A.V. SAVKIN AND I.R. PETERSEN uncertain systems in which the uncertainty satisæes a certain integral quadratic constraint; e.g., see ë5, 6, 7ë. The ad

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1 Journal of Mathematical Systems, Estimation, and Control Vol. 6, No. 3, 1996, pp. 1í14 cæ 1996 Birkhíauser-Boston Robust H 1 Control of Uncertain Systems with Structured Uncertainty æ Andrey V. Savkin y Ian R. Petersen y Abstract This paper considers a problem of absolute stabilization with a speciæed level of disturbance attenuation for a class of uncertain systems in which the uncertainty is structured and satisæes a certain integral quadratic constraint. The paper shows that an uncertain system is absolutely stabilizable with a speciæed level of disturbance attenuation if and only if there exists a solution to a corresponding H 1 control problem. The paper also shows that if an uncertain system can be absolutely stabilized with a speciæed level of disturbance attenuation via nonlinear output feedback control, then it can be absolutely stabilized with a speciæed level of disturbance attenuation via linear output feedback control. Key words: robust H 1 control, absolute stabilizability, uncertain systems, integral quadratic constraint, structured uncertainty, nonlinear control AMS Subject Classiæcations: 93B36 1 Introduction An important idea to emerge in recent years is the connection between the Riccati equation approachtoh 1 control and the problem of stabilizing an uncertain system containing norm bounded uncertainty; e.g., see ë1, 2, 3ë. In these papers, the notion of stabilizability considered is that of quadratic stabilizability. This notion can be extended to consider the problem of robustly stabilizing a system and also giving a speciæed level of disturbance attenuation; e.g., see ë4ë. In contrast to the quadratic stabilizability approach mentioned above, an alternative approach involves the notion of absolute stabilizability for æ Received June 6, 1994; received in ænal form December 12, Summary appeared in Volume 6, Number 3, y This work was supported by the Australian Research Council. 1

2 A.V. SAVKIN AND I.R. PETERSEN uncertain systems in which the uncertainty satisæes a certain integral quadratic constraint; e.g., see ë5, 6, 7ë. The advantage of this approach is that it allows for non-conservative results to be obtained for the case of uncertain systems with structured uncertainty. Furthermore, using this framework, it was shown in ë5ë that if an uncertain system with structured uncertainty can be absolutely stabilized using nonlinear output feedback control, then it can also be absolutely stabilized using linear output feedback control. The main contribution of this paper is to extend the results of ë5ë to consider a problem of absolute stabilization with a speciæed level of disturbance attenuation. Also, the results of this paper enable some of the strong assumptions made in ë5ë to be weakened. As in ë5ë, a key technical result used in this paper is an extension of the ës-procedure" result of ë8ë. An important feature of the nonlinear S- procedure result of this paper is that the assumptions required are signiæcantly weaker than the assumptions required in the nonlinear S-procedure of ë5ë. It is this nonlinear S-procedure result which enables us to consider the problem of absolute stabilization with a speciæed level of disturbance attenuation for the case of uncertain systems with structured uncertainty. Furthermore, this result also allows us to consider the possible use of nonlinear controllers. The remainder of the paper proceeds as follows. In Section 2, we deæne the class of uncertain systems under consideration. For this class of uncertain systems, we deæne our notion of absolute stabilizability with a speciæed level of disturbance attenuation. In Section 3, we establish our technical result concerning the extension of the ës-procedure" result of ë8ë to the case of nonlinear systems. Section 4 of the paper presents our main result which is a necessary and suæcient condition for absolute stabilizability with a speciæed level of disturbance attenuation. This condition is given in terms of the existence of solutions to a pair of parameter dependent algebraic Riccati equations of the game type. 2 Problem Statement We consider an output feedback H 1 control problem for an uncertain system of the following form: _xètè = Axètè+B 1 wètè+b 2 uètè+ zètè = C 1 xètè+d 12 uètè; ç 1 ètè = K 1 xètè+g 1 uètè;. D s ç s ètè; 2

3 ROBUST H 1 CONTROL ç k ètè = K k xètè+g k uètè; yètè = C 2 xètè+d 21 wètè è2.1è where xètè 2 R n is the state, wètè 2 R p is the disturbance input, uètè 2 R m is the control input, zètè 2 R q is the error output, ç 1 ètè 2 R h1 ;:::;ç k ètè 2 R h k are the uncertainty outputs, ç 1 ètè 2 R r1 ;:::;ç k ètè 2 R r k are the uncertainty inputs and yètè 2 R l is the measured output. The uncertainty in this system is described by a set of equations of the form ç 1 ètè = ç 1 èt; ç 1 èæèj t è ç 2 ètè = ç 2 èt; ç 2 èæèj t è. ç k ètè = ç k èt; ç k èæèj t è è2.2è where the following Integral Quadratic Constraint is satisæed. Deænition 2.1 èintegral Quadratic Constraint; see ë9, 1, 11, 12, 13, 14, 15, 5, 6, 16, 7ë.è An uncertainty of the form è2.2è is an admissible uncertainty for the system è2.1è if the following conditions hold: Given any locally square integrable control input uèæè and locally square integrable disturbance input wèæè, and any corresponding solution to the system è2.1è, è2.2è, let è;t æ è be the interval on which this solution exists. Then there exist constants d 1 ç ;:::;d k çand a sequence ft i g 1 i=1 such that t i! t æ, t i ç and Z ti kç s ètèk 2 dt ç Z ti kç s ètèk 2 dt + d s 8i 8s =1;:::;k: è2.3è Here kæk denotes the standard Euclidean norm and L 2 ë; 1è denotes the Hilbert space of square integrable vector valued functions deæned on ë; 1è. Note that t i and t? may be equal to inænity. The class of all such admissible uncertainties çèæè =ëç 1 èæè;:::;ç k èæèë is denoted æ. In references ë14ë and ë15ë,a number of examples are given of physical systems in which the uncertainty naturally æts into the above framework. Also, note that the above uncertainty description allows for uncertainties in which the uncertainty input ç s depends dynamically on the uncertainty output ç s. In this case, the constant d s may beinterpreted as a measure of the size of the initial condition on the uncertainty dynamics. Also, it is clear that the uncertain system è2.1è, è2.3è allows for uncertainty satisfying a norm bound condition. In this case, the uncertain system would be described by the state equations _xètè = ëa + D s æ s ètèk s ëxètè+ëb D s æ s ètèg s ëuètè+b 1 wètè;

4 A.V. SAVKIN AND I.R. PETERSEN yètè = C 2 xètè+d 21 wètè; sup kæ s ètèk ç1 è2.4è where æ s ètè are the uncertainty matrices èsee e.g. ë2, 3ëè. Indeed, let ç s ètè =æ s ètèëk s xètè+g s uètèë. Then the uncertainties ç s èæè satisfy conditions è2.3è with d s = and with any t i. For the uncertain system è2.1è, è2.3è, we consider a problem of absolute stabilization with a speciæed level of disturbance attenuation. The class of controllers considered are nonlinear output feedback controllers of the form x_ c ètè = æèx c ètè;yètèè; uètè = çèx c ètè;yètèè; è2.5è where æèx c ;yè and çèx c ;yè are continuous vector functions. Note that the dimension of the controller state vector x c ètè in è2.5è may be arbitrary. Deænition 2.2 The uncertain system è2.1è, è2.3è is said to be absolutely stabilizable with disturbance attenuation æ èvia nonlinear output feedback controlè if there exists an output feedback controller è2.5è and constants c 1 é and c 2 é such that the following conditions hold: èiè For any initial condition ëxèè;x c èèë, any admissible uncertainty inputs çèæè and any disturbance input wèæè 2 L 2 ë; 1è, then ëxèæè;x c èæè;uèæè;ç 1 èæè;:::;ç k èæèë 2 L 2 ë; 1è èhence, t æ = 1 è and kxèæèk kx c èæèk kuèæèk ç c 1 ëkxèèk 2 + kx c èèk 2 + kwèæèk kç s èæèk 2 2 d s ë: è2.6è èiiè The following H 1 norm bound condition is satisæed: If xèè = and x c èè=, then J = æ sup sup wèæè2l 2ë;1è çèæè2æ kzèæèk 2 2, c 2 P k d s kwèæèk 2 2 éæ 2 : è2.7è Here, kqèæèk R 2 denotes the L 2 ë; 1è norm of a function qèæè. That is, kqèæèk 2 æ 1 2 = kqètèk 2 dt. Observation 2.1 It follows from the above deænition that if the uncertain system è2.1è, è2.3è is absolutely stabilizable with disturbance attenuation æ, then the corresponding closed loop system è2.1è, è2.3è, è2.5è 4

5 ROBUST H 1 CONTROL with wèæè 2 L 2 ë; 1è, has the property that xètè! as t! 1. Indeed, since ëxèæè;uèæè;çèæè;wèæèë 2 L 2 ë; 1è, we can conclude from è2.1è that _xèæè 2 L 2 ë; 1è. However, using the fact that xèæè 2 L 2 ë; 1è and _xèæè 2 L 2 ë; 1è, it now follows that xètè! ast!1. 3 S-Procedure for Nonlinear Systems In this section, we present a result which extends the ës-procedure" of ë8ë. The result is closely related to the result presented in ë5ë. However, the assumptions required for this result are considerably weaker than the assumptions required in ë5ë. The main result of this section applies to a nonlinear, time-invariant system of the form _hètè =æèhètè;èètèè è3.1è where hètè 2 R N is the state and èètè 2 R M is the input. Associated with the system è3.1è is the following set of functionals: f èhèæè;èèæèè = f 1 èhèæè;èèæèè =. f k èhèæè;èèæèè = ç èhètè;èètèèdt; ç 1 èhètè;èètèèdt; ç k èhètè;èètèèdt: Assumptions The system è3.1è and associated set of functionals satisfy the following assumptions: 3.1 The functions æèæ; æè;ç èæ;æè;:::;ç k èæ;æè are continuous. 3.2 For all èèæè 2 L 2 ë; 1è and all initial conditions hèè 2 R N, the corresponding solution hèæè belongs to L 2 ë; 1è and the corresponding quantities f èhèæè;èèæèè, f 1 èhèæè;èèæèè, :::, f k èhèæè;èèæèè are ænite. 3.3 Given any " é, there exists a constant æ é such that the following condition holds: For any input function è èæè 2 fè èæè 2 L 2 ë; 1è; kè èæèk 2 2 ç æg and any h 2 fh 2 R N : kh k ç æg, let h ètè denotes the corresponding solution to è3.1è with initial condition h èè = h. Then jf s èh èæè;è èæèè j é"for s =;1;:::;k. Note, Assumption 3.3 is a stability type assumption on the system è3.1è. 5

6 A.V. SAVKIN AND I.R. PETERSEN Notation For the system è3.1è satisfying the above assumptions, we deæne æ ç L 2 ë; 1è as follows: æ is the set of fhèæè;èèæèg such that èèæè 2 L 2 ë; 1è and hèæè is the corresponding solution to è3.1è with initial condition hèè =. Theorem 3.1 Consider the system è3.1è and associated functionals and suppose the Assumptions 3.1 í 3.3 are satisæed. If f èhèæè;èèæèè ç for all fhèæè;èèæèg2æsuch that f 1 èhèæè;èèæèè ç ;:::;f k èhèæè;èèæèè ç, then there exist constants ç ç ;ç 1 ç;:::;ç k ç such that P k s= ç s é and ç f èhèæè;èèæèè ç ç 1 f 1 èhèæè;èèæèè + ç 2 f 2 èhèæè;èèæèè + æææ+ç k f k èhèæè;èèæèè è3.2è for all fhèæè;èèæèg2æ. In order to prove this theorem, we will use the following convex analysis result, the proof of whichwas given in ë5ë. However, we ærst introduce some notation. Notation Given S ç R n and T ç R n, then S + T := fx + y : x 2 S; y 2 T g. Also, clèsè denotes the closure of the set S. Lemma 3.1 èsee ë5ë for proofè Consider a set M ç R k+1 with the property that a + b 2 clèmè for all a; b 2 M. If x ç for all vectors æ æ x æææ x k 2 M such that x1 ç ;:::x k ç, then there exist constants ç ç ;:::;ç æ P k k ç such that s= ç s é and ç x ç ç 1 x 1 + ç 2 x 2 + æ æææ+ç k x k for all x æææ x k 2 M: Proof of Theorem 3.1: the following claim. In the order to prove this theorem we establish Claim: Given any æ é and any input è èæè 2 L 2 ë; 1è, then there exists a constant æ é such that the following condition holds: for any h 2 fh 2 R N : kh k ç æ g, let h 1 ètè denotes the corresponding solution to è3.1è with initial condition h 1 èè = and let h 2 ètè denotes the corresponding solution to è3.1è with initial condition h 2 èè = h. Then jf s èh 1 èæè;è èæèè, f s èh 2 èæè;è èæèè j éæ for s =;1;:::;k. Indeed, let æ é be some constant and let æ be the constant from Assumption 3.3 corresponding to " = æ 4. According to Assumption 3.2, ëh 1 èæè;è èæèë 2 L 2 ë; R 1è and, therefore, there exists a T é such that kh 1 èt èk ç æ 2 and 1 T kè ètèk 2 dt ç æ. Assumption 3.1 implies that there exists a constant æ é such that for all kh k éæ, the solution h 2 èæè of the system è3.1è with input è èæè and initial condition h 2 èè = h satisæes condition Z T jç s èh 1 ètè;è ètèè, ç s èh 2 ètè;è ètèèjdt é æ 2 6 è3.3è

7 ROBUST H 1 CONTROL for all s and kh 2 èt è, h 1 èt èk é æ 2 : è3.4è Since kh 1 èt èk ç æ 2,wehave from è3.4è that kh 2èT èk çæ. Furthermore, Assumption 3.3 and time invariance of the system è3.1è imply that jç s èh 1 ètè;è ètèè, ç s èh 2 ètè;è ètèèjdt ç T T èjç s èh 1 ètè;è ètèèj + jç s èh 2 ètè;è ètèèjèdt ç æ 4 + æ 4 = æ 2 : From this and the inequality è3.3è the claim follows immediately. Now suppose f èhèæè;èèæèè ç for all fhèæè;èèæèg2æ such that and let f 1 èhèæè;èèæèè ç ;:::;f k èhèæè;èèæèè ç næ æ o M := f èhèæè;èèæèè ::: f k èhèæè;èèæèè 2 R k+1 : fhèæè;èèæèg2æ It æ follows fromæthe assumption on the set æ that x ç for all vectors x æææ x k 2 M such that x1 ç ;:::;x k ç. Let fh a èæè;è a èæèg2æ and fh b èæè;è b èæèg2æ be given. Since h a èæè 2 L 2 ë; 1è, then there exists a sequence ft i g 1 i=1 such that T i é for all i, T i!1and h a èt i è! as i!1. Now consider the corresponding sequence fh i èæè;è i èæèg 1 i=1 ç æ, where ç èa ètè t2ë;t è i ètè = i è; è b èt, T i è t ç T i : We will establish that f s èh i èæè;è i èæèè! f s èh a èæè;è a èæèè + f s èh b èæè;è b èæèè as i! 1 for s = ; 1;:::;k. Indeed, let s 2 f; 1;:::;kg be given and æx i. Now suppose h ~ i b èæè is the solution to è3.1è with input èèæè =è bèæè and initial condition h ~ i b èè = h aèt i è. It follows from the time invariance of the system è3.1è that h i ètè ç h ~ i b èt, T iè. Hence, f s èh i èæè;è i èæèè = ç s èh i ètè;è i ètèèdt Z Ti = ç s èh a ètè;è a ètèèdt + ç s èh i ètè;è b èt,t i èèdt T Z i Ti = ç s èh a ètè;è a ètèèdt + f s è h ~ i bèæè;è b èæèè: Using the fact that h a èt i è!, the R above claim implies f s è h ~ i b ètè;è bètèè! T f s èh b èæè;è b èæèè as i! 1. Also, i ç s èh a ètè;è a ètèèdt! f s èh a èæè;è a èæèè. Hence, f s èh i èæè;è i èæèè! f s èh a èæè;è a èæèè + f s èh b èæè;è b èæèè: 7 :

8 A.V. SAVKIN AND I.R. PETERSEN From the above, it follows that the set M has the property that a + b 2 clèm è for all a; b 2 M. Hence, Lemma 3.1 implies that there exist constants ç ç ;:::;ç 1 ç such that P k s= ç s é and ç x ç ç 1 x 1 + æææ+ç k x k for all æ x æææ x k æ 2M: That is, condition è3.2è is satisæed. 2 4 The Main Results In this section, we present the main result of this paper which establishes a necessary and suæcient condition for the uncertain system è2.1è, è2.3è to be absolutely stabilizable with a speciæed level of disturbance attenuation. This condition is given in terms of the existence of solutions to a pair of parameter dependent algebraic Riccati equations. The Riccati equations under consideration are deæned as follows: Let ç 1 é, :::,ç k ébegiven constants and consider the algebraic Riccati equations èa, B 2 E,1 1 ^D 12 ^C 1 è X + XèA, B 2 E,1 1 ^D 12 ^C 1 è + Xè ^B1 ^B 1, B 2 E,1 1 B 2èX + ^C 1 èi, ^D12 E,1 1 ^D 12è ^C1 =; è4.1è èa, ^B1 ^D 21 E 2,1 C 2èY + Y èa, ^B1 ^D 21 E 2,1 C 2è + Y è ^C 1 ^C1, C 2E 2,1 C 2èY + ^B1 èi, ^D 12 E 2,1 ^D 21 è ^B 1 = è4.2è where ^C 1 = C 1 p ç1 K 1. p çk K k ; ^D12 = D 12 p ç1 G 1. p çk G k ^D 21 = æ æ,1 D 21 ::: æ ; ; E 1 = ^D 12 ^D12 ; E 2 = ^D21 ^D 21 ; ^B 1 = æ æ,1 B 1 p ç1,1 D1 ::: p çk,1 Dk æ : è4.3è Assumptions The uncertain system è2.1è, è2.3è will be required to satisfy the following additional assumptions: 4.1 The pair èa; C 1 è is observable. 4.2 E 1 é : 4.3 The pair èa; B 1 è is controllable. 8

9 ROBUST H 1 CONTROL 4.4 E 2 é : Theorem 4.1 Consider the uncertain system è2.1è, è2.3è and suppose that Assumptions are satisæed. Then the following statements are equivalent: èiè The uncertain system è2.1è,è2.3è is absolutely stabilizable with disturbance attenuation æ via the nonlinear output feedback control è2.5è. èiiè There exist constants ç 1 é ;:::;ç k é such that the Riccati equations è4.1è and è4.2è have solutions Xéand Y é and such that the spectral radius of their product satisæes çèxy è é 1. If condition èiiè holds, then the uncertain system è2.1è, è2.3è is absolutely stabilizable with disturbance attenuation æ via a linear controller of the form where _x c ètè = A c x c ètè+b c yètè; uètè = C c x c ètè è4.4è A c = A + B 2 C c, B c C 2 +è^b1,b c^d21 è ^B 1 X B c = èi, YXè,1 èyc 2+ ^B1^D 21 èe 2,1 C c =,E 1,1 2X + ^D 12 ^C1 è: è4.5è Proof: èiè è èiiè Consider the set æ of vector functions çèæè =ëxèæè;x c èæè;çèæè;wèæèë in L 2 ë; 1è connected by è2.1è, è2.5è and the initial condition ëxèè;x c èèë =. Condition è2.7è implies that there exists a constant æ 1 é such that Jé æ 2,2æ 1.Let æ 2 = minëè2c 1 kè,1 ;æ 1 è2c 2 kèc 1 +1èè,1 ;æ 1 è2èc 1 +1èè,1 ë, where c 1 and c 2 are the constants from Deænition 2.2. Consider the functionals f ;f 1 ;:::;f k from æ to R where f èçèæèè =,èkzèæèk 2 2, æ2 kwèæèk æ 2 kçèæèk 2 è; f 1 èçèæèè = kç 1 èæèk 2 2,kç 1 èæèk 2 2+æ 2 kçèæèk 2 ;. f k èçèæèè = kç k èæèk 2 2,kç kèæèk 2 2+æ 2 kçèæèk 2 : è4.6è Here kçèæèk 2 = kxèæèk kx c èæèk kçèæèk kwèæèk 2 2. We will prove that f èçèæèè ç for all çèæè 2 æ such that f s èçèæèè ç for s =1;:::;k. Indeed, if f s èçèæèè ç for s =1;:::;k, then the vector function çèæè satisæes the 9

10 A.V. SAVKIN AND I.R. PETERSEN constraints è2.3è with d s = æ 2 kçèæèk 2 2 and t i = 1. Hence, condition èiè of Deænition 2.2 implies that kçèæèk 2 ç èc 1 +1èkwèæèk 2 2+c 1 kæ 2 kçèæèk 2. Since æ 2 ç è2c 1 kè,1, then kçèæèk 2 ç 2èc 1 +1èkwèæèk 2 2 : è4.7è Condition èiiè of Deænition 2.2 implies that,èkzèæèk 2 2,èæ 2,2æ 1 èkwèæèk 2 2è+ c 2 kæ 2 kçèæèk 2 ç for all çèæè 2 æ such that f s èçèæèè ç for s =1;:::;k. Since æ 2 ç æ 1 è2c 2 kèc 1 +1èè,1 and æ 2 ç æ 1 è2èc 1 +1èè,1, inequality è4.7è implies that æ 1 kwèæèk 2 2 ç c 2 kæ 2 kçèæèk 2 and æ 1 kwèæèk 2 2 ç æ 2 kçèæèk 2. Therefore, f èçèæèè =,èkzèæèk 2 2, æ2 kwèæèk æ 2 kçèæèk 2 è ç ç,ç kzèæèk 2 2, æ 2 kwèæèk æ 2 kçèæèk 2 + æ 1 kwèæèk 2 2,c 2 kæ 2 kçèæèk 2 + æ 1 kwèæèk 2 2, æ 2 kçèæèk 2 =,èkzèæèk 2 2, èæ2, 2æ 1 èkwèæèk 2 2è+c 2 kæ 2 kçèæèk 2 ç : Now the closed loop system è2.1è, è2.5è can be rewritten as the system è3.1è where hèæè =ëxèæè;x c èæèë and èèæè =ëç 1 èæè;:::;ç k èæè;wèæèë. Using the continuity of the coeæcients in the controller è2.5èè, it follows immediately that the closed loop system satisæes Assumption 3.1. Furthermore, given any èèæè =ëç 1 èæè;:::;ç k èæè;wèæèë 2 L 2 ë; 1è, it follows that the uncertainty inputs ëç 1 èæè;:::;ç k èæèë satisfy the constraints è2.3è with d s = kç s èæèk 2 2 and t i = 1. Hence, condition èiè of Deænition 2.2 implies that Assumption 3.2 is satisæed. Also, given any èèæè =ëç 1 èæè;:::;ç k èæè;wèæèë 2 L 2 ë; 1è such that kèèæèk 2 2 ç æ, then the constraints è2.3è are satisæed with d s ç æ. Thus, condition èiè of Deænition 2.2 also implies that Assumption 3.3 is satisæed. We can now apply Theorem 3.1 to the above closed loop system. Using this theorem, P it follows that there exist constants ç ç ;ç 1 ç;:::;ç k ç k such that s= ç s é and the inequality è3.2è is satisæed for all çèæè 2 æ. Now we prove that ç s é for all s =;1;:::;k. Condition è3.2è for the functionals è4.6è implies that ç èkzèæèk 2 2, æ 2 kwèæèk 2 2è+ ç,æ è kç s èæèk kwèæèk 2 2è ç s èkç s èæèk 2 2,kç s èæèk 2 2è è4.8è where æ = æ 2 P k s= ç s é. If ç j = for some j =1;:::;k then we can take wèæè ç ;ç s èæè ç for all s 6= j and ç j èæè 6=. Then, the inequality è4.8è is not satisæed, because the left side of è4.8è is non-negative and the right side of è4.8è is negative. èanalogously, ifç =,we can take wèæè 6= 1

11 ROBUST H 1 CONTROL and çèæè ç.è Therefore, ç s é for s =;1;:::;k. In this case, we can take in è4.8è ç =1. Consider the following linear system where _xètè = Axètè+ ^B1^wètè+B 2 uètè; ^zètè = ^C1 xètè+ ^D12 uètè; yètè = C 2 xètè+ ^D21 ^wètè è4.9è ^wèæè = ëæwèæè; p ç 1 ç 1 èæè;:::; p ç k ç k èæèë; ^zèæè = ëzèæè; p ç 1 ç 1 èæè;:::; p ç k ç k èæèë and the matrix coeæcients ^B1 ; ^C1 ; ^D12 and ^D21 are deæned by è4.3è. In this system, ^w is the disturbance input and ^z is the controlled output. The inequality è4.8è with ç =1may be rewritten as k^zèæèk 2 2,k^wèæèk 2 2ç,æ k^wèæèk 2 2 : è4.1è That is, ^J æ = k^zèæèk 2 2 sup ^wèæè2l 2ë;1è;xèè=;x cèè= k ^wèæèk 2 é 1: 2 è4.11è Therefore, the controller è2.5è with the initial condition x c èè = solves a standard output feedback H 1 control problem for the system è4.9è. Hence, condition èiiè of the theorem follows directly using Theorems 5.5 and 5.6 of ë17ë. This completes the proof of this part of the theorem. èiiè è èiè It is a standard result from H 1 control theory that if condition èiiè holds then the linear controller è4.4è solves a standard output feedback H 1 control problem deæned by the system system è4.9è and H 1 cost bound è4.11è; e.g., see ë17, 18ë. Furthermore, condition è4.11è implies that there exists a constant æ é such that the inequality è4.1è is satisæed for all the solutions of the closed loop system è4.9è, è4.4è with ^wèæè 2 L 2 ë; 1è and the initial condition ëxèè;x c èèë =. Now the closed loop uncertain system deæned by è2.1è and è4.4è may be rewritten as _hètè =Phètè+Q^wètè; è4.12è where h = ç ç ç x ;P = x c A B c C 2 ç ç C c ^B1 ;Q= A c ^D 12 Since the controller è4.4è solves the H 1 control problem described above, the matrix P is stable. Furthermore, condition è2.3è implies that any disturbance input wèæè 2 L 2 ë; 1è and admissible uncertainty inputs ç 1 èæè;:::;ç k èæè satisfy the following integral quadratic constraint: 11 ç :

12 A.V. SAVKIN AND I.R. PETERSEN There exists a constant d ç and a sequence ft i g 1 i=1 such that t i! t æ èwhere ë;t æ è is the interval of existence for the corresponding solutionè and where Z ti k ^wètèk 2 dt ç Moreover, the constant d is given by Z ti T = æ ^C1 ^D12 C c æ : d = æ 2 kwèæèk kthètèk 2 dt + d 8i ç s d s : è4.13è However, since ^z = Th, condition è4.1è and stability of the matrix P imply that there exists a constant æé such that èkthètèk 2,k^wètèk 2 èdt ç,æ èkhètèk 2 +k^wètèk 2 èdt è4.14è for all ëhèæè; ^wèæèë 2 L 2 ë; 1è connected by è4.12è with hèè =. Using Theorem 1 of ë11ë, condition è4.14è and the stability of the matrix P imply absolute stability of the uncertain system è4.12è, è4.13è. Thus, there exists a constant c é such that for any initial condition ëxèè;x c èèë and any uncertainty ^wèæè described by è4.13è, then ëxèæè;x c èæè; ^wèæèë 2 L 2 ë; 1è and èkxètèk 2 + kx c ètèk 2 + k ^wètèk 2 èdt ç c ëkxèèk 2 + kx c èèk 2 + dë: è4.15è Since uètè =C c x c ètèincontroller è4.4è, ^wèæè =ëæwèæè; p ç 1 ç 1 èæè;:::; p ç k ç k èæèë P and d = æ 2 kwèæèk 2 k 2 + ç sd s, condition èiè of Deænition 2.2 follows from the inequality è4.15è. We now establish condition èiiè of Deænition 2.2. We have established that condition è4.1è is satisæed for all the solutions of the closed loop system è2.1è, è4.4è with zero initial condition. This may be rewritten as condition è4.8è with ç = 1. Furthermore, all solutions to the closed loop system è2.1è, è4.4è from L 2 ë; 1è satisfy condition è2.3è with d s = maxë;,èkz s èæèk 2 2,kç s èæèk 2 2èë and t i = 1. Therefore, it follows from è4.8è that condition è2.7è holds with c 1 = maxëç 1 ;:::;ç k ë. This completes the proof of the theorem. 2 The following corollary is an immediate consequence of the above theorem and the remarks following the deænition of the uncertain system è2.1è, è2.3è. 12

13 ROBUST H 1 CONTROL Corollary 4.1 The uncertain system with norm bounded uncertainties, è2.4è will be absolutely stabilizable with disturbance attenuation æ via the linear controller è4.4è,è4.5è if condition èiiè of Theorem 4.1 is satisæed. References ë1ë I. R. Petersen. A stabilization algorithm for a class of uncertain systems, Systems and Control Letters 8 è1987è, 181í188. ë2ë M. A. Rotea and P. P. Khargonekar. Stabilization of uncertain systems with norm bounded uncertainty A control Lyapunov approach, SIAM Journal of Control and Optimization 27è6è è1989è, 1462í1476. ë3ë P. P. Khargonekar, I. R. Petersen, and K. Zhou. Robust stabilization of uncertain systems and H 1 optimal control, IEEE Transactions on Automatic Control AC-35è3è è199è, 356í361. ë4ë L. Xie and C.E. de Souza. Robust H 1 control for linear systems with norm-bounded time-varying uncertainty, IEEE Transactions on Automatic Control 37è8è è1992è, 1188í1191. ë5ë A. V. Savkin and I. R. Petersen. Nonlinear versus linear control in the absolute stabilizability of uncertain linear systems with structured uncertainty, IEEE Transactions on Automatic Control 4è1è è1995è, 122í127. ë6ë A. V. Savkin and I. R. Petersen. A connection between H 1 control and the absolute stabilizability of uncertain systems, Systems and Control Letters 23è3è è1994è, 197í23. ë7ë A.V. Savkin and I.R. Petersen. Time-varying problems of H 1 control and absolute stabilizability of uncertain systems, In Proceedings of the 1994 American Control Conference, Baltimore, MD, July ë8ë A. Megretsky and S. Treil. Power distribution inequalities in optimization and robustness of uncertain systems, Journal of Mathematical Systems, Estimation and Control 3è3è è1993è, 31í319. ë9ë V. A. Yakubovich. Minimization of quadratic functionals under the quadratic constraints and the necessity of a frequency condition in the quadratic criterion for absolute stability of nonlinear control systems, Soviet Mathematics Doklady 14 è1974è, 593í597. ë1ë V. A. Yakubovich. Abstract theory of absolute stability of nonlinear systems, Vestnik Leningrad University, Series 1 41è13è è1977è, 99í

14 A.V. SAVKIN AND I.R. PETERSEN ë11ë V. A. Yakubovich. Absolute stability of nonlinear systems with a periodically nonstationary linear part, Soviet Physics Doklady 32è1è è1988è, 5í7. ë12ë V. A. Yakubovich. Dichotomy and absolute stability of nonlinear systems with periodically nonstationary linear part, Systems and Control Letters 11è3è è1988è, 221í228. ë13ë A. V. Savkin. Absolute stability of nonlinear control systems with nonstationary linear part, Automation and Remote Control 41è3è è1991è, 362í367. ë14ë A. A. Voronov. Stability, Controllability, and Observability. Moscow: Nauka, ë15ë A. A. Krasovskii, editor. Moscow: Nauka, Handbook of Automatic Control Theory. ë16ë A. V. Savkin and I. R. Petersen. Minimax optimal control of uncertain systems with structured uncertainty, International Journal of Robust and Nonlinear Control 5è2è è1995è, 119í137. ë17ë T. Basar and P. Bernhard. H 1 -Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach. Boston: Birkhíauser, ë18ë I. R. Petersen, B. D. O. Anderson, and E. A. Jonckheere. A ærst principles solution to the non-singular H 1 control problem, International Journal of Robust and Nonlinear Control 1è3è è1991è, 171í185. Department of Electrical and Electronic Engineering and Cooperative Research Center for Sensor Signal and Information Processing, University of Melbourne, Parkville, Victoria 352, Australia Department of Electrical Engineering, Australian Defence Force Academy, Campbell, 26, Australia W.P. Dayawansa 14