (II) is sufficient for robustness of the system(2) on A x jj with respect to the absolute stability on N o(L).

Regretfully, like many other multi variable robustness results, the proposedapproach has its limitations. Namely, in proving Theorems 4 and 5, we assumed thatthe only information we possess about the model uncertainty or model error is a singlenumber which measures the size or magnitude of the model error. It is known thatresults that use only error magnitude information, called unstructured information,are very restrictive because they guard against absolutely all types of structure in thesystem model. However, although the guaranteed margin provided by Theorems 3and 4, is a conservative one, the knowledge of the structure of the perturbed matricescan be used to reduce this conservatism.

To mitigate these difficulties we shall combine the information concerning thenature of the perturbations which can physically occur, and mathematical character­ization of the perturbation matrices. That is, we define the perturbations in certaindirections which are more appropriate from a physical standpoint. This leads todecomposition of the perturbations into two components, one of which lies along thegiven direction in the space of all perturbation matrices. In particular, define thematrices AA and AB as

AA = y(t)A + l!A

AB = y(t)B + l!B

(12 a)

(12 b)

where y(t) is a scalar function, and l!A and l!B represent the perturbations in systemdynamics, which lie out of the directions A and B, respectively.

Notice that the case y(t) = 0, for all t E [0, 00), which corresponds to unstructuredinformation, is included in the above formulation.

The decomposition of the perturbation matrices must be made on the basic ofengineering judgement about the type of model uncertainty and model error that are

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Robust absolutely stable Lurie systems 363

reasonable for the nominal design model representing the physical system. In otherwords, the designer must be able to identify the class of perturbations which maypossibly occur in the physical system.

Now, suppose that the perturbation matrices IiA and liB lie entirely along thedirections A and B, respectively, that is lJ.A = 0 and lJ.B = O. The following theoremsgive the sector (Ym;n, Ymax)' i.e., the bounds on the scalar function y(t) such that theperturbed system remains stable.

Theorem 6

Let A be stable, (A, B) be controllable, H satisfy (4), and the positive diagonalmatrix M and the non-negative diagonal matrix 0 be such that T( - jw,jw), (6), ispositive semi-definite for every w;;o O. Then, if the following inequalities are satisfied:

A';;;~(U) = Ymin < y(t) < Ymax = iz: (U)

where y(t) is a memoryless, time-varying non-linear function,

(13)

U = Q-l{[A+ BL(I- Dl)-lC]T[H +!CT0L(I- Dl)-IC]

+ [H + !CT0L(I- Dl)-l C]T . [A + BL(I + DL)-l C]} (14)

and IiA = y(t)A, liB = y(t)B, where (IiA, liB, L) E A x B x L, then perturbed system(2) is robust on A x 8 with respect to the absolute stability on N oiL).

It can be easily proved that if the algebraic approach to absolute stability of Luriesystem is used then the following theorem holds.

Theorem 7If F(L),(9 b) is stable for every L ELand if [Q + FT(L)P(L) + p T(L)F(L)] is negative

definite for every L E L, then the condition (13), (14) is sufficient for robustness of thesystem (2) on A x 8, where IiA = y(t)A, and liB = y(t)B, with respect to the absolutestability on N oiL).

In essence, Theorem 6 and Theorem 7 show that if y(t) E (Ymin, Ymax) then theperturbed system remains absolutely stable. Hence, the directions

{y(t)A; y(t)B:y(t) E (Ymin, Ymax)}

are termed stability directions.The next corollaries are easy to prove but have an interesting interpretation.

Corollary I

If Amin(U) is not negative then the bound Ymin ceases to exist and

y(t) E ( - 00, Ymax)

Corollary 2If Ama•(U) is not positive then the bound Yma• ceases to exist and

y(t) E (Ymin, 00)

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364 L. T. Grujic and D. B. Petkovski

ExampleA Lurie- Postnikov system

~=(-I: _1~)X+(_:)f(L)

L = (I, 2)x + !f(L)

where f(L) E No(L) and L= [0, I] will be considered. Evidently,

E(L) = 5~L(I,2)

and

~5L

-18+-­5-L

F(L) =5L

8-­5-L

2(1 + 5~\) J-2(6+~)5- L

It is easy to verify stability of F(L), VL E L. Let a(L) = IOL. Then, for H = I and5- L

e = 4, which were determined in Grujic (1981 b),

[

1+ a(L)P(L) =

2a(L)

2a(L) ]

1+ 4a(L)

In this case P(L) == PT(L) is positive definite for every L E L. Besides, the matrix[ql + FT(L)P(L) + PT(L)F(L)] is negative definite for q = 18 for every L E L, whichcan be easily verified.

Now we are ready to establish the bounds on the perturbation matrices ~A and~B such that the perturbed Lurie system remains absolutely stable.

Case I. A = 0, B = 0.This case corresponds to the approaches suggested so far in the literature, when

the perturbation matrices are characterized simply by their norms.The results of Theorem 5 have been employed for establishing the bounds on

perturbation matrices. The bounds are defined by the inequality

II~AII +2'795084811~BII < 1·1481481

It is known that the robustness criteria in terms of a matrix norm tend to beconservative. The measures of the 'size' of the perturbations used in this case do notdistinguish the 'direction' of the perturbations and, in practice, many such perturb­ations simply cannot occur physically.

To evaluate realistically the robustness of the closed-loop Lurie system, modellinguncertainties about the design model and parameter variations must be known a

priori. In what follows we consider two cases.

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Robust absolutely stable Lurie systems

Case 2. A=[ -~ -~J. B=[~JIn this case

;,,;;.~ (U) = - 0'008308

;'';;;~(U) = -0'10036

365

From Theorem 6, i.e. Corollary 2 it follows that the perturbed system will remainstable as long as

y(t) E (-0'08308,00)

Case 3. A = [~

In this case

2J' B = [0'5J.o 0'5

;'';;.'. (U) = 1·3656048

;'';;i~(U) = -0'43209

From Theorem 6 it follows that the perturbed system will remain stable if

y(t) E (-0-43209, 1-3656048)

It is important to emphasize that the robustness results based on Theorem 6 arefar less conservative than the results obtained via the matrix norm characterization.Furthermore, the perturbation matrices !1Aand !18 considered in Case 2 and Case 3do not satisfy the condition (10) required by Theorems 4 and 5.

Finally, in order to compare the frequency domain and algebraic domainrobustness criteria and to give the reader some indication of how conservative thefrequency domain criteria are, consider Case 3. It can be shown, after intensivenumerical calculations, that using the Popov-like approach to the robustness of Luriesystems (6), the upper bound of y(t) is given by

I'm ax = 0·33875

which is 24·80% of the upper bound obtained from the algebraic criteria based onTheorem 6.

7. Summary and conclusionsAn approach for stability robustness analysis of Lurie systems with multiple non­

linearities subject to model variations has been proposed. Results have been generatedin both the frequency and algebraic domain which quantitatively characterize a wideclass of variations in the system model which will not destabilize the system. Theproblem of minimizing the conservativeness of robustness criteria has also beenaddressed. The reduction is based on a priori knowledge of model uncertainties and

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model variations about the design model. The results indicate that robustness criteriacoupled with physical understanding of system dynamics are essential to a successfulapplication of the robustness theory. Computationally, the method is very feasible; itinvolves a very simple numerical algorithm. The robustness results have beenillustrated by a second-order numerical example.

ACKNOWLEDGMENT

This work was supported in part by the U.S.-Yugoslav Joint Fund for Scientificand Technological Cooperation in cooperation with DoE under Grant PP-727.

AppendixProof of Theorem 2

Let El E R" x m be non-negative diagonal, H = HT E R" X" and

Along a motion of the system (I) the eulerian derivative of vf (A I) is

li(x) = xT(ATH + HA)x + PBTHx

+ xTHBf +t!TElC(Ax + Bx) +!(Ax + Bf)TCTElf (A 2)

Let ME R'" xm be positive diagonal, the choice of which is free. Then Qf,

satisfies

Qf (E) = [ET - r (E)K - 1]Mf(E) (A 3)

(A 4)

(A 8)

It now results that, by adding to and subtracting from the right-hand side of (A 2) theterms Qf (E) and XTXXTx,

ri(x) = xT(ATH + HA + XXT)x _ XTXXTX _ Qf

+ If[BTH +!(ElCA + MC)]x + xT[HB +HATCTEl + CTM)]f

-P[MK-'-!(DTM+MD+BTCTEl+ElCB]f (A 5)

Since T( - jw,jw) is positive definite for all to ;;;, 0, A is stable and (A, B) is controllablethen there exist H, X and Y obeying (4). Since H is chosen to satisfy (4) then 1/ (A 5)can be set in the form

li(x) = -xTQx -IIXTx + Vfl1 2- Qf (A 6)

Using this result we easily verify that the eulerian derivative of vJ(x) (A I) along amotion of the perturbed system (2) is

r/(x) = - xT{Q - (L\AT+ CTl:TI\iL\BT)(H + !CTElI\il:C)

+ (H + !CTl:TI\iElC)(L\A + L\BI\il:C)}x - \IXTx + V!\I2 - QJ (A 7)

It is now evident that (7) and (A 7) imply

l/(X) = - xTR(L\A, L\ B, I\i),,, - II XTx + V!11 2- QJ

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Robust absolutely stable Lurie systems 367

For every x E R", i.e. for every f E R m, and for every f e No(L), i.e. for every N E L,

there is L E L such that

R("'A, "'8, N) = R("'A, "'8, L), 'if("'A, "'8) E A x B (A 9)

Positive definiteness of R("'A, "'8, L) for every ("'A, "'8, L) E A x B x L together with(A 9), non-negativeness of both IIXTx+ V/l1 2 and nI for aliiENo(L) prove that ,/ isobviously negative definite for every ("'A, "'8,1) E A x B x N o(L). This result andpositive definiteness together with radial unboundedness of vI (A I) due to positivedefiniteness of H, non-negativeness of diagonal e, lENo(L) and (4)-(6) proveabsolute stability of x = 0 of the system (2) on A x B x No(L) (Narendra and Taylor1973). Hence, the system (2) is robust over A x fj with respect to the absolute stabilityon No(L) in view of Definition 3, which proves Theorem 2. 0

Proof of Theorem 3

Notice that stability of F(L) and negative definiteness of [FT(L)P(L) + P" (L)F(L)]that is implied by positive definiteness of 0 and negative definiteness of [0 +FT(L)P(L) + PT(L)F(L)] imply positive definiteness of [P(L) + PT(L)] due to theLiapunov matrix theorem. For every (f,1) E R'" x No(L) there is L E L for whichF(N) = F(L) and P(N) = P(L). This result, positive definiteness of [P(L) + PT(L)] forevery L ELand

prove positive definiteness and radial unboundedness of vi (x) (A I). Along a motion of(2) the eulerian derivative of vl(x) (A I) is

(A 10)

Negative definiteness of [FT(L) . P(L) + pT(L) . F(L) + OJ for every L ELand positivesemi-definiteness of R("'A, "'8, L) for every ("'A, "'8, L) E A x B x L imply, respec­tively, the same properties of [FT(N)' P(N) + PT(N)' F(N) + OJ and R("'A, "'8, N)for every ("'A, "'8, N)E A x B x L. Hence, ,;! (A 10) is negative definite for all("'A, "'8,/) E A x B x N o(L) which together with the properties of vi proves robust­ness of the system (2)over A x fj with respect to the absolute stability on N o(L) in viewof Definition 3. Thus, Theorem 3 holds. 0

Proof of Theorem 6

To prove the conditions (14) recall the following lemma.

Lemma I (Trail and Thornheim 1957)

If Rand S are symmetric matrices and R is positive definite, there exists a non­singular matrix W such that

W T• (R + S)W = I + G

where matrix G is a diagonal matrix whose elements are eigenvalues of R - I S.

Therefore, using the results of Lemma I, and Theorem 2, it can easily be concludedthat the perturbed system will remain absolutely stable if the following inequality is

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satisfied:

Robust absolutely stable Lurie systems

1- y(t)Aj{O-I([A + BL(I- OL)-I . C]T[H + -tCT8L(I- OL)-I . C]

+ [H +-tCT. 8L ·(1- OL)-I . C]T. [A + BL(I- Oq-' . C])} >0

i.e.,

I - y(t)Aj(U) > 0, j = 1,2, ..., n

Now, under the assumption that Am,x (U) > 0, which is usual, it follows that

Ym,x = A';;,'x (U)

In a similar way, if Amin(U) < 0 then

Ymin = ).';;i~(U)

i.e., the perturbed system remains absolutely stable if

y(t) E (Ymin, Ym,xl

for all t E [0, 00). o

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