3 Robust Mixed H2?H? Filtering The filtering problem has been playing an important role in signal processing and control engineering. Among various filtering schemes, the celebrated Kalman filtering (also known as H2 filtering) approach minimizes the H2 norm of the estimation error, under the assumptions that an exact model is available and the noise processes have exactly known statistical properties. However, this is seldom the case in practical applications. As an alternative, the H? filtering method has been proposed, which provides an upper bound for the worst-case estimation error without the need for knowledge of noise statistics. The H? filter has also been proven to be more robust than the traditional Kalman filter, when model uncertainties exist in the system. However, there is no provision in H? filtering to ensure that the variance of the state estimation error lies within acceptable bounds. In this respect, it is natural to consider combining the performance requirements of the Kalman filter and the H? filter into a mixed H2 ?H? filtering problem. For deterministic systems, the mixed H2 ?H? filtering problems have been extensively studied. Various methods have been proposed to solve the mixed H2 ?H? filtering problems, such as the algebraic equation approach, time-domain Nash game approach, and convex optimization approach. It should be pointed out that, results of H2 ?H? filtering for nonlinear systems are relatively few. On the other hand, as for the stochastic setting, recently there has been growing research interest in the stochastic filtering problems with or without H? performance constraints. The main reason is that stochastic modeling has been applied in many practical systems, such as image processing, communication systems, biological systems, and aerospace systems. However, the filtering problem for nonlinear stochastic systems with or without H2 ?H? performance constraints has received little attention, which is still open and remains challenging. It is, therefore, our intention in this chapter to cope with the robust H2 ?H? filtering problem for a class of nonlinear stochastic systems. Motivated by the above discussion, in this chapter we will fully investigate the mixed H2 ?H? filtering problem for systems with deterministic uncertainties and Variance-Constrained Multi-Objective Stochastic Control and Filtering, First Edition. Lifeng Ma, Zidong Wang and Yuming Bo. Е 2015 John Wiley & Sons, Ltd. Published 2015 by John Wiley & Sons, Ltd. 46 Variance-Constrained Multi-Objective Stochastic Control and Filtering stochastic nonlinearities, where the nonlinearities are characterized by statistical means. Our aim is to design a filter such that, for all admissible stochastic nonlinearities and deterministic uncertainties, the overall filtering process is exponentially mean-square quadratically stable, the H2 filtering performance is achieved, and the prescribed disturbance attenuation level is guaranteed in an H? sense. New formulas are derived for exponential mean-square quadratic stability, H2 performance, and H? performance. In particular, due to the introduction of the stochastic nonlinearities, a new lemma about the relations between the stability of the system and the non-negative definite solution to matrix inequality is developed for deriving our H2 performance, which is actually the extension of results for linear systems. The solution to the H2 ?H? filtering problem is enforced within a unified LMI framework. In order to demonstrate the flexibility of the proposed framework, we will examine two types of optimization problems that optimize either the H2 performance or the H? performance, and a numerical example is provided to illustrate the design procedures and performances of the proposed method. The remainder of this chapter is organized as follows: In Section 3.1, a class of uncertain discrete-time nonlinear stochastic systems is described and the robust H2 ?H? filtering problem for the systems is formulated. The solution to the robust H2 ?H? filtering problem is derived based on the notion of exponential mean-square quadratic stability, characterizations of the H2 performance, and H? performance in Section 3.2. An LMI algorithm is developed in Section 3.3 for designing the H2 ?H? filter for the systems with stochastic nonlinearities and deterministic norm-bounded parameter uncertainties. An illustrative example is presented in Section 3.4 to demonstrate the applicability of the method and some concluding remarks are provided in Section 3.5. 3.1 System Description and Problem Formulation Consider the following class of discrete-time systems with stochastic nonlinearities and deterministic norm-bounded parameter uncertainties: ?x(k + 1) = (A + H FE)x(k) + f (x(k)) + B ?(k), 1 1 ? ?y(k) = (C + H2 FE)x(k) + g(x(k)) + D11 ?(k), ? ?z2 (k) = L2 x(k), ?z? (k) = L? x(k), ? (3.1) where x(k) ? ?n is the state, y(k) ? ?m is the measured output, z2k ? ?p1 is a combination of the states to be estimated (with respect to H2 -norm constraints), z? (k) ? ?p2 is another combination of the states to be estimated (with respect to H? -norm Robust Mixed H2 ?H? Filtering 47 constraints), ?(k) ? ?r is a zero mean Gaussian white noise sequence with covariance ? > 0, and A, B1 , C, D11 , L2 , L? , H1 , H2 and E are known real matrices with appropriate dimensions. The matrix F ? ?iОj represents the deterministic norm-bounded parameter uncertainties satisfying FF T ? I. (3.2) The deterministic uncertainty F is said to be admissible if the condition (3.2) is met. The functions f (x(k)): ?n ? ?n and g(x(k)): ?n ? ?m are stochastic nonlinear functions of the states, which are assumed to have the following first moments for all x(k): {[ ] } f (x(k)) ? |x(k) = 0, (3.3) g(x(k)) with the covariance given by } {[ ] ] f (x(k)) [ T T f (xj ) g(xj ) |x(k) = 0, ? g(x(k)) and ? k ? j, } ] [ ]T ] q [ ? ] ?1i ?1i f (x(k)) [ T T f (x(k)) g(x(k)) |x(k) = x(k)T ?i x(k), ?2i ?2i g(x(k)) (3.4) {[ (3.5) i=1 where ?1i ? ?nО1 and ?2i ? ?mО1 (i = 1,...,q) are known column vectors with compatible dimensions of f (x(k)) and g(x(k)), and ?i (i = 1,...,q) are known positive definite matrices with appropriate dimensions. Now consider the following filter for the system (3.1): ?x? (k + 1) = A? ? x(k) + Ky(k), ? ? ?z? 2 (k) = L? 2 x? (k), ?z? (k) = L? x? (k), ? ?? (3.6) where x? (k) is the state estimate, z? 2 (k) is an estimate for z2 (k), and z? ? (k) is an estimate ? K, ? L? 2 and L? ? are the filter parameters to be determined. for z? (k), and A, The augmented system is described as follows by combining (3.6) with (3.1): ?x? (k + 1) = A? ? ? x(k) + Be h(x(k)) + B?(k), ? ?e2 (k) ? z2 (k) ? z? 2 (k) = L2 x(k) ? L? 2 x? (k) = C2 x? (k), ?e (k) ? z (k) ? z? = L x(k) ? L? x? (k) = C x? (k), ? ?k ? ? ? ? ? (3.7) Variance-Constrained Multi-Objective Stochastic Control and Filtering 48 where [ ] x(k) x? (k) = , x? (k) [ ] f (x(k)) h(x(k)) = , g(x(k)) [ ] A 0 A? = A? + ?A, A? = ? ? , KC A ] [ [ ] H ?A = ? 1 F E 0 ? He FEe , KH2 ] [ ] [ I 0 B1 ? , , B= ? Be = 0 K? KD11 [ [ ] ] C? = L? ?L? ? , C2 = L2 ?L? 2 . (3.8) (3.9) (3.10) (3.11) (3.12) Since the augmented system (3.7) contains both deterministic and stochastic entries, we need to introduce the notion of stochastic stability (in the mean-square sense) for the augmented system (3.7). Definition 3.1.1 The system (3.7) is said to be exponentially mean-square quadratically stable if with ?(k) = 0 there exist constants ? ? 1 and ? ? (0, 1) such that ?{??x(k)?2 } ? ?? k ?{??x0 ?2 }, ? x? 0 ? ?n , k ? ?+ , (3.13) for all uncertainties F satisfying the condition (3.2). In this chapter, our objective is to design the filter (3.6) for the system (3.1) such that, for all stochastic nonlinearities and all admissible deterministic uncertainties, the augmented system (3.7) is exponentially mean-square quadratically stable, the estimation error e2 (k) satisfies the H2 performance constraint, and the estimation error e? (k) satisfies the H? performance constraint. More specifically, we aim to design the filter (3.6) such that the following requirements are satisfied simultaneously: (R1) For a given ? > 0, the system (3.7) is exponentially mean-square quadratically stable and (3.14) J2 = lim ?{?e2 (k)?2 } < ?. k?? (R2) For a given ? > 0, the system (3.7) is exponentially mean-square quadratically stable and ? ? ? ? ?{?e? (k)?2 } < ? 2 ?{??(k)?2 }, (3.15) k=0 k=0 for all nonzero ?(k) under zero initial condition. The above design problem will be referred to as the robust H2 ?H? filtering problem. Robust Mixed H2 ?H? Filtering 49 3.2 Algebraic Characterizations for Robust H2 ?H? Filtering In this section, we will present the algebraic characterizations for robust H2 filtering and robust H? filtering. Under the framework of a common Lyapunov function, the solution to the robust H2 ?H? filtering problem is provided for the system (3.7). 3.2.1 Robust H2 Filtering Before proceeding, we denote [ [ ] [ ]T ] ??i ? ?i 0 , ?i ? ?1i ?1i . ?2i ?2i 0 0 (3.16) In order to derive the characterization guaranteeing the robust H2 filtering performance, we need the following technical results. Lemma 3.2.1 Let V(?x(k)) = x? (k)T P?x(k) be a Lyapunov functional, where P > 0. If there exist positive real scalars ?, ?, ?, and 0 < ? < 1 such that ???x(k)?2 ? V(?x(k)) ? ???x(k)?2 (3.17) ?{V(?x(k + 1))|?x(k)} ? V(?x(k)) ? ? ? ?V(?x(k)), (3.18) and then the sequence x? (k) satisfies ?{??x(k)?2 } ? ? ? ??x0 ?2 (1 ? ?)k + . ? ?? (3.19) Lemma 3.2.2 If the system (3.7) is exponentially mean-square quadratically stable, then ?{A? ? A? + q ? st(Be ?i BTe )stT (?? i )} < 1 (3.20) i=1 or, equivalently, ?{A? T ? A? T + q ? st(?? i )stT (Be ?i BTe )} < 1. i=1 We are now ready to give the stability conditions. (3.21) Variance-Constrained Multi-Objective Stochastic Control and Filtering 50 Theorem 3.2.3 The system (3.7) is exponentially mean-square quadratically stable if, for all admissible uncertainties, there exists a positive definite matrix P satisfying A? T PA? ? P + q ? [?? i tr(Be ?i BTe P)] < 0. (3.22) i=1 Proof. Define the Lyapunov functional V(?x(k)) = x? (k)T P?x(k), where P > 0 is the solution to (3.22). By using the super-Martingale property for the system (3.7) with ?(k) = 0, we obtain ?{V(?x(k + 1))|?x(k)} ? V(?x(k)) ? x(k) + ?{hT (x(k))BTe PBe h(x(k))} ? x? (k)T P?x(k) = x? (k)T A? T PA? = x? (k)T (A? T PA? ? P)?x(k) + tr[?{Be h(x(k))h(x(k))T BTe }P] = x? (k) (A? T PA? ? P)?x(k) + tr[Be T q ? {?i x(k)T ?i x(k)}BTe P] i=1 = x? (k)T (A? T PA? ? P)?x(k) + q ? [x(k)T ?i x(k)tr[Be ?i BTe P]] i=1 = x? (k) (A? T PA? ? P)?x(k) + T [ ] q ? T ?i 0 [?x(k) x? (k)tr[Be ?i BTe P]] 0 0 i=1 = x? (k) (A? T PA? ? P)?x(k) + x? (k)T T q ? [?? i tr[Be ?i BTe P]]?x(k) i=1 = x? (k)T {A? T PA? ? P + q ? [?? i tr[Be ?i BTe P]]}?x(k). (3.23) i=1 We know from (3.21) that there must exist a sufficiently small scalar ? satisfying 0 < ? < ?max (P) and A? T PA? ? P + q ? ?? i tr[Be ?i BTe P] < ??I. (3.24) i=1 Therefore, we obtain ?{V(?x(k + 1))|?x(k)} ? V(?x(k)) ? ???x(k)T x? (k) ? ? and the proof follows immediately from Lemma 3.2.1. ? V(?x(k)) ?max (P) (3.25) ? Robust Mixed H2 ?H? Filtering 51 Let us proceed to compute the H2 performance J2 that is used in the constraint (3.14). Define the state variance by Q(k) ? ?{?x(k)?x(k)T } {[ ][ ]T } . = ? x? 1 (k) x? 2 (k) ? x? n (k) x? 1 (k) x? 2 (k) ? x? n k (3.26) The Lyapunov-type equation that governs the evolution of the state variance matrix Q(k) can be derived from the system (3.7) as follows: ? ? B? T Q(k + 1) = AQ(k) A? T + ?[Be h(x(k))h(x(k)T )BTe ] + B? { ? = AQ(k) A? T + ? [ Be [ ? = AQ(k) A? T + ? Be f (x(k)) g(x(k)) q ? ][ ]T } f (x(k)) ? B? T BTe + BR? g(x(k)) ] ? B? T (?i x(k)T ?i x(k))BTe + B? i=1 ? = AQ(k) A? T + q ? i=1 ? = AQ(k) A? T + q ? { [ } ] T ?i 0 T ? B? T Be ?i ?(?x(k) x? (k))Be + B? 0 0 ? B? T {Be ?i BTe tr[?(?x(k)?x(k)T )?? i ]} + B? i=1 ? = AQ(k) A? T + q ? ? B? T . [Be ?i BTe tr(Q(k)?? i )] + B? (3.27) i=1 Rewrite (3.27) in the form of a stack matrix as follows: ? B? T ), st(Q(k + 1)) = ?st(Q(k)) + st(B? (3.28) where ? ? A? ? A? + q ? st(Be ?i BTe )stT (?? i ). (3.29) i=1 If the system (3.7) is exponentially mean-square quadratically stable, it then follows from Lemma 3.2.2 that ?(?) < 1 and Q(k) in (3.28) will converge to Q when k ? ?, i.e., Q = lim Q(k). k?? (3.30) 52 Variance-Constrained Multi-Objective Stochastic Control and Filtering Therefore, the H2 performance can be written by J2 = lim ?{?e2 (k)?2 } k?? = lim ?{?x(k)T C2T C2 x? (k)} k?? = lim ?{tr[C2 x? (k)?x(k)T C2T ]} k?? = lim tr[C2 Q(k)C2T ] k?? = tr[C2 QC2T ]. (3.31) ? such that the following backNow suppose that there exists a symmetric matrix P(k) ward recursion is satisfied: ? + 1)A? + ? P(k) = A? T P(k q ? ? + 1)] + CT C2 . ?? i tr[Be ?i BTe P(k 2 (3.32) i=1 We rearrange (3.32) in the form of a stack matrix as follows: ? = ?st(P(k ? + 1)) + st(CT C2 ), st(P(k)) 2 (3.33) where ? ? A? T ? A? T + q ? st(?? i )stT (Be ?i BTe ). (3.34) i=1 If the system (3.7) is exponentially mean-square quadratically stable, it follows from ? Lemma 3.2.2 that ?(?) < 1 and P(k) in (3.33) will converge to P? when k ? ?, i.e., ? P? = lim P(k). k?? (3.35) Hence, in the steady state, (3.32) becomes P? = A? T P? A? + q ? ? + CT C2 . ?? i tr[Be ?i BTe P] 2 (3.36) i=1 Summarizing the above results, we obtain the following theorem, which gives an alternative to compute the H2 performance. Theorem 3.2.4 If the system (3.7) is exponentially mean-square quadratically stable, the H2 performance can be expressed in terms of P? as follows: ? J2 = tr[?B? T P? B], where P? > 0 is the solution to (3.36). (3.37) Robust Mixed H2 ?H? Filtering 53 Proof. Note that ? + 1) ? Q(k)P(k)} ? lim tr{Q(k + 1)P(k k?? ? A? T + = lim tr{[AQ(k) k?? q ? ? B? T ]P(k ? + 1) Be ?i BTe tr(Q(k)?? i ) + B? i=1 ? + 1)A? + ?Q(k)[A? T P(k q ? ? + 1)) + CT C2 ]} ?? i tr(Be ?i BTe P(k 2 i=1 = 0. (3.38) Using the properties of tr[AB] = tr[BA] and tr[A + B] = tr[A] + tr[B] to (3.38), we have ? tr[C2 QC2T ] = tr[?B? T P? B]. (3.39) ? This completes the proof. Remark 3.1 We use (3.34) to compute the H2 performance instead of (3.29). The reason is that the H2 filtering performance and the H? filtering performance need to be characterized as a similar structure so that the solution to the addressed H2 ?H? filtering problem can be obtained by using a unified LMI approach. Although we have already discussed the H2 filtering performance and H? filtering performance, we still need to put in much effort because it is not a trivial task to merge two performance objectives into a unified LMI framework. One will see in the next subsection that the structure of (3.37) is similar to that for the H? filtering performance. Notice that the system model in (3.1) involves parameter uncertainties, and hence the exact H2 performance (3.37) cannot be obtained by simply solving the equation (3.36). One way to deal with this problem is to provide an upper bound for the actual H2 performance. Suppose that there exists a positive definite matrix P such that the following matrix inequality is satisfied: A? T PA? ? P + q ? ?? i tr[Be ?i BTe P] + C2T C2 < 0. (3.40) i=1 Before proving that the solution P > 0 to (3.40) is an upper bound for P? in Theorem 3.2.6, we develop the following lemma, which is very important to reveal the relationship between the stochastic stability and the non-negative-definite solution to a matrix inequality. Lemma 3.2.5 Consider the system ?(k + 1) = M?(k) + Be f (x(k)), (3.41) Variance-Constrained Multi-Objective Stochastic Control and Filtering 54 [ ] x(k) where ?(k) = ? ?2n , x(k) ? ?n , ?{f (x(k))|x(k)} = 0 and ?{f (x(k))f T x? (k) ?q (x(k))|x(k)} = i=1 ?i (x(k)T ?i x(k)). Here, ?i = ?i ?iT , ?i (i = 1,...,q) are column vectors, ?i (i = 1,...,q) are known positive definite matrices with appropriate dimensions. If the system (3.41) is exponentially mean-square stable, and there exists a symmetric matrix Y satisfying T M YM ? Y + q ? ?? i tr[Be ?i BTe Y] < 0 (3.42) i=1 [ ] ? 0 i ?? i = , 0 0 where (3.43) then Y ? 0. Proof. It follows from (3.42) that T M YM ? Y + q ? ?? i tr[Be ?i BTe Y] = ??, (3.44) i=1 for some ? > 0. Define a functional W(?(k)) = ?(k)T Y?(k). Applying the super-Martingale property to system (3.41) yields ?{W(?(k + 1))|?(k)} ? W(?(k)) = ?(k)T (M T YM ? Y)?(k) + ?{f T (x(k))BTe YBe f (x(k))} = ?(k)T (M T YM ? Y)?(k) + x(k)T ?i x(k)tr(Be ?i BTe Y) = ?(k) [M YM ? Y + T T q ? ?? i tr[Be ?i BTe Y]]?(k) i=1 T = ??(k) ??(k). (3.45) Summing (3.45) from 0 to n with respect to k, we obtain ?{?nT Y?n } ? ?0T Y?0 = ? n ? ?(k)T ??(k). (3.46) k=0 Let n ? ? in (3.46); it then follows from the exponential mean-square stability of the system (3.41) and the fact lim ?{?nT Y?n } ? ?Y? lim ?{?nT ?n } n?? n?? Robust Mixed H2 ?H? Filtering 55 that lim ?{?nT Y?n } = 0. Hence, we have from (3.46) that n?? ?0T Y?0 = ? ? ?(k)T ??(k) ? 0. (3.47) k=0 Since (3.47) holds for any nonzero initial state ?0 , we arrive at the conclusion that Y ? 0. ? Remark 3.2 For linear time-invariant stochastic systems, it is well known that the stability of the system implies the existence of a positive definite solution to a Lyapunov matrix equation. Lemma 3.2.5 provides a ?nonlinear version? of such a property, which would help establish the upper bound for the H2 performance. ? Comparing (3.36) to (3.40), we Now we are ready to give the upper bound for P. obtain the following main result in this subsection. Theorem 3.2.6 If there exists a positive definite matrix P satisfying (3.40), then the system (3.7) is exponentially mean-square quadratically stable, and P? ? P, (3.48) ? ? ? tr[?B? T PB], tr[?B? T P? B] (3.49) where P? satisfies (3.36). Proof. It is obvious that (3.40) implies (3.22). Hence it follows directly from Theorem 3.2.3 that the system (3.7) is exponentially mean-square quadratically stable and that, P? exists and meets (3.36). Subtracting (3.40) from (3.36) yields ? A? ? (P ? P) ? + A? T (P ? P) q ? ? < 0, ?? i tr[Be ?i BTe (P ? P)] (3.50) i=1 which indicates from Lemma 3.2.5 that P ? P? ? 0. Also, (3.48) implies (3.49). This completes the proof. ? The corollary given below follows immediately from Theorem 3.2.6 and (3.14). Corollary 3.2.7 If there exists a positive definite matrix P satisfying (3.40) and ? < ?, then the system (3.7) is exponentially mean-square quadratically tr[?B? T PB] stable and (3.14) is satisfied for some ?. Variance-Constrained Multi-Objective Stochastic Control and Filtering 56 3.2.2 Robust H? Filtering In this subsection, we present the algebraic characterization for the H? filtering performance in (R2). Note that the expression defined in (3.15) is used to describe the H? filtering performance of the stochastic system, where the expectation operator is utilized on both the filtering error and the disturbance input. In the following theorem, the sufficient conditions are provided for establishing the H? -norm performance. Theorem 3.2.8 For a given ? > 0, the system (3.7) is exponentially mean-square quadratically stable and achieves the H? -norm constraint (3.15) for all nonzero ?(k) if there exists a positive definite matrix P satisfying q ? ??T ? ? TC P A ? P + ?? i tr[Be ?i BTe P] + C? A A? T PB? ? ? ? < 0, i=1 ? T PA T PB 2I? ? ? ? ? B ? ? B ? ? (3.51) for all admissible uncertainties. Proof. It is obvious that (3.51) implies (3.22); hence it follows from Theorem 3.2.3 that the system (3.7) is exponentially mean-square quadratically stable. Next, for any nonzero ?(k), it follows from (3.51) that ?{V(?x(k + 1))|?x(k)} ? ?{V(?x(k))} + ?{e? (k)T e? (k)} ? ? 2 ?{?(k)T ?(k)} = ?{?x(k) [A? T PA? ? P + T q ? ? ? x(k) + ?(k)T B? T PA? ?? i tr[Be ?i BTe P]]?x(k) + x? (k)T A? T PB?(k) i=1 T ? +?(k) B PB?(k) + x? (k)T C? C? x? (k) ? ? 2 ?(k)T ?(k)} T ?T = ?{?x(k) [A? T PA? ? P + T q ? T C? ]?x(k) ?? i tr[Be ?i BTe P] + C? i=1 ? ? x(k) + ?(k)T (B? T PB? ? ? 2 I)?(k)} +?x(k) A PB?(k) + ?(k)T B? T PA? T ?T q ?[ ]T ? ? ? TC ? x? (k) ?A? T PA? ? P + ?? i tr[Be ?i BTe P] + C? A? T PB? ? ? =?? i=1 ?(k) ? ? ? B? T PB? ? ? 2 I ? B? T PA? ? ? [ ]} x? (k) О < 0. ?(k) (3.52) Now, summing (3.52) from 0 to ? with respect to k yields ? ? [?{V(?x(k + 1))|?x(k)} ? ?{V(?x(k))} + ?{e? (k)T e?k } ? ? 2 ?{?(k)T ?(k)}] < 0, k=0 (3.53) Robust Mixed H2 ?H? Filtering 57 i.e., ? ? ?{?e? (k)? } < ? 2 k=0 2 ? ? ?{??(k)?2 } + ?{V(?x0 )} ? ?{V(?x? )}. (3.54) k=0 Since x? 0 = 0 and the system (3.7) is exponentially mean-square quadratically stable, it is straightforward to see that ? ? ?{?e? (k)? } < ? 2 k=0 2 ? ? ?{??(k)?2 }. (3.55) k=0 ? This ends the proof. So far, the solutions to the H2 filtering problem (R1) and H? filtering problem (R2) have been developed separately. In the next section, we will focus on the mixed H2 ?H? filtering problem. 3.3 Robust H2 ?H? Filter Design Techniques In this section, we will present the solution to the robust H2 ?H? filtering design problem for discrete-time systems with stochastic nonlinearities and deterministic norm-bounded parameter uncertainty, i.e., we design the filter that satisfies the performance requirements (R1) and (R2) simultaneously. According to the results obtained in the previous section and the conditions (3.22), (3.40), and (3.51), we conclude that (3.40) and (3.51) imply (3.22). Hence (3.22) becomes redundant and is not involved in the filter design. Therefore, in order to achieve our design objectives (R1) and (R2), we can cast the original robust H2 ?H? filtering problem as follows: Problem (Pa). Design the filter (3.6) such that there exists a positive definite matrix P satisfying the following inequalities: ? < ?, tr[?B? T PB] A? T PA? ? P + ? q ?? i tr[Be ?i BTe P] + C2T C2 < 0, (3.56) (3.57) i=1 q ? ??T ? ? TC A P A ? P + ?? i tr[Be ?i BTe P] + C? A? T PB? ? ? ? < 0. i=1 ? ? T T 2 ? ? ? ? B PB ? ? I ? B PA ? (3.58) The problem (Pa) is to find the filter (3.6) to ensure that (3.56) to (3.58) are satisfied for all admissible uncertainties. Note that, at this stage, such a problem is not solved yet, since the matrix trace terms and the uncertainty F are involved in (3.56) to (3.58), which make the problem very complicated. Our goal is therefore to transform (3.56) to (3.58) into LMIs, in order to obtain solutions to the aforementioned filtering problem. Variance-Constrained Multi-Objective Stochastic Control and Filtering 58 In order to transform problem (Pa) into a convex optimization problem, we first deal with the matrix trace terms in (3.56) to (3.58) by introducing new variables. The following theorem presents a sufficient condition for solving the problem (Pa). Theorem 3.3.1 Given ? > 0 and ? > 0. If there exist positive definite matrices P > 0, ? > 0, and positive scalars ?i > 0 (i = 1, ? , q) such that the following matrix inequalities tr[?] < ?, [ ] T ? ?? ? B P < 0, 1 ? 2 PB? ?P [ T ] T T ? ? ? ?? i 1i 2i Be P? [ ] ? < 0 (i = 1, ? , q), ? ?1i ?P ? ?PBe ?2i ? ? ? ? ?P ? A? 1 ? ??1 ?? 12 ?иии 1 ? ??q ?? q2 ? C ? 2 ? ? ?P ? A? ? ? 21 ??1 ?1 ?иии 1 ? ??q ?? q2 ? C? ? ? 0 1 2 1 1 A? T ?P?1 ?1 ?? 12 0 иии иии ?q ?? q2 0 0 иии ??1 I иии иии иии 0 иии 0 0 0 0 иии иии ??q I 0 1 ? C2T ? 0? ? 0 ? < 0, и и и? ? 0? ?I ?? 1 A? T ?P?1 ?1 ?? 12 0 иии иии ?q ?? q2 0 T C? 0 0 иии ??1 I иии иии иии 0 иии 0 иии 0 0 B? T 0 0 0 иии иии иии ??q I 0 0 0 ?I 0 ? 0 ? B? ? ? 0 ? и и и ? < 0, ? 0 ? 0 ? ? ?? 2 I ? (3.59) (3.60) (3.61) (3.62) (3.63) hold, then (3.56) to (3.58) are satisfied. Proof. We define new variables ?i > 0 (i = 1, ? , q) satisfying tr[Be ?i BTe P] < ?i (i = 1, ? , q), (3.64) i.e., [ ] [ ]T ? ?1i tr[Be 1i BTe P] < ?i ?2i ?2i (i = 1, ? , q). (3.65) Robust Mixed H2 ?H? Filtering 59 Using the properties of trace and the Schur Complement Lemma, we obtain [ ] [ ]T [ ] [ T T] T ?1i ?1i ? T ? ? tr[Be Be P] = 1i 2i Be PBe 1i < ?i (3.66) ?2i ?2i ?2i [ T T] T ? ?? [ i ] ?1i ?2i Be ? ? ? < 0 (i = 1, ? , q), ?? (3.67) ? ?P?1 ? ?Be 1i ?2i ? ? which is equivalent to (3.61). Next, let us prove that (3.62) is equivalent to A? T PA? ? P + q ? ?i ?? i + C2T C2 < 0. (3.68) i=1 By using the Schur Complement Lemma again in (3.68), we have q ? ? ?i ?? i + C2T C2 ?P + ? i=1 ? A? ? ? A? T ? <0 ? ?P?1 ? (3.69) ?? ??P + CT C 2 2 ? A? ? 1 ? ?2 ? 1 ? иии ? 1 ? ? ?? q2 1 1 A? T ?P?1 ?? 12 0 иии иии ?? q2 0 0 иии ??1?1 I иии иии иии 0 иии 0 0 иии ??q?1 I ? ? ? ? < 0, ? ? ? ? (3.70) which is equivalent to (3.62). Moreover, it follows from (3.64) and (3.68) that A? T PA? ? P + q ? ?? i tr[Be ?i BTe P] + C2T C2 i=1 < A? T PA? ? P + q ? ?i ?? i + C2T C2 < 0, (3.71) i=1 which indicates (3.57). Similarly, by using the Schur Complement Lemma, (3.63) results in (3.58). At last, we need to prove that (3.59) and (3.60) imply (3.56). Since (3.60) is equivalent to ? 2 < ?, ? 2 B? T PB? 1 1 (3.72) it follows from (3.59) and (3.72) that ? = tr[? 2 B? T PB? ? 2 ] < tr[?] < ?. tr[?B? T PB] 1 This completes the proof. 1 (3.73) ? Variance-Constrained Multi-Objective Stochastic Control and Filtering 60 In the following, we will carry on to ?eliminate? the uncertainty F contained in (3.62) and (3.63) using the S-procedure technique. Then, a convex optimization problem will be formulated, and the robust H2 ?H? filter can be designed by using an LMI approach. Theorem 3.3.2 Given ? > 0 and ? > 0. If there exist positive definite matrices S > 0, R > 0, and ? > 0, real matrices Qi (i = 1, 2, 3, 4), positive scalars ?i > 0 (i = 1, ? , q) and ?i > 0 (i = 1, 2) such that the following linear matrix inequalities are feasible: ?1? ?0? [ ] ?0? [ ] ?0? 1 0 и и и 0 ? ? ? + и и и + 0 и и и 0 1 ? ? ? < ?, ??? ??? ?0? ?1? ? ? ? ? ? ?? ? 1 B1 ? 2 ? 1 ? ?(RB1 + Q2 D11 )? 2 ??i ? ? ?1i ? ?R?1i + Q2 ?2i 1 1 ? 2 BT1 ? 2 (BT1 R + DT11 QT2 )? ? ?S ?S ? < 0, ? ? ?S ?R ?1iT ?1iT R + ?2iT QT2 )? ?<0 ?S ?S ? ? ?S ?R [ ] G GT11 < 0, G11 G22 [ ] G GT33 < 0, G33 G44 (i = 1, ? , q), (3.75) (3.76) (3.77) (3.78) where ?S ?S ? ? ?S ?R ? ? SA SA ? C + Q RA + Q2 C RA + Q 2 1 ? 1 1 ? ?12 ?12 G=? ? 0 0 ? иии иии ? 1 1 ? ? ?q2 ?q2 ? 0 0 ? (3.74) AT S AT S ?S ?S AT R + CT QT2 + QT1 AT R + CT QT2 ?S ?R 0 0 иии 0 0 иии 0 0 0 0 Robust Mixed H2 ?H? Filtering 61 1 1 ?12 0 иии ?q2 иии иии иии иии иии иии иии иии ?q2 0 0 0 0 иии ??q I 0 1 1 0 ?12 0 0 0 0 ??1 I 0 0 ??1 I иии иии 0 0 0 0 G11 ?L2 ? Q3 =? 0 ? ? ?1 E G22 ??I =?0 ? ?0 G33 ?L? ? Q4 ? 0 =? ? 0 ? ? ?2 E G44 ??I ?0 =? ?0 ? ?0 0 ??1 I 0 0 ?? 2 I 0 0 L2 0 ?1 E 0 H1T S 0 H1T R 0 ?? ? 0 ? 0 ? ? 0 ? , 0 ? ? 0 ? и и и ?? 0 ? ? ??q I ? 0 + H2T QT2 0 0 0 0 (3.79) 0 0 0 иии иии иии 0 0 0 0? 0? , (3.80) ? 0? 0 ? 0 ?, ? ??1 I ? (3.81) L? 0 0 BT1 S BT1 R 0 + DT11 QT2 0 0 0 0 иии иии 0 0 0 ?2 E H1T S 0 H1T R + H2T QT2 0 0 0 0 0 иии иии 0 0 0 0 ??2 I 0 0 ? 0 ?? , 0 ? ? ??2 I ? 0? 0? ? , (3.82) 0? ? 0? (3.83) then there exists a filter of the form (3.6) such that the requirements (R1) and (R2) are satisfied for all stochastic nonlinearities and all admissible deterministic uncertainties. Moreover, if LMIs (3.74) to (3.78) are feasible, the desired filter parameters can be determined by ?1 Q1 (S ? R)?1 X12 , A? = X12 ?1 Q2 , K? = X12 L? 2 = Q3 (S ? R)?1 X12 , L? ? = Q4 (S ? R)?1 X12 , T < 0. where the matrix X12 comes from the factorization I ? RS?1 = X12 Y12 (3.84) Variance-Constrained Multi-Objective Stochastic Control and Filtering 62 Proof. Rewrite the condition in the following form: M + HFE + ET F T H T < 0, (3.85) where ? ? ?P ? A? 1 ? ?2 ? ? ? 1 M= 1 ? иии 1 ? ??q ?? q2 ? C ? 2 [ H = 0 HeT [ E = Ee 0 1 ?1 ?? 12 0 иии иии ?q ?? q2 0 0 иии ??1 I иии иии иии 0 иии 0 0 0 0 0 иии 0 иии ? C2T ? 0? ? 0 ?, и и и? ? 0? ?I ?? 1 A? T ?P?1 и и и ??q I иии 0 ]T 0 0 , ] 0 0 . Applying Lemma 2.3.1 to (3.85), it follows that (3.85) holds if and only if there exists a positive scalar parameter ?1 such that the following LMI holds: ? ?P ? ? ? A1 ? ?2 ??1 ?1 ?иии 1 ? 2 ??q ?? q ? C2 ? 0 ? ? ?1 Ee 1 1 A? T ?P?1 ?1 ?? 12 0 иии иии ?q ?? q2 0 C2T 0 0 He 0 иии ??1 I иии 0 иии 0 иии 0 иии 0 иии 0 0 HeT 0 0 0 0 0 иии иии иии иии ??q I 0 0 0 0 ?I 0 0 0 0 ??1 I 0 ?1 EeT ? ? 0 ? ? 0 ? и и и ? < 0. ? 0 ? 0 ? 0 ?? ??1 I ? (3.86) Applying the congruence transformation diag{I, P, I, ? , I, I, I, I} to (3.86), we obtain ? ?P ? ? ? PA1 ? ?2 ??1 ?1 ?иии 1 ? 2 ??q ?? q ? C2 ? 0 ? ? ?1 Ee 1 1 A? T P ?P ?1 ?? 12 0 иии иии ?q ?? q2 0 C2T 0 0 PHe 0 иии ??1 I иии 0 иии 0 иии 0 иии 0 иии 0 0 HeT P 0 0 0 0 0 иии иии иии иии ??q I 0 0 0 0 ?I 0 0 0 0 ??1 I 0 ?1 EeT ? ? 0 ? ? 0 ? и и и ? < 0. ? 0 ? 0 ? 0 ?? ??1 I ? (3.87) Robust Mixed H2 ?H? Filtering 63 Recall that our goal is to derive the expression of the filter parameters from (3.6). To do this, we partition P and P?1 as [ [ ?1 ] ] R X12 S Y12 ?1 P= T , P = T , (3.88) X12 X22 Y12 Y22 where the partitioning of P and P?1 is compatible with that of A? defined in (3.10), i.e., R ? RnОn , X12 ? RnОn , X22 ? RnОn , S ? RnОn , Y12 ? RnОn , and Y22 ? RnОn . Define [ ?1 [ ] ] S I R I T1 = T (3.89) , T2 = T , 0 X12 Y12 0 which imply that PT1 = T2 and T1T PT1 = T1T T2 . Again, define the change of filter parameters as follows: ? T S, Q1 = X12 AY 12 T S, Q3 = L? 2 Y12 ? Q2 = X12 K, T Q4 = L? ? Y12 S. (3.90) Further applying the congruence transformations diag{T1 , T1 , I, ? , I, I, I, I} to (3.89), we obtain [ ] T J11 J12 < 0, (3.91) J12 J22 where J11 ?I S?1 AT S?1 (AT R + CT QT2 + QT1 ) ?S?1 ? ? ?I ?R AT AT R + CT QT2 ? ?1 ?1 AS A ?S ?I ? ?1 RA + Q C C + Q )S ?I ?R (RA + Q ? 2 1 2 1 1 ? 2 ?1 2 ?1 S ?1 0 0 =? ? 0 0 0 0 ? иии иии иии иии ? 1 1 ? ?q2 0 0 ?q2 S?1 ? 0 0 0 0 ? 1 S?1 ?12 1 2 1 0 0 ?1 0 0 0 0 0 ??1 I 0 ??1 I иии иии 0 0 0 0 иии иии иии иии иии иии иии иии иии S?1 ?q2 1 2 ?q 0 0 0 0 иии ??q I 0 ? 0 ? 0 ? ? 0 ? 0 ? , 0 ? ? 0 ? иии ? 0 ? ? ??q I ? (3.92) Variance-Constrained Multi-Objective Stochastic Control and Filtering 64 J12 ?(L2 ? Q3 )S?1 L2 0 0 0 0 иии 0 0 H1T S H1T R + H2T QT2 0 0 и и и =? ? ?1 ?1 E 0 0 0 0 иии ? ?1 ES 0 0? 0 0? , ? 0 0? ??I 0 0 ? 0 ?. J22 = ? 0 ??1 I ? ? 0 ??1 I ? ?0 (3.93) (3.94) Also, performing the congruence transformation diag{S, I, S, I, I, I, ? , I, I, I, I} to (3.86), we obtain (3.77). Similarly, the condition (3.63) can be written in the following form: M + HFE + ET F T H T < 0, (3.95) where ? ? ?P ? A? ? ? 12 ??1 ?1 M =? иии 1 ? ??q ?? q2 ? C? ? ? 0 [ H = 0 HeT [ E = Ee 0 1 1 A? T ?P?1 ?1 ?? 12 0 иии иии ?q ?? q2 0 T C? 0 0 иии ??1 I иии иии иии 0 иии 0 иии 0 0 B? T 0 0 0 иии иии иии 0 иии 0 иии ??q I 0 0 ]T 0 0 0 , ] 0 0 0 . 0 ?I 0 ? 0 ? B? ? ? 0 ? и и и ?, ? 0 ? 0 ? ? ?? 2 I ? By applying Lemma 2.3.1 to (3.95) to eliminate the uncertainty F, we know that (3.95) holds if and only if there exists a positive scalar parameter ?2 such that the following LMI holds: ? ?P ? A? ? 1 ?? ?? 2 1 1 ? ? и и и1 ? ?2 ??q ?q ? C? ? 0 ? ? 0 ? ?2 Ee 1 1 A? T ?P?1 ?1 ?? 12 0 иии иии ?q ?? q2 0 T C? 0 0 B? 0 He 0 иии ??1 I иии иии иии 0 иии 0 иии 0 и и ии и и 0 иии 0 0 B? T HeT 0 0 0 0 0 0 иии иии иии иии иии ??q I 0 0 0 0 0 ?I 0 0 0 0 0 ?? 2 I 0 0 0 0 0 ??2 I 0 ?2 EeT ? 0 ?? 0 ? ? ? ? < 0. (3.96) 0 ? 0 ? 0 ? ? 0 ? ??2 I ? Robust Mixed H2 ?H? Filtering 65 Performing three congruence transformations diag{I, P, I, ? , I}, diag{T1 , T1 , I, ? , I, I}, and diag{S, I, S, I, I, I, ? , I, I} to (3.96), we get (3.78). Moreover, (3.75) and (3.76) are obtained from (3.60) and (3.61) by applying the congruence transformations diag{I, T1 } and using the property of trace, and (3.74) is derived from (3.59). [ ] ?S ?S Furthermore, if the LMIs (3.74) to (3.78) are feasible, then we have < ?S ?R [ ?1 ] S I T 0 , i.e., < 0. > 0. It follows directly from XX ?1 = I that I ? RS?1 = X12 Y12 I R Hence, one can always find square and nonsingular X12 and Y12 . Therefore, (3.84) is obtained from (3.90), which concludes the proof. ? Remark 3.3 The addressed robust H2 ?H? filter can be obtained by solving the LMIs (3.74) to (3.78) in Theorem 3.3.2. Note that the feasibility of LMIs can be checked efficiently via the interior point method. Up to now, the filter has been designed to satisfy the requirements (R1) and (R2). As a by-product, the results in Theorem 3.3.2 also suggest the following two optimization problems: (P1) The optimal H? filtering problem with H2 performance constraints for uncertain nonlinear stochastic systems: min S>0,R>0,?>0,Q1 ,Q2 ,Q3 ,Q4 ,?1 ,иии ,?q ,?1 ,?2 ? subject to (3.74) to (3.78), for some given ?. (P2) The optimal H2 filtering problem with H? performance constraints for uncertain nonlinear stochastic systems: min S>0,R>0,?>0,Q1 ,Q2 ,Q3 ,Q4 ,?1 ,иии ,?q ,?1 ,?2 ? subject to (3.65) to (3.69) for some given ?. On the other hand, in view of (3.84), we make the linear transformation on the state estimate x? (k) = X12 x? (k), (3.97) and then obtain a new representation form of the filter as follows: ? x? (k + 1) = A? ? x(k) + Ky(k), ? ? ? z? 2 (k) = L2 x? (k), ? ? z? ? (k) = L? ? x? (k), ? (3.98) 66 Variance-Constrained Multi-Objective Stochastic Control and Filtering where A? = Q1 (S ? R)?1 , K? = Q2 , L? 2 = Q3 (S ? R)?1 , L? ? = Q4 (S ? R)?1 . (3.99) We can now see from (3.98) that the filter parameters can be obtained directly by T for X12 in (3.84). solving LMIs (3.74) to (3.78) without solving I ? RS?1 = X12 Y12 Remark 3.4 In many engineering applications, the performance constraints are often specified a priori. For example, in Theorem 3.3.2, the filter is designed after H? performance and H2 performance are prescribed. In fact, however, we can obtain an improved performance by the optimization method. The aim of problem (P1) is to exploit the design freedom to meet the optimal H? performance under a prescribed ?, while the purpose of the problem (P2) is to search an optimal solution among all solutions achieving the H2 performance under a prescribed ? 2 . These are certainly attractive because the addressed multi-objective problems can be solved while a local optimal performance can also be achieved, and the computation is efficient by using the Matlab LMI Toolbox. 3.4 An Illustrative Example Consider a discrete-time system described by (3.1) with stochastic nonlinearities and deterministic norm-bounded parameter uncertainties as follows: ??0.5 0 ?0.1? ?0.3? ? 0.6 0.3 0.2 ? A=? , B1 = ? 0 ? , ? 0.1 0.4 0.1 ? ? ? ? ?0.2? ? ? [ ] C = 1 ?0.6 2 , D11 = 1, [ ] [ ] L? = 1 0.3 0.5 , L2 = 1 0 2 , ?0.5? H1 = ?0.6? , ? ? ?0? H2 = 0.6, [ ] E = 0.8 0 0 , where ?(k) is a zero mean Gaussian white noise sequence with covariance ? = 1. The deterministic uncertainty F satisfies the condition (3.2), and the stochastic nonlinear Robust Mixed H2 ?H? Filtering 67 functions f (x(k)) and g(x(k)) satisfy the following assumptions: ?{f (x(k))|x(k)} = 0, ?{g(x(k))|x(k)} = 0, T ?1? ?1? ?0.5 0 0 ? ?{f (x(k))f (x(k))T |x(k)} = ?0? ?0? x(k)T ? 0 0.8 0 ? x(k) ? ?? ? ? ? ?0? ?0? ? 0 0 0.6? T ?1 0 0 ? ?0.1? ?0.1? + ? 0 ? ? 0 ? x(k)T ?0 0.5 0 ? x(k), ? ? ? ?? ? ?0 0 0.8? ? 0 ?? 0 ? ?0.5 0 0 ? ?1 0 0 ? ?{g(x(k))g(x(k))T |x(k)} = x(k)T ? 0 0.8 0 ? x(k) + 0.1x(k)T ?0 0.5 0 ? x(k). ? ? ? ? ? 0 0 0.6? ?0 0 0.8? Now let us examine the following three cases. Case 1: ? 2 = 6, ? = 5. This case is exactly concerned with the addressed robust H2 ?H? filtering problem and hence can be tackled by using Theorem 3.3.2 with q = 2. In fact, there are many solutions for this case. We provide one solution by employing the Matlab LMI toolbox, given by ??0.1110 A? = ??0.0056 ? ??0.5967 [ L? 2 = ?0.7268 [ L? ? = ?0.9257 ? 0.6632 ? ?0.0086 0.0758 ? 0.2307 ?0.0638? , K? = ??0.8490? , ? ? ? 1.9788 0.2797 ? ??0.0875? ] 0.0730 ?0.1260 , ] 0.0696 ?0.0311 . Case 2: ? = 5. In this case, we wish to design the filter that minimizes the H? performance under the H2 performance constraints. That is, we want to solve the problem (P1). Solving the optimization problem (3.97) using the LMI toolbox yields the minimum value 2 = 5.8003, and ?min ??0.1013 A? = ? 0.1081 ? ??0.6725 [ L? 2 = ?0.7344 [ L? ? = ?0.9134 ? 0.6655 ? ?0.0319 0.0558 ? 0.2080 ?0.0457? , K? = ??0.8767? , ? ? ? 1.9790 0.2148 ? ??0.4613? ] 0.0843 ?0.1176 , ] 0.0804 ?0.0453 . 68 Variance-Constrained Multi-Objective Stochastic Control and Filtering Case 3: ? 2 = 6. We now deal with the problem (P2). Solving the optimization problem (3.97), we obtain the minimum H2 performance ?min = 3.6776, and ??0.2006 A? = ? 0.1929 ? ??0.6979 [ L? 2 = ?0.6454 [ L? ? = ?0.8115 ? 0.6159 ? ?0.0161 0.0674 ? 0.2332 ?0.0077? , K? = ??0.3776? , ? ? ? 1.9733 0.0835 ? ??0.7593? ] 0.0977 ?0.1511 , ] 0.0700 ?0.0551 . The results show that the designed system can satisfy the H2 filtering performance and the H? disturbance rejection performance simultaneously. In Case 2, in order to achieve a better disturbance rejection performance, the optimization algorithm (P1) is employed to obtain the optimal solution. Similarly, to get a better H2 filtering performance, the optimization algorithm (P2) is applied to obtain the optimal solution in Case 3. Remark 3.5 Within the LMI framework developed in this chapter, we can show that there are some trade offs that can be used for satisfying specific performance requirements. For example, the H? performance will be improved if the H2 performance constraints become more relaxed (larger). Also, if the value of the H? performance constraint is allowed to be increased, then the H2 performance can be further reduced. Hence, the proposed approach allows much flexibility in making compromises between the H2 performance and the H? performance, while the essential multiple objectives can all be met simultaneously. 3.5 Summary A robust H2 ?H? filter has been designed in this chapter for a class of uncertain discrete time nonlinear stochastic systems. A key technology is used to convert the matrix trace terms into linear matrix inequalities and to eliminate the uncertainty in the matrix inequalities. The filter is obtained under a unified flexible LMI framework. Sufficient conditions for the solvability of the H2 ?H? filtering problem are given in terms of a set of feasible LMIs. Two types of optimization problems are proposed by either optimizing the H2 performance or the H? performance. Our method can be extended to robust H2 ?H? output control.

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