Robust finite horizon minimax filtering for discrete-time stochastic ...
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Systems & Control Letters 52 (2004) 99 – 112
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Robust !nite horizon minimax !ltering for discrete-time stochastic uncertain systems Myung-Gon Yoona;∗ , Valery A. Ugrinovskiib , Ian R. Petersenb a School
b School
of Electrical Engineering and Telecommunications, The University of New South Wales, NSW 2052, Australia of Information Technology and Electrical Engineering, Australian Defence Force Academy, Canberra, ACT 2600, Australia Received 19 June 2002; received in revised form 4 November 2003; accepted 14 November 2003
Abstract We study a !nite-horizon robust minimax !ltering problem for time-varying discrete-time stochastic uncertain systems. The uncertainty in the system is characterized by a set of probability measures under which the stochastic noises, driving the system, are de!ned. The optimal minimax !lter has been found by applying techniques of risk-sensitive LQG control. The structure and properties of resulting !lter are analyzed and compared to H∞ and Kalman !lters. c 2003 Elsevier B.V. All rights reserved. Keywords: Robust !ltering; Kalman !lter; H∞ !lter; Risk-sensitive control
1. Introduction From the viewpoint of robustness, the standard Kalman !lter may su@er from poor performance. Indeed, the standard Kalman !lter does not incorporate information on the uncertainty present in a systematic way. This fact has been a major motivation for current research on a robust version of Kalman !lter. For example, Petersen, McFarlane, Theodor, Shaked and de Souza [3,8,14] studied a guaranteed cost 'lter that leads to a bound on the estimation error variances when applied to a class of systems having norm-bounded uncertainty. Also in [9], the H∞ !ltering method was extended to deal with systems with deterministic noise inputs and uncertainty satisfying integral quadratic constraints (IQC). Similarly for a class of uncertain systems with bounded parameter perturbations, a robust version of the H∞ estimation problem was studied in [15]. A standard assumption used in Kalman !lter design requires that the stochastic noise driving the nominal system is white and Gaussian. It is plausible, however, that in a practical system the noise can be perturbed by some uncertainty and as a result it is not white or Gaussian anymore. Hence, the performance of Kalman !lter may be compromised due to the lack of robustness.
∗
This work was supported by The Australian Research Council. Corresponding author. Tel.: +61-2-9385-6576; fax: +61-2-9385-5993/5997. E-mail address: [email protected] (M.-G. Yoon).
c 2003 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2003.11.004
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Perturbations of the reference noise can be elegantly modeled by introducing a new probability measure under which the noise has a di@erent probability distribution. This approach to uncertainty modeling, introduced in [7,10,16,17] and adopted in this paper, serves as an alternative to a commonly used uncertainty modeling that considers sets of unknown parameters and external disturbances. An important feature of this approach is that a set of probability measures acting on a given nominal system produces a collection of perturbed systems. A !lter is then designed that ensures an acceptable performance for all systems of the considered class of perturbed systems. After introducing in Section 2 the notion of uncertain systems modeled using probability measure perturbations, we formulate a robust version of the Kalman !ltering problem in Section 3. This formulation is based on a widely used worst-case design interpretation of robust design. This motivates our use of the name ‘minimax optimal !lter’ for our robust version of the Kalman !lter. Also in Section 3, a complete solution to the minimax optimal !ltering problem is given, which is derived from existing results on risk-sensitive control and properties of the relative entropy functional. In Section 4, the structure and properties of the minimax optimal !lter are analyzed and compared to those of the Kalman and H∞ !lters. Finally, a numerical example is presented in Section 5 to illustrate these properties. 2. Stochastic uncertain system 2.1. Nominal system We follow the problem formulation of [7,16,17,19]. Consider the following time-varying discrete-time stochastic system on a !nite-time interval: xk+1 = Ak xk + wk+1 ;
(1)
zk = Dk xk ;
(2)
k = Lk xk ;
(3)
yk+1 = Ck xk + Ek wk+1 + vk+1 ;
k = 0; : : : ; N;
(4)
where xk ∈ Rn is the state, zk ∈ Rm denotes the uncertainty output, k ∈ Rp is the output to be estimated, and yk ∈ Rq is the measurement. With a !xed probability measure P de!ned on the noise space, := {(x0 ; w1 ; v1 ; : : : ; : : : ; wN ; vN )};
(5)
the process noise wk ∈ Rn and the measurement noise vk ∈ Rq are independent white Gaussian processes with zero mean and covariances k ¿ 0 and k ¿ 0, respectively. The initial state x0 ∈ Rn is a Gaussian random vector with mean xO0 and covariance X0 ¿ 0. We assume {x0 ; wk ; vk } are mutually independent and have the Gaussian joint distribution: d x0 exp − 12 (x0 − xO0 ) X0−1 (x0 − xO0 ) P(d x0 × dw1 × · · · × dvN ) = n (2) |X0 | N −1 1 dwk+1 dvk+1 −1 −1 × exp − 2 (wk+1 k+1 wk+1 + vk+1 k+1 vk+1 ) ; n+q | (2) || | k+1 k+1 k=0 (6) where | · | denotes the determinant of a matrix.
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2.2. Uncertain systems We consider the class of uncertain dynamics that can be generated using Eqs. (1)–(4) with a di@erent probability measure Q. We will make use of a dependence structure for Q given by Q(d x0 × dw1 × dv1 × · · · × dwN × dvN ) = Q(d x0 )Q(dw1 |x0 )Q(dv1 |x0 )Q(dw1 |x0 ; w1 ; v1 )Q(dv1 |x0 ; w1 ; v1 ) · · · Q(dwN |x0 ; w1 ; v1 ; : : : ; wN −1 ; vN −1 )Q(dvN |x0 ; w1 ; v1 ; : : : ; wN −1 ; vN −1 )
(7)
which can be regarded as a perturbation of the nominal joint probability measure (6). In order to characterize the class of admissible perturbations, we need a measure of discrepancy between the probability measures P and Q. For this we choose the relative entropy de!ned by dQ E Q log dQ if QP and log ∈ L1 (dQ); dP dP h(QP) := (8) +∞ otherwise: Here QP denotes the fact that the perturbed measure Q is absolutely continuous with respect to the reference measure P. For a detailed study on relative entropy, see [4]. Let P denote the set of probability measures Q such that h(QP) ¡ ∞. Let us consider a special case of Q ∈ P to illustrate the notion of uncertain systems generated by measure perturbations. De!ne Fk to be the -algebra generated by the sequence {x0 ; y0 ; : : : ; xk ; yk }, assuming y0 = 0, and completed by including all corresponding sets of probability zero. For a given Q ∈ P, we assume that there exist Fk -adapted random sequences { k } and {!k } such that the restriction of the measure Q on the -algebra F‘ has the following Radon–Nikodym derivative:
dQ (x0 ; w1 ; v1 ; : : : ; wN ; vN )
:= exp − 12 # X0−1 # + # X0−1 (x0 − xO0 ) dP F‘ ‘ × exp − 12 k=0
−1 k k+1
k
−1 + wk+1 k+1
k
−1 −1 !k + vk+1 k+1 !k + − 12 !k k+1
(9)
with # ∈ Rn , 0 6 ‘ 6 N − 1. Then, from (6) it follows that the joint distribution of x0 , wk and vk under the measure Q is also Gaussian and is given by Q(d x0 × · · · × dvN )|F‘ =
d x0 (2)n |X0 |
exp − 12 (x0 − xO0 − #) X0−1 (x0 − xO0 − #)
‘ dwk+1 dvk+1 × exp − 12 (wk+1 − n+q (2) |k+1 k+1 | k=0
k)
−1 k+1 (wk+1 −
k)
−1 − 12 (vk+1 − !k ) k+1 (vk+1 − !k ) :
(10)
Hence, the initial state x0 and noise variables {wk ; vk } can be represented by x0 = # + x0Q ;
(11)
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wk =
k−1
+ wkQ ;
(12)
vk = !k−1 + vkQ ;
(13)
where # ∈ Rn , wkQ and vkQ are Gaussian-independent sequences with zero means and covariances k and k under Q, respectively. This results in the following representation of system (1) and (4) under the perturbed probability measure Q: Q xk+1 = Ak xkQ +
Q + wk+1 ;
k
Q yk+1 = Ck xkQ + Ek
k
(14)
Q Q + !k + Ek wk+1 + vk+1 ;
(15)
x0Q
where the initial state is Gaussian with mean xO0 + # and covariance X0 . See [5] for general results on this kind of measure change. Moreover, from the chain rule [4], we can explicitly obtain the relative entropy h(QP) as follows: N −1 1 Q −1 −1 1 −1 h(QP) = 2 # X0 # + (16) E k k+1 k + !k k+1 !k : 2 k=0
Eq. (16) allows us to quantify the energy brought into the system by the perturbation associated with the perturbed measure Q ∈ P de!ned by (10) or, equivalently, the input signals { k ; !k }. This energy expressed as the quadratic sum of signals { k ; !k } can be evaluated by the value of relative entropy h(QP). This motivates us to use the relative entropy functional as a measure of the size of uncertainty in the system. This idea serves as a natural extension of that behind the sum quadratic constraints (SQCs) [6] and integral quadratic constraints (IQCs) [12,22]. The de!nition given below was introduced in [7]; also, see [16–19] where extensions of this uncertainty description are introduced in the context of continuous-time and in!nite-horizon problems. Denition 1. Given a constant d ¿ 0, a probability measure Q ∈ P is said to de!ne an admissible uncertainty if Q satis!es N 1 h(QP) 6 E Q (Cu ) + d where Cu = zk 2 (17) 2 k=1
and {zk } is the uncertainty output (2). The set of all admissible uncertainty measures in P will be denoted by &. It should be noted that the admissible uncertainty of De!nition 1 is more general than that de!ned by the SQC: N −1 1 Q −1 −1 Q 1 −1 # X # + E k k+1 k + !k k+1 !k 6 E (Cu ) + d 0 2 2 k=0
which is constructed by replacing the relative entropy functional on the right-hand side of (17) with expression (16) involving the inputs k , !k . This is because not every measure Q ∈ & admits a representation of the form (10); e.g., see [11]. 3. Minimax optimal ltering problem In this paper, we will focus on the class of !lters u = {u0 ; : : : ; uN } having a representation uk = fk (x0 ; u0 ; : : : ; uk−1 ; y1 ; : : : ; yk );
k = 1; : : : ; N
(18)
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103
for some nonanticipative function fk . Filters of this form will be termed as feasible 'lters. From the above de!nition, the output variable uk of a feasible !lter is Yk -measurable; here, Yk is a -algebra generated by the measurements {x0 ; y0 ; : : : ; yk }. Thus, feasible !lters are to produce their output using the information in the measured output y0 ; : : : ; yk and initial condition x0 of the system. Given a feasible !lter u, de!ne a !ltering error associated with particular noise and measurement samples as follows: N
N
k=0
k=0
1 1 Ce := k − uk 2 = Lk xk − uk 2 : 2 2
(19)
Then, the expected error achieved when this !lter is used to estimate the output k of the uncertain system (1)–(4) with the uncertainty Q ∈ P is given by E Q (Ce ). According to the above de!nitions, the worst-case error over the set of admissible perturbations Q ∈ & can be expressed as V (u) := sup E Q (Ce ):
(20)
Q∈&
Denition 2 (Robust minimax optimal !ltering problem). Find the feasible !lter uopt minimizing the worstcase error, i.e., V (uopt ) = inf V (u) = inf sup E Q (Ce ) u
u Q∈&
(21)
From De!nition 2, one can see that the minimax optimal !ltering problem de!ned above is closely related to the general minimax optimal output feedback control problem considered in Ref. [7]. This problem was formulated in [7] for a general case where the underlying uncertain system and control cost functional were nonlinear. Also, a specialization of the general theory to a class of minimax LQG control problems was presented. However, the results presented in [7] did not directly allow for cost functionals of the form (19). In this paper, we extend the theory of [7] to solve the minimax !ltering problem of De!nition 2. Our aim is to obtain a tractable solution to the minimax optimal !ltering problem similar to that obtained in [7] for the minimax optimal LQG control. This will lead to a tractable minimax optimal !lter design procedure based on a pair of matrix Riccati recursions. This result is presented in this section. The main technical di@erence that prompted us to develop an extension of the results obtained in [7] is due to the presence of the cross-term in the !ltering error functional (19). Our second aim will be to establish connections between the minimax optimal !lter and !lters obtained for the same system using H∞ !ltering and Kalman !ltering approaches. These results will be presented in the next section. We now proceed with the extension of the results of [7] to the minimax !ltering problem posed in De!nition 2. In [7], it is shown that the constrained minimax problem (21) can be indirectly solved via an unconstrained version of the problem as follows: V* := inf sup E Q {Ce − *[h(QP) − Cu − d]}; u Q∈P
= *{inf sup E Q *−1 C* − h(QP) + d}; u Q∈P
(22) (23)
where C* is de!ned by N
C* := Ce + *Cu =
1 [xk (Lk Lk + *Dk Dk )xk − 2xk Lk uk + uk uk ]: 2 k=0
(24)
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Lemma 1 (Petersen et al. [7, Theorem 2.1]). De'ne + := {* ¿ 0; V* ¡ ∞}. Then V (uopt ) is 'nite if and only if + = ∅. Moreover if + = ∅, then we have V (uopt ) = inf V* ;
(25)
*∈+
:= V*∗ ;
(26)
when the in'mum in (25) is achieved. This means that the minimax optimal !lter uopt is given by the minimizer of the minimax game (23) corresponding to * = *∗ . The solution of the minimax game (23) is based on the fact that for each * ¿ 0, the minimax problem (23) can be converted into a risk-sensitive control problem associated with the nominal system (1), thus leaving us with a collection of *-parameterized risk-sensitive control problems. Such conversion exploits a result from the theory of large deviations referred to as the duality relation between the free energy and relative entropy [4,11]. Using this duality relation, we obtain inf sup E Q [*−1 C* − h(QP)] = inf log E P (exp*−1 C* ) := W* : u Q∈P
(27)
u
The risk-sensitive control problem on the right-hand side of (27) is a so-called partial information linearquadratic-exponential Gaussian (LEQG) control problem which is known to admit a solution in a closed form. Remark 1. Since the quadratic cost C* itself depends on *, the parameter * cannot be interpreted as the risk-sensitive parameter, cf. [2,20]. 3.1. Design of a minimax optimal 'lter In this section, we present a complete solution to the minimax optimal !ltering problem. This solution is derived from a known solution to the risk-sensitive problem developed in [21]. It also involves a generalization of the results of [12]. In order to make use of the solution to the risk-sensitive control problem presented in [21], we recast the !ltering problem in De!nition 2 into a partial information LEQG control problem associated with the nominal system xk+1 = Ak xk + wk+1 ;
(28)
yk+1 = Ck xk + Ek wk+1 + vk+1
(29)
and an exponential cost N 1 P exp [xk (Lk Lk + *Dk Dk )xk − 2xk Lk uk + uk uk ] ; J* := E 2*
* ¿ 0:
(30)
k=0
Consider matrices Rk ¿ 0; Pk ¿ 0, de!ned by the following Riccati equations: −1 ˜ −1 −1 ˜ Rk+1 = ˜ k+1 + A˜ k (R−1 k − * Lk Lk − Dk Dk + Ck k+1 Ck ) Ak ;
Pk = *Dk Dk + Ak Pk+1 (I − *−1 k+1 Pk+1 )−1 Ak ; −1 −1 ˜ −1 k+1 := k+1 + Ek k+1 Ek ;
R0 = X0 ;
PN +1 = 0;
˜ k+1 := k+1 + Ek k+1 Ek ;
−1 A˜ k := Ak − ˜ k+1 Ek k+1 Ck ;
(31) (32) (33)
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and satisfying the conditions 0{Rk (Lk Lk + *Dk Dk )} ¡ *;
(34)
−1 Ak ¡ *; 0 Rk *Dk Dk + Ak Pk+1 I − *−1 k+1 Pk+1
(35)
where 0(·) denotes the spectral radius of a matrix. Also, consider a sequence {1k } de!ned by −1 1k+1 = A˜ k 1k + A˜ k Kk [Ck ˜ −1 k+1 (yk+1 − Ck 1k ) + Dk Dk 1k − * Lk (uk − Lk 1k )] −1 +˜ k+1 Ek k+1 yk+1 ;
10 = xO0 ;
(36)
Kk−1 := Rk − *−1 Lk Lk − Dk Dk + Ck ˜ −1 k+1 Ck :
(37)
Lemma 2. Suppose there exist the matrices Rk ¿ 0; Pk ¿ 0, de'ned by (31), (32) and satisfying conditions (34), (35) for some * ¿ 0. Then the optimal risk-sensitive controller solving problem (27) is given by the output feedback control law: uk* = Lk (I − *−1 Rk Pk )−1 1k :
(38)
Proof. The claim easily follows from Theorems 7.3.2, 8.2.1 and 8.3.1 of [21] by substituting the cost (30), system matrices (28)–(29) and noise property described in Section 2.1. This result shows that the optimal !lter minimizing the cost log E P (exp *−1 C* ) is simply a minimum stress estimator in the risk-sensitive control problem (28), (30); see [21, Theorem 8.3.1]. Moreover, !lter (38) is identical to the solution of the optimal H ∞ !ltering problem with the same cost function C* , e.g., see, [21]. Consider the following dynamic game problem: inf sup [C* (x; u) − * U(x0 ; w; v)];
(39)
u x0 ;w;v
where 1 U(x0 ; w; v) = 2
N −1 k=0
−1 wk k+1
wk +
−1 vk k+1 vk
+ (x0 − xO0 )
X0−1 (x0
− xO0 ) ;
(40)
subject to Eqs. (1)–(4) driven by deterministic disturbance inputs wk , vk and initial state x0 ∈ Rn . It is worth noting that game (39) is not a standard H∞ -type problem since the cost C* itself is *-dependent. In spite of this fact, it is obvious that the solution technique of the game-type H∞ problem in [1] works for problem (39). This allows us to apply Theorem 6.8 of [1] to problem (39) as follows. Let *b be the in!mum of * ∈ (0; ∞) such that the dynamic game (39) has a saddle point solution. Lemma 3 (Basar and Bernhard [1]). Given * ¿ 0, if both Riccati recursions (31) and (32) have solutions satisfying conditions (34) and (35), then * ¿ *b and the optimal minimizing strategy uk in the dynamic game (39) is given by (38). In addition, if any of conditions (34) or (35) fails, then * 6 *b and game (39) does not have a 'nite value. Lemma 3 summarizes a well-known fact [21] that the dynamic game (39) is equivalent to the risk-sensitive problem (27) in the sense that they share the same optimal output feedback solution. Also, the conditions for this optimal feedback to exist are the same since they are formulated in terms of the same Riccati recursions.
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M.-G. Yoon et al. / Systems & Control Letters 52 (2004) 99 – 112
We are in a position to present our main result on the minimax optimal !ltering problem under consideration. For this, we need an explicit representation of the minimal cost W* of the risk-sensitive problem (27) since the minimization of V* in (25) makes use of an explicit form of W* , i.e., V* = *(W* + d). Lemma 4. With the optimal risk-sensitive control uk* in (38), the minimal cost W* de'ned in (27) is given by W* = 12 xO0 Q0 xO0 −
N
1 log|I − Rk (*−1 Lk Lk + Dk Dk )| |I − Qk+1 Gk |; 2
(41)
k=0
where, for k = 0; : : : ; N , Qk = (I − *−1 Pk Rk )−1 *−1 Pk ;
(42)
−1 −1 ˜ −1 −1 ˜ ˜ −1 Gk = (A˜ k Kk Ck ˜ −1 k+1 Ck + Ek Ck )[(I − * Rk Mk ) Rk + (Ck k+1 Ck ) ](Ak Kk Ck k+1 Ck + Ek Ck ) :
(43)
Proof. This is a straightforward generalization of the result in Section 2 of [2] to the case where the cost function has a cross-product term and the measurement is correlated with the process noise. Required algebraic manipulations are basic but too lengthy to be included here. Theorem 1. Suppose *b ¡ ∞. Let *∗ be de'ned from the following condition: V*∗ = inf V* ;
(44)
*¿*b
where the value V* of the unconstrained dynamic game (23) is given by N * xO0 Q0 xO0 + 2d − log|(I − *−1 Rk Lk Lk − Rk Dk Dk )(I − Qk+1 Gk )| : V* = 2
(45)
k=0
Then, 'lter (36)–(38) corresponding to * = *∗ is the minimax optimal 'lter uopt that attains the minimum of the worst-case error functional V (u). Conversely, if *b = +∞, then the minimax optimal 'lter uopt does not exist and V (u) = ∞ for all feasible 'lters. Proof. The !rst part of the theorem follows from Lemma 4. The second claim follows from the observation that the condition *b = +∞ implies that the corresponding problem (39) does not have a !nite upper value. Hence it follows that, for any * ¿ 0, V* = ∞. The proof of this fact follows along the lines similar to those proving the corresponding claim in [17]. The claim now follows from Lemma 1. Remark 2. Solving the minimization problem (44) only requires that inequalities (34) and (35) be veri!ed for each * ∈ (0; ∞) in order to !nd the value *b ¿ 0. Therefore those conditions should not be read as constraints of the minimization problem (44). Remark 3. Although the parameter d ¿ 0 that de!nes the class of admissible uncertainties in De!nition 1 does not appear in the !lter structure, irrespective of *, it has an e@ect on the resulting minimax optimal !lter. Indeed, being a parameter of V* as seen from (45), this parameter d ¿ 0 a@ects the value of *∗ determined in (44).
M.-G. Yoon et al. / Systems & Control Letters 52 (2004) 99 – 112
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Note that the recursion of Pk in (32) and condition (35) can be rewritten with a new matrix Pˆ k := *−1 Pk as follows: Pˆ k = Dk Dk + Ak Pˆ k+1 (I − k+1 Pˆ k+1 )−1 Ak ;
Pˆ N +1 = 0;
−1 0 Rk Dk Dk + Ak Pˆ k+1 I − k+1 Pˆ k+1 Ak ¡ 1:
(46) (47)
Hence, if the solution Pˆ k to recursion (46) satisfying (47) exists, then, for all * ¿ 0, recursion (32) has a solution given by Pk = *Pˆ k . Equivalently, if there exists any *p such that Pk exists, then Pk exists for all *. The above observation has an important implication. If either the condition Pˆ k ¿ 0 or (47) fails, then the minimax optimal !lter does not exist and V (u) = ∞. Furthermore, if recursion (46) has a positive de!nite solution satisfying (47), then the existence of a minimax optimal !lter depends entirely on recursions (31) and (34). This situation will be further investigated in the next section. 3.2. Remarks on the asymptotic behavior of V* In this section, we investigate the properties of V* as * → *b and * → +∞. This will give further insight into the structure of the value of the unconstrained game (23). First, consider recursion (31) and condition (34) in the case where * → 0. Suppose that for any arbitrarily small * ¿ 0 there exists a positive semide!nite solution to recursion (31) satisfying condition (34). Observe that (31) implies Rk+1 → ˜ k+1 ¿ 0 for all k as * → 0. Also, condition (34) forces Rk → 0 as * → 0, provided that the generic condition Lk Lk ¿ 0 holds. This contradiction means that there must exist a suTciently small *˜ ¿ 0 such that condition (34) fails for * 6 *. ˜ This implies that, under the assumption that Lk Lk ¿ 0, we have *b ¿ 0. Also, from (41) it follows that W* := log inf u J* → ∞ as * approaches the critical value that causes condition (34) to fail. Next, consider the case where * → +∞. Note that as * → +∞, the matrix {Rk } in (31) converges to the *-independent matrix Rˆ k ¿ 0 obtained from the following recursion: ˜ −1 −1 ˜ Rˆ k+1 := ˜ k+1 + A˜ k (Rˆ −1 k − Dk Dk + Ck k+1 Ck ) Ak ;
Rˆ 0 = X0 ;
0{Rˆ k Dk Dk } ¡ 1:
(48) (49)
Also as * → +∞, the matrix Gk converges to a *-independent matrix Gˆ k de!ned by −1 −1 ˜ ˆ ˜ −1 ˆ −1 + (Ck ˜ −1 Gˆ k := (A˜ k Kˆ k Ck ˜ −1 k+1 Ck + Ek Ck )[(Rk − Dk Dk ) k+1 Ck ) ](Ak K k Ck k+1 Ck + Ek Ck ) ;
(50)
˜ −1 −1 Kˆ k := (Rˆ −1 k − Dk Dk + Ck k+1 Ck ) :
(51)
Furthermore, assuming the existence of Pˆ k in (46) and (47), we have Qk = (I − *−1 Pk Rk )−1 *−1 Pk → Qˆ k := (I − Pˆ k Rˆ k )−1 Pˆ k ;
(52)
where Qˆ k is also *-independent. In particular, we have Q0 = Qˆ 0 since Pˆ k is *-independent and Rˆ 0 = X0 . Corollary 1. Suppose recursions (46) and (48) generate the matrices Pˆ k and Rˆ k satisfying conditions (47) and (49), respectively. Then, as * → +∞, it follows that V* ≈ *c1 + c2 ;
(53)
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M.-G. Yoon et al. / Systems & Control Letters 52 (2004) 99 – 112
where 1 c1 := 2
xO0 Qˆ 0 xO0 −
N
log (I − Rˆ k Dk Dk )(I − Qˆ k+1 Gˆ k ) + 2d
;
k=0 N
c2 :=
1 Trace[(I − Rˆ k Dk Dk )−1 Rˆ k Lk Lk ]: 2 k=0
Proof. From the equality I − *−1 Rk Mk = (I − Rk Dk Dk )[I − *−1 (I − Rk Dk Dk )−1 Rk Lk Lk ]
(54)
and the fact that lim*→∞ (−* log|I − *−1 7|) = Trace(7) for any symmetric matrix 7 = 7 ¿ 0, we have the constant term in (53). Consequently, the corollary follows from Lemma 4 with *-independent matrices Rˆ k , Qˆ k and Gˆ k . Corollary 2. Suppose that the uncertainty matrix Dk satis'es the condition 0(Rˆ k Dk Dk ) ¿ 1 for some k. Then, the minimax optimal 'lter does not exist and V (u) = ∞ for all feasible 'lters u. Proof. This proof is by establishing a contradiction. Suppose, that there exists a minimax optimal !lter, and hence from Theorem 1, it follows that *b ¡ ∞. Then, we have from the de!nition of the constant *b that for all * ¿ *b , the covariance matrix Rk de!ned by recursion (31) satis!es condition (34). In particular, since Rk → Rˆ k as * → ∞, then for a suTciently large *, (34) implies that 0(Rˆ k Dk Dk ) ≤ 1. This leads to the contradiction with the assumption 0(Rˆ k Dk Dk ) ¿ 1. Thus, the minimax optimal !lter does not exist. 4. Interpretations of minimax optimal lter In this section, we establish connections between the minimax optimal !lter constructed in Theorem 1 and !lters obtained using the H∞ !ltering and Kalman !ltering approaches. First, consider the structure of the minimax optimal !lter as *∗ → ∞. In this case, the minimax optimal !lter becomes ∗
uk* = Lk (I − Rˆ k Pˆ k )−1 1k ;
(55)
˜ −1 ˜ 1k+1 = A˜ k 1k + A˜ k Kˆ k Dk Dk 1k + A˜ k Kˆ k Ck ˜ −1 k+1 (yk+1 − Ck 1k ) + k+1 Ek k+1 yk+1 ;
(56)
˜ −1 −1 ˜ Rˆ k+1 = ˜ k+1 + A˜ k (Rˆ −1 k − Dk Dk + Ck k+1 Ck ) Ak ;
(57)
Rˆ 0 = X0 ;
where Kˆ k was de!ned in (51). This !lter can be referred to as a modi!ed Kalman !lter. Indeed, setting Dk ≡ 0 leads to Pˆ k = 0 in (46). Hence, the substitution Dk ≡ 0 and Lk ≡ I into (55) shows that !lter (55) becomes a standard Kalman !lter. This is an expected result since we know that the minimum stress or suboptimal H∞ !lter converges to the standard Kalman !lter (Dk ≡ 0) as the input-to-error H∞ gain of the plant-!lter interconnection approaches in!nity [21]. Next, let us consider the case where Dk → 0. From (46), it follows that Pˆ k → 0. Then, !lter (38) corresponding to some !xed value * has the following form: uk* = Lk 1k ;
(58)
M.-G. Yoon et al. / Systems & Control Letters 52 (2004) 99 – 112
109
1k+1 = A˜ k 1k + A˜ k K˜ k Ck ˜ k+1 (yk+1 − Ck 1k ) + ˜ k+1 Ek ˜ −1 k+1 yk+1 ;
(59)
−1 ˜ −1 −1 K˜ k := (R˜ −1 k − * Lk Lk + Ck k+1 Ck ) ;
(60)
−1 ˜ −1 −1 ˜ R˜ k+1 = ˜ k+1 + A˜ k (R˜ −1 k − * Lk Lk + Ck k+1 Ck ) Ak ;
R˜ 0 = X0 :
(61)
That is, !lter (38) becomes a standard H∞ !lter [13]. Furthermore, √ the minimax optimal !lter that corresponds to *=*∗ , becomes a suboptimal H∞ !lter whose attenuation level *∗ is determined by the design parameter d. This observation reveals that the minimax optimal !lter can be viewed as a variation of H∞ !lter, characterized by Dk and d. Combined with previous case, this observation illustrates that the !lter proposed in this paper provides a “trade-o=” between the robustness of H∞ !lter and the performance of Kalman !lter. Now consider the case where both Dk → 0 and d → 0. Then from (46), Pˆ k → 0 and consequently from (52), Qˆ k → 0. Thus, the coeTcient c1 de!ned in Corollary 1 converges to zero. Recall that this coeTcient determines the slope of V* at large *; see (53). From this fact and the monotonicity of V* with Dk = 0, d = 0 in (45), we conclude that *∗ → ∞, i.e. the minimax optimal !lter approaches the standard Kalman !lter. This situation is very plausible since no uncertainty is left in the system as Dk → 0 and d → 0 and consequently, h(QP) → 0. Thus, the above observation reveals that the minimax !lter will restore the performance of Kalman !lter in this case. As a counterpart of this fact, we are tempted to conjecture that the minimax optimal !lter will have the structure of modi!ed H∞ !lter when Dk is very large. Surprisingly, however, this is not the case. Once again the minimax !lter approaches the modi'ed Kalman !lter as it did with small Dk . This is because as was shown in the proof of Corollary 2, if Dk increases to its upper limit then we have *b → ∞ and inevitably *∗ → ∞. At a glance, this result seems to be contradictory with the fact that H∞ !lter has better robustness than Kalman !lter. In fact we simply have {Dk ; d}-modi!cations of these two !lters. Finally, we take a di@erent viewpoint on the problem of robust minimax !ltering. Recall that the robust minimax !ltering is associated with the game inf sup (C* − *U) = inf sup (Ce − *U + *Cu ); u x0 ;w;v
u x0 ;w;v
(62)
where Ce , Cu and U are de!ned in (19), (17) and (40), respectively. By setting Cu ≡ 0 in (62), we obtain a standard H ∞ !ltering problem [13] in which the attenuation level is given by the ratio Ce =U, i.e., the induced operator norm of L2 space. However, since in the robust minimax !ltering problem under consideration, Cu corresponds to the L2 norm of the uncertainty output, and therefore Cu = 0, the resulting minimax !lter has to minimize not only Ce but Cu as well. This is obvious since a large Cu may increases the value of game (62). Therefore in robust minimax !ltering, H ∞ -optimality of the !lter is sacri!ced due to the necessity to properly bound the uncertainty output. Furthermore, an attempt to minimize Cu only might cause a large Ce , resulting in a large H ∞ !ltering error. Thus, the minimax optimal !lter will aim to achieve a tradeo@ between these two terms.
5. An illustrative example We consider a numerical example to illustrate previous analysis. The system matrices are chosen to be, for all k, 0:6 0 Ak = ; Ck = [1 1]; Ek = 01×2 ; k+1 = k+1 = X0 = I2×2 : (63) 0 −0:6
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Table 1 Worst-case cost V*∗ (d; 9). 9
*b
*∗ (d = 0)
0 0.01 0.10 0.20 0.40 0.413 0.4132 0.413211
2.0 2.0 2.1 2.3 4.8 46.1 626.2 2520.0
∞ 82.6 9.6 5.9 6.6 47.4 641.8 2578.0
*∗ (d = 10−3 ) 150.0 72.7 9.6 5.9 6.6 47.4 641.8 2578.0
V*∗ (d = 0) 26.4 26.9 33.0 43.6 138.4 885.1 1.30E4 5.59E4
V*∗ (d = 10−3 ) 26.7 27.0 33.0 43.6 138.4 885.1 1.30E4 5.59E4
Also, the initial condition is x0 = [1 1] and the uncertainty matrix Dk and output matrix Lk are given by Lk = I2×2 ;
Dk (9) = 9 · I2×2 ;
(64)
where 9 is a parameter. Assume N = 20. Numerical computations reveal that the upper bound on 9, for which one can !nd * such that conditions (34) and (35) are satis!ed, is 9max ≈ 0:413211, as summarized in Table 1 with two choices of d ∈ {0; 10−3 }. Note that the condition 0(Rˆ k Dk Dk ) ¡ 1 in Corollary 2 provides a conservative bound 9 ¡ 0:6274 for the existence of the minimax !lter. Table 1 shows that, as discussed before, if d = 0 and 9 ≈ 0 then the minimax optimal !lter e@ectively acts as a Kalman !lter (*∗ = ∞). As the uncertainty parameter 9 increases, the gap between *∗ and *b reduces thus causing the minimax optimal !lter to perform like the modi!ed H∞ !lter. Furthermore, as 9 → 9max , we observe that both *∗ and *b approach ∞. Therefore, excessively large uncertainty will cause both the minimax optimal !lter and modi!ed H∞ !lter to perform in a manner similar to that of the modi!ed Kalman !lter (55). In order to compare the robustness property of the minimax optimal !lter with that of the standard Kalman !lter, a set of parametric perturbations A: =
0:6
0
0
−0:6
+ : · I2×2 ;
: ∈ [ − 0:4; 0:4]
(65)
was considered. We considered two di@erent !lters, the Kalman !lter designed for the nominal system and the minimax optimal !lter designed with 9 = 0:4 and d = 10−3 . For each !lter, we computed the error variance N P ;(:) := E { k=0 Lk x˜k − uk 2 =2}, where x˜k was generated using the state equation with A = A: . For every : among 300 points evenly spaced over the interval [ − 0:4; 0:4], we performed 10,000 numerical simulations and obtained the error variance graphs shown in Fig. 1. This !gure shows that the minimax optimal !lter is less sensitive to perturbations in the system, as compared to the Kalman !lter. Although the Kalman !lter has a superior performance when the perturbations of the nominal system are small |:| 6 0:3, its performance substantially deteriorates when perturbations are large, 0:3 6 |:| 6 0:4. In contrast, performance of the minimax optimal !lter remains essentially the same across entire range of uncertain parameter :. Finally, note that the experimental value 26:37 indicated in Fig. 1, which is the average of all ;(:), |:| 6 0:01, corresponds to the cost V*∗ = 26:4 predicted in the case 9 = d = 0; see Table 1.
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Fig. 1. Error variances ;(:) of perturbed systems.
6. Conclusion We proposed a !nite horizon minimax !lter for a discrete-time systems with uncertainties in the underlying probability measure. We found the optimal minimax !lter from a known solution to a risk-sensitive control problem. The properties and structure of the minimax !lter have been investigated and this analysis revealed that the minimax !lter is a variation on both the Kalman !lter and the H ∞ !lter. To illustrate the computation of the !lter and its properties, a numerical example was presented. References [1] T. BasUar, P. Bernhard, H ∞ -Optimal Control and Related Minimax Design Problems, 2nd Edition, BirkhVauser, Boston, 1995. [2] I.B. Collings, M.R. James, J.B. Moore, An information-state approach to risk-sensitive tracking problems, J. Math. Systems, Estimat. Control 6 (3) (1996) 1–24. [3] C.E. de Souza, U. Shaked, Robust H2 !ltering for uncertain systems with measurable inputs, IEEE Trans. Signal Process. 47 (1999) 2286–2292. [4] P. Dupuis, R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley, New York, 1997. [5] R.J. Elliott, L. Aggoun, J.B. Moore, Hidden Markov Models; Estimation and Control, 2nd Edition, Application of Mathematics, Vol. 29, Springer, Berlin, 1995. [6] S.O.R. Moheimani, A.V. Savkin, I.R. Petersen, Robust !ltering, prediction, smoothing and observability of uncertain systems, IEEE Trans. Circuits Systems Part 1, Fund. Theory Appl. 45 (4) (1998) 446–457. [7] I.R. Petersen, M.R. James, P. Dupuis, Minimax optimal control of stochastic uncertain systems with relative entrophy constrains, IEEE Trans. Automat. Control 45 (3) (2000) 398–412. [8] I.R. Petersen, D.C. McFarlane, Optimal guaranteed cost control and !ltering for uncertain linear systems, IEEE Trans. Automat. Control 39 (1993) 1971–1977. [9] I.R. Petersen, A.V. Savkin, Robust Kalman Filtering for Signals and Systems with Large Uncertainties, BirkhVauser, Boston, 1999. [10] I.R. Petersen, V.A. Ugrinovskii, A.V. Savkin, Robust Control Design Using H ∞ Methods, Springer, London, 2000. [11] P.D. Pra, L. Meneghini, W.J. Ruggaldier, Connections between stochastic control and dynamic games, Math. Control, Signals, Systems 9 (1996) 303–326. [12] A.V. Savkin, I.R. Petersen, A connection between H ∞ control and the absolute stabilizability of uncertain systems, Systems and Control Lett. 23 (3) (1994) 197–203. [13] U. Shaked, Y. Theodor, H∞ -optimal estimation: A tutorial, in: Proceedings of The 31th IEEE Conference on Decision and Control, Tucson, AR, 1992, pp. 2278–2286. [14] Y. Theodor, U. Shaked, Robust discrete-time minimum-variance !ltering, IEEE Trans. Signal Process. 44 (1996) 181–189. [15] Y. Theodor, U. Shaked, C.E. de Souza, A game theory approach to robust discrete-time H∞ -estimation, IEEE Trans. Signal Process. 42 (1994) 1486–1495. [16] V.A. Ugrinovskii, I.R. Petersen, Guaranteed cost LQG !ltering for stochastic discrete time uncertainty systems via risk-sensitive control, in: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AR, 1999, pp. 564 –569.
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[17] V.A. Ugrinovskii, I.R. Petersen, Finite horizon minimax optimal control of stochastic partially observed time varying uncertain system, Math. Control, Signals, Systems 12 (1999) 1–23. [18] V.A. Ugrinovskii, I.R. Petersen, Minimax LQG control of stochastic partially observed uncertain systems, SIAM J. Control Optim. 40 (4) (2001) 1189–1226. [19] V.A. Ugrinovskii, I.R. Petersen, Robust output feedback stabilization via risk-sensitive control, Automatica 38 (6) (2002) 945–955. [20] P. Whittle, Risk-sensitive Linear/quadratic/Gaussian control, Adv. Appl. Prob. 13 (1981) 764–777. [21] P. Whittle, Risk-sensitive Optimal Control, Wiley, Chichester, 1990. [22] V.A. Yakubovich, Nonconvex optimization problem: the in!nite-horizon linear-quadratic control problem with quadratic constraints, Systems Control Lett. 19 (1) (1992) 13–22.
Systems & Control Letters 52 (2004) 99 – 112
www.elsevier.com/locate/sysconle
Robust !nite horizon minimax !ltering for discrete-time stochastic uncertain systems Myung-Gon Yoona;∗ , Valery A. Ugrinovskiib , Ian R. Petersenb a School
b School
of Electrical Engineering and Telecommunications, The University of New South Wales, NSW 2052, Australia of Information Technology and Electrical Engineering, Australian Defence Force Academy, Canberra, ACT 2600, Australia Received 19 June 2002; received in revised form 4 November 2003; accepted 14 November 2003
Abstract We study a !nite-horizon robust minimax !ltering problem for time-varying discrete-time stochastic uncertain systems. The uncertainty in the system is characterized by a set of probability measures under which the stochastic noises, driving the system, are de!ned. The optimal minimax !lter has been found by applying techniques of risk-sensitive LQG control. The structure and properties of resulting !lter are analyzed and compared to H∞ and Kalman !lters. c 2003 Elsevier B.V. All rights reserved. Keywords: Robust !ltering; Kalman !lter; H∞ !lter; Risk-sensitive control
1. Introduction From the viewpoint of robustness, the standard Kalman !lter may su@er from poor performance. Indeed, the standard Kalman !lter does not incorporate information on the uncertainty present in a systematic way. This fact has been a major motivation for current research on a robust version of Kalman !lter. For example, Petersen, McFarlane, Theodor, Shaked and de Souza [3,8,14] studied a guaranteed cost 'lter that leads to a bound on the estimation error variances when applied to a class of systems having norm-bounded uncertainty. Also in [9], the H∞ !ltering method was extended to deal with systems with deterministic noise inputs and uncertainty satisfying integral quadratic constraints (IQC). Similarly for a class of uncertain systems with bounded parameter perturbations, a robust version of the H∞ estimation problem was studied in [15]. A standard assumption used in Kalman !lter design requires that the stochastic noise driving the nominal system is white and Gaussian. It is plausible, however, that in a practical system the noise can be perturbed by some uncertainty and as a result it is not white or Gaussian anymore. Hence, the performance of Kalman !lter may be compromised due to the lack of robustness.
∗
This work was supported by The Australian Research Council. Corresponding author. Tel.: +61-2-9385-6576; fax: +61-2-9385-5993/5997. E-mail address: [email protected] (M.-G. Yoon).
c 2003 Elsevier B.V. All rights reserved. 0167-6911/$ - see front matter doi:10.1016/j.sysconle.2003.11.004
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Perturbations of the reference noise can be elegantly modeled by introducing a new probability measure under which the noise has a di@erent probability distribution. This approach to uncertainty modeling, introduced in [7,10,16,17] and adopted in this paper, serves as an alternative to a commonly used uncertainty modeling that considers sets of unknown parameters and external disturbances. An important feature of this approach is that a set of probability measures acting on a given nominal system produces a collection of perturbed systems. A !lter is then designed that ensures an acceptable performance for all systems of the considered class of perturbed systems. After introducing in Section 2 the notion of uncertain systems modeled using probability measure perturbations, we formulate a robust version of the Kalman !ltering problem in Section 3. This formulation is based on a widely used worst-case design interpretation of robust design. This motivates our use of the name ‘minimax optimal !lter’ for our robust version of the Kalman !lter. Also in Section 3, a complete solution to the minimax optimal !ltering problem is given, which is derived from existing results on risk-sensitive control and properties of the relative entropy functional. In Section 4, the structure and properties of the minimax optimal !lter are analyzed and compared to those of the Kalman and H∞ !lters. Finally, a numerical example is presented in Section 5 to illustrate these properties. 2. Stochastic uncertain system 2.1. Nominal system We follow the problem formulation of [7,16,17,19]. Consider the following time-varying discrete-time stochastic system on a !nite-time interval: xk+1 = Ak xk + wk+1 ;
(1)
zk = Dk xk ;
(2)
k = Lk xk ;
(3)
yk+1 = Ck xk + Ek wk+1 + vk+1 ;
k = 0; : : : ; N;
(4)
where xk ∈ Rn is the state, zk ∈ Rm denotes the uncertainty output, k ∈ Rp is the output to be estimated, and yk ∈ Rq is the measurement. With a !xed probability measure P de!ned on the noise space, := {(x0 ; w1 ; v1 ; : : : ; : : : ; wN ; vN )};
(5)
the process noise wk ∈ Rn and the measurement noise vk ∈ Rq are independent white Gaussian processes with zero mean and covariances k ¿ 0 and k ¿ 0, respectively. The initial state x0 ∈ Rn is a Gaussian random vector with mean xO0 and covariance X0 ¿ 0. We assume {x0 ; wk ; vk } are mutually independent and have the Gaussian joint distribution: d x0 exp − 12 (x0 − xO0 ) X0−1 (x0 − xO0 ) P(d x0 × dw1 × · · · × dvN ) = n (2) |X0 | N −1 1 dwk+1 dvk+1 −1 −1 × exp − 2 (wk+1 k+1 wk+1 + vk+1 k+1 vk+1 ) ; n+q | (2) || | k+1 k+1 k=0 (6) where | · | denotes the determinant of a matrix.
M.-G. Yoon et al. / Systems & Control Letters 52 (2004) 99 – 112
101
2.2. Uncertain systems We consider the class of uncertain dynamics that can be generated using Eqs. (1)–(4) with a di@erent probability measure Q. We will make use of a dependence structure for Q given by Q(d x0 × dw1 × dv1 × · · · × dwN × dvN ) = Q(d x0 )Q(dw1 |x0 )Q(dv1 |x0 )Q(dw1 |x0 ; w1 ; v1 )Q(dv1 |x0 ; w1 ; v1 ) · · · Q(dwN |x0 ; w1 ; v1 ; : : : ; wN −1 ; vN −1 )Q(dvN |x0 ; w1 ; v1 ; : : : ; wN −1 ; vN −1 )
(7)
which can be regarded as a perturbation of the nominal joint probability measure (6). In order to characterize the class of admissible perturbations, we need a measure of discrepancy between the probability measures P and Q. For this we choose the relative entropy de!ned by dQ E Q log dQ if QP and log ∈ L1 (dQ); dP dP h(QP) := (8) +∞ otherwise: Here QP denotes the fact that the perturbed measure Q is absolutely continuous with respect to the reference measure P. For a detailed study on relative entropy, see [4]. Let P denote the set of probability measures Q such that h(QP) ¡ ∞. Let us consider a special case of Q ∈ P to illustrate the notion of uncertain systems generated by measure perturbations. De!ne Fk to be the -algebra generated by the sequence {x0 ; y0 ; : : : ; xk ; yk }, assuming y0 = 0, and completed by including all corresponding sets of probability zero. For a given Q ∈ P, we assume that there exist Fk -adapted random sequences { k } and {!k } such that the restriction of the measure Q on the -algebra F‘ has the following Radon–Nikodym derivative:
dQ (x0 ; w1 ; v1 ; : : : ; wN ; vN )
:= exp − 12 # X0−1 # + # X0−1 (x0 − xO0 ) dP F‘ ‘ × exp − 12 k=0
−1 k k+1
k
−1 + wk+1 k+1
k
−1 −1 !k + vk+1 k+1 !k + − 12 !k k+1
(9)
with # ∈ Rn , 0 6 ‘ 6 N − 1. Then, from (6) it follows that the joint distribution of x0 , wk and vk under the measure Q is also Gaussian and is given by Q(d x0 × · · · × dvN )|F‘ =
d x0 (2)n |X0 |
exp − 12 (x0 − xO0 − #) X0−1 (x0 − xO0 − #)
‘ dwk+1 dvk+1 × exp − 12 (wk+1 − n+q (2) |k+1 k+1 | k=0
k)
−1 k+1 (wk+1 −
k)
−1 − 12 (vk+1 − !k ) k+1 (vk+1 − !k ) :
(10)
Hence, the initial state x0 and noise variables {wk ; vk } can be represented by x0 = # + x0Q ;
(11)
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wk =
k−1
+ wkQ ;
(12)
vk = !k−1 + vkQ ;
(13)
where # ∈ Rn , wkQ and vkQ are Gaussian-independent sequences with zero means and covariances k and k under Q, respectively. This results in the following representation of system (1) and (4) under the perturbed probability measure Q: Q xk+1 = Ak xkQ +
Q + wk+1 ;
k
Q yk+1 = Ck xkQ + Ek
k
(14)
Q Q + !k + Ek wk+1 + vk+1 ;
(15)
x0Q
where the initial state is Gaussian with mean xO0 + # and covariance X0 . See [5] for general results on this kind of measure change. Moreover, from the chain rule [4], we can explicitly obtain the relative entropy h(QP) as follows: N −1 1 Q −1 −1 1 −1 h(QP) = 2 # X0 # + (16) E k k+1 k + !k k+1 !k : 2 k=0
Eq. (16) allows us to quantify the energy brought into the system by the perturbation associated with the perturbed measure Q ∈ P de!ned by (10) or, equivalently, the input signals { k ; !k }. This energy expressed as the quadratic sum of signals { k ; !k } can be evaluated by the value of relative entropy h(QP). This motivates us to use the relative entropy functional as a measure of the size of uncertainty in the system. This idea serves as a natural extension of that behind the sum quadratic constraints (SQCs) [6] and integral quadratic constraints (IQCs) [12,22]. The de!nition given below was introduced in [7]; also, see [16–19] where extensions of this uncertainty description are introduced in the context of continuous-time and in!nite-horizon problems. Denition 1. Given a constant d ¿ 0, a probability measure Q ∈ P is said to de!ne an admissible uncertainty if Q satis!es N 1 h(QP) 6 E Q (Cu ) + d where Cu = zk 2 (17) 2 k=1
and {zk } is the uncertainty output (2). The set of all admissible uncertainty measures in P will be denoted by &. It should be noted that the admissible uncertainty of De!nition 1 is more general than that de!ned by the SQC: N −1 1 Q −1 −1 Q 1 −1 # X # + E k k+1 k + !k k+1 !k 6 E (Cu ) + d 0 2 2 k=0
which is constructed by replacing the relative entropy functional on the right-hand side of (17) with expression (16) involving the inputs k , !k . This is because not every measure Q ∈ & admits a representation of the form (10); e.g., see [11]. 3. Minimax optimal ltering problem In this paper, we will focus on the class of !lters u = {u0 ; : : : ; uN } having a representation uk = fk (x0 ; u0 ; : : : ; uk−1 ; y1 ; : : : ; yk );
k = 1; : : : ; N
(18)
M.-G. Yoon et al. / Systems & Control Letters 52 (2004) 99 – 112
103
for some nonanticipative function fk . Filters of this form will be termed as feasible 'lters. From the above de!nition, the output variable uk of a feasible !lter is Yk -measurable; here, Yk is a -algebra generated by the measurements {x0 ; y0 ; : : : ; yk }. Thus, feasible !lters are to produce their output using the information in the measured output y0 ; : : : ; yk and initial condition x0 of the system. Given a feasible !lter u, de!ne a !ltering error associated with particular noise and measurement samples as follows: N
N
k=0
k=0
1 1 Ce := k − uk 2 = Lk xk − uk 2 : 2 2
(19)
Then, the expected error achieved when this !lter is used to estimate the output k of the uncertain system (1)–(4) with the uncertainty Q ∈ P is given by E Q (Ce ). According to the above de!nitions, the worst-case error over the set of admissible perturbations Q ∈ & can be expressed as V (u) := sup E Q (Ce ):
(20)
Q∈&
Denition 2 (Robust minimax optimal !ltering problem). Find the feasible !lter uopt minimizing the worstcase error, i.e., V (uopt ) = inf V (u) = inf sup E Q (Ce ) u
u Q∈&
(21)
From De!nition 2, one can see that the minimax optimal !ltering problem de!ned above is closely related to the general minimax optimal output feedback control problem considered in Ref. [7]. This problem was formulated in [7] for a general case where the underlying uncertain system and control cost functional were nonlinear. Also, a specialization of the general theory to a class of minimax LQG control problems was presented. However, the results presented in [7] did not directly allow for cost functionals of the form (19). In this paper, we extend the theory of [7] to solve the minimax !ltering problem of De!nition 2. Our aim is to obtain a tractable solution to the minimax optimal !ltering problem similar to that obtained in [7] for the minimax optimal LQG control. This will lead to a tractable minimax optimal !lter design procedure based on a pair of matrix Riccati recursions. This result is presented in this section. The main technical di@erence that prompted us to develop an extension of the results obtained in [7] is due to the presence of the cross-term in the !ltering error functional (19). Our second aim will be to establish connections between the minimax optimal !lter and !lters obtained for the same system using H∞ !ltering and Kalman !ltering approaches. These results will be presented in the next section. We now proceed with the extension of the results of [7] to the minimax !ltering problem posed in De!nition 2. In [7], it is shown that the constrained minimax problem (21) can be indirectly solved via an unconstrained version of the problem as follows: V* := inf sup E Q {Ce − *[h(QP) − Cu − d]}; u Q∈P
= *{inf sup E Q *−1 C* − h(QP) + d}; u Q∈P
(22) (23)
where C* is de!ned by N
C* := Ce + *Cu =
1 [xk (Lk Lk + *Dk Dk )xk − 2xk Lk uk + uk uk ]: 2 k=0
(24)
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Lemma 1 (Petersen et al. [7, Theorem 2.1]). De'ne + := {* ¿ 0; V* ¡ ∞}. Then V (uopt ) is 'nite if and only if + = ∅. Moreover if + = ∅, then we have V (uopt ) = inf V* ;
(25)
*∈+
:= V*∗ ;
(26)
when the in'mum in (25) is achieved. This means that the minimax optimal !lter uopt is given by the minimizer of the minimax game (23) corresponding to * = *∗ . The solution of the minimax game (23) is based on the fact that for each * ¿ 0, the minimax problem (23) can be converted into a risk-sensitive control problem associated with the nominal system (1), thus leaving us with a collection of *-parameterized risk-sensitive control problems. Such conversion exploits a result from the theory of large deviations referred to as the duality relation between the free energy and relative entropy [4,11]. Using this duality relation, we obtain inf sup E Q [*−1 C* − h(QP)] = inf log E P (exp*−1 C* ) := W* : u Q∈P
(27)
u
The risk-sensitive control problem on the right-hand side of (27) is a so-called partial information linearquadratic-exponential Gaussian (LEQG) control problem which is known to admit a solution in a closed form. Remark 1. Since the quadratic cost C* itself depends on *, the parameter * cannot be interpreted as the risk-sensitive parameter, cf. [2,20]. 3.1. Design of a minimax optimal 'lter In this section, we present a complete solution to the minimax optimal !ltering problem. This solution is derived from a known solution to the risk-sensitive problem developed in [21]. It also involves a generalization of the results of [12]. In order to make use of the solution to the risk-sensitive control problem presented in [21], we recast the !ltering problem in De!nition 2 into a partial information LEQG control problem associated with the nominal system xk+1 = Ak xk + wk+1 ;
(28)
yk+1 = Ck xk + Ek wk+1 + vk+1
(29)
and an exponential cost N 1 P exp [xk (Lk Lk + *Dk Dk )xk − 2xk Lk uk + uk uk ] ; J* := E 2*
* ¿ 0:
(30)
k=0
Consider matrices Rk ¿ 0; Pk ¿ 0, de!ned by the following Riccati equations: −1 ˜ −1 −1 ˜ Rk+1 = ˜ k+1 + A˜ k (R−1 k − * Lk Lk − Dk Dk + Ck k+1 Ck ) Ak ;
Pk = *Dk Dk + Ak Pk+1 (I − *−1 k+1 Pk+1 )−1 Ak ; −1 −1 ˜ −1 k+1 := k+1 + Ek k+1 Ek ;
R0 = X0 ;
PN +1 = 0;
˜ k+1 := k+1 + Ek k+1 Ek ;
−1 A˜ k := Ak − ˜ k+1 Ek k+1 Ck ;
(31) (32) (33)
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and satisfying the conditions 0{Rk (Lk Lk + *Dk Dk )} ¡ *;
(34)
−1 Ak ¡ *; 0 Rk *Dk Dk + Ak Pk+1 I − *−1 k+1 Pk+1
(35)
where 0(·) denotes the spectral radius of a matrix. Also, consider a sequence {1k } de!ned by −1 1k+1 = A˜ k 1k + A˜ k Kk [Ck ˜ −1 k+1 (yk+1 − Ck 1k ) + Dk Dk 1k − * Lk (uk − Lk 1k )] −1 +˜ k+1 Ek k+1 yk+1 ;
10 = xO0 ;
(36)
Kk−1 := Rk − *−1 Lk Lk − Dk Dk + Ck ˜ −1 k+1 Ck :
(37)
Lemma 2. Suppose there exist the matrices Rk ¿ 0; Pk ¿ 0, de'ned by (31), (32) and satisfying conditions (34), (35) for some * ¿ 0. Then the optimal risk-sensitive controller solving problem (27) is given by the output feedback control law: uk* = Lk (I − *−1 Rk Pk )−1 1k :
(38)
Proof. The claim easily follows from Theorems 7.3.2, 8.2.1 and 8.3.1 of [21] by substituting the cost (30), system matrices (28)–(29) and noise property described in Section 2.1. This result shows that the optimal !lter minimizing the cost log E P (exp *−1 C* ) is simply a minimum stress estimator in the risk-sensitive control problem (28), (30); see [21, Theorem 8.3.1]. Moreover, !lter (38) is identical to the solution of the optimal H ∞ !ltering problem with the same cost function C* , e.g., see, [21]. Consider the following dynamic game problem: inf sup [C* (x; u) − * U(x0 ; w; v)];
(39)
u x0 ;w;v
where 1 U(x0 ; w; v) = 2
N −1 k=0
−1 wk k+1
wk +
−1 vk k+1 vk
+ (x0 − xO0 )
X0−1 (x0
− xO0 ) ;
(40)
subject to Eqs. (1)–(4) driven by deterministic disturbance inputs wk , vk and initial state x0 ∈ Rn . It is worth noting that game (39) is not a standard H∞ -type problem since the cost C* itself is *-dependent. In spite of this fact, it is obvious that the solution technique of the game-type H∞ problem in [1] works for problem (39). This allows us to apply Theorem 6.8 of [1] to problem (39) as follows. Let *b be the in!mum of * ∈ (0; ∞) such that the dynamic game (39) has a saddle point solution. Lemma 3 (Basar and Bernhard [1]). Given * ¿ 0, if both Riccati recursions (31) and (32) have solutions satisfying conditions (34) and (35), then * ¿ *b and the optimal minimizing strategy uk in the dynamic game (39) is given by (38). In addition, if any of conditions (34) or (35) fails, then * 6 *b and game (39) does not have a 'nite value. Lemma 3 summarizes a well-known fact [21] that the dynamic game (39) is equivalent to the risk-sensitive problem (27) in the sense that they share the same optimal output feedback solution. Also, the conditions for this optimal feedback to exist are the same since they are formulated in terms of the same Riccati recursions.
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We are in a position to present our main result on the minimax optimal !ltering problem under consideration. For this, we need an explicit representation of the minimal cost W* of the risk-sensitive problem (27) since the minimization of V* in (25) makes use of an explicit form of W* , i.e., V* = *(W* + d). Lemma 4. With the optimal risk-sensitive control uk* in (38), the minimal cost W* de'ned in (27) is given by W* = 12 xO0 Q0 xO0 −
N
1 log|I − Rk (*−1 Lk Lk + Dk Dk )| |I − Qk+1 Gk |; 2
(41)
k=0
where, for k = 0; : : : ; N , Qk = (I − *−1 Pk Rk )−1 *−1 Pk ;
(42)
−1 −1 ˜ −1 −1 ˜ ˜ −1 Gk = (A˜ k Kk Ck ˜ −1 k+1 Ck + Ek Ck )[(I − * Rk Mk ) Rk + (Ck k+1 Ck ) ](Ak Kk Ck k+1 Ck + Ek Ck ) :
(43)
Proof. This is a straightforward generalization of the result in Section 2 of [2] to the case where the cost function has a cross-product term and the measurement is correlated with the process noise. Required algebraic manipulations are basic but too lengthy to be included here. Theorem 1. Suppose *b ¡ ∞. Let *∗ be de'ned from the following condition: V*∗ = inf V* ;
(44)
*¿*b
where the value V* of the unconstrained dynamic game (23) is given by N * xO0 Q0 xO0 + 2d − log|(I − *−1 Rk Lk Lk − Rk Dk Dk )(I − Qk+1 Gk )| : V* = 2
(45)
k=0
Then, 'lter (36)–(38) corresponding to * = *∗ is the minimax optimal 'lter uopt that attains the minimum of the worst-case error functional V (u). Conversely, if *b = +∞, then the minimax optimal 'lter uopt does not exist and V (u) = ∞ for all feasible 'lters. Proof. The !rst part of the theorem follows from Lemma 4. The second claim follows from the observation that the condition *b = +∞ implies that the corresponding problem (39) does not have a !nite upper value. Hence it follows that, for any * ¿ 0, V* = ∞. The proof of this fact follows along the lines similar to those proving the corresponding claim in [17]. The claim now follows from Lemma 1. Remark 2. Solving the minimization problem (44) only requires that inequalities (34) and (35) be veri!ed for each * ∈ (0; ∞) in order to !nd the value *b ¿ 0. Therefore those conditions should not be read as constraints of the minimization problem (44). Remark 3. Although the parameter d ¿ 0 that de!nes the class of admissible uncertainties in De!nition 1 does not appear in the !lter structure, irrespective of *, it has an e@ect on the resulting minimax optimal !lter. Indeed, being a parameter of V* as seen from (45), this parameter d ¿ 0 a@ects the value of *∗ determined in (44).
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Note that the recursion of Pk in (32) and condition (35) can be rewritten with a new matrix Pˆ k := *−1 Pk as follows: Pˆ k = Dk Dk + Ak Pˆ k+1 (I − k+1 Pˆ k+1 )−1 Ak ;
Pˆ N +1 = 0;
−1 0 Rk Dk Dk + Ak Pˆ k+1 I − k+1 Pˆ k+1 Ak ¡ 1:
(46) (47)
Hence, if the solution Pˆ k to recursion (46) satisfying (47) exists, then, for all * ¿ 0, recursion (32) has a solution given by Pk = *Pˆ k . Equivalently, if there exists any *p such that Pk exists, then Pk exists for all *. The above observation has an important implication. If either the condition Pˆ k ¿ 0 or (47) fails, then the minimax optimal !lter does not exist and V (u) = ∞. Furthermore, if recursion (46) has a positive de!nite solution satisfying (47), then the existence of a minimax optimal !lter depends entirely on recursions (31) and (34). This situation will be further investigated in the next section. 3.2. Remarks on the asymptotic behavior of V* In this section, we investigate the properties of V* as * → *b and * → +∞. This will give further insight into the structure of the value of the unconstrained game (23). First, consider recursion (31) and condition (34) in the case where * → 0. Suppose that for any arbitrarily small * ¿ 0 there exists a positive semide!nite solution to recursion (31) satisfying condition (34). Observe that (31) implies Rk+1 → ˜ k+1 ¿ 0 for all k as * → 0. Also, condition (34) forces Rk → 0 as * → 0, provided that the generic condition Lk Lk ¿ 0 holds. This contradiction means that there must exist a suTciently small *˜ ¿ 0 such that condition (34) fails for * 6 *. ˜ This implies that, under the assumption that Lk Lk ¿ 0, we have *b ¿ 0. Also, from (41) it follows that W* := log inf u J* → ∞ as * approaches the critical value that causes condition (34) to fail. Next, consider the case where * → +∞. Note that as * → +∞, the matrix {Rk } in (31) converges to the *-independent matrix Rˆ k ¿ 0 obtained from the following recursion: ˜ −1 −1 ˜ Rˆ k+1 := ˜ k+1 + A˜ k (Rˆ −1 k − Dk Dk + Ck k+1 Ck ) Ak ;
Rˆ 0 = X0 ;
0{Rˆ k Dk Dk } ¡ 1:
(48) (49)
Also as * → +∞, the matrix Gk converges to a *-independent matrix Gˆ k de!ned by −1 −1 ˜ ˆ ˜ −1 ˆ −1 + (Ck ˜ −1 Gˆ k := (A˜ k Kˆ k Ck ˜ −1 k+1 Ck + Ek Ck )[(Rk − Dk Dk ) k+1 Ck ) ](Ak K k Ck k+1 Ck + Ek Ck ) ;
(50)
˜ −1 −1 Kˆ k := (Rˆ −1 k − Dk Dk + Ck k+1 Ck ) :
(51)
Furthermore, assuming the existence of Pˆ k in (46) and (47), we have Qk = (I − *−1 Pk Rk )−1 *−1 Pk → Qˆ k := (I − Pˆ k Rˆ k )−1 Pˆ k ;
(52)
where Qˆ k is also *-independent. In particular, we have Q0 = Qˆ 0 since Pˆ k is *-independent and Rˆ 0 = X0 . Corollary 1. Suppose recursions (46) and (48) generate the matrices Pˆ k and Rˆ k satisfying conditions (47) and (49), respectively. Then, as * → +∞, it follows that V* ≈ *c1 + c2 ;
(53)
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where 1 c1 := 2
xO0 Qˆ 0 xO0 −
N
log (I − Rˆ k Dk Dk )(I − Qˆ k+1 Gˆ k ) + 2d
;
k=0 N
c2 :=
1 Trace[(I − Rˆ k Dk Dk )−1 Rˆ k Lk Lk ]: 2 k=0
Proof. From the equality I − *−1 Rk Mk = (I − Rk Dk Dk )[I − *−1 (I − Rk Dk Dk )−1 Rk Lk Lk ]
(54)
and the fact that lim*→∞ (−* log|I − *−1 7|) = Trace(7) for any symmetric matrix 7 = 7 ¿ 0, we have the constant term in (53). Consequently, the corollary follows from Lemma 4 with *-independent matrices Rˆ k , Qˆ k and Gˆ k . Corollary 2. Suppose that the uncertainty matrix Dk satis'es the condition 0(Rˆ k Dk Dk ) ¿ 1 for some k. Then, the minimax optimal 'lter does not exist and V (u) = ∞ for all feasible 'lters u. Proof. This proof is by establishing a contradiction. Suppose, that there exists a minimax optimal !lter, and hence from Theorem 1, it follows that *b ¡ ∞. Then, we have from the de!nition of the constant *b that for all * ¿ *b , the covariance matrix Rk de!ned by recursion (31) satis!es condition (34). In particular, since Rk → Rˆ k as * → ∞, then for a suTciently large *, (34) implies that 0(Rˆ k Dk Dk ) ≤ 1. This leads to the contradiction with the assumption 0(Rˆ k Dk Dk ) ¿ 1. Thus, the minimax optimal !lter does not exist. 4. Interpretations of minimax optimal lter In this section, we establish connections between the minimax optimal !lter constructed in Theorem 1 and !lters obtained using the H∞ !ltering and Kalman !ltering approaches. First, consider the structure of the minimax optimal !lter as *∗ → ∞. In this case, the minimax optimal !lter becomes ∗
uk* = Lk (I − Rˆ k Pˆ k )−1 1k ;
(55)
˜ −1 ˜ 1k+1 = A˜ k 1k + A˜ k Kˆ k Dk Dk 1k + A˜ k Kˆ k Ck ˜ −1 k+1 (yk+1 − Ck 1k ) + k+1 Ek k+1 yk+1 ;
(56)
˜ −1 −1 ˜ Rˆ k+1 = ˜ k+1 + A˜ k (Rˆ −1 k − Dk Dk + Ck k+1 Ck ) Ak ;
(57)
Rˆ 0 = X0 ;
where Kˆ k was de!ned in (51). This !lter can be referred to as a modi!ed Kalman !lter. Indeed, setting Dk ≡ 0 leads to Pˆ k = 0 in (46). Hence, the substitution Dk ≡ 0 and Lk ≡ I into (55) shows that !lter (55) becomes a standard Kalman !lter. This is an expected result since we know that the minimum stress or suboptimal H∞ !lter converges to the standard Kalman !lter (Dk ≡ 0) as the input-to-error H∞ gain of the plant-!lter interconnection approaches in!nity [21]. Next, let us consider the case where Dk → 0. From (46), it follows that Pˆ k → 0. Then, !lter (38) corresponding to some !xed value * has the following form: uk* = Lk 1k ;
(58)
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109
1k+1 = A˜ k 1k + A˜ k K˜ k Ck ˜ k+1 (yk+1 − Ck 1k ) + ˜ k+1 Ek ˜ −1 k+1 yk+1 ;
(59)
−1 ˜ −1 −1 K˜ k := (R˜ −1 k − * Lk Lk + Ck k+1 Ck ) ;
(60)
−1 ˜ −1 −1 ˜ R˜ k+1 = ˜ k+1 + A˜ k (R˜ −1 k − * Lk Lk + Ck k+1 Ck ) Ak ;
R˜ 0 = X0 :
(61)
That is, !lter (38) becomes a standard H∞ !lter [13]. Furthermore, √ the minimax optimal !lter that corresponds to *=*∗ , becomes a suboptimal H∞ !lter whose attenuation level *∗ is determined by the design parameter d. This observation reveals that the minimax optimal !lter can be viewed as a variation of H∞ !lter, characterized by Dk and d. Combined with previous case, this observation illustrates that the !lter proposed in this paper provides a “trade-o=” between the robustness of H∞ !lter and the performance of Kalman !lter. Now consider the case where both Dk → 0 and d → 0. Then from (46), Pˆ k → 0 and consequently from (52), Qˆ k → 0. Thus, the coeTcient c1 de!ned in Corollary 1 converges to zero. Recall that this coeTcient determines the slope of V* at large *; see (53). From this fact and the monotonicity of V* with Dk = 0, d = 0 in (45), we conclude that *∗ → ∞, i.e. the minimax optimal !lter approaches the standard Kalman !lter. This situation is very plausible since no uncertainty is left in the system as Dk → 0 and d → 0 and consequently, h(QP) → 0. Thus, the above observation reveals that the minimax !lter will restore the performance of Kalman !lter in this case. As a counterpart of this fact, we are tempted to conjecture that the minimax optimal !lter will have the structure of modi!ed H∞ !lter when Dk is very large. Surprisingly, however, this is not the case. Once again the minimax !lter approaches the modi'ed Kalman !lter as it did with small Dk . This is because as was shown in the proof of Corollary 2, if Dk increases to its upper limit then we have *b → ∞ and inevitably *∗ → ∞. At a glance, this result seems to be contradictory with the fact that H∞ !lter has better robustness than Kalman !lter. In fact we simply have {Dk ; d}-modi!cations of these two !lters. Finally, we take a di@erent viewpoint on the problem of robust minimax !ltering. Recall that the robust minimax !ltering is associated with the game inf sup (C* − *U) = inf sup (Ce − *U + *Cu ); u x0 ;w;v
u x0 ;w;v
(62)
where Ce , Cu and U are de!ned in (19), (17) and (40), respectively. By setting Cu ≡ 0 in (62), we obtain a standard H ∞ !ltering problem [13] in which the attenuation level is given by the ratio Ce =U, i.e., the induced operator norm of L2 space. However, since in the robust minimax !ltering problem under consideration, Cu corresponds to the L2 norm of the uncertainty output, and therefore Cu = 0, the resulting minimax !lter has to minimize not only Ce but Cu as well. This is obvious since a large Cu may increases the value of game (62). Therefore in robust minimax !ltering, H ∞ -optimality of the !lter is sacri!ced due to the necessity to properly bound the uncertainty output. Furthermore, an attempt to minimize Cu only might cause a large Ce , resulting in a large H ∞ !ltering error. Thus, the minimax optimal !lter will aim to achieve a tradeo@ between these two terms.
5. An illustrative example We consider a numerical example to illustrate previous analysis. The system matrices are chosen to be, for all k, 0:6 0 Ak = ; Ck = [1 1]; Ek = 01×2 ; k+1 = k+1 = X0 = I2×2 : (63) 0 −0:6
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Table 1 Worst-case cost V*∗ (d; 9). 9
*b
*∗ (d = 0)
0 0.01 0.10 0.20 0.40 0.413 0.4132 0.413211
2.0 2.0 2.1 2.3 4.8 46.1 626.2 2520.0
∞ 82.6 9.6 5.9 6.6 47.4 641.8 2578.0
*∗ (d = 10−3 ) 150.0 72.7 9.6 5.9 6.6 47.4 641.8 2578.0
V*∗ (d = 0) 26.4 26.9 33.0 43.6 138.4 885.1 1.30E4 5.59E4
V*∗ (d = 10−3 ) 26.7 27.0 33.0 43.6 138.4 885.1 1.30E4 5.59E4
Also, the initial condition is x0 = [1 1] and the uncertainty matrix Dk and output matrix Lk are given by Lk = I2×2 ;
Dk (9) = 9 · I2×2 ;
(64)
where 9 is a parameter. Assume N = 20. Numerical computations reveal that the upper bound on 9, for which one can !nd * such that conditions (34) and (35) are satis!ed, is 9max ≈ 0:413211, as summarized in Table 1 with two choices of d ∈ {0; 10−3 }. Note that the condition 0(Rˆ k Dk Dk ) ¡ 1 in Corollary 2 provides a conservative bound 9 ¡ 0:6274 for the existence of the minimax !lter. Table 1 shows that, as discussed before, if d = 0 and 9 ≈ 0 then the minimax optimal !lter e@ectively acts as a Kalman !lter (*∗ = ∞). As the uncertainty parameter 9 increases, the gap between *∗ and *b reduces thus causing the minimax optimal !lter to perform like the modi!ed H∞ !lter. Furthermore, as 9 → 9max , we observe that both *∗ and *b approach ∞. Therefore, excessively large uncertainty will cause both the minimax optimal !lter and modi!ed H∞ !lter to perform in a manner similar to that of the modi!ed Kalman !lter (55). In order to compare the robustness property of the minimax optimal !lter with that of the standard Kalman !lter, a set of parametric perturbations A: =
0:6
0
0
−0:6
+ : · I2×2 ;
: ∈ [ − 0:4; 0:4]
(65)
was considered. We considered two di@erent !lters, the Kalman !lter designed for the nominal system and the minimax optimal !lter designed with 9 = 0:4 and d = 10−3 . For each !lter, we computed the error variance N P ;(:) := E { k=0 Lk x˜k − uk 2 =2}, where x˜k was generated using the state equation with A = A: . For every : among 300 points evenly spaced over the interval [ − 0:4; 0:4], we performed 10,000 numerical simulations and obtained the error variance graphs shown in Fig. 1. This !gure shows that the minimax optimal !lter is less sensitive to perturbations in the system, as compared to the Kalman !lter. Although the Kalman !lter has a superior performance when the perturbations of the nominal system are small |:| 6 0:3, its performance substantially deteriorates when perturbations are large, 0:3 6 |:| 6 0:4. In contrast, performance of the minimax optimal !lter remains essentially the same across entire range of uncertain parameter :. Finally, note that the experimental value 26:37 indicated in Fig. 1, which is the average of all ;(:), |:| 6 0:01, corresponds to the cost V*∗ = 26:4 predicted in the case 9 = d = 0; see Table 1.
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Fig. 1. Error variances ;(:) of perturbed systems.
6. Conclusion We proposed a !nite horizon minimax !lter for a discrete-time systems with uncertainties in the underlying probability measure. We found the optimal minimax !lter from a known solution to a risk-sensitive control problem. The properties and structure of the minimax !lter have been investigated and this analysis revealed that the minimax !lter is a variation on both the Kalman !lter and the H ∞ !lter. To illustrate the computation of the !lter and its properties, a numerical example was presented. References [1] T. BasUar, P. Bernhard, H ∞ -Optimal Control and Related Minimax Design Problems, 2nd Edition, BirkhVauser, Boston, 1995. [2] I.B. Collings, M.R. James, J.B. Moore, An information-state approach to risk-sensitive tracking problems, J. Math. Systems, Estimat. Control 6 (3) (1996) 1–24. [3] C.E. de Souza, U. Shaked, Robust H2 !ltering for uncertain systems with measurable inputs, IEEE Trans. Signal Process. 47 (1999) 2286–2292. [4] P. Dupuis, R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, Wiley, New York, 1997. [5] R.J. Elliott, L. Aggoun, J.B. Moore, Hidden Markov Models; Estimation and Control, 2nd Edition, Application of Mathematics, Vol. 29, Springer, Berlin, 1995. [6] S.O.R. Moheimani, A.V. Savkin, I.R. Petersen, Robust !ltering, prediction, smoothing and observability of uncertain systems, IEEE Trans. Circuits Systems Part 1, Fund. Theory Appl. 45 (4) (1998) 446–457. [7] I.R. Petersen, M.R. James, P. Dupuis, Minimax optimal control of stochastic uncertain systems with relative entrophy constrains, IEEE Trans. Automat. Control 45 (3) (2000) 398–412. [8] I.R. Petersen, D.C. McFarlane, Optimal guaranteed cost control and !ltering for uncertain linear systems, IEEE Trans. Automat. Control 39 (1993) 1971–1977. [9] I.R. Petersen, A.V. Savkin, Robust Kalman Filtering for Signals and Systems with Large Uncertainties, BirkhVauser, Boston, 1999. [10] I.R. Petersen, V.A. Ugrinovskii, A.V. Savkin, Robust Control Design Using H ∞ Methods, Springer, London, 2000. [11] P.D. Pra, L. Meneghini, W.J. Ruggaldier, Connections between stochastic control and dynamic games, Math. Control, Signals, Systems 9 (1996) 303–326. [12] A.V. Savkin, I.R. Petersen, A connection between H ∞ control and the absolute stabilizability of uncertain systems, Systems and Control Lett. 23 (3) (1994) 197–203. [13] U. Shaked, Y. Theodor, H∞ -optimal estimation: A tutorial, in: Proceedings of The 31th IEEE Conference on Decision and Control, Tucson, AR, 1992, pp. 2278–2286. [14] Y. Theodor, U. Shaked, Robust discrete-time minimum-variance !ltering, IEEE Trans. Signal Process. 44 (1996) 181–189. [15] Y. Theodor, U. Shaked, C.E. de Souza, A game theory approach to robust discrete-time H∞ -estimation, IEEE Trans. Signal Process. 42 (1994) 1486–1495. [16] V.A. Ugrinovskii, I.R. Petersen, Guaranteed cost LQG !ltering for stochastic discrete time uncertainty systems via risk-sensitive control, in: Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AR, 1999, pp. 564 –569.
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