STOCHASTIC STABILITY OF EXTENDED FILTERING FOR NONLINEAR SYSTEMS WITH MEASUREMENT PACKET LOSSES

Transcription

1 Proceedings of the IASTED International Conference Modelling, Identification and Control (AsiaMIC 013) April 10-1, 013 Phuet, Thailand STOCHASTIC STABILITY OF EXTENDED FILTERING FOR NONLINEAR SYSTEMS WITH MEASUREMENT PACKET LOSSES Gang Wang, Jie Chen, Jian Sun School of Automation Beijing Institute of Technology Beijing 10081, China ABSTRACT This paper is concerned with stochastic stability of a new extended filtering for nonlinear systems subject to measurement pacet losses. The measurements sensored are transmitted to remote estimator through a pacet-dropping networ. By introducing a time-stamped pacet arrival indicator sequence, the measurement loss process is modeled as an independent, identically distributed (i.i.d.) and therefore a Bernoulli process. Boundedness of estimation error covariance matrices is studied by verifying existence of a critical threshold for measurement pacet arrival probability. It is also shown that, under appropriate assumptions, the estimation error remains bounded as long as noise covariance matrices and initial estimation error can be ensured small enough. KEY WORDS Extended filtering, nonlinear systems, stochastic stability, measurement pacet losses. 1. Introduction In recent years, considerable attention has been paid to networed control systems where the communication between sensors, controllers, and actuators is often realized through a shared networ medium [, [1, [3. Networed control systems possess many advantages over the traditional point-to-point control systems, such as low cost, reduced power requirements, easy installation, simple maintenance and high reliability. However, transmitting the observed data and control signals across the networ inevitably results in some newly emerged problems including, but not limited to, random time-delay, pacet loss, quantization and channel fading, which render the estimation and control for networed control systems significantly challenging [4, [6, [5. In particular, the state estimation problem across a networ has gained recurring research attention in the literature. Pioneering wor on linear estimation with measurement losses can be traced bac to [7, [9 and [8. In [7, the pacet loss process was modeled as an i.i.d. Bernoulli process and then the linear minimum mean square error (LMMSE) estimator with filter iterations similar to the standard Kalman filter by only utilizing the statistics of pacet arrival indicator sequence was derived. In [9, a sufficient and necessary condition for the existence of the linear recursive estimator when the pacet loss sequence was not necessarily i.i.d. was established. Moreover, the condition for the uniformly asymptotic stability of the LMMSE estimator was developed for linear systems with measurements corrupted by white multiplicative noise in [8. It s worth mentioning that in this case the error covariance matrix is governed by a deterministic equation iteration and therefore an equivalent linear system without measurement losses can be constructed to facilitate the asymptotic stability analysis of the LMMSE estimator. Since widely application of the clairvoyant Kalman filter, ranging from tracing, detection to control, Kalman filtering with intermittent observations has attracted relative attention in last decade; see [3, [10, [11, [13, [1. In [10, the effects of pacet loss resulted from the unreliability of the networ on the stability and performance of Kalman filtering were investigated for Bernoulli pacet loss process. In addition, inspired by the uncertainty threshold principle, it s also been shown there exists a critical value for the pacet arrival probability such that the averaged estimation error covariance matrix will be bounded for any initial condition if the probability exceeds the critical probability; otherwise, the averaged estimation error covariance matrix will diverge for some initial condition. To capture the possible transient correlation of networ channel variations, the pacet loss process was modelled as a timehomogenous two-state Marov chain in [11 and sufficient conditions for the introduced pea covariance stability for general vector systems were established therein. Followed by [13 and [14, necessary and sufficient conditions guaranteeing stability of the averaged estimation error covariance matrix were developed for second-order systems with respect to different system structures and for certain classes of higher-order systems. In cases above, Kalman filtering with intermittent observations under different pacet loss models is studied by only utilizing the time-stamped pacet loss indicator sequence, which renders the estimation error covariance matrix iteration stochastic and therefore poses significant challenges to analyze its stability. More recently, however, a suboptimal estimator for linear systems with Bernoulli pacet losses was proposed in [15, by combining the use of time-stamped measurement innovation sequence DOI: /P

2 and statistics of the indicator stochastic variable when designing the filter, to balance the performance and the stability analysis. In the presence of nonlinearities, extended Kalman filtering with intermittent observations (called intermittent extended Kalman filter for short) and unscented Kalman filtering with intermittent observations (similarly, called intermittent unscented Kalman filter for short) have recently been studied in [16, [0 and [17, respectively. Followed the spirit of [18, conditions for guaranteeing stochastic stability of the intermittent extended Kalman filter and the intermittent unscented Kalman filter are developed, respectively. Additionally, it has been shown in [16 and [17 that the existence of a critical value for pacet arrival probability to ensure statistical convergence of the averaged error covariance matrix. The present paper, which can be seen as a further complementary to the prior wor mentioned above, is concerned with the issue of estimating the state of a general nonlinear system, where the state estimate is based upon the observed data provided by an unreliable sensor networ. The contributions of this paper include: Similar to the intermittent extended Kalman filter, we first extend the recently proposed linear suboptimal filter in [15 to the nonlinear case and derive a extended filter for which, comparing with the intermittent extended Kalman filter in [16, the filter gain can be computed offline. We then generalize the concept of uniform observability to the case considered in this paper. Unlie the statistical convergence properties of the averaged error covariance matrices for the intermittent extended Kalman filter, the asymptotic convergence properties of the error covariance matrices can be derived for the proposed estimator under appropriate assumptions. Moreover, conditions for guaranteeing stochastic stability of the estimation error of the proposed extended filter are also obtained. The remainder of this paper is organized as follows. We briefly introduce the nonlinear system to be considered in Section. and establish the proposed extended filter in Section 3., which is followed by our main results on the boundedness of the error covariance matrices and the s- tochastic stability of the estimation error in Section 4. and Section 5., respectively. Finally, the conclusion is drawn in Section 6.. Notations: Throughout this paper, the set of all nonnegative integers is denoted by N; the Euclidean norm for real vectors or the spectral norm for real matrices is denoted by. Furthermore, We use P > 0 ( 0) to represent the positive definite (positive semi-definite) matrix P, use E{x} to denote the expectation value of x, and use C 1 to denote all the continuously differentiable functions.. System Description and Preliminaries Consider the following nonlinear discrete-time stochastic system with measurement pacet losses x +1 = f(x ) + ω (1) y = γ h(x ) + υ () where = 0, 1,... is the time instant, x R n is the state vector with the initial state x 0, y R p is the measurement vector, and the process noise ω R n, measurement noise υ R p are both zero-mean white Gaussian vectors with covariance matrices E{ω ωj T } = Q δ j, E{υ υj T } = R δ j, respectively, where δ j is the Kronecer delta function. The initial state x 0 is also assumed to be a zeromean white Gaussian random vector with covariance matrix E{x 0 x T 0 } = P 0 > 0. Moreover, the nonlinear functions f, h are assumed to be C 1 functions (C 1 denotes all the continuously differentiable functions). For clarity, system (1) is considered autonomous, however, the results p- resented in this paper can be readily generalized to the controlled nonlinear systems. The pacet arrival indicator variable γ is assumed to tae binary values on 0 and 1. Moreover, the random process is characterized by parameter λ with Pr{γ = 1} = λ (3) Pr{γ = 0} = 1 λ (4) where the pacet arrival probability λ [0, 1 is nown, and the sequence {γ }, the noise processes ω, υ and the initial state x 0 are assumed to be mutually independent for all N. 3. Derivation of Extended Filter In this section, the extended filter for general nonlinear systems with measurement pacet losses will be constructed. To proceed, the following assumption on pacet arrival indicator sequence is stated. Assumption 1 γ is supposed to be time-stamped, i.e., the value of γ at every time instant can be observed. So the information γ together with observation y are available in the estimator design. Linearize nonlinear functions f and h at points ˆx, ˆx, respectively. f(x ) =f(x ) + f (x ˆx ) + ϕ(x, ˆx ) ˆx h(x ) =h(ˆx ) + h (x ˆx ) ˆx + ψ(x, ˆx ). Denote by F = f and H ˆx = h the Jacobian ˆx matrices of nonlinear functions f, h at points ˆx, ˆx 18

3 respectively. Then the linearized approximation of the o- riginal nonlinear system becomes x +1 =F x + ω + [ f(ˆx ) F ˆx + ϕ(x, ˆx ) y =γ H x + υ + [ γ h(ˆx ) γ H ˆx + γ ψ(x, ˆx ). The linear suboptimal filter proposed in [15 is ˆx =ˆx + λp H T ˆx +1 =F ˆx (λhp H T + R) (y γ H ˆx ) P =P λ P H T P +1 =F P F T + Q. (λhp H T + R) HP So the proposed extended filter can be stated as follows ˆx =ˆx + λp H T (λh P H T + R ) (y γ h(ˆx )) (5) ˆx +1 = f(ˆx ) (6) P =P λ P H T (λh P H T + R ) H P (7) P +1 = F P F T + Q. (8) In order to compare the extended filter proposed in this paper and the intermittent extended Kalman filter proposed in [16, the intermittent extended Kalman filter is presented as follows for completeness. ˆx =ˆx + γ P H T (H P H T + R ) (y h(ˆx )) (9) ˆx +1 = f(ˆx ) (10) P = P γ P H T (H P H T + R ) H P (11) P +1 = F P F T + Q. (1) Remar 1 As pointed out in Section 1., the intermittent extended Kalman filter iteration (see (11)-(1)) is very much involved with the stochastic pacet arrival indicator sequence {γ }, so the intermittent extended Kalman filter iteration is inherently stochastic and therefore cannot be determined offline. Hence, only statistical properties can be derived. Note, however that the extended filter iteration proposed in this paper (see (7)-(8)) is readily expressed in terms of expectation value of random variable γ, is deterministic and asymptotic convergence can be derived, which can be seen as one characteristic property of our proposed estimator. 4. Boundedness of Estimation Error Covariance Matrices In this section, some lemmas and assumptions are introduced before the boundedness of the error covariance matrices resulted from the proposed filter (7) and (8) is verified. Lemma 1 ([16) Suppose that UCU T < A holds for matrix U R n n and symmetric positive definite matrices A, C R n n, then U T A U < C holds. Lemma ([16) For symmetric positive definite matrices A, C R n n, (A + C) > A A CA holds. Before moving on, the observability condition for general linear time-varying systems is discussed for the proposed extended filter. Consider the following linear time-varying system missing measurement pacet losses x +1 = F x + ω (13) y = γ H x + υ (14) where system parameter matrices F, H depend on the estimates ˆx, ˆx, respectively and Bernoulli random variable γ is time-stamped with E{γ } = λ. Similarly to [16, the concept of uniform observability for linear timevarying systems can be also extended to systems with measurement pacet losses discussed in this paper. Definition 1 Let the generalized observability Gramian be given by +s ˇM +s, = λφ T i,( λh i ) T ( λh i )Φ i, (15) i= where the transition matrix Φ i, = F i F i+1 F with Φ i,i = I n. Then the matrix pair (F, H ) is said to be uniformly observable if there exist some integer s > 0 and two positive real constants m, m > 0 such that the generalized observability Gramian ˇM +s, satisfies for every N. 0 < mi n ˇM +s, mi n (16) Assumption There exist positive real constants f, h, h, q, q, r, r such that the following bounds on system parameter matrices hold for every N F f (17) h I n H T H h I n (18) qi n Q qi n (19) ri p R ri p. (0) Remar Observe, that if f < 1, the desired bounds for the error covariance matrices (1) subject to (7)-(8) can be derived directly from Assumption, even for the worst case λ = 0, i.e., the desired bounds are independent of the measurement process if f < 1. 19

4 In the sequel, the Bernoulli measurement pacet loss process with a nonzero parameter λ will be discussed and the bounds for the error covariance matrices will be derived. Theorem 1 Under Assumption and assumption that (F, H ) is uniformly observable, that H exists for every N, then there exists a critical value for pacet arrival probability λ c = 1 1 such that the error covariance matrices are bounded provided that λ > λ c, i.e., there exists a f positive real constant pair p, p such that pi n P P +1 pi n. (1) Proof: Since H is invertible for every N, given that (F, H ) is uniformly observable and thereby detectable, even for λ = 1, the lower bound can be obtained directly from [19. Clearly, according to (7)-(8), we have P +1 = F [ P λ P H T ( λh P H T + R ) H P F T + Q. Setting A = λh P H T, C = R and using Lemma to the inverse term above yields P +1 (1 λ)f P F T + F H R H T F T + Q. Considering the bounds on matrices H, R, Q, we have P +1 (1 λ)f P F T + r h F F T + qi n ( rf (1 λ)f ) P + h + q I n. Recursively, it follows that P +1 { [(1 λ)f P1 0 ( rf + h ) + q [ (1 λ)f } i I n. () i=0 Denote p = max { ( P1 0, rf h + q )} and rewrite () as follows [ P +1 p (1 λ)f i In, 0. (3) i=0 It s noteworthy that under the assumption λ > 1 1, the f p sum in (3) converges to p = and therefore the 1 (1 λ)f upper bound in (1) follows directly from (3). Remar 3 It should be noted that, similarly to the intermittent extended Kalman filter for general nonlinear systems, Riccati-lie iteration of estimation error covariance P can be just seen as a first-order approximation to the true error covariance which, nevertheless, does not generally possess a linear iteration. Therefore, boundedness of error covariances P, P do not necessarily imply stochastic stability of the estimation error e. Then study of the behavior of the averaged estimation error E{e } in Section 5. are of significant implications. 5. Stochastic Stability of Estimation Error In this section, some appropriate assumptions are given before the estimation error resulted from the proposed extended filter will be shown to be bounded. Assumption 3 There exist positive real constants f, f, h, h, q, q, r, r such that the following bounds on system parameter matrices hold for every N f I n F T F f I n (4) H h (5) qi n Q qi n (6) ri p R ri p. (7) Assumption 4 For any positive real constant pair ɛ ϕ, ɛ ψ, there exists another positive real constant pair δ ϕ, δ ψ such that the following two inequalities hold for all x ˆx δ ϕ and x ˆx δ ψ, respectively, ϕ(x, ˆx ) ɛ ϕ x ˆx (8) ψ(x, ˆx ) ɛ ψ x ˆx. (9) Assumption 5 There exist positive real constants p, p such that the error covariances P, P are bounded, i.e., pi n P P pi n. (30) In other words, there exists a set D for λ such that P, P are bounded for λ D. Remar 4 Notice, that Theorem 1 derived explicit condition for pacet arrival probability λ to ensure the boundedness of error covariances for systems where H exists. However, no explicit condition for λ for more general timevarying systems has been given in the literature. To lessen this strict requirement, we directly assume the boundedness of error covariances in Assumption 5, also see [16, [17, which will be used in Theorem. Lemma 3 Let f(λ) be f(λ) = (1 + θ) [ 1 + δ(λ λ ) where positive constants θ = q ( ), δ = 1 + pf h pf pf d and, d are defined in (34), (37), respectively. Then there at least exists a set of the form S = (λ n, 1 such that f(λ) < 1 holds for all λ S. Proof: Note, that f(λ) < 1 holds at least for both λ = 1 and 0. However, for λ = 0, the assumption on the boundedness of the covariance matrices P and P +1 can be easily violated except the very case f < 1. Moreover, f(λ) is continuous over [0, 1 and reaches its maximum value at point λ = 1/. From λ = 1/ to λ = 1, f(λ) decreases. Considering f(1) = (1+θ) < 1 and the continuity of f(λ), then there always exists a neighborhood of point λ = 1, denoted by S = (λ n, 1, in which f(λ) < 1 holds. 130

5 Theorem Consider the nonlinear stochastic system described by (1)-() and the extended filter given by (5)-(8). Under Assumption 3-4 and assumption that λ D S, if there exists a real constant ɛ > 0 such that E{ e 1 0 } ɛ, then the state estimation error e as stated in (31) is exponentially bounded in mean square and bounded with probability one. Proof: Since λ D S, according to Assumption 5, there exists positive real constants p, p such that pi n P P pi n. From (5)-(6), the state estimation error is defined by e +1 =x +1 ˆx +1 =F (I γ K )e + r + s (31) where, we define r = ϕ γ F K ψ and s = ω F K υ. Define V (e ) = e T P e, then V +1 (e +1 ) = e T +1 P +1 e +1 = [ F (I γ K H )e + s + r T P +1 [ F (I γ K H )e + s + r = e T (I γ K H ) T F T P +1 F (I γ K H ) e + s T P +1 s + s T P [ +1 F (I γ K H )e + r + r T P [ +1 F (I γ K H )e + r. (3) For the first term above, it yields { } E (F γ F K H ) T P +1 (F γ F K H ) = (F λf K H ) T P +1 (F λf K H ) + (λ λ )(F K H ) T P +1 (F K H ) [1 + (λ λ )pf h pf d (I n λk H ) T F T P +1 F (I n λk H ) (33) where we denote by, d the upper bound on the matrix norm K and the lower bound on the matrix norm D = I n λk H, respectively, which are calculated as follows. Since K = λp H T (λh P H T + R ) and (4)-(7), considering λh P H T 0, it easily follows K λph r ph r =. (34) Now d is calculated as follows. In (7), it states that (I n λk H )P = (1 λ)p + λ [ P P ( λh ) T ( λh P λh T + R ) ( λh )P, (35) then using the matrix inversion lemma [1 to the second term in last equation yields P = (I n λk H )P ( = (1 λ)p + λ P + λht R ) H. (36) Owing to the positive definiteness, therefore the invertibility of covariance matrices P, P, it can be obtained that I n λk H = [ (1 λ)p + λ ( P + λht R H ) P, where, P > 0, (P + λht R H ) > 0, so [(1 λ)p + λ(p + λht R H ) > 0, P > 0. According to matrix theory on eigenvalue and singular value estimates of product of two positive definite matrices, see [, Theorem 3.1, it yields I n λk H = [ (1 λ)p + λ ( P 1 p p p [ (1 λ)p + ( 1 λ + + λht R λ H ) P p + λh r λr ) p = d. (37) r + λh p p On the other hand, from (8), it follows P +1 =F P F T + Q (1 + q ) F P F T pf, substituting the last equation into (7) yields P +1 (1 + q ) [(F λf K H )P pf (F λf K H ) T + F K R (F K ) T + (λ λ )F K H P (F K H ) T (1 + q ) (F λf K H )P pf (F λf K H ) T (38) where, (λ λ )F K H P (F K H ) T + F K R (F K ) T 0. Set A = P +1, U = F λf K H, C = (1 + q P in (38) pf ) and use Lemma 1 to show that the following inequality holds (F λf K H ) T P +1 (F λf K H ) < (1 + q ) P pf. 131

6 Combining (33) and (39) yields { } E (F γ F K H ) T P +1 (F γ F K H ) [1 + (λ λ )pf h pf d (F λf K H ) T P (F λf K H ) < (1 + q ) [1 + (λ λ )pf h pf pf d P. +1 According to Lemma 3, it follows f(λ) < 1 for λ S. So there exists a real number 0 < α < 1 such that f(λ) = 1 α holds for λ S. Therefore, we can derive { } E (F γ F K H ) T P +1 (F γ F K H ) < (1 α)p. (39) Notice, { that the expectation value of the second term E s T P [ } +1 F (I γ K H )e + r in (3) becomes zero owing to zero-mean Gaussian noises ω, υ in the term s and mutual uncorrelation between the noises and the sequence γ. The remaining two terms can be proved to be bounded using the similar methods in [18 under Assumption 3-4. So (3) can be readily shown to be such that E { V +1 (e +1 ) } < (1 α)e { V (e ) } + ρ 1 e + ρ (40) for two real constants ρ 1, ρ > 0 depending on parameters δ ϕ, δ ψ, ɛ ϕ, ɛ ψ and the bounds of system parameter matrices as stated in (4)-(7). Combing results in Theorem 1 and Theorem, the following corollary can be readily stated. Corollary 1 Consider the nonlinear stochastic system described by (1)-() and the extended filter given by (5)-(8). Under Assumption 3-4 and assumption that H exists for every N, that (F, H ) is uniformly observable, and that max{λ c, λ n } < λ 1, if there exists a real constant ɛ > 0 such that E{ e 1 0 } ɛ, then the state estimation error e as stated in (31) is exponentially bounded in mean square and bounded with probability one. Remar 5 Since the assumptions stated above satisfy those in Theorem 1, then based on Theorem 1, it can be concluded that there exist two positive real constants p, p such that pi n P P +1 pi n. The remaining proof follows directly from that of Theorem. 6. Conclusion In this paper, the problem of state estimation of nonlinear systems with Bernoulli measurement pacet losses has been considered by extending the recently proposed suboptimal estimator in [15 to the nonlinear case. By generalizing the concept of uniform observability, it has been shown that, for certain classes of nonlinear systems, the existence of a critical value for pacet loss rate such that the estimation error covariance matrices are bounded for any initial condition. Furthermore, the behavior of the estimation error has also been investigated and certain conditions for ensuring stochastic stability have been established. Acnowledgments This wor was supported in part by the Natural Science Foundation of China under Grant , National Science Foundation for Distinguished Young Scholars of China under Grant , Projects of Major International (Regional) Joint Research Program NSFC under Grant , Beijing Education Committee Cooperation Building Foundation Project XK , and Research Fund for the Doctoral Program of Higher Education of China References [1 J. Hespanha, P. Naghshtabrizi, and Y. Xu, A survey of recent results in networed control systems, Proceedings of the IEEE, 95(1), 007, [ W. Zhang, M. Branicy, and S. Phillips, Stability of networed control systems, Control Systems, IEEE, 1(1), 001, [3 L. Schenato, B. Sinopoli, M. Franceschetti, K. Poolla, and S. S. Sastry, Foundations of control and estimation over lossy networs, Proceedings of the IEEE, 95(1), 007, [4 H. Zhang and L. Xie, Control and estimation of systems with input/output delays. (Springer Publishing Company, Incorporated, 007). [5 S. Sun, L. Xie, W. Xiao, and Y. Soh, Optimal linear estimation for systems with multiple pacet dropouts, Automatica, 44(5), 008, [6 M. Sahebsara, T. Chen, and S. Shah, Optimal H filtering in networed control systems with multiple pacet dropout, Automatic Control, IEEE Transactions on, 5(8), 007, [7 N. Nahi, Optimal recursive estimation with uncertain observation, Information Theory, IEEE Transactions on, 15(4), 1969, [8 J. Tugnait, Stability of optimum linear estimators of stochastic signals in white multiplicative noise, Automatic Control, IEEE Transactions on, 6(3), 1981, [9 M. Hadidi and S. Schwartz, Linear recursive state estimators under uncertain observations, Automatic Control, IEEE Transactions on, 4(6), 1979,

7 [10 B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. Jordan, and S. Sastry, Kalman filtering with intermittent observations, Automatic Control, IEEE Transactions on, 49(9), 004, [11 M. Huang and S. Dey, Stability of Kalman filtering with marovian pacet losses, Automatica, 43(4), 007, [1 K. You and L. Xie, Minimum data rate for mean square stabilizability of linear systems with marovian pacet losses, Automatic Control, IEEE Transactions on, 56(4), 011, [13 K. You, M. Fu, and L. Xie, Mean square stability for Kalman filtering with marovian pacet losses, Automatica, 47(1), 011, [14 K. You, M. Fu, and L. Xie, Necessary and suffficient conditions for stability of Kalman filtering with marovian pacet losses, Proceedings of 18th IFAC World Congress, Milano, Italy, 011, [15 H. Zhang, X. Song, and L. Shi, Convergence and mean square stability of suboptimal estimator for systems with measurement pacet dropping, Automatic Control, IEEE Transactions on, 57(5), 01, [16 S. Kluge, K. Reif, and M. Broate, Stochastic stability of the extended Kalman filter with intermittent observations, Automatic Control, IEEE Transactions on, 55(), 010, [17 L. Li and Y. Xia, Stochastic stability of the unscented Kalman filter with intermittent observations, Automatica, 48(5), 01, [18 K. Reif, S. Gunther, E. Yaz, and R. Unbehauen, Stochastic stability of the discrete-time extended Kalman filter, Automatic Control, IEEE Transactions on, 44(4), 1999, [19 B. Anderson and J. Moore, Detectability and stabilizability of time-varying discrete-time linear systems, SIAM Journal on Control and Optimization, 19(1), 1981, 0-3. [0 Z. Jin, C. Ko, and R. Murray, Estimation for nonlinear dynamical systems over pacet-dropping networs, Procedings of IEEE American Control Conference (ACC 07), 007, [1 R. Horn and C. Johnson, Review and miscellanea, Matrix analysis (Cambridge university press, 1990). [ L. Lu and C. Pearce, Some new bounds for singular values and eigenvalues of matrix products, Annals of Operations Research, 98(1), 000,