Robust H Filtering for Nonlinear Stochastic Systems

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 2, FEBRUARY 2005

Robust

589

H

Filtering for Nonlinear Stochastic Systems

Weihai Zhang, Bor-Sen Chen, Fellow, IEEE, and Chung-Shi Tseng, Member, IEEE

Abstract—This paper describes the robust filtering analysis and synthesis of nonlinear stochastic systems with state and exogenous disturbance-dependent noise. We assume that the state and measurement are corrupted by stochastic uncertain exogenous disturbance and that the system dynamic is modeled by Itô-type stochastic differential equations. For general nonlinear stochastic systems, the filter can be obtained by solving second-order nonlinear Hamilton–Jacobi inequalities. When the worst-case disturbance is considered in the design procedure, a mixed 2 filtering problem is also solved by minimizing the total estimation error energy. It is found that for a class of special nonlinear stofiltering design can be given via solving chastic systems, the several linear matrix inequalities instead of Hamilton–Jacobi inequalities. A few examples show that the proposed methods are effective. Index Terms—Hamilton–Jacobi inequality, matrix inequality, nonlinear stochastic systems.

filtering, linear

I. INTRODUCTION

I

N the past few decades, the control, since it was first formulated by [1], has been extensively developed (see the celebrated paper [2] for its state-space treatment), and the discontrol can be found in [3] cussion on output feedback for nonlinear or [4] and [5] for linear uncertain systems. The filtering problem is to design an estimator to esso-called timate the unknown state combination via output measurement, gain (from the external disturbance to which guarantees the ; the estimation error) to be less than a prescribed level see [6] and [7] for a discrete-time investigation and [8]–[10] for filtering investigation. In contrast with a general nonlinear the well-known Kalman filter, one of the main advantages of filtering is that it is not necessary to know exactly the statistical properties of the external disturbance but only assumes the external disturbance to have bounded energy. See [11]–[13] filtering in signal processing. for practical applications of filtering and control problems In recent years, stochastic with system models expressed by Itô-type stochastic differential Manuscript received June 16, 2003; revised January 20, 2004. This work was supported by National Science Council under Contract NSC 91-2213-E007-014 and by the Chinese Natural Science Foundation under Grant 60474013. The associate editor coordinating the review of this paper and approving it for publication was Dr. Yuan-Pei Lin. W. Zhang is with the Shenzhen Graduate School, Harbin Insitute of Technology, HIT Campus, Shenzhen 518055 China (e-mail: [email protected]). B.-S. Chen is with the Department of Electrical Engineering, National Tsing Hua University, 30043 Hsin-Chu, Taiwan, R.O.C. (e-mail: [email protected]). C.-S. Tseng is with the Department of Electrical Engineering, Ming Hsin University of Science and Technology, 30401 Hsin Feng, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2004.840724

equations have become a popular research topic and has gained extensive attention; see [14]–[19] and the references therein. We filsummarize below the recent development on stochastic tering problem. In [15], a bounded real lemma was presented for linear continuous-time stochastic systems, according to which full- and estimation problems for stationary reduced-order robust continuous-time linear stochastic uncertain systems were discussed by [14] and [17], respectively. All the above works are limited to the linear stationary stochastic systems, whereas [18] investigated the same problem for a class of special nonlinear stochastic systems. In [16] and [19], the linear and nonlinear control problems have been discussed. stochastic control There have been a lot of studies on nonlinear or state estimation in deterministic systems; see, e.g., [3], [6], [8], and [9], and the references therein. However, it should be noted that up to now, there is little corresponding work on the general nonlinear stochastic systems governed by Itô-type stochastic differential equations, which has many applications in practice [20]. Unlike the deterministic case, the Hamilton–Jacobi inequality (HJI) associated with the nonlinear stochastic filter is a second-order (not first-order) nonlinear partial differential inequality due to the effect of the diffusion term, which filtering problem more complex. makes the stochastic In the present paper, we discuss the infinite horizon robust state estimation for nonlinear stochastic uncertain systems, assuming that the system state is corrupted not only by white noise, but also by exogenous disturbance signal, and the measurement output is also corrupted by exogenous disturbance. Our goal in this paper is to construct an asymptotically stable (in some sense) observer that leads to a stable estimation error -gain with respect to uncertain disturbance process whose signal is less than a prescribed level. In Section II, for nonlinear perturbed stochastic systems with state and external disturbance-dependent noise, the robust estimation problem is investigated. As in deterministic case, filtering problem, stawhen we study the infinite horizon bility is an essential requirement. That is, we should search for filter, which makes the augmented a stable (in some sense) system to be asymptotically stable in probability. Unlike most previous work on nonlinear filtering problem (e.g., [9] and [10]), a very general form of the filter is considered, which needs us , and via solving to determine three parameter functions a second-order nonlinear HJI. An exact form of HJI associated filtering is derived. Meanwhile, a subopwith nonlinear filter is also studied, that is, of all the timal mixed filters, we seek one to minimize an upper bound of the total es-

1053-587X/$20.00 © 2005 IEEE

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timation error energy when the worst-case disturbance (from to ) is considered in the design procedure. For a class of special nonlinear stochastic systems, Section III presents a linear matrix inequality (LMI)-based algorithm for filtering design, and solving the LMIs is much its robust easier than solving the HJI. Section IV presents some examples to illustrate our proposed methods. Section V concludes this paper. For convenience, we adopt the following notations: Trace (transpose) of matrix . Tr Positive semidefinite (positive definite) matrix . Identity matrix. Euclidean 2-norm of the -dimensional real vector . space of nonanticipative stowith rechastic processes spect to filtration satisfying . Class of functions twice continuously differential with re, except possibly spect to at the origin. II. NONLINEAR

filtering problem, it is inevitably related to stostochastic chastic stability, see [21] for the definitions of “globally asymptotic stability in probability” and “exponentially mean square stability.” The following proposition is a special case of [16], which plays an important role in this paper. Proposition 1: For system (1), if the state information is completely available and there exists a positive function solving the following HJI:

(3) then

(4) with initial state . holds for some For convenience, we give its proof as follows. be the infinitesimal generator of (1), which Proof: Let is defined as

FILTERING

Consider the following nonlinear stochastic system (the time variable is suppressed): Applying “completing the square,” we have (1)

is called the system state, In the above, is the measurement, is the state combination to stands for the exogenous be estimated, and disturbance signal. , and are smooth functions . is a standard onewith dimensional (1-D) Wiener process defined on the probability relative to an increasing family of space -algebras . In (1), the state equation, in engineering terminology, can be written as [20]

(2) which means that the state and external disturbance are dependent on the same noise, where is a stationary white noise. In particular, along the lines of this paper, it is easy to treat with filtering problem for the system that the state the nonlinear and external disturbance are dependent on uncorrelated noise. When the system state is not completely available, to estimate from the observable information with an -gain (from to the estimation error) less than a pre, a nonlinear stochastic filter should scribed level be constructed. Since this paper deals with the infinite horizon

ZHANG et al.: ROBUST

FILTERING FOR NONLINEAR STOCHASTIC SYSTEMS

Considering (3), we immediately have

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one can see that, for any

with , where

That is, achieves the maximal possible energy gain from the disturbance input to the controlled output . Thereefore, in can be viewed as the worst case disturbance. this case, In what follows, we construct the following estimator equation for the estimation of (5)

By integrating and taking expectation from 0 to , we have

, and , which are to be determined, are where matrices of appropriate dimensions with sufficient smoothness. ; then, we get the following augmented system: Set (6) where

Because

, therefore

In addition, let

Letting in the above, (4) is followed, accordingly, by Proposition 1. Remark 1: From the Proof of Proposition 1, it is easily seen and can be weakened to become that in (3), and , respectively. Here, we take a stronger and ) only for the purpose of condition (i.e., simplicity in applications. Remark 2: Proposition 1 can be called a bounded real lemma (BRL) of nonlinear stochastic systems; a linear stochastic BRL can be found in [15]. In addition, one anonymous reviewer pointed out that a more general BRL for nonlinear stochastic systems was derived in [22]. Within the frame of stochastic game theory, [23] studied a certain type of minimax dynamic game for stochastic nonlinear systems, and an Hamilton-Jacobi equation, which contains an Hamiltonian function, was derived. Remark 3: If we let

denote the estimator error; then, the nonlinear stochastic filtering can be stated as follows: Find filter gain matrices , and in (5) such that we have the following. of the augmented system (6) 1) The equilibrium point is globally asymptotically stable in probability in the case . , the fol2) For a given disturbance attenuation level lowing relation holds: (7) Some main results of this section are listed as follows. Theorem 1: For given disturbance attenuation level if there exists a positive Lyapunov function solving the following HJI:

,

(8)

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for some matrices , and of suitable dimensions, then the filtering problem is solved by (5), where stochastic

Corollary 2: If there exists a positive constant and such that the a positive Lyapunov function conditions where filter gain matrices , and of suitable , then dimensions hold for some disturbance attenuation filtering problem is solved by (5). the stochastic )

To prove Theorem 1, the following lemma on the globally of asymptotic stability of the equilibrium point

(12) )

(9) is needed. Lemma 1 [21]: Assume there exists a positive Lyapunov function satisfying for all nonzero ; then, the equilibrium point of (9) is globally asymptotically stable in probability. Proof of Theorem 1: We first show that (7) holds. Applying Proposition 1 to the system (6), we immediately have that (7) holds if

(13) )

(10) (14)

for admits solutions and . By a series of computations, (10) is equivalent to (8). Second, we show that of the augmented system (6) to be globally asymptotically stable in probability in the case . By Lemma 1, we only need to prove for some , i.e.,

Proof of Corollary 2: Applying the following well-known fact: (15) it follows that

(11)

is defined as the infinitesimal generator of system (6). where While (11) is obvious because of (8), the Proof of Theorem 1 is complete. More specifically, if , i.e., only the state-dependent noise, Theorem 1 yields the following corollary. in (1), if the following HJI Corollary 1: For

By condition

(16)

admits solutions for and for some , then the stochastic problem is solved by (5), where in this case

filtering

by Since it is easy to test that and , this corollary is shown. In general, (13) and (14) are a pair of coupled HJIs, but if as the form of , then (13) we take and (14) become decoupled and can be solved independently. , Corollary 2 yields the following. Especially, for



Corollary 3: The consequence of Corollary 2 still there exists a positive Lyapunov function holds if solving the HJI

(17)

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Generally speaking, the design of robust filter is not filters, we seek one that minimizes the unique. Of all the when the worst-case disturtotal error energy bance (from to ) is considered in the design procedure with . This is the initial state filtering design problem. In the sense so-called mixed of Remark 3, it can be seen that the worst-case disturbance from to is of the following form:

also solves HJI

(18) with some matrices , and of suitable dimensions. . AddiProof: Note that in this case, we can take tionally

(22) where the above

is an admissible solution of (10) or (8). Substituting into (6) yields

(23) Theorem 2: For any prescribed disturbance attenuation level , if there exists a positive Lyapunov function solving the following HJI: The rest is omitted. Under a standard assumption [24] , repeating the same procedure as in Corollary 2, we have the following result. – of Corollary 2 are replaced by Corollary 4: If , and , as shown below, respectively, then for , the consequence of Corollary 2 still holds. any )

(24) for some filter gain matrices , and of suitfilter can able dimensions, then a suboptimal mixed be synthesized by solving the following constraint optimization problem:

(19) )

s.t.

(20) )

(25)

from (24); therefore, the Proof: First, we have filtering problem is solved by (5). when of (22) Second, we assert . To prove is implemented in (23), i.e., , by Itô’s formula, this assertion, we first note that for any we have

(21) Proof: We only need to note that under the condition of because of the assumption . Remark 4: In most literature (e.g., [3]) on deterministic nonlinear control or filtering, one often assumes for simplicity, which implies that the disturbances and to state and measurement , respectively, are independent. Under the above assumption, (19) comes down to (26)

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In addition, still using Itô’s formula, for that

, it follows

or mixed (24). From Theorem 2, to synthesize nonlinear filter, one should solve HJI (8) or constraint optimization problem (25); neither is easy. Maybe, only in some special cases, such as for linear time-invariant systems, one can give an applicable design algorithm. Moreover, from the Proof of Theis a tighter upper bound of due orem 2, . Especially when , then to from (26), it follows that

(27) Letting for

Therefore, with respect to

, (26) follows

We find that for general nonlinear stochastic system (1), to filter, one needs to solve HJI (8), which is design its robust not an easy thing. However, for a class of special nonlinear stochastic systems, the above-mentioned problem can be converted into solving LMIs, as done in [18]; therefore, a numerical algorithm is admissible. It is well known that using the LMI-based control for both deterministic and stotechnique to study chastic systems has become a popular approach in recent years; see [15], [22], [27], and the references therein. We consider below the following special nonlinear stochastic system governed by Itô differential equation:

(28)

(30)

. Moreover

a.s.

III. LMI-BASED APPROACH FOR A CLASS OF FILTERING NONLINEAR

is a non-negative supermartingale . Additionally, from (24), we have

where

with measurement output (31)

(29) Hence, (23) is globally asymptotically stable in probability. By Doob’s convergence theorem [25], . Moreover, . in (26) and applying the above asserFinally, taking tion yields

and are 1-D standard Wiener processes. where and to Without loss of generality, we also assume , , and are conbe mutually uncorrelated. stant matrices of suitable dimensions, and still represents the exogenous disturbance signal. Equation (30) is a special case of the state equation of (1) with only state-dependent noise, for the purpose of simplicity. As a matter of , and regards fact, in (1), if one takes and as the Taylor’s series expansion of and , respectively, then the state equation of (1) comes down to (30). In (31), we assume that the measurement output, as in [14], is expressed by a stochastic differential equation. For the special nonlinear systems (30) and (31), we take the (see [26] for the following linear filter for the estimation of treatment of discrete-time nonlinear systems): (32)

Theorem 2 is concluded, i.e., by solving (25), a suboptimal filter is obtained. mixed It should be mentioned that HJI (8) for nonlinear filtering is implied by (24); therefore, for a suboptimal mixed filtering problem, we only need to minimize subject to

where

. Still, letting

, then

(33)



where

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Applying (15) together with (37) and (39), we have the estimafor simplicity) tions as (taking

(41) (34) For a prescribed disturbance attenuation level and such that to find constant matrices

(42) (43)

, we want Substituting (41)–(43) into (40), we have (35)

. Define the

holds for any index as

(44)

performance Obviously, if (38) holds, then there exists

such that

(36) Obviously, the filtering performance (35) holds iff . -based robust state estimation problems As in [14], the , are formulated as follows: Given a prescribed value find an estimator in (32) leading (33) to be exponentially mean ; Moreover, for all square stable in the case of with . nonzero In this section, a sufficient condition is given for stochastic filtering design of systems (30) and (31). Now, we first give a lemma as follows. Lemma 2: Suppose there exists a scalar such that

Therefore, , which yields (33) being exby Lemma 2. ponentially mean square stable for Second, we prove for all nonzero with . Note that for any

(37) If the following matrix inequalities

(38) (39) have solutions square stable when where

, then (33) is exponentially mean , and the performance , .

Remark 5: Equation (37) is, in fact, a very loose condition, satisfies the globally Lipschitz which means that filcondition at the origin. If, in the definition of stochastic tering, we demand (33) to be locally exponentially mean square , then it is only necessary that (37) holds in the stable neighborhood of the origin. Proof: We first prove (33) to be exponentially mean . Take the Lyapunov candidate as square stable when , where is a solution to (38) and (39), and be the infinitesimal operator of (33); then let

Therefore, if

(45) (40)

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then there exists for any nonzero

, which yields . As (45) is equivalent to (38) by Schur’s complement [27], the Proof of Lemma 2 is completed. Lemma 2 is only of theoretical value because it is inconvefilter. The following result is more nient for designing the suitable to use in practice. Theorem 3: Under the conditions of Lemma 2, if the following LMIs

diag , then (39) is equivalent to (46). If we take Substituting (34) into (49), we have

(50)

where we have the the equation at the bottom of the page, which is equivalent to

(46)

(47) (51)

have solutions , then (33) is exponentially mean square stable for filtering performance holds, and the

,

(48) filter. is the corresponding Proof: By Schur’s complement, (38) is equivalent to

(49)

Letting , (51) becomes (47). From , and therefore, an our assumption, filter is constructed as in the form of (48), and the proof of Theorem 3 is completed. Based on the above discussion, we summarize the following design algorithm. Design Algorithm: and by solving Step i) Obtain solutions , and . LMIs (46) and (47), , and substitute Step ii) Set the just obtained into (32); then, (32) is the filter. desired Remark 6: If, in (30) and (31), the external disturbance signal is regarded as a white noise, following the line of [14], one can further discuss the mixed filtering problem for this filkind of special nonlinear system. That is, of all the ters, we select one that minimizes the estimation error variance .



IV. SOME ILLUSTRATIVE EXAMPLES Below, we give some examples to illustrate our developed theory in the above sections. Example 1 (1-D Nonlinear Filtering Design): Suppose a stochastic signal is generated by the following nonlinear stochastic system driven by a Wiener process and corrupted by a stochastic external disturbance ; we therefore construct an filter to estimate from the measurement signal .

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and solving these two LMIs together with constraints and , we have

Then

Assuming that the disturbance attenuation is prescribed as , and the state estimator is as the form of (5), the augmented system takes the form of (6) with

(52) filter.

is the corresponding

V. CONCLUSION

Setting

, it is then easy to test that HJI if we take . filter is obtained as the form of Therefore, the robust

Clearly, there may be more than one solution to HJI (8). In genstate estimator is not unique. eral, the robust Filtering Design for Special Nonlinear Example 2 ( System): Consider the filtering design of nonlinear stochastic signal processes (30) and (31) with

filtering In this study, we have discussed the robust problem for affine nonlinear stochastic systems with state and external disturbance-dependent noise, and a bounded real lemma for stochastic nonlinear systems is derived. To obtain filter of nonlinear stochastic systems, one should solve the an HJI, which is a second-order nonlinear partial differential filtering analysis inequalities. Meanwhile, the mixed is also discussed, where the performance is minimized when the effect of the worst-case disturbance is considered. filtering design for a class of special nonlinear Finally, the systems is also solved via the LMI Toolbox technique [28] instead of solving the HJI, giving a more convenient algorithm for practical applications [27]. REFERENCES [1] G. Zames, “Feedback and optimal sensitivity: Model reference transformation, multiplicative seminorms, and approximative inverses,” IEEE Trans. Autom. Control, vol. AC-26, no. 2, pp. 301–320, Apr. 1981. [2] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. Francis, “State-space solutions to standard and problems,” IEEE Trans. Autom. Control, vol. 34, no. 8, pp. 831–847, Aug 1989. [3] A. Isidori and A. Astolfi, “Disturbance attenuation and -control via measurement feedback in nonlinear systems,” IEEE Trans. Autom. Control, vol. 37, no. 9, pp. 1283–1293, Sep. 1992. [4] L. Xie, “Output feedback control of systems with parameter uncertainty,” Int. J. Control, vol. 63, pp. 741–750, 1996. [5] L. Xie, M. Fu, and C. E. de Souza, “ control and quadratic stabilization of systems with parameter uncertainty via output feedback,” IEEE Trans. Autom. Control, vol. 37, no. 8, pp. 1253–1256, Aug. 1992. [6] C.-S. Tseng and B.-S. Chen, “ fuzzy estimation for a class of nonlinear discrete-time dynamic systems,” IEEE Trans. Signal Process., vol. 49, no. 11, pp. 2605–2619, Nov. 2001. [7] M. J. Grimble and A. El-Sayed, “Solution of the optimal linear filtering problem for discrete-time systems,” IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 8, pp. 1092–1104, Jul. 1990. [8] W. M. McEneaney, “Robust/ filtering for nonlinear systems,” Syst. Control Lett., vol. 33, pp. 315–325, 1998. [9] C. F. Yung, Y. F. Li, and H. T. Sheu, “ filtering and solution bound for nonlinear systems,” Int. J. Control, vol. 74, pp. 565–570, 2001.

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Obviously, satisfies (37) with Substituting the above data into (46) and (47) with

.

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Weihai Zhang was born in 1965 in Shandong Province, China. He received the M.Sc. degree from Hangzhou University, Hangzhou, China, and the Ph.D. degree from Zhejiang University, Hangzhou, in 1994 and 1998, respectively. From May 2001 to July 2003, he was a Postdoctoral Researcher at National Tsing Hua University, Hsin-Chu, Taiwan, R.O.C. He is now a Research Fellow with Shenzhen Graduate School, Harbin Insitute of Technology, HIT Campus, Shenzhen, China. His research interests are in linear and nonlinear stochastic control, robust filtering, and stochastic stability.

Bor-Sen Chen (M’82–SM’89–F’01) received the B.S. degree from Tatung Institute of Technology, Taipei, Taiwan, R.O.C., the M.S. degree from National Central University, Chungli, Taiwan, and the Ph.D degree from the University of Southern California, Los Angeles, in 1970, 1973 and 1982, respectively. He was a Lecturer, Associate Professor, and Professor at the Tatung Institute of Technology from 1973 to 1987. He is currently a Tsing Hua Professor of electrical engineering and computer science at National Tsing Hua University, Hsinchu, Taiwan. His current research interests are in control engineering, signal processing and system biology. He is a member of the Editorial Advisory Board of two journals (the International Journal of Fuzzy Systems and the International Journal of Control, Automation and Systems) and editor of the Asian Journal of Control. Dr. Chen has received the Distinguished Research Award from National Science Council of Taiwan four times. He is a Research Fellow of the National Science Council of Taiwan and holds the excellent scholar Chair in engineering. He has also received the Automatic Control medal from the Automatic Control Society of Taiwan in 2001. He is an associate editor of IEEE TRANSACTIONS ON FUZZY SYSTEMS He is a Fuzzy Systems Technical Committee member of the IEEE Neural Network Council.

Chung-Shi Tseng (M’01) received the B.S. degree from the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., the M.S. degree from the Department of Electrical Engineering and Computer Engineering, University of New Mexico, Albuquerque, NM, and the Ph.D. degree in the electrical engineering from National Tsing-Hua University, Hsin-Chu, Taiwan. He is now an Associate Professor at Ming Hsin University of Science and Technology, Hsin-Feng, Taiwan. His research interests are in signal processing, nonlinear robust control, fuzzy control, and robotics.