Event-Triggered Finite-Time <i>H</i>_∞ Filtering for Discrete-Time Nonlinear Stochastic Systems

This paper addresses the problem of event-triggered finite-time H _∞ filter design for a class of discrete-time nonlinear stochastic systems with exogenous disturbances. The stochastic Lyapunov-Krasoviskii functional method is adopted to design a filter such that the filtering error system is stochastic finite-time stable (SFTS) and preserves a prescribed performance level according to the pre-defined event-triggered criteria. Based on stochastic differential equations theory, some sufficient conditions for the existence of H _∞ filter are obtained for the suggested system by employing linear matrix inequality technique. Finally, the desired H _∞ filter gain matrices can be expressed in an explicit form.

• Event-Triggered Scheme
• Discrete-Time Nonlinear Stochastic Systems
• Stochastic Finite-Time Stable
• Linear Matrix Inequalities (LMIS)

1. Introduction

During the past few decades, there has been a rapidly growing interest in nonlinear stochastic systems. Based on the fundamental stochastic stability theory [1] and Lyapunov-Krasovskii functional [2], some results can be found in the literature [3] - [15]. Specifically, problems of stochastic stabilization and destabilization were studied for nonlinear differential equations by noise and impulsive stochastic nonlinear systems respectively in [4] [6] [10] [14]. References [5] [7] [11] [12] [15] investigated state-feedback and output feedback stabilization problems for stochastic nonlinear systems and stochastic delay nonlinear systems. Fault detection filter and full-order H_∞ filter were provided for nonlinear stochastic systems and nonlinear switched stochastic systems in terms of second-order nonlinear Hamilton-Jacobi inequalities and T-S fuzzy framework respectively in references [3] [8]. Dissipativity and tracking control problems were presented for nonlinear stochastic dynamical systems in references [9] [13].

On the other hand, increasing effort has been paid to the study of event-triggered control (ETC) of nonlinear stochastic systems due to their significance in science and engineering applications. Many important results have been presented for event-triggered control of nonlinear stochastic systems in references [16] - [24]. Dynamic event-triggered control, dynamic self-triggered control and event-triggered stability were investigated for a class of nonlinear stochastic systems by introducing an additional internal dynamic variable in [16] [17]. Based on event-triggered predictive control (ETPC) scheme, a novel discrete-time feedback law was designed for the stabilization of continuous-time stochastic systems with output delay in [24]. The input-to-state practically exponential mean-square stability of stochastic nonlinear delay systems with exogenous disturbances was provided and a framework of event-triggered stabilization was received for the stochastic systems without applying the well-known Lyapunov theorem respectively in [18] and [21]. Periodic event-generators and continuous event-enerators were studied in both static and dynamic cases in [19]. Reference [20] addressed the dynamic event-based fault detection problem of nonlinear stochastic systems influenced by random nonlinearity, data transmission delays and packet dropout. Based on fuzzy technique, the problem of event-triggered optimized control for uncertain nonlinear Itô-type stochastic systems with time-delay was addressed in [22]. The modified unscented Kalman filter was proposed for stochastic nonlinear system with Markov packet dropout in [23].

Although the problem of event-triggered control for nonlinear stochastic systems has been investigated, there has little literature on filtering problem for discrete-time nonlinear stochastic systems. With above inspirations, we aim to propose an event-triggered finite-time filtering scheme for discrete-time nonlinear stochastic systems with exogenous disturbance. We present the definition of SFTS into a class of discrete-time nonlinear stochastic systems. By employing the event-triggered strategy, we construct a detection filter such that the resulting filter error augmented system is SFTS. Sufficient conditions for SFTS of the filter error system is established by constructing the Lyapunov-Krasovskii functional candidate combined with LMIs. The desired event-triggered finite-time filter can be constructed by solving a set of LMIs.

This paper is organized as the following. First, some preliminaries and the problem formulation are introduced in Section 2. In Section 3, in terms of event-triggered technique, a sufficient condition for SFTS of the filter error system is established and a method for designing the corresponding filter is presented. Finally, some conclusions are drawn in Section 4.

Notation: Throughout this paper, the notations used are quite standard. We use R n to denote the n-dimensional Euclidean space. R > 0 denotes a symmetric positive definite matrix. The symbol * in a matrix denotes a term that is defined by symmetry of the matrix. I and 0 denote the identity and zero matrices with appropriate dimensions. λ max ( R ) and λ min ( R ) denote the maximum and the minimum of the eigenvalues of a real symmetric matrix R. The superscript T denotes the transpose for vectors or matrices. Ξ ( P ) is the mathematical expectation of P. Matrices, if not explicitly stated, are with compatible dimensions.

2. Problem Formulation and Preliminaries

We shall consider the following discrete-time nonlinear stochastic system:

{ x ( k + 1 ) = A x ( k ) + f ( k , x ( k ) ) + D 1 v ( k ) + g ( k , x ( k ) ) ϖ ( k ) y ( k ) = C x ( k ) + D 2 v ( k ) z ( k ) = L x ( k ) , x ( 0 ) = x 0 ∈ R n (1)

where x ( k ) ∈ R n , y ( k ) ∈ R m , v ( k ) ∈ R p , z ( k ) ∈ R q , are state vector, measurement output, external disturbance, and controlled output respectively, ϖ ( k ) is a one-dimensional zero-mean process which satisfies

Ξ [ ω ( k ) ] = 0 , Ξ [ ω ( i ) ω ( j ) ] = 0 , i ≠ j , Ξ [ ω 2 ( k ) ] = δ (2)

where Ξ is the expected value. Here δ > 0 is a known scalar. The matrices A , C , D 1 D 2 , L are constant matrices with appropriate dimensions.

Assumption 1 The nonlinear functions f ( k , x ( k ) ) and g ( k , x ( k ) ) satisfy the following quadratic inequalities:

| f ( k , x ( k ) ) − f ( k , x ˜ ( k ) ) | 2 ≤ ε 1 2 | x ( k ) − x ˜ ( k ) | 2

| g ( k , x ( k ) ) − g ( k , x ˜ ( k ) ) | 2 ≤ ε 2 2 | x ( k ) − x ˜ ( k ) | 2

for all x ( k ) , x ˜ ( k ) ∈ R n , where ε 1 , ε 2 > 0 are constants related to the function f ( k , x ( k ) ) , g ( k , x ( k ) ) .

Assume that { t k } k ∈ N denotes the triggered instants and there is no time-delay in sampler and actuator, t 0 < t 1 < t 2 < ⋯ < t k < t k + 1 , t k ≤ k < t k + 1 . x ( t k ) is the current sampled system state, t k + 1 is the next sampled instant, which can be determined by the event-trigger, and x ( t 0 ) = x 0 is chosen as the initial sampled state.

In this paper, the event-triggering schemes are described by

t k + 1 = inf { k ≥ t k | e y ( k ) T Q e y ( k ) − η y T ( k ) Q y ( k ) > 0 } , (3)

where e y ( k ) = y ( k ) − y ( t k ) , η is a constant and Q = Π T Π is a symmetric and positive definite matrix with appropriate dimension to be determined.

We now consider the following filter:

{ x ^ ( k + 1 ) = A f x ^ ( k ) + B f y ( t k ) z ^ ( k ) = L f x ^ ( k ) (4)

where x ^ ( k ) ∈ R n is the filter state, and matrices A f , B f , L f are filter parameters with compatible dimensions to be determined.

Define x ¯ ( k ) = [ x T ( k ) x ^ T ( k ) ] , z ¯ ( k ) = z ( k ) − z ^ ( k ) . Then the filtering error system is

{ x ¯ ( k + 1 ) = A ¯ x ¯ ( k ) + F ¯ ( k , x ( k ) ) + D ¯ v ¯ ( k ) − B ¯ f K e ¯ y ( k ) + G ¯ ( k , x ( k ) ) ϖ ( k ) z ¯ ( k ) = L ¯ x ¯ ( k ) (5)

where

A ¯ = [ A 0 B f C A f ] , F ¯ ( k , x ( k ) ) = [ f ( k , x ( k ) ) 0 ] , G ¯ ( k , x ( k ) ) = [ g ( k , x ( k ) ) 0 ] ,

D ¯ = [ D 1 0 B f D 2 0 ] , B ¯ f = [ 0 B f ] , L ¯ = [ L − L f ] , e ¯ y ( k ) = [ e y ( k ) 0 ] ,

v ¯ ( k ) = [ v ( k ) 0 ] , K = [ I 0 ] .

Before providing the main results, we summarize several needed definitions and lemmas from the literature.

Definition 2.1 The filtering error system (5) with event-triggered scheme (3) and v ( k ) = 0 is said to be stochastic finite-time stable (SFTS) with respect to ( c 1 , c 2 , P , N ) , where P > 0 , 0 < c 1 < c 2 , if the following relation holds:

Ξ [ x T ( 0 ) P x ( 0 ) ] < c 1 ⇒ Ξ [ x T ( k ) P x ( k ) ] < c 2 for all k ∈ 1 , 2 , ⋯ , N .

Definition 2.2 For γ > 0 , suppose the event-triggered residual system in (5) is stochastic finite-time stable (SFTS) with respect to ( c 1 , c 2 , P , N ) , then system (5) is said to have a weighted H_∞ attenuation level γ for all nonzero v ( k ) ∈ l 2 [ 0 , ∞ ) , if the following inequality holds:

Ξ { ∑ k = k 0 ∞ z ¯ T ( k ) z ¯ ( k ) } < γ 2 ∑ k = k 0 ∞ v ¯ T ( k ) v ¯ ( k ) (6)

Lemma 2.1 ( [25]). Let Ω ∈ R n × n be a symmetric matrix, and let x ∈ R n , then the following inequality holds

λ min ( Ω ) x T x ≤ x T Ω x ≤ λ max ( Ω ) x T x . (7)

Lemma 2.2 (Schur complement [26] [27]) Given a symmetric matrix ϕ = [ ϕ 11 ϕ 12 ϕ 21 ϕ 22 ] , the following three conditions are equivalent to each other:

1) ϕ < 0 ;

2) ϕ 11 < 0 , and ϕ 22 − ϕ 12 T ϕ 11 − 1 ϕ 12 < 0 ;

3) ϕ 22 < 0 , and ϕ 11 − ϕ 12 ϕ 22 − 1 ϕ 12 T < 0 .

3. Main Results

In this section, we focus on stochastic finite-time stable (SFTS) with respect to ( c 1 , c 2 , P , N ) of the event-triggered residual system in (5), and propose sufficient conditions of system performance analysis.

Theorem 3.1 For given constants γ , η , δ > 0 and μ > 1 , suppose that there exist symmetric positive definite matrices R = P 1 2 Ω P 1 2 such that the following LMIs hold

Θ < 0 (8)

where

Θ = [ Θ 11 Θ 12 Θ 13 Θ 14 Θ 15 Θ 16 * Θ 22 Θ 23 0 0 0 * * − γ 2 I 0 0 0 * * * − η 1 2 I 0 0 * * * * − I 0 * * * * * − Ω − 1 ]

μ N − k 0 λ max ( Ω ) c 1 λ min ( Ω ) ≤ c 2 ,

Θ 11 = A ¯ T R A ¯ + A ¯ T R Ψ 1 + Ψ 1 T R A ¯ + Ψ 1 T R Ψ 1 + δ   Ψ 2 T R Ψ 2 − μ R ,

Θ 12 = − A ¯ T R B ¯ f K − Ψ 1 T R B ¯ f K , Θ 13 = A ¯ T R D ¯ + Ψ 1 T R D ¯ , Ψ i = [ ε i 0 0 0 ] , i = 1 , 2 ,

Θ 14 = [ Π ¯ C ¯ 0 Π ¯ D ¯ 2 ] T , Θ 15 = [ L ¯ 0 0 ] T , Θ 15 = [ 0 0 P 1 2 D ¯ ] T ,

Θ 22 = K T B ¯ f T R B ¯ f K , Θ 23 = − K T B ¯ f T R D ¯ , Q ¯ = [ Q 0 0 0 ] = Π ¯ T Π ¯ ,

Q ¯ 1 = C ¯ T Π ¯ T Π ¯ C ¯ , Q ¯ 2 = C ¯ T Π ¯ T Π ¯ D ¯ 2 , Q ¯ 3 = D ¯ 2 T Π ¯ T Π ¯ D ¯ 2 ,

Π ¯ = [ Π 0 ] , C ¯ = [ C 0 ] , D ¯ 2 = [ D 2 0 ] .

Proof: Consider the following Lyapunov function candidate for system (5):

V ( x ¯ ( k ) ) = x ¯ T ( k ) R x ¯ ( k ) (9)

Then, based on assumption 1, (8) and Schur complement, it follows that

Γ ( k ) ≜ Ξ [ V ( x ¯ ( k + 1 ) ) − μ V ( x ¯ ( k ) ) ] = [ A ¯ x ¯ ( k ) + F ¯ ( k , x ( k ) ) − B ¯ f K e ¯ y ( k ) + G ¯ ( k , x ( k ) ) ϖ ( k ) ] T R [ A ¯ x ¯ ( k )       + F ¯ ( k , x ( k ) ) − B ¯ f K e ¯ y ( k ) + G ¯ ( k , x ( k ) ) ϖ ( k ) ] − μ x ¯ T ( k ) R x ¯ ( k ) = [ x ¯ T ( k ) e ¯ y ( k ) T ] [ Γ 11 Γ 12 * Γ 22 ] [ x ¯ ( k ) e ¯ y ( k ) ] < 0 (10)

where Γ 11 = Θ 11 , Γ 12 = Θ 12 , Γ 22 = Θ 22 .

Then for ∀ k ∈ [ t k , t k + 1 ) , we have

Ξ [ V ( x ¯ ( k ) ) ] ≤ Ξ [ μ V ( x ¯ ( k − 1 ) ) ] . (11)

Proceeding in an iterative fashion, we obtain the following inequality:

Ξ [ V ( x ¯ ( k ) ) ] < μ k − k 0 Ξ [ V ( x ¯ ( k 0 ) ) ] = μ k − k 0 Ξ [ V ( x ¯ ( 0 ) ) ] ≤ μ N − k 0 λ max ( Ω ) c 1 (12)

On the other hand, it can be derived from (9) and lemma 2.1 that

Ξ [ V ( x ¯ ( k ) ) ] ≥ λ min ( Ω ) Ξ [ x ¯ T ( k ) P x ¯ ( k ) ] . (13)

Thus we have that

Ξ [ x ¯ T ( k ) P x ¯ ( k ) ] ≤ μ N − k 0 λ max ( Ω ) c 1 λ min ( Ω ) ≤ c 2 . (14)

According to Definition 2.1, the filter error systems (5) with v ( k ) = 0 is SFTS.

Next, we prove the event-based residual system in (5) satisfies H_∞ performance value.

In view of event condition (3) (8), together with Lemma 2.2, the following inequality can be deduced:

T ( k ) ≜ Ξ [ V ( x ¯ ( k + 1 ) ) − μ V ( x ¯ ( k ) ) + η y T ( k ) Q y ( k ) − e y ( k ) T Q e y ( k )     + z ¯ T ( k ) z ¯ ( k ) − γ 2 v ¯ T ( k ) v ¯ ( k ) ] = [ A ¯ x ¯ ( k ) + F ¯ ( k , x ( k ) ) + D ¯ v ¯ ( k ) − B ¯ f K e ¯ y ( k ) + G ¯ ( k , x ( k ) ) ϖ ( k ) ] T     × R [ A ¯ x ¯ ( k ) + F ¯ ( k , x ( k ) ) + D ¯ v ¯ ( k ) − B ¯ f K e ¯ y ( k ) + G ¯ ( k , x ( k ) ) ϖ ( k ) ]

    − μ x ¯ T ( k ) R x ¯ ( k ) + η [ C x ( k ) + D 2 v ( k ) ] T R [ C x ( k ) + D 2 v ( k ) ]     − e y ( k ) T Q e y ( k ) + z ¯ T ( k ) z ¯ ( k ) − γ 2 v ¯ T ( k ) v ¯ ( k ) = [ x ¯ T ( k ) e ¯ y ( k ) T v ¯ T ( k ) ] [ T 11 T 12 T 13 * T 22 T 23 * * T 33 ] [ x ¯ ( k ) e ¯ y ( k ) v ¯ ( k ) ] < 0 (15)

where

T 11 = A ¯ T R A ¯ + A ¯ T R Ψ 1 + Ψ 1 T R A ¯ + Ψ 1 T R Ψ 1 + δ   Ψ 2 T R Ψ 2 − μ R + η Q ¯ 1 + L ¯ T L ¯ ,

T 12 = − A ¯ T R B ¯ f K − Ψ 1 T R B ¯ f K , T 13 = A ¯ T R D ¯ + Ψ 1 T R D ¯ + η Q ¯ 2 ,

T 22 = K T B ¯ f T R B ¯ f K − Q ¯ , T 23 = Θ 23 , T 33 = D ¯ T R D ¯ + η Q ¯ 3 − γ 2 I ,

Q ¯ = [ Q 0 0 0 ] , Q ¯ 1 = [ C T Q C 0 0 0 ] , Q ¯ 2 = [ C T Q D 2 0 0 0 ] , Q ¯ 3 = [ D 2 T Q D 2 0 0 0 ] .

Then we can conclude that (6) holds. It can be concluded that the event-triggered residual system in (5) possesses a prescribed H_∞ performance index proposed in Definition 2.2. Thus the proof is completed.

The following theorem will set forth our filter design method for the system (1).

Theorem 3.2 For given constants γ , η , δ > 0 and μ > 1 , the filtering error system (5) with the event-triggering strategy (3) is SFTS with respect to ( c 1 , c 2 , P , N ) and the error signal satisfies (6), if there exist positive definite matrix

R = [ R 11 R 12 R 12 T R 22 ] and matrices G 1 , G 2 , G 3 with appropriate dimensions satisfying:

Θ ˜ < 0 , (16)

where

Θ ˜ = [ Θ ˜ 11 Θ ˜ 12 Θ ˜ 13 Θ ˜ 14 Θ ˜ 15 Θ ˜ 16 * Θ ˜ 22 Θ ˜ 23 0 0 0 * * − γ 2 I 0 0 0 * * * − η 1 2 I 0 0 * * * * − I 0 * * * * * − P 1 2 R − 1 P 1 2 ] , (17)

Θ ˜ 11 = A ˜ T R A ˜ + A ˜ T R Ψ 1 + Ψ 1 T R A ˜ + Ψ 1 T R Ψ 1 + δ   Ψ 2 T R Ψ 2 − μ R ,

Θ ˜ 12 = − A ˜ T R B ˜ f K − Ψ 1 T R B ˜ f K , Θ ˜ 13 = A ˜ T R D ¯ + Ψ 1 T R D ¯ , Θ ˜ 14 = Θ 14 ,

Θ ˜ 15 = [ L ˜ 0 0 ] , Θ ˜ 16 = Θ 16 , Θ ˜ 22 = B ˜ f T R B ˜ f K ,

Θ ˜ 23 = − K T B ˜ f T R D ¯ A ˜ = [ A 0 R 22 − 1 G 2 C R 12 − 1 G 1 ] ,

B ˜ f = [ 0 R 22 − 1 G 2 ] , L ˜ = [ L − G 3 ] .

Moreover, the suitable filter parameters A f , B f , L f in system (4) can be given by

A f = R 12 − 1 G 1 , B f = R 22 − 1 G 2 , L f = G 3 . (18)

Proof By Theorem 3.1, let A f = R 12 − 1 G 1 , B f = R 22 − 1 G 2 , L f = G 3 , then the condition (8) is equivalent to (16).

4. Conclusion

In this paper, we have introduced the concept of SFTS into a class of discrete-time nonlinear stochastic systems with exogenous disturbances. We have addressed the event-triggered finite-time filter designing problem. A sufficient condition is provided to guarantee the SFTS of the filter error system. For the presented event-triggering schemes, the criteria for the event-based filter residual systems with a prescribed performance level γ were established by adopting Lyapunov-Krasovski function method. Sufficient conditions for H_∞ performance analysis and corresponding filter designing technique have been provided in a given finite-time interval in terms of LMIs technique, respectively.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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