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H∞ control and filtering of discrete-time stochastic systems with multiplicative noise
Automatica 37 (2001) 409}417
Brief Paper
H control and "ltering of discrete-time stochastic systems with multiplicative noise夽 E. Gershon *, U. Shaked , I. Yaesh Department of Electrical Engineering-Systems, Tel-Aviv University, Tel Aviv, Israel Taas Israel Industries, Advanced Systems Division, P.O. Box 1044/77 Ramat Hasharon, Israel Received 17 September 1998; revised 1 December 1999; received in "nal form 2 July 2000
Abstract Linear discrete-time systems with stochastic uncertainties in their state-space matrices are considered. The problems of "nitehorizon "ltering and output-feedback control are solved, taking into account possible cross-correlations between the uncertain parameters. In both problems, a cost function is de"ned which is the expected value of the relevant standard H performance index with respect to the uncertain parameters. A solution to the "ltering problem is obtained "rst by applying the adjoint system and deriving a bounded real lemma for this system. This solution guarantees a prescribed estimation level of accuracy while minimizing an upper bound on the covariance of the estimation error. The solution of the "ltering problem is also extended to the in"nite-horizon case. The results of the "ltering problem are used to solve the corresponding output-feedback problem. A "ltering example is given where a comparison is made with the results obtained using bounded uncertainty design techniques. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Stochastic H control; Stochastic H "ltering; Multiplicative noise; Bilinear systems
1. Introduction The analysis and design of controllers for systems with stochastic uncertainties have received much attention in the past (Dragan, Morozan, & Halanay, 1992; Halanay & Morozan, 1992; Willems & Willems, 1976; Wonham, 1967), where mainly robust stability has been considered. Recently, a renewed interest in this problem has been encountered and solutions to the stochastic control problem have been derived that ensure a worst-case performance bound in the H style (Costa & Kubrusly, 1996; Dragan & Morozan, 1997; El Ghaoui, 1995; Hinrichsen & Pritchard, 1998). Systems whose parameter uncertainties are modeled as white noise processes in a linear setting have been treated in Boyd, El Ghaoui, Feron, and Balakrishnan (1994), 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. Sugie under the direction of Editor Roberto Tempo. This work was supported by the C&M Maus Chair at Tel Aviv University and the British EPSRC Grant no. GR/M69418. * Corresponding author. Tel.: #972-3-640-8764; fax: #972-6407095. E-mail address: [email protected] (E. Gershon).
Costa and Kubrusly (1996), Dragan and Morozan (1997), El Ghaoui (1995), Gershon, Shaked, and Yaesh (1999), Hinrichsen and Pritchard (1998), Yasuda, Kherat, Skelton, and Yaz (1990). Such models of uncertainties are encountered in many areas of applications (see Costa & Kubrusly, 1996 and the references therein) such as nuclear "ssion and heat transfer, population models and immunology (Mohler & Kolodziej, 1980). In control theory such models are encountered in gain scheduling when the scheduling parameters are corrupted with measurement noise. Recently, the output-feedback control problem for these systems has been solved in the continuous-time setting in the stationary case (Hinrichsen & Pritchard, 1998), where two coupled non-linear inequalities were obtained. The corresponding discrete-time problem has been considered in Dragan and Stoica (1998). The solution there refers only to the in"nite-time-horizon problem and, therefore, does not deal with transients that stem from a non-zero initial condition and "nal time weights. The drawback of the latter solution is that it requires a solution of an in"nite number of LMI sets. In the present paper we are mainly concerned with the "nite-horizon case. In this setting we obtain solutions in
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 1 6 4 - 3
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E. Gershon et al. / Automatica 37 (2001) 409}417
terms of recursive Riccati equations that are easy to use. This recursions do not naturally appear when the in"nitehorizon solution is derived in LMI forms as in Dragan and Stoica (1998). In this paper we also deal with the case where the measured output matrix contains a stochastic statedependent uncertainty. Such models, which appear in cases where the measurements include state derivatives (e.g. acceleration control of an aircraft or missile), have not been considered in Dragan and Stoica (1998) or anywhere else. A "nite-horizon version of the bounded real lemma (BRL) is required if one tries to extend the results and the methods of the stationary problem to the "nite-horizon case. In the stochastic state-feedback control problem, a &control-type' BRL has been obtained in Gershon et al. (1999) using standard completing to square arguments. The solution of the output-feedback and the "ltering problems require, however, a dual BRL which has not been derived before. In the present paper such a BRL is developed and, consequently, the "ltering and the output-feedback problems are solved, in the "nite-horizon H setting, in presence of multiplicative white-noise in the system dynamics and the measurement. The paper is organized as follows: In Section 2, we formulate the problems and we bring the state-feedback solution. In Section 3, we present the BRL that is based on the adjoint system. We solve the "ltering problem by applying this new BRL and obtain a solution which guarantees a prescribed level of estimation accuracy, in the H -norm sense, that also minimizes an upper bound on the covariance of the estimation error. In Section 4, we apply the state-feedback results and transform the resulting system to "t the "ltering model. On the latter we apply the "ltering solution of Section 3 and derive a simple recursion that provides the required outputfeedback controller. The theory developed is demonstrated in Section 5 by a "ltering example whose results are compared to those obtained by applying the deterministic technique of Xie and de Souza (1992). Notation. Throughout the paper the superscript &T' describes transposition, RL denotes the n-dimensional Euclidean space and the notation P'0, for P3RL"L means that the matrix P is symmetric and positive de"nite. We also denote by ET + ) , expectation with respect to v. The space of square summable functions over [0 N!1] is denoted by l [0 N!1], and ""."" stands for the standard l -norm, ""u"" "( ,\ I uI2uI ) . We also denote by "" fI ""0 the product fI2RfI and by dGH the Kronecker delta function. 2. Problem formulation We consider the following system: xI> "(AI #DI vI )xI #B I wI #(B I #GI gI )uI , x(0)"x ,
yI "(CI #FI fI )xI #nI , k"0, 1,2, N!1,
zI "¸I xI #D I uI , (1)
where xI 3RL is the state vector, wI 3RO is a exogenous disturbance, uI 3RQ is the control input signal, x is an unknown initial state, yI 3RP is the measured output, nI is a measurement noise and zI 3RK is the objective vector and where the sequences +vI ,, +mI , and +gI , are standard random scalar sequences with zero mean that satisfy E+vI vH ,"dIH , E+gI gH ,"dIH , E+g v ,"b d , "b "(1, I H I IH I E+fI fH ,"dIH , E+fI gH ,"pI dIH , E+fI vH ,"aI dIH , "aI "(1, "pI "(1. Assuming, for simplicity, that
(2)
[¸I2 D I 2 ]D I "[0 RI I ], RI I '0, (3) we consider the following problems: (i) The discrete-time state-feedback problem: This problem was solved by Costa and Kubrusly (1996) for in"nite dimensions (Dragan & Stoica, 1998; Boyd et al., 1994), for the stationary case, and (Gershon et al., 1999) for the "nite-horizon case. We consider (1) where we exclude yI and where we look for a state-feedback rule uI "KI xI that achieves JOE +""zI "" !c""wI "" ,#E +x 2(N)QM , x(N), E T E T !cx2QM x (0, QM , 50, QM '0 for all nonzero (+wI ,, x ) where +wI ,3l [0 N!1] and x 3RL. The state-feedback solution is obtained by requiring that the following Riccati-type recursion (see Gershon et al., 1999) QI "AI2M M I AI #DI2QI> DI #¸I2¸I !*I2'\ I *I , Q, "QM ,
(4)
will satisfy RI OcI!B I 2 QI> B I '0, ∀k3[0 N!1] and Q (cQM ,
(5a,b)
where M M OQ [I!c\B B 2 Q ]\, I I> I I I> 'I OB I 2 M M I B I #GI2QI> GI #RI I , *I OB2 I M M I AI #bI GI2QI> DI , and KI O!'\ I *I . (6a}d) Remark. The above result is based on the BRL of Gershon et al. (1999). An alternative derivation for
E. Gershon et al. / Automatica 37 (2001) 409}417
the state-feedback control problem is brought in the Appendix. There, the optimal strategies of the control signal uI is obtained. This strategy is used in Section 4 to derive a solution to the output-feedback control problem. (ii) Stochastic H -xltering: We consider (1) where B I , GI and D I are zero. We look for ¸I x( I , the "ltered estimate of zI , where x( "A x( #K (y !C x( ), x( "0. I> I I I I I I
(7)
We de"ne e "x !x( , w "[w 2 n 2] 2 I I I I I I and we consider the following cost function: J OE +""¸I> eI> "" !c""w I "" ,!cx2P x , T D P '0.
(8a}b)
411
(i) The adjoint system: We consider the following system: xI> "(AI #DI vI )xI #BI wI , zI "CI xI , (11) where +vI , is a standard white sequence and x is an unknown initial condition. Our aim is to "nd, for S'0, where S\ re#ects the amount of uncertainty in x , a necessary and su$cient condition for J$ "E +""zI "" !c""wI "" ,!cx2Sx (0, T ∀(+wI ,, x )I0,
(12)
where +wI ,3l [0 N!1] and x 3RL. For simplicity reasons we start with c"1. In order to "nd the adjoint of (11) we derive zI " I\ CI '(k,0)x #CI '(k, j#1)BH wH , 0(k4N, H C x , k"0, (13) where '(k, j) is the transition matrix given by
(9)
'(k, j)"
(A #D v ) (A #D v )(A #D v ), j(k, I\ I\ I\ 2 H> H> H> H H H I, k"j.
Given c'0, we look for ¸I xI for which J is negative for all nonzero (+w I ,, x ) where +w I ,3l [0 N!1] and x 3RL. (iii) Stochastic H output-feedback control: We look for an output-feedback controller for (1) that achieves, for a given c'0, J O E +""z "" !c""w "" #x 2QM x , T E D I I , , , !cx2QM x (0, QM , 50, QM '0
(10)
for all nonzero (+w ,, x ) where +w ,3l [0, N!1] and I I x 3RL. Similar to the standard case (Green & Limebeer, 1995), this problem involves the estimation of an appropriate combination of the states and the application of the state-feedback results with a proper modi"cation. We note that problems (i) and (ii) are extensions, to the state-multiplicative noise case, of the "nite-horizon discrete-time results of Yaesh and Shaked (1991).
3. Stochastic H 5ltering The problem of the stochastic H "ltering cannot be solved using the BRL of Dragan and Stoica (1998) which aimes at the stationary case only. We thus consider the adjoint system for which we derive a BRL. We later use this lemma to solve the "ltering problem.
We de"ne G to be the linear operator that maps (x , wI ) to zI as given by (13). By using the following inner product: 1(x , w I ),(x , w I )2"E +x 2 Sx ,# T 1w I , w I 2, where S is de"ned in (12) and where 1w I , w I 2"E + ,\ w 2 w ,, we obtain the adjoint I I I T GH of G. Denoting GHzI "[x? 2 w? I 2 ] 2 we apply the following equality for any wI and zI in l [0 N!1] and x 3RL: 1G(x , wI ), zI 2"1(x , wI ),GHzI 2"E T +x2Sx ? ,#E + ,\ wI2w? I ,. I T We thus obtain
,\ I\ E z 2(C '(k,0)x #C '(k, j#1)B w ) I I I H H I T H
,\ "E +x2Sx ? ,#E wI2w? I . T T I From the latter we get x ? "S\ ,\ H '( j,0)CH2zH , and w? I "BI2 ,\ ' 2( j, k#1)C 2 z . De"ning j? I " HI> H H ,\ ' 2( j, k#1)C 2 z we obtain the state-space repHI> H H resentation of GH as j "(A 2#D 2v )j #C 2z , ? I\ I I I ? I I ? I
j
w "B 2j , ? I I ? I
Sx "j . ? ? \ (14)
k"0,1,2, N!1,
? ,\
"0,
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E. Gershon et al. / Automatica 37 (2001) 409}417
(ii) A BRL for the xltering problem: Consider the adjoint system (14). The corresponding index of performance for (14) is J? "E +""w? I "" !""zI "" #x ? 2 Sx ? ,. T
(15)
Since the adjoint system is anti-causal with respect to k, the reversal of time according to the de"nitions below will lead to a causal system on which we can apply the "nite-horizon BRL of Gershon et al. (1999): We de"ne kM "N!1!k and we obtain the following causal system: jIM > "(AM IM2#DM IM2v IM )jIM #CM IM2z IM , j "0, w ? IM "BM IM2jIM ,
kM "N!1, N!2,2,0,
Sx? "j, , (16)
z IM Oz,\\IM , where we denote jIM Oj? ,\\IM , w ? IM Ow? ,\\IM , AM IM OA,\\IM , BM IM OB,\\IM , and CM IM OC,\\IM . We obtain that J? of (15) is given by J "E +""w "" !""z "" #j 2 S\j ,. The result ? ? I I ? \ ? \ T that corresponds to any c'0 is obtained by absorbing the factor c\ in B of (11) and transferring it to CM M2 I I of (16). The corresponding J then becomes J " ? ? E +""w ? I "" !c""z I "" #j? \ 2 S\j? \ ,. We can thus apT the results of Gershon et al. (1999) ply and obtain that the H -norm of the adjoint system is less than c i! there exists QIM that satis"es the following recursion: QIM "AM IM QIM > AM I2M #AM IM QIM > CM I2M #M \ IM CM IM QIM > AM IM2 #BM IM BM IM2#DM IM QIM > DM IM2,
Q, "S\,
(17)
where #M IM OcI!CM IM QIM > CM I2M '0. De"ning M K I "Q,\I the latter becomes M K I> "AI M K I AI2#AI M K I CI2'I \ K I AI2#BI BI2 I CI M #D M K D 2, M K "S\, I I I
(18)
in which it is required that 'I I OcI!CI M K I CI2'0, k"0,2, N!1.
(19)
We thus have the following: Theorem 3.1. Consider the system of (11) and (12). Given c'0, a necessary and suzcient condition for (10) to hold is that M K I of (18) satisxes 'I I '0, ∀k"0,1,2, N!1. The proof readily follows from the fact that the condition of the theorem is necessary and su$cient for achieving J? (0 for all nonzero +z? I ,. Since the adjoint operator is linear and bounded its norm equals the one of G, where the latter is the operator of the system of (11) (Kreyszig, 1978).
Remark. The above result is based on adjoint system considerations. While in the deterministic case the system and its adjoint lead to the same results, our experience shows that in the stochastic case the worst-case minimum attenuation levels (i.e. c) that are achieved by applying the BRL of Dragan et al. (1998) and the result of Theorem 3.1 do not perfectly coincide. This phenomenon is in line with the fact that both BRL's achieve a minimum level which is somewhat higher than the worst attenuation level that is obtained by simulating the behaviour of the system and its adjoint. However, we "nd that both systems achieve the same norm for the operator that relates the disturbance sequence +wI , and the objective sequence +zI ,. (iii) Stochastic H -xltering: In the sequel we also address the problem of bounding the covariance of the states of a system of the type of (11) where +wI , and x are considered to be random processes. We bring "rst the following result: Lemma 3.1. The solution to (18) is an upper-bound to the covariance E +xI xI2, of the states of the system (11), T V U where vI and wI are modeled as zero-mean uncorrelated standard white noise sequences independent of x and where the latter is a zero mean random vector. Proof. Considering the term of xI> xI> 2 , and taking the expectation with respect to vI , wI and x , we "nd that PM I>O E +xI> xI> 2 ,"AI PM I AI2#DI PM I DI2#BI BI2, T V U PM "E +x x2,. V
(20)
Subtracting (20) from (18) we obtain *M I> OM K I> !PM I> "AI *M I A2I #DI *M I DI2#AI M K I C2I 'I \ C M K I I I AI2, *M "M K !PM . It follows by induction that for M K 5PM we achieve *M I 50, ∀k'0. 䊐 We consider next the H -"ltering problem of Section 2. De"ning mI "[xI2 eI2] 2 we obtain mI> "[AI I #DI I vI #DI I fI ]mI #BI I w I ,
zI "CI I mI , (21)
where
B AI I "diag+AI , AI !K I CI ,, BI I " I B I
D DI I " I DI
0
0 , DI I " 0 !K I FI
CI I "[0 ¸I ].
0
,
!K I
0 0
, (22a}e)
E. Gershon et al. / Automatica 37 (2001) 409}417
Like in (7), a gain observer K I is looked for which achieves a negative J for all nonzero (+wI ,, x ), where J is de"ned in (9). We obtain the following theorem: Theorem 3.2. Consider the system of (21) and (9). Given c'0, a necessary and suzcient condition for J to be negative for all nonzero (+w I ,, x ) where w I 3l [0 N!1] and x 3RL is that MI "+MGH I ,, which is obtained by the following recursion: MI> "AI I MI AI I2#AI I MI CI 2I #I \ I CI I MI AI 2I #BI I BI I2 #DI I MI DI I 2 #DI I MI DI I 2 # aI DI I MI DI 2 I #aI DI I MI DI I 2 , M "[I I] 2P\[I I]
(23a,b)
satisxes # I I OcI!CI I MI CI I2'0, ∀k"0,1,2, N!1.
(23c)
(24)
where &I OM I [cI!¸2I ¸I M I ]\ and (M I OI#FI M I FI2#CI &I CI2.
(25)
Proof. The recursion of (23a) readily follows from Theorem 3.1. In order to comply with the weighting on the initial condition in (9), and with the choice of x( "0, nature should be forced to select the component e in m to be equal to x . We choose, therefore, the weighting on the initial condition on the performance index to be [x2 e2!x2]diag+P , mI,[x2 e2!x2] 2 where m is a large positive scalar. Since [x2 e2!x2]" [x2 e2]B 2, where
BO
I
0
!I
I
(22e) #I I "cI!¸I M I ¸I2, it follows from (23) that the equation for M I is M I> "(AI !K I CI )&I (A2I !CI2K I 2) #K I (I#FI M I FI2)K I 2 #DI M I DI2 # B I B I 2 !(K I FI M I DI2 #DI M I FI2K I 2 )aI .
(26)
From Lemma 3.1 we have that MI of (23a) is an upper bound on E +m m 2, and thus ¸ M ¸ 2 is a bound I I I T K D U I I on E +zI zI2,. For each k"0,1,2, N!1, K I can T K D U thus be derived to minimize M I> . We "nd from (26) that M I> "AI &I AI2#DI M I DI2 #[K I !KH I ]( M I [K I !KH I ] 2#B I B I 2 ! [AI &I CI2#aI DI M I FI2]( M \ I
If (23c) is satisxed, an observer gain K I that achieves the estimation error level of c and minimizes, at each instant, an upper bound on the covariance of the estimation error with respect to the parameters vI , fI , w I and x , where +w I , and x are modeled as uncorrelated zero-mean white noise sequence and vector, respectively, is given by KH "[A & C2#D M F 2a ](M \, I I I I I I I I I
413
,
we obtain the weight S"B 2diag+P , mI,B. The corresponding weight in the adjoint system is then S\"[I I] 2P\ [I I]#m\[0 I] 2[0 I] and (23b) is obtained when m tends to in"nity. Since by (23c) and
;[CI &I AI2#aI FI M I DI2], where KH is de"ned in (24) and & and ( M are de"ned in I I I (25). The observer gain KH clearly minimizes, at time k, I the upper bound on the covariance of z . 䊐 I> The observer gain K is derived at each instant i, by G (24) based on the value of M that was obtained from the G recursion in (23a). Although the right-hand side of (23a) is nonlinear in M , once M , and therefore K , were I G\ G\ obtained the matrix M can be readily calculated. G (iv) Inxnite-horizon stochastic xltering: In order to prove the mean-square stability of the "lter that is obtained above, in the case where the system matrices are all constant and N tends to in"nity, we consider (21) where AI , DI , DI , BI and CI are constant. Since the dy namic matrix of the adjoint of (21) is the transpose of AI #DI v #DI f , the mean-square stability of (21) is I I assured by the stability of the adjoint system j "(AI 2#DI 2v #DI 2f )j . I> I I I Applying the arguments of Boyd et al. (1994) we obtain: Lemma 3.2. System (21) is mean-square stable if there exists a positive-dexnite matrix Q3RL"L that satisxes trace+! PK ,(0, ∀i, H G where PK 3RL"L satisxes the following recursion: G PK "AI 2PK AI #DI 2 PK DI #DI 2 PK DI G> G G G #a DI 2 PK DI #a DI 2PK DI G G G G with
PM PK " PM
PM , PM '0 PM
(27)
(28)
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E. Gershon et al. / Automatica 37 (2001) 409}417
and where !H (Q)OAI QAI 2!Q#DI QDI 2#DI QDI 2# aDI QDI 2#aDI QDI 2.
by the method of completing to squares. The solution of the output-feedback problem is, then, obtained by transforming the problem to a "ltering one. Denoting rI OwI !wH I when wHI is de"ned in (A.2), and using uI "KI x( I , where KI is de"ned in (6) and where x( I is yet to be found, we obtain from (1) that
Since !H (M)#AI MCI 2#I \CI MAI 2#BI BI 2"0 (29) is the stationary version of (23a), and since we choose K to satisfy the latter, we obtain, using the fact that a(1, that the requirement of (27) is satis"ed if trace+BI BI 2PK G , is positive for all i'0. Using the de"nition of BI we "nd that trace+BI BI 2PK G ,"0 for some i only if
x "(A #D v #B K )x #B r I> I I I I VI I I I #(B I KSI #B I #GI gI )KI x( I , yI "(CI #FI fI )xI #nI .
[B2 0]PK G [B2 0] 2"0. (30) Eq. (28) implies that PK G !+AI 2,GPK AI G is positive semide"nite. Since AI has a diagonal block structure, the result of (30) can occur only if AGB "0. Denoting the system xI> "AxI #BwI , zI "HxI by S(A, B, H) the latter result implies that the system S(A, B, I) is of the moving average type (a FIR system). Excluding such systems we arrive at the following result.
Substituting in (A.3) we look for x( for which
SI "cQM !Q
x( I> "(AI #B I KVI )x( I #(B I KSI #B I )uI #K I (yI !CI x( I ).
e "[A #B K !K C ]e #B r !K n I> I I VI I I I I I I I #DI xI vI #GI KI xI gI !K I FI xI fI !GI KI eI gI . De"ning mI "[xI2 eI2] 2 and w I "[rI2 nI2] 2 we obtain the following system, which is equivalent to the one in (21): mI> "[AI I #DI I vI #DI I fI #DI I gI ]mI #BI I w I ,
To solve the "nite-horizon output-feedback control problem a careful derivation of the expression of J of (10) is required in order to obtain the optimal strategies for u and w. These are needed for deriving an equivalent to the separation principle and they are obtained in the Appendix by solving the state-feedback control problem
zI "CI I mI ,
0 0
, DI
I
0
"
!K F I I
(34)
where
A #(B K #B )K #B K I I SI I I I VI 0
(33)
Using the de"nition of (8b) for xI> of (31) and x( I> of (33) we obtain
4. Stochastic output-feedback control
D DI " I I D I
(32a,b)
is less then zero for all nonzero (+wI ,,+nI ,, x ), where zI "KI (xI !x( I ). We consider the following state estimator:
Remark. (i) In the case where S(A, B, I) is of the FIR type, stability in the mean-square sense, is guaranteed if we solve (29) where the term eI is added to the left-hand L side of the equation, with 0(e;1. (ii) When stability is guaranteed, the stationary "lter gain K might have been derived by solving the station ary version of (23) and (24). This, however, may not be feasible due to the non-linearities involved. An alternative method is to solve the corresponding "nite-horizon problem taking a very large N.
,\ ""zI ""I !""rI ""0I !c""nI "" #(I !x2SI x , JO E E T D I
Theorem 3.3. Consider the system of (21) with constant matrices, where S(A, B, I) is of the inxnite impulse response (IIR) type, and where N tends to inxnity. This system is stable in the mean-square sense, for any constant K that solves (29).
AI " I
(31)
![B K #B ]K B 0 I SI I I , BI " I , I A #B K !K C B !K I I VI I I I I 0 G K !G K I I and CI "[0 K ]. , DI " I I I I I 0 G K !G K I I I I
E. Gershon et al. / Automatica 37 (2001) 409}417
Applying the results of Section 3 we arrive at the following theorem: Theorem 4.1. Consider the system of (1) where x( is dexned I in (33). There exists a controller that achieves (10) for a given c iw the recursion of (4) satisxes (5) and there exists QI "+QI ,, that is obtained by the following recursion: I GH I QI "BM BM 2#AI QI AI 2#DI QI DI 2 #DI QI DI 2 I> I I I I I I I I I I I #DI QI DI 2 #(DI QI DI 2 #DI QI DI 2 )a I I I I I I I I I I # (DI QI DI 2 #DI QI DI 2 )b #(DI QI DI 2 I I I I I I I I I I #DI QI DI 2 )p #AI QI CM 2)\CM QI AI 2 (35) I I I I I I I I I I I which satisxes ) OcI!CM QI CM 2'0, ∀k"0,1,2, N!1, I I I I where SI \ SI \ QI "c , SI \ SI \
(36)
CM O[0 ' K ] I I I
cB R\ 0 I I and BM O , I cB R\ !K I I I and where ' and R are dexned in (6) and (5a), respectiveI I ly, and SI is dexned in (32b). If the latter holds, the controller that achieves (10) and minimizes an upper bound on the covariance of z I of (34), with respect to the stochastic parameters v , g , f , x , +r , and +n ,, when the latter two sequences I I I I I are modeled as uncorrelated zero-mean white noise sequences, is given by u "K x( . In the latter, K is given by I I I I (6d) and x( is given by (33) with I K "!#HRM \, (37) I I I where RM OC QI C 2#C QI K 2'2 )\' K QI C2 I I I I I I I I I I I I I #I#F QI F 2, AM OA #B K , I I I I I I VI #HO!AM QI C 2!AM QI K 2'2 )\' K QI C2 I I I I I I I I I I I !a D QI F 2!p G K (QI !QI 2 )F 2. I I I I I I I I I I Proof. Since the output-feedback control problem has been transformed to one of "ltering it is possible to apply the "ltering BRL, thus arriving at the recursion for QI . Similar to the "ltering problem, we "nd that ) is I given by cI!' K QI K 2'2 . We thus obtain that I I I I I QI "!I #K RM K 2 ##HK 2 #K #H2 where I> I I I I I I I I
415
Rearranging the latter and completing to squares, with respect to K , we obtain I QI "!I #[K ##HRM \]RM [K I> I I I I I I #RM \#H]2!#HRM \#H2 I I I I I and the observer gain that minimizes QI at the kth I instant is thus given by (37). 䊐 It follows from the above that the solution to the output-feedback control problem, at each instant i, is obtained by deriving "rst the state-feedback gain K usI ing (4)}(6). This gain is substituted in (35) and (36) and Q is thus obtained from (35) using the previous values of G Q and K . Once Q which satis"es (36) is derived, G\ G\ G the gain matrix K is obtained from (37). G The stability of the resulting output-feedback control system, in the case where the system matrices are all constant and the horizon N tends to in"nity, is implied by the stability, in the mean-square sense, of the statefeedback and the "ltering subsystems. This stability is guaranteed if S(A, B, ¸) is of the IIR type. 5. Example In this section we solve a stochastic H -"ltering prob lem using the theory of Section 3, and we compare our results with those obtained by the design of Xie and de Souza (1992). We consider the following IIR system:
0.1 0.6 2 0 x " x # w , I> I !1.65 !0.5#a 0 0 I I y "[2 2#c ]x #[0 1]n , I I I I where +a , and +c , are zero-mean uncorrelated whiteI I noise Gaussian sequences with standard deviations of 0.06 and 0.1, respectively. Using the theory of Section 3 we have D"[0 I] 2[0 0.06] and F"[0 0.1]. We want to obtain a H -estimate of z "[!0.225 0.45]x I I where we apply a very large initial value P and consider the case where N is very large. We "nd that c"1.4 is close to the minimum possible value. For this value of c, the recursion of (23) converges, up to a tolerance of 10\, within 80 steps. The observer gain that we obtain from (24) is K "[0.1877 !0.4704] 2. Our method is aimed at dealing with white multiplicative uncertainties. It is not, therefore, suitable to tackle constant deterministic uncertainties. Nevertheless it is of interest to compare our results with those obtained by the method of Xie and de Souza (1992), we choose
!I OAM QI AM 2 #AM QI K 2'2 )\' K QI AM 2 #D QI D 2#cB R\B 2 I I I I I I I I I I I I I I I I I I I # b [D (QI !QI )K 2G 2#G K (QI !QI 2 )D 2]#G K [QI #QI !QI !QI 2 ]K 2G 2. I I I I I I I I I I I I I I I I I I I
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E. Gershon et al. / Automatica 37 (2001) 409}417
the uncertainty interval in the latter to yield the same estimation level of 1.4. We choose, for example, a 3[!0.015 0.015] and c 3[!0.029 0.029]. We note I I that in our stochastic approach 80% of the values of a and 77% of c lie outside these intervals. Using the I I method of Xie and de Souza (1992) we search for one free parameter e and obtain e"1.396. The resulting K is then [0.3114 !0.2424] 2. Comparing the two designs it is found that ours achieves the estimation level of 1.29, which is lower than the level of c"1.4 that achieved by the other design.
6. Conclusions The problem of H -optimal control and "ltering of discrete-time linear systems with multiplicative stochastic uncertainties has been solved in the "nite-horizon case. Requiring a Luenberger-type H estimator a neces sary and su$cient condition for the existence of a solution is obtained. This solution is not unique, however, a speci"c selection of the observer gain leads, in addition to compliance with a prescribed estimation level, also to a minimization of an upper bound on the covariance of the estimation error when the disturbance and the initial condition are modeled as uncorrelated zero-mean whitenoise signals. In this sense, our solution resembles the situation in the standard H "ltering problem where the solution to the corresponding Riccati equation is an upper bound to the covariance of the estimation error (Bernstein & Haddad, 1989). An example is given which shows that the results obtained by the new method favorably compare with those achieved by the norm-bounded design techniques. It should be pointed out that in the later techniques the prescribed bound on the index of performance is guaranteed for all the parameters in the uncertainty interval. In our approach, on the other hand, there is a "nite nonzero probability of violating the index of performance. There is a trade-o! here between a better attenuation and estimation level and certainty in achieving these levels. In many practical problems the designer may prefer achieving better performance on the average, knowing that in a small proportions of the cases, the design may not achieve its goals.
"rst for w and then for u , we obtain I I E +JI ,"!""w !wH"" I #""u !uH"" I I I 0 I I T E I #x 2RM (Q )x !z 2z #cw 2w #E +( ,, I I I I I I I T E I (A.1) where ( consists of terms that are a$ne in the stochasI tic parameters, M M , ' , * and K are de"ned in (6a}d): I I I I uHOK x , wHOK x #K u , I I I I VI I SI I K OR\B 2 Q A , K OR\B 2 Q B , VI I I I> I SI I I I> I RM (Q )OA 2M M A #D 2Q D #¸ 2¸ !* 2'\* . I I I I I I> I I I I I I (A.2a}e) Taking the sum of both sides of the above JI , from zero I to N!1, we obtain using (A.1) ,\ E +JI ,"E ""x "" , !""x "" / T E I I T E , / ,\ " E +!""w !wH"" I #""u !uH"" I , I I 0 I I I T P ,\ # E +x 2RM (Q )x , I I I I T E ,\ # E +!z 2z #cw 2w ,#E +( ,. I I I I T E I I T E Hence, ,\ J" E +!""w !wH"" I #""u !uH"" I , I I 0 I I I T E ,\ # E ""x ""M I #""x "" M I 0/ / \A / I T E #E +( ,#E ""x ""M L , . I T P T E , / \/
(A.3)
Clearly the optimal strategy for u is given by u "uH I I I where Q satis"es (4)}(6). I References
Appendix We consider JI "x 2 Q x !x 2Q x I I> I> I> I I I and we substitute for x of (1). Taking the expectation I> with respect to g and v , and completing the squares, I I
Bernstein, D., & Haddad, W. M. (1989). LQG control with an H performance bound: A Riccati equation approach. IEEE Transactions on Automatic Control, AC-34, 293}305. Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequality in systems and control theory. SIAM Frontier Series. Philadelphia: SIAM. Costa, O. L. V., & Kubrusly, C. S. (1996). State-feedback H -control for discrete-time in"nite-dimensional stochastic bilinear systems. Journal of Mathematical Systems, Estimation and Control, 6, 1}32.
E. Gershon et al. / Automatica 37 (2001) 409}417 Dragan, V., Morozan, T., & Halanay, A. (1992). Optimal stability compensator for linear systems with state dependent noise. Stochastic Analysis and Application, 10, 557}572. Dragan, V., & Morozan, T. (1997). Mixed input}output optimization for time-varying Ito systems with state dependent noise. Dynamics of Continuous, Discrete and Impulsive Systems, 3, 317}333. Dragan, V., & Stoica, A. (1998). A c attenuation problem for discretetime-varying stochastic systems with multiplicative noise. Reprint Series of the Institute of Mathematics of the Romanian Academy, No 10. El Ghaoui, L. (1995). State-feedback control of systems with multiplicative noise via linear matrix inequalities. Systems and Control Letters, 23, 223}228. Gershon, E., Shaked, U., & Yaesh, I. (1999). H control and "ltering of discrete-time stochastic bilinear systems. Proceedings of the European Control Conference (ECC99), Karlsruhe, Germany, 1999. Green, M., & Limebeer, D. J. N. (1995). Linear robust control. Englewood Cli!s, NJ: Prentice-Hall. Halanay, A., & Morozan, T. (1992). Optimal stability compensator for linear discrete-time systems under independent random perturbations. Revue Romaine de Mathematiques Pures Appliquees, 3, 213}223. Hinrichsen, D., & Pritchard, A. J. (1998). Stochastic H . SIAM Journal of Control Optimization, 36(5), 1504}1538. Kreyszig, E. (1978). Introductory functional analysis with applications. New York: Wiley. Mohler, M. M., & Kolodziej, W. J. (1980). An overview of stochastic bilinear control processes. IEEE Transactions on Systems, Man and Cybernetics, 10, 913}919. Willems, J. L., & Willems, J. C. (1976). Feedback stabilizability for stochastic systems with state and control dependent noise. Automatica, 12, 277}283. Wonham, W. M. (1967). Optimal stationary control of a linear system with state-dependent noise. SIAM Journal of Control and Optimization, 5, 486}500. Xie, L., & de Souza, C. E. (1992). Robust H control for linear systems with norm-bounded time-varying uncertainty. IEEE Transactions on Automatic Control, AC-37, 1188}1192. Yaesh, I., & Shaked, U. (1991). H -optimal state estimation of discrete-time systems. Recent advances in mathematical theory of systems, control, networks and signal processing, Proceedings of MTNS-91, Kobe, Japan, vol. 1 (pp. 261}267), Mita Press. Yasuda, K., Kherat, S., Skelton, R. E., & Yaz, E. (1990). Covariance control and robustness of bilinear systems Proceedings of the 29th CDC.
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Uri Shaked (M'79}SM'91}F'93) was born in Israel in 1943. He received the B.Sc. and M.Sc. degrees in physics from the Hebrew University, Jerusalem, Israel, in 1964 and 1968, respectively, and the Ph.D degree in applied mathematics from the Weizmann Institute, Rehovot, Israel in 1975. From 1974 to 1976 he was a Senior Visiting Fellow at the Control and Management Science Division of the Faculty of Engineering at Cambridge University UK. In 1976 he jointed the Faculty of Engineering at Tel Aviv University, Israel, and in 1985}1989 he was the Chairman of the Department of Electrical Engineering-Systems there. In 1993}1998 he was the Dean of the Faculty of Engineering at Tel Aviv University. He is the incumbent of Celia and Macros Chair of Computer Systems Engineering there since 1989. During the 1983}1984, 1989}1990 and 1998}1999 academic years he spent his sabbatical year in the Electrical Engineering Departments at the University of California, Berkeley, Yale University, and Imperial College in London, respectively. His research interests include linear optimal control and "ltering, robust control, H -optimal control, and digital implementations of control lers and "lters. Eli Gershon was born in Israel in 1955. He received the B.Sc. and M.Sc. degrees in chemistry and biochemistry from Tel Aviv University, Tel Aviv, Israel, in 1981 and 1984, respectively, and the Ph.D. degree in electrophysiology from Tel Aviv University in 1992. From 1996 to the present he is a Ph.D. student at the Department of Electrical Engineering-Systems in the Faculty of Engineering at Tel Aviv University. His research interests include linear H -opti mal control and estimation of stochastic state-multiplicative systems and polytopic designs. Isaac Yaesh received his B.Sc. EE from the Israel Institute of Technology. At 1981 and 1992, respectively, he received his M.Sc. EE and Ph.D. from Tel Aviv University from the System department at the faculty of Engineering. In 1981}1986 he was with the IDF and served as a research engineer and since 1986 he is with IMI/Advanced System Division. His current interest are robust control and estimation, H theory, di!erential games and their applications in #ight control and guidance systems. Dr. Yaesh is a co-author of the book `Advances in Missile Guidance Theorya.
Brief Paper
H control and "ltering of discrete-time stochastic systems with multiplicative noise夽 E. Gershon *, U. Shaked , I. Yaesh Department of Electrical Engineering-Systems, Tel-Aviv University, Tel Aviv, Israel Taas Israel Industries, Advanced Systems Division, P.O. Box 1044/77 Ramat Hasharon, Israel Received 17 September 1998; revised 1 December 1999; received in "nal form 2 July 2000
Abstract Linear discrete-time systems with stochastic uncertainties in their state-space matrices are considered. The problems of "nitehorizon "ltering and output-feedback control are solved, taking into account possible cross-correlations between the uncertain parameters. In both problems, a cost function is de"ned which is the expected value of the relevant standard H performance index with respect to the uncertain parameters. A solution to the "ltering problem is obtained "rst by applying the adjoint system and deriving a bounded real lemma for this system. This solution guarantees a prescribed estimation level of accuracy while minimizing an upper bound on the covariance of the estimation error. The solution of the "ltering problem is also extended to the in"nite-horizon case. The results of the "ltering problem are used to solve the corresponding output-feedback problem. A "ltering example is given where a comparison is made with the results obtained using bounded uncertainty design techniques. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Stochastic H control; Stochastic H "ltering; Multiplicative noise; Bilinear systems
1. Introduction The analysis and design of controllers for systems with stochastic uncertainties have received much attention in the past (Dragan, Morozan, & Halanay, 1992; Halanay & Morozan, 1992; Willems & Willems, 1976; Wonham, 1967), where mainly robust stability has been considered. Recently, a renewed interest in this problem has been encountered and solutions to the stochastic control problem have been derived that ensure a worst-case performance bound in the H style (Costa & Kubrusly, 1996; Dragan & Morozan, 1997; El Ghaoui, 1995; Hinrichsen & Pritchard, 1998). Systems whose parameter uncertainties are modeled as white noise processes in a linear setting have been treated in Boyd, El Ghaoui, Feron, and Balakrishnan (1994), 夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. Sugie under the direction of Editor Roberto Tempo. This work was supported by the C&M Maus Chair at Tel Aviv University and the British EPSRC Grant no. GR/M69418. * Corresponding author. Tel.: #972-3-640-8764; fax: #972-6407095. E-mail address: [email protected] (E. Gershon).
Costa and Kubrusly (1996), Dragan and Morozan (1997), El Ghaoui (1995), Gershon, Shaked, and Yaesh (1999), Hinrichsen and Pritchard (1998), Yasuda, Kherat, Skelton, and Yaz (1990). Such models of uncertainties are encountered in many areas of applications (see Costa & Kubrusly, 1996 and the references therein) such as nuclear "ssion and heat transfer, population models and immunology (Mohler & Kolodziej, 1980). In control theory such models are encountered in gain scheduling when the scheduling parameters are corrupted with measurement noise. Recently, the output-feedback control problem for these systems has been solved in the continuous-time setting in the stationary case (Hinrichsen & Pritchard, 1998), where two coupled non-linear inequalities were obtained. The corresponding discrete-time problem has been considered in Dragan and Stoica (1998). The solution there refers only to the in"nite-time-horizon problem and, therefore, does not deal with transients that stem from a non-zero initial condition and "nal time weights. The drawback of the latter solution is that it requires a solution of an in"nite number of LMI sets. In the present paper we are mainly concerned with the "nite-horizon case. In this setting we obtain solutions in
0005-1098/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 1 6 4 - 3
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E. Gershon et al. / Automatica 37 (2001) 409}417
terms of recursive Riccati equations that are easy to use. This recursions do not naturally appear when the in"nitehorizon solution is derived in LMI forms as in Dragan and Stoica (1998). In this paper we also deal with the case where the measured output matrix contains a stochastic statedependent uncertainty. Such models, which appear in cases where the measurements include state derivatives (e.g. acceleration control of an aircraft or missile), have not been considered in Dragan and Stoica (1998) or anywhere else. A "nite-horizon version of the bounded real lemma (BRL) is required if one tries to extend the results and the methods of the stationary problem to the "nite-horizon case. In the stochastic state-feedback control problem, a &control-type' BRL has been obtained in Gershon et al. (1999) using standard completing to square arguments. The solution of the output-feedback and the "ltering problems require, however, a dual BRL which has not been derived before. In the present paper such a BRL is developed and, consequently, the "ltering and the output-feedback problems are solved, in the "nite-horizon H setting, in presence of multiplicative white-noise in the system dynamics and the measurement. The paper is organized as follows: In Section 2, we formulate the problems and we bring the state-feedback solution. In Section 3, we present the BRL that is based on the adjoint system. We solve the "ltering problem by applying this new BRL and obtain a solution which guarantees a prescribed level of estimation accuracy, in the H -norm sense, that also minimizes an upper bound on the covariance of the estimation error. In Section 4, we apply the state-feedback results and transform the resulting system to "t the "ltering model. On the latter we apply the "ltering solution of Section 3 and derive a simple recursion that provides the required outputfeedback controller. The theory developed is demonstrated in Section 5 by a "ltering example whose results are compared to those obtained by applying the deterministic technique of Xie and de Souza (1992). Notation. Throughout the paper the superscript &T' describes transposition, RL denotes the n-dimensional Euclidean space and the notation P'0, for P3RL"L means that the matrix P is symmetric and positive de"nite. We also denote by ET + ) , expectation with respect to v. The space of square summable functions over [0 N!1] is denoted by l [0 N!1], and ""."" stands for the standard l -norm, ""u"" "( ,\ I uI2uI ) . We also denote by "" fI ""0 the product fI2RfI and by dGH the Kronecker delta function. 2. Problem formulation We consider the following system: xI> "(AI #DI vI )xI #B I wI #(B I #GI gI )uI , x(0)"x ,
yI "(CI #FI fI )xI #nI , k"0, 1,2, N!1,
zI "¸I xI #D I uI , (1)
where xI 3RL is the state vector, wI 3RO is a exogenous disturbance, uI 3RQ is the control input signal, x is an unknown initial state, yI 3RP is the measured output, nI is a measurement noise and zI 3RK is the objective vector and where the sequences +vI ,, +mI , and +gI , are standard random scalar sequences with zero mean that satisfy E+vI vH ,"dIH , E+gI gH ,"dIH , E+g v ,"b d , "b "(1, I H I IH I E+fI fH ,"dIH , E+fI gH ,"pI dIH , E+fI vH ,"aI dIH , "aI "(1, "pI "(1. Assuming, for simplicity, that
(2)
[¸I2 D I 2 ]D I "[0 RI I ], RI I '0, (3) we consider the following problems: (i) The discrete-time state-feedback problem: This problem was solved by Costa and Kubrusly (1996) for in"nite dimensions (Dragan & Stoica, 1998; Boyd et al., 1994), for the stationary case, and (Gershon et al., 1999) for the "nite-horizon case. We consider (1) where we exclude yI and where we look for a state-feedback rule uI "KI xI that achieves JOE +""zI "" !c""wI "" ,#E +x 2(N)QM , x(N), E T E T !cx2QM x (0, QM , 50, QM '0 for all nonzero (+wI ,, x ) where +wI ,3l [0 N!1] and x 3RL. The state-feedback solution is obtained by requiring that the following Riccati-type recursion (see Gershon et al., 1999) QI "AI2M M I AI #DI2QI> DI #¸I2¸I !*I2'\ I *I , Q, "QM ,
(4)
will satisfy RI OcI!B I 2 QI> B I '0, ∀k3[0 N!1] and Q (cQM ,
(5a,b)
where M M OQ [I!c\B B 2 Q ]\, I I> I I I> 'I OB I 2 M M I B I #GI2QI> GI #RI I , *I OB2 I M M I AI #bI GI2QI> DI , and KI O!'\ I *I . (6a}d) Remark. The above result is based on the BRL of Gershon et al. (1999). An alternative derivation for
E. Gershon et al. / Automatica 37 (2001) 409}417
the state-feedback control problem is brought in the Appendix. There, the optimal strategies of the control signal uI is obtained. This strategy is used in Section 4 to derive a solution to the output-feedback control problem. (ii) Stochastic H -xltering: We consider (1) where B I , GI and D I are zero. We look for ¸I x( I , the "ltered estimate of zI , where x( "A x( #K (y !C x( ), x( "0. I> I I I I I I
(7)
We de"ne e "x !x( , w "[w 2 n 2] 2 I I I I I I and we consider the following cost function: J OE +""¸I> eI> "" !c""w I "" ,!cx2P x , T D P '0.
(8a}b)
411
(i) The adjoint system: We consider the following system: xI> "(AI #DI vI )xI #BI wI , zI "CI xI , (11) where +vI , is a standard white sequence and x is an unknown initial condition. Our aim is to "nd, for S'0, where S\ re#ects the amount of uncertainty in x , a necessary and su$cient condition for J$ "E +""zI "" !c""wI "" ,!cx2Sx (0, T ∀(+wI ,, x )I0,
(12)
where +wI ,3l [0 N!1] and x 3RL. For simplicity reasons we start with c"1. In order to "nd the adjoint of (11) we derive zI " I\ CI '(k,0)x #CI '(k, j#1)BH wH , 0(k4N, H C x , k"0, (13) where '(k, j) is the transition matrix given by
(9)
'(k, j)"
(A #D v ) (A #D v )(A #D v ), j(k, I\ I\ I\ 2 H> H> H> H H H I, k"j.
Given c'0, we look for ¸I xI for which J is negative for all nonzero (+w I ,, x ) where +w I ,3l [0 N!1] and x 3RL. (iii) Stochastic H output-feedback control: We look for an output-feedback controller for (1) that achieves, for a given c'0, J O E +""z "" !c""w "" #x 2QM x , T E D I I , , , !cx2QM x (0, QM , 50, QM '0
(10)
for all nonzero (+w ,, x ) where +w ,3l [0, N!1] and I I x 3RL. Similar to the standard case (Green & Limebeer, 1995), this problem involves the estimation of an appropriate combination of the states and the application of the state-feedback results with a proper modi"cation. We note that problems (i) and (ii) are extensions, to the state-multiplicative noise case, of the "nite-horizon discrete-time results of Yaesh and Shaked (1991).
3. Stochastic H 5ltering The problem of the stochastic H "ltering cannot be solved using the BRL of Dragan and Stoica (1998) which aimes at the stationary case only. We thus consider the adjoint system for which we derive a BRL. We later use this lemma to solve the "ltering problem.
We de"ne G to be the linear operator that maps (x , wI ) to zI as given by (13). By using the following inner product: 1(x , w I ),(x , w I )2"E +x 2 Sx ,# T 1w I , w I 2, where S is de"ned in (12) and where 1w I , w I 2"E + ,\ w 2 w ,, we obtain the adjoint I I I T GH of G. Denoting GHzI "[x? 2 w? I 2 ] 2 we apply the following equality for any wI and zI in l [0 N!1] and x 3RL: 1G(x , wI ), zI 2"1(x , wI ),GHzI 2"E T +x2Sx ? ,#E + ,\ wI2w? I ,. I T We thus obtain
,\ I\ E z 2(C '(k,0)x #C '(k, j#1)B w ) I I I H H I T H
,\ "E +x2Sx ? ,#E wI2w? I . T T I From the latter we get x ? "S\ ,\ H '( j,0)CH2zH , and w? I "BI2 ,\ ' 2( j, k#1)C 2 z . De"ning j? I " HI> H H ,\ ' 2( j, k#1)C 2 z we obtain the state-space repHI> H H resentation of GH as j "(A 2#D 2v )j #C 2z , ? I\ I I I ? I I ? I
j
w "B 2j , ? I I ? I
Sx "j . ? ? \ (14)
k"0,1,2, N!1,
? ,\
"0,
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(ii) A BRL for the xltering problem: Consider the adjoint system (14). The corresponding index of performance for (14) is J? "E +""w? I "" !""zI "" #x ? 2 Sx ? ,. T
(15)
Since the adjoint system is anti-causal with respect to k, the reversal of time according to the de"nitions below will lead to a causal system on which we can apply the "nite-horizon BRL of Gershon et al. (1999): We de"ne kM "N!1!k and we obtain the following causal system: jIM > "(AM IM2#DM IM2v IM )jIM #CM IM2z IM , j "0, w ? IM "BM IM2jIM ,
kM "N!1, N!2,2,0,
Sx? "j, , (16)
z IM Oz,\\IM , where we denote jIM Oj? ,\\IM , w ? IM Ow? ,\\IM , AM IM OA,\\IM , BM IM OB,\\IM , and CM IM OC,\\IM . We obtain that J? of (15) is given by J "E +""w "" !""z "" #j 2 S\j ,. The result ? ? I I ? \ ? \ T that corresponds to any c'0 is obtained by absorbing the factor c\ in B of (11) and transferring it to CM M2 I I of (16). The corresponding J then becomes J " ? ? E +""w ? I "" !c""z I "" #j? \ 2 S\j? \ ,. We can thus apT the results of Gershon et al. (1999) ply and obtain that the H -norm of the adjoint system is less than c i! there exists QIM that satis"es the following recursion: QIM "AM IM QIM > AM I2M #AM IM QIM > CM I2M #M \ IM CM IM QIM > AM IM2 #BM IM BM IM2#DM IM QIM > DM IM2,
Q, "S\,
(17)
where #M IM OcI!CM IM QIM > CM I2M '0. De"ning M K I "Q,\I the latter becomes M K I> "AI M K I AI2#AI M K I CI2'I \ K I AI2#BI BI2 I CI M #D M K D 2, M K "S\, I I I
(18)
in which it is required that 'I I OcI!CI M K I CI2'0, k"0,2, N!1.
(19)
We thus have the following: Theorem 3.1. Consider the system of (11) and (12). Given c'0, a necessary and suzcient condition for (10) to hold is that M K I of (18) satisxes 'I I '0, ∀k"0,1,2, N!1. The proof readily follows from the fact that the condition of the theorem is necessary and su$cient for achieving J? (0 for all nonzero +z? I ,. Since the adjoint operator is linear and bounded its norm equals the one of G, where the latter is the operator of the system of (11) (Kreyszig, 1978).
Remark. The above result is based on adjoint system considerations. While in the deterministic case the system and its adjoint lead to the same results, our experience shows that in the stochastic case the worst-case minimum attenuation levels (i.e. c) that are achieved by applying the BRL of Dragan et al. (1998) and the result of Theorem 3.1 do not perfectly coincide. This phenomenon is in line with the fact that both BRL's achieve a minimum level which is somewhat higher than the worst attenuation level that is obtained by simulating the behaviour of the system and its adjoint. However, we "nd that both systems achieve the same norm for the operator that relates the disturbance sequence +wI , and the objective sequence +zI ,. (iii) Stochastic H -xltering: In the sequel we also address the problem of bounding the covariance of the states of a system of the type of (11) where +wI , and x are considered to be random processes. We bring "rst the following result: Lemma 3.1. The solution to (18) is an upper-bound to the covariance E +xI xI2, of the states of the system (11), T V U where vI and wI are modeled as zero-mean uncorrelated standard white noise sequences independent of x and where the latter is a zero mean random vector. Proof. Considering the term of xI> xI> 2 , and taking the expectation with respect to vI , wI and x , we "nd that PM I>O E +xI> xI> 2 ,"AI PM I AI2#DI PM I DI2#BI BI2, T V U PM "E +x x2,. V
(20)
Subtracting (20) from (18) we obtain *M I> OM K I> !PM I> "AI *M I A2I #DI *M I DI2#AI M K I C2I 'I \ C M K I I I AI2, *M "M K !PM . It follows by induction that for M K 5PM we achieve *M I 50, ∀k'0. 䊐 We consider next the H -"ltering problem of Section 2. De"ning mI "[xI2 eI2] 2 we obtain mI> "[AI I #DI I vI #DI I fI ]mI #BI I w I ,
zI "CI I mI , (21)
where
B AI I "diag+AI , AI !K I CI ,, BI I " I B I
D DI I " I DI
0
0 , DI I " 0 !K I FI
CI I "[0 ¸I ].
0
,
!K I
0 0
, (22a}e)
E. Gershon et al. / Automatica 37 (2001) 409}417
Like in (7), a gain observer K I is looked for which achieves a negative J for all nonzero (+wI ,, x ), where J is de"ned in (9). We obtain the following theorem: Theorem 3.2. Consider the system of (21) and (9). Given c'0, a necessary and suzcient condition for J to be negative for all nonzero (+w I ,, x ) where w I 3l [0 N!1] and x 3RL is that MI "+MGH I ,, which is obtained by the following recursion: MI> "AI I MI AI I2#AI I MI CI 2I #I \ I CI I MI AI 2I #BI I BI I2 #DI I MI DI I 2 #DI I MI DI I 2 # aI DI I MI DI 2 I #aI DI I MI DI I 2 , M "[I I] 2P\[I I]
(23a,b)
satisxes # I I OcI!CI I MI CI I2'0, ∀k"0,1,2, N!1.
(23c)
(24)
where &I OM I [cI!¸2I ¸I M I ]\ and (M I OI#FI M I FI2#CI &I CI2.
(25)
Proof. The recursion of (23a) readily follows from Theorem 3.1. In order to comply with the weighting on the initial condition in (9), and with the choice of x( "0, nature should be forced to select the component e in m to be equal to x . We choose, therefore, the weighting on the initial condition on the performance index to be [x2 e2!x2]diag+P , mI,[x2 e2!x2] 2 where m is a large positive scalar. Since [x2 e2!x2]" [x2 e2]B 2, where
BO
I
0
!I
I
(22e) #I I "cI!¸I M I ¸I2, it follows from (23) that the equation for M I is M I> "(AI !K I CI )&I (A2I !CI2K I 2) #K I (I#FI M I FI2)K I 2 #DI M I DI2 # B I B I 2 !(K I FI M I DI2 #DI M I FI2K I 2 )aI .
(26)
From Lemma 3.1 we have that MI of (23a) is an upper bound on E +m m 2, and thus ¸ M ¸ 2 is a bound I I I T K D U I I on E +zI zI2,. For each k"0,1,2, N!1, K I can T K D U thus be derived to minimize M I> . We "nd from (26) that M I> "AI &I AI2#DI M I DI2 #[K I !KH I ]( M I [K I !KH I ] 2#B I B I 2 ! [AI &I CI2#aI DI M I FI2]( M \ I
If (23c) is satisxed, an observer gain K I that achieves the estimation error level of c and minimizes, at each instant, an upper bound on the covariance of the estimation error with respect to the parameters vI , fI , w I and x , where +w I , and x are modeled as uncorrelated zero-mean white noise sequence and vector, respectively, is given by KH "[A & C2#D M F 2a ](M \, I I I I I I I I I
413
,
we obtain the weight S"B 2diag+P , mI,B. The corresponding weight in the adjoint system is then S\"[I I] 2P\ [I I]#m\[0 I] 2[0 I] and (23b) is obtained when m tends to in"nity. Since by (23c) and
;[CI &I AI2#aI FI M I DI2], where KH is de"ned in (24) and & and ( M are de"ned in I I I (25). The observer gain KH clearly minimizes, at time k, I the upper bound on the covariance of z . 䊐 I> The observer gain K is derived at each instant i, by G (24) based on the value of M that was obtained from the G recursion in (23a). Although the right-hand side of (23a) is nonlinear in M , once M , and therefore K , were I G\ G\ obtained the matrix M can be readily calculated. G (iv) Inxnite-horizon stochastic xltering: In order to prove the mean-square stability of the "lter that is obtained above, in the case where the system matrices are all constant and N tends to in"nity, we consider (21) where AI , DI , DI , BI and CI are constant. Since the dy namic matrix of the adjoint of (21) is the transpose of AI #DI v #DI f , the mean-square stability of (21) is I I assured by the stability of the adjoint system j "(AI 2#DI 2v #DI 2f )j . I> I I I Applying the arguments of Boyd et al. (1994) we obtain: Lemma 3.2. System (21) is mean-square stable if there exists a positive-dexnite matrix Q3RL"L that satisxes trace+! PK ,(0, ∀i, H G where PK 3RL"L satisxes the following recursion: G PK "AI 2PK AI #DI 2 PK DI #DI 2 PK DI G> G G G #a DI 2 PK DI #a DI 2PK DI G G G G with
PM PK " PM
PM , PM '0 PM
(27)
(28)
414
E. Gershon et al. / Automatica 37 (2001) 409}417
and where !H (Q)OAI QAI 2!Q#DI QDI 2#DI QDI 2# aDI QDI 2#aDI QDI 2.
by the method of completing to squares. The solution of the output-feedback problem is, then, obtained by transforming the problem to a "ltering one. Denoting rI OwI !wH I when wHI is de"ned in (A.2), and using uI "KI x( I , where KI is de"ned in (6) and where x( I is yet to be found, we obtain from (1) that
Since !H (M)#AI MCI 2#I \CI MAI 2#BI BI 2"0 (29) is the stationary version of (23a), and since we choose K to satisfy the latter, we obtain, using the fact that a(1, that the requirement of (27) is satis"ed if trace+BI BI 2PK G , is positive for all i'0. Using the de"nition of BI we "nd that trace+BI BI 2PK G ,"0 for some i only if
x "(A #D v #B K )x #B r I> I I I I VI I I I #(B I KSI #B I #GI gI )KI x( I , yI "(CI #FI fI )xI #nI .
[B2 0]PK G [B2 0] 2"0. (30) Eq. (28) implies that PK G !+AI 2,GPK AI G is positive semide"nite. Since AI has a diagonal block structure, the result of (30) can occur only if AGB "0. Denoting the system xI> "AxI #BwI , zI "HxI by S(A, B, H) the latter result implies that the system S(A, B, I) is of the moving average type (a FIR system). Excluding such systems we arrive at the following result.
Substituting in (A.3) we look for x( for which
SI "cQM !Q
x( I> "(AI #B I KVI )x( I #(B I KSI #B I )uI #K I (yI !CI x( I ).
e "[A #B K !K C ]e #B r !K n I> I I VI I I I I I I I #DI xI vI #GI KI xI gI !K I FI xI fI !GI KI eI gI . De"ning mI "[xI2 eI2] 2 and w I "[rI2 nI2] 2 we obtain the following system, which is equivalent to the one in (21): mI> "[AI I #DI I vI #DI I fI #DI I gI ]mI #BI I w I ,
To solve the "nite-horizon output-feedback control problem a careful derivation of the expression of J of (10) is required in order to obtain the optimal strategies for u and w. These are needed for deriving an equivalent to the separation principle and they are obtained in the Appendix by solving the state-feedback control problem
zI "CI I mI ,
0 0
, DI
I
0
"
!K F I I
(34)
where
A #(B K #B )K #B K I I SI I I I VI 0
(33)
Using the de"nition of (8b) for xI> of (31) and x( I> of (33) we obtain
4. Stochastic output-feedback control
D DI " I I D I
(32a,b)
is less then zero for all nonzero (+wI ,,+nI ,, x ), where zI "KI (xI !x( I ). We consider the following state estimator:
Remark. (i) In the case where S(A, B, I) is of the FIR type, stability in the mean-square sense, is guaranteed if we solve (29) where the term eI is added to the left-hand L side of the equation, with 0(e;1. (ii) When stability is guaranteed, the stationary "lter gain K might have been derived by solving the station ary version of (23) and (24). This, however, may not be feasible due to the non-linearities involved. An alternative method is to solve the corresponding "nite-horizon problem taking a very large N.
,\ ""zI ""I !""rI ""0I !c""nI "" #(I !x2SI x , JO E E T D I
Theorem 3.3. Consider the system of (21) with constant matrices, where S(A, B, I) is of the inxnite impulse response (IIR) type, and where N tends to inxnity. This system is stable in the mean-square sense, for any constant K that solves (29).
AI " I
(31)
![B K #B ]K B 0 I SI I I , BI " I , I A #B K !K C B !K I I VI I I I I 0 G K !G K I I and CI "[0 K ]. , DI " I I I I I 0 G K !G K I I I I
E. Gershon et al. / Automatica 37 (2001) 409}417
Applying the results of Section 3 we arrive at the following theorem: Theorem 4.1. Consider the system of (1) where x( is dexned I in (33). There exists a controller that achieves (10) for a given c iw the recursion of (4) satisxes (5) and there exists QI "+QI ,, that is obtained by the following recursion: I GH I QI "BM BM 2#AI QI AI 2#DI QI DI 2 #DI QI DI 2 I> I I I I I I I I I I I #DI QI DI 2 #(DI QI DI 2 #DI QI DI 2 )a I I I I I I I I I I # (DI QI DI 2 #DI QI DI 2 )b #(DI QI DI 2 I I I I I I I I I I #DI QI DI 2 )p #AI QI CM 2)\CM QI AI 2 (35) I I I I I I I I I I I which satisxes ) OcI!CM QI CM 2'0, ∀k"0,1,2, N!1, I I I I where SI \ SI \ QI "c , SI \ SI \
(36)
CM O[0 ' K ] I I I
cB R\ 0 I I and BM O , I cB R\ !K I I I and where ' and R are dexned in (6) and (5a), respectiveI I ly, and SI is dexned in (32b). If the latter holds, the controller that achieves (10) and minimizes an upper bound on the covariance of z I of (34), with respect to the stochastic parameters v , g , f , x , +r , and +n ,, when the latter two sequences I I I I I are modeled as uncorrelated zero-mean white noise sequences, is given by u "K x( . In the latter, K is given by I I I I (6d) and x( is given by (33) with I K "!#HRM \, (37) I I I where RM OC QI C 2#C QI K 2'2 )\' K QI C2 I I I I I I I I I I I I I #I#F QI F 2, AM OA #B K , I I I I I I VI #HO!AM QI C 2!AM QI K 2'2 )\' K QI C2 I I I I I I I I I I I !a D QI F 2!p G K (QI !QI 2 )F 2. I I I I I I I I I I Proof. Since the output-feedback control problem has been transformed to one of "ltering it is possible to apply the "ltering BRL, thus arriving at the recursion for QI . Similar to the "ltering problem, we "nd that ) is I given by cI!' K QI K 2'2 . We thus obtain that I I I I I QI "!I #K RM K 2 ##HK 2 #K #H2 where I> I I I I I I I I
415
Rearranging the latter and completing to squares, with respect to K , we obtain I QI "!I #[K ##HRM \]RM [K I> I I I I I I #RM \#H]2!#HRM \#H2 I I I I I and the observer gain that minimizes QI at the kth I instant is thus given by (37). 䊐 It follows from the above that the solution to the output-feedback control problem, at each instant i, is obtained by deriving "rst the state-feedback gain K usI ing (4)}(6). This gain is substituted in (35) and (36) and Q is thus obtained from (35) using the previous values of G Q and K . Once Q which satis"es (36) is derived, G\ G\ G the gain matrix K is obtained from (37). G The stability of the resulting output-feedback control system, in the case where the system matrices are all constant and the horizon N tends to in"nity, is implied by the stability, in the mean-square sense, of the statefeedback and the "ltering subsystems. This stability is guaranteed if S(A, B, ¸) is of the IIR type. 5. Example In this section we solve a stochastic H -"ltering prob lem using the theory of Section 3, and we compare our results with those obtained by the design of Xie and de Souza (1992). We consider the following IIR system:
0.1 0.6 2 0 x " x # w , I> I !1.65 !0.5#a 0 0 I I y "[2 2#c ]x #[0 1]n , I I I I where +a , and +c , are zero-mean uncorrelated whiteI I noise Gaussian sequences with standard deviations of 0.06 and 0.1, respectively. Using the theory of Section 3 we have D"[0 I] 2[0 0.06] and F"[0 0.1]. We want to obtain a H -estimate of z "[!0.225 0.45]x I I where we apply a very large initial value P and consider the case where N is very large. We "nd that c"1.4 is close to the minimum possible value. For this value of c, the recursion of (23) converges, up to a tolerance of 10\, within 80 steps. The observer gain that we obtain from (24) is K "[0.1877 !0.4704] 2. Our method is aimed at dealing with white multiplicative uncertainties. It is not, therefore, suitable to tackle constant deterministic uncertainties. Nevertheless it is of interest to compare our results with those obtained by the method of Xie and de Souza (1992), we choose
!I OAM QI AM 2 #AM QI K 2'2 )\' K QI AM 2 #D QI D 2#cB R\B 2 I I I I I I I I I I I I I I I I I I I # b [D (QI !QI )K 2G 2#G K (QI !QI 2 )D 2]#G K [QI #QI !QI !QI 2 ]K 2G 2. I I I I I I I I I I I I I I I I I I I
416
E. Gershon et al. / Automatica 37 (2001) 409}417
the uncertainty interval in the latter to yield the same estimation level of 1.4. We choose, for example, a 3[!0.015 0.015] and c 3[!0.029 0.029]. We note I I that in our stochastic approach 80% of the values of a and 77% of c lie outside these intervals. Using the I I method of Xie and de Souza (1992) we search for one free parameter e and obtain e"1.396. The resulting K is then [0.3114 !0.2424] 2. Comparing the two designs it is found that ours achieves the estimation level of 1.29, which is lower than the level of c"1.4 that achieved by the other design.
6. Conclusions The problem of H -optimal control and "ltering of discrete-time linear systems with multiplicative stochastic uncertainties has been solved in the "nite-horizon case. Requiring a Luenberger-type H estimator a neces sary and su$cient condition for the existence of a solution is obtained. This solution is not unique, however, a speci"c selection of the observer gain leads, in addition to compliance with a prescribed estimation level, also to a minimization of an upper bound on the covariance of the estimation error when the disturbance and the initial condition are modeled as uncorrelated zero-mean whitenoise signals. In this sense, our solution resembles the situation in the standard H "ltering problem where the solution to the corresponding Riccati equation is an upper bound to the covariance of the estimation error (Bernstein & Haddad, 1989). An example is given which shows that the results obtained by the new method favorably compare with those achieved by the norm-bounded design techniques. It should be pointed out that in the later techniques the prescribed bound on the index of performance is guaranteed for all the parameters in the uncertainty interval. In our approach, on the other hand, there is a "nite nonzero probability of violating the index of performance. There is a trade-o! here between a better attenuation and estimation level and certainty in achieving these levels. In many practical problems the designer may prefer achieving better performance on the average, knowing that in a small proportions of the cases, the design may not achieve its goals.
"rst for w and then for u , we obtain I I E +JI ,"!""w !wH"" I #""u !uH"" I I I 0 I I T E I #x 2RM (Q )x !z 2z #cw 2w #E +( ,, I I I I I I I T E I (A.1) where ( consists of terms that are a$ne in the stochasI tic parameters, M M , ' , * and K are de"ned in (6a}d): I I I I uHOK x , wHOK x #K u , I I I I VI I SI I K OR\B 2 Q A , K OR\B 2 Q B , VI I I I> I SI I I I> I RM (Q )OA 2M M A #D 2Q D #¸ 2¸ !* 2'\* . I I I I I I> I I I I I I (A.2a}e) Taking the sum of both sides of the above JI , from zero I to N!1, we obtain using (A.1) ,\ E +JI ,"E ""x "" , !""x "" / T E I I T E , / ,\ " E +!""w !wH"" I #""u !uH"" I , I I 0 I I I T P ,\ # E +x 2RM (Q )x , I I I I T E ,\ # E +!z 2z #cw 2w ,#E +( ,. I I I I T E I I T E Hence, ,\ J" E +!""w !wH"" I #""u !uH"" I , I I 0 I I I T E ,\ # E ""x ""M I #""x "" M I 0/ / \A / I T E #E +( ,#E ""x ""M L , . I T P T E , / \/
(A.3)
Clearly the optimal strategy for u is given by u "uH I I I where Q satis"es (4)}(6). I References
Appendix We consider JI "x 2 Q x !x 2Q x I I> I> I> I I I and we substitute for x of (1). Taking the expectation I> with respect to g and v , and completing the squares, I I
Bernstein, D., & Haddad, W. M. (1989). LQG control with an H performance bound: A Riccati equation approach. IEEE Transactions on Automatic Control, AC-34, 293}305. Boyd, S., El Ghaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequality in systems and control theory. SIAM Frontier Series. Philadelphia: SIAM. Costa, O. L. V., & Kubrusly, C. S. (1996). State-feedback H -control for discrete-time in"nite-dimensional stochastic bilinear systems. Journal of Mathematical Systems, Estimation and Control, 6, 1}32.
E. Gershon et al. / Automatica 37 (2001) 409}417 Dragan, V., Morozan, T., & Halanay, A. (1992). Optimal stability compensator for linear systems with state dependent noise. Stochastic Analysis and Application, 10, 557}572. Dragan, V., & Morozan, T. (1997). Mixed input}output optimization for time-varying Ito systems with state dependent noise. Dynamics of Continuous, Discrete and Impulsive Systems, 3, 317}333. Dragan, V., & Stoica, A. (1998). A c attenuation problem for discretetime-varying stochastic systems with multiplicative noise. Reprint Series of the Institute of Mathematics of the Romanian Academy, No 10. El Ghaoui, L. (1995). State-feedback control of systems with multiplicative noise via linear matrix inequalities. Systems and Control Letters, 23, 223}228. Gershon, E., Shaked, U., & Yaesh, I. (1999). H control and "ltering of discrete-time stochastic bilinear systems. Proceedings of the European Control Conference (ECC99), Karlsruhe, Germany, 1999. Green, M., & Limebeer, D. J. N. (1995). Linear robust control. Englewood Cli!s, NJ: Prentice-Hall. Halanay, A., & Morozan, T. (1992). Optimal stability compensator for linear discrete-time systems under independent random perturbations. Revue Romaine de Mathematiques Pures Appliquees, 3, 213}223. Hinrichsen, D., & Pritchard, A. J. (1998). Stochastic H . SIAM Journal of Control Optimization, 36(5), 1504}1538. Kreyszig, E. (1978). Introductory functional analysis with applications. New York: Wiley. Mohler, M. M., & Kolodziej, W. J. (1980). An overview of stochastic bilinear control processes. IEEE Transactions on Systems, Man and Cybernetics, 10, 913}919. Willems, J. L., & Willems, J. C. (1976). Feedback stabilizability for stochastic systems with state and control dependent noise. Automatica, 12, 277}283. Wonham, W. M. (1967). Optimal stationary control of a linear system with state-dependent noise. SIAM Journal of Control and Optimization, 5, 486}500. Xie, L., & de Souza, C. E. (1992). Robust H control for linear systems with norm-bounded time-varying uncertainty. IEEE Transactions on Automatic Control, AC-37, 1188}1192. Yaesh, I., & Shaked, U. (1991). H -optimal state estimation of discrete-time systems. Recent advances in mathematical theory of systems, control, networks and signal processing, Proceedings of MTNS-91, Kobe, Japan, vol. 1 (pp. 261}267), Mita Press. Yasuda, K., Kherat, S., Skelton, R. E., & Yaz, E. (1990). Covariance control and robustness of bilinear systems Proceedings of the 29th CDC.
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Uri Shaked (M'79}SM'91}F'93) was born in Israel in 1943. He received the B.Sc. and M.Sc. degrees in physics from the Hebrew University, Jerusalem, Israel, in 1964 and 1968, respectively, and the Ph.D degree in applied mathematics from the Weizmann Institute, Rehovot, Israel in 1975. From 1974 to 1976 he was a Senior Visiting Fellow at the Control and Management Science Division of the Faculty of Engineering at Cambridge University UK. In 1976 he jointed the Faculty of Engineering at Tel Aviv University, Israel, and in 1985}1989 he was the Chairman of the Department of Electrical Engineering-Systems there. In 1993}1998 he was the Dean of the Faculty of Engineering at Tel Aviv University. He is the incumbent of Celia and Macros Chair of Computer Systems Engineering there since 1989. During the 1983}1984, 1989}1990 and 1998}1999 academic years he spent his sabbatical year in the Electrical Engineering Departments at the University of California, Berkeley, Yale University, and Imperial College in London, respectively. His research interests include linear optimal control and "ltering, robust control, H -optimal control, and digital implementations of control lers and "lters. Eli Gershon was born in Israel in 1955. He received the B.Sc. and M.Sc. degrees in chemistry and biochemistry from Tel Aviv University, Tel Aviv, Israel, in 1981 and 1984, respectively, and the Ph.D. degree in electrophysiology from Tel Aviv University in 1992. From 1996 to the present he is a Ph.D. student at the Department of Electrical Engineering-Systems in the Faculty of Engineering at Tel Aviv University. His research interests include linear H -opti mal control and estimation of stochastic state-multiplicative systems and polytopic designs. Isaac Yaesh received his B.Sc. EE from the Israel Institute of Technology. At 1981 and 1992, respectively, he received his M.Sc. EE and Ph.D. from Tel Aviv University from the System department at the faculty of Engineering. In 1981}1986 he was with the IDF and served as a research engineer and since 1986 he is with IMI/Advanced System Division. His current interest are robust control and estimation, H theory, di!erential games and their applications in #ight control and guidance systems. Dr. Yaesh is a co-author of the book `Advances in Missile Guidance Theorya.