_{1}

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This paper investigates the problem of event-triggered finite-time
H
_{∞} control for a class of switched stochastic systems. The main objective of this study is to design an event-triggered state feedback
H
_{∞} controller such that the resulting closed-loop system is finite-time bounded and satisfies a prescribed
H
_{∞} level in some given finite-time interval. Based on stochastic differential equations theory and average dwell time approach, sufficient conditions are derived to ensure the finite-time stochastic stability with the prescribed
H
_{∞} performance for the relevant closed-loop system by employing the linear matrix inequality technique. Finally, the desired state feedback
H
_{∞} controller gain matrices can be expressed in an explicit form.

In the last few decades, switched systems have attracted much attention in the field of control systems [

On the other hand, the periodic and aperiodic control strategies are presented on digital platforms. The conventional sampled-data scheme is the so-called periodic sampling or time-triggered control (TTC) mechanism. In the time-triggered control scheme, all the sampled data are transmitted and updated and the actuator state is adjusted at each sampling instant which leads to some unnecessary sampling and inefficient waste of communication resource. To this end, the event-triggered control (ETC) is introduced which is a typical aperiodic sampling scheme and capable of efficiently utilizing the communication bandwidth and significantly reducing the number of unnecessary data transmission for some networked control systems with limited communication bandwidth. Over the past few years, many worthy results have been provided for event-triggered control of switched systems [

Recently, increasing attention has been paid to the ETC of stochastic systems due to their significance in science and engineering applications. In the past few years, some contribution has been reported for ETC of stochastic systems in the literature [

This paper will study the event-triggered finite-time H ∞ control problem for switched stochastic systems. The main contributions can be summarized as follows. The coupling between the switching signals and triggered signals is analyzed. A novel framework of finite-time stability for augmented closed-loop switched stochastic system is established. The sufficient condition for event-triggered finite-time H ∞ controller of switched stochastic systems is obtained by adopting the average dwell time technique and multiple Lyapunov-Krasovskii functional method with LMIs. The design of controller parameters are presented which can guarantee the mentioned system is finite-time bounded and satisfies a weighted H ∞ disturbance attenuation performance, which can avoid some unnecessary data transmission.

The rest of this paper is arranged as follows. In Section 2, the problem formulation and necessary preliminaries are presented. We give a sufficient condition for finite-time ETC of the mentioned augmented system in terms of LMIs in Section 3. Moreover, a designing approach of an event-triggered finite-time H ∞ controller is presented. Finally, some conclusions are summarized in Section 4.

Notation: The notations used in this paper are quite standard. R n stands for the n-dimensional Euclidean space. The notation X > Y (respectively, X ≥ Y , where X and Y are real symmetric matrices) means that the matrix X − Y is positive definite (respectively, positive semi-definite). I and 0 denote the identity and zero matrices with appropriate dimensions. λ max ( Q ) and λ min ( Q ) denotes the maximum and the minimum of the eigenvalues of a real symmetric matrix Q. The superscript T denotes the transpose for vectors or matrices. The symbol * in a matrix denotes a term that is defined by symmetry of the matrix.

Consider the following continuous-time switched stochastic system:

d x ( t ) = [ A σ ( t ) x ( t ) + B σ ( t ) u ( t ) + D 1 σ ( t ) v ( t ) ] d t + [ A 1 σ ( t ) x ( t ) + B 1 σ ( t ) u ( t ) + D 2 σ ( t ) v ( t ) ] d ω ( t ) (1)

z ( t ) = C σ ( t ) x ( t ) + D σ ( t ) u ( t ) , x ( 0 ) = x 0 ∈ R n (2)

where x ( t ) ∈ R n , u ( t ) ∈ R m , v ( t ) ∈ R p , z ( t ) ∈ R q are state vector, control input vector, external disturbance, and controlled output respectively, where v ( t ) satisfies the constraint condition with respect to the finite-time interval [ 0 , T ]

∫ 0 T v T ( t ) v ( t ) d t ≤ d , d > 0 (3)

and ω ( t ) ∈ R is a standard Wiener process satisfying Ξ { d ω ( t ) } = 0 , Ξ { d ω 2 ( t ) } = d t , where Ξ is the expected value, which is assumed to be independent of the system mode σ ( t ) . σ ( t ) : [ 0 , + ∞ ) → S = { 1 , 2 , 3 , ⋯ , p } is the switching signal which is a piecewise constant function depending on t, and p is the number of subsystems. A σ ( t ) , B σ ( t ) , D 1 σ ( t ) , A 1 σ ( t ) , B 1 σ ( t ) , D 2 σ ( t ) , C σ ( t ) , D σ ( t ) are known constant matrices of appropriate dimensions.

In this paper, the finite-time H ∞ controller is event-triggered, and the state-feedback sub-controllers are determined on the sampled states of the sub-system.

Assume that { t k } k ∈ N denotes the triggered instants and there is no time-delay in sampler and actuator. Then, the state is sampled and the control input is computed at instant t k simultaneously such that

u ( t ) = K σ ( t ) x ( t k ) , t ∈ [ t k , t k + 1 ) (4)

where K σ ( t ) are the sub-controller gains, 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.

Remark 2.1 At sampling time instant t k , the controller (4) will receive the sampled state x ( t k ) , which will be held constant until next event is generated at time instant t k + 1 . The sampled state x ( t k ) is used to update the control input in (4) which keep the control signal continuous on the interval [ t k , t k + 1 ) by zero order holder.

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

t k + 1 = inf { t > t k | e t k T ( t ) Ω e t k ≥ μ 2 x T ( t ) Ω x ( t ) } (5)

where

e t k ( t ) = x ( t k ) − x ( t ) (6)

μ is a constant and Ω is a symmetric and positive definite matrix with appropriate dimension to be determined.

Remark 2.2 when the equality e t k T ( t ) Ω e t k = μ 2 x T ( t ) Ω x ( t ) is satisfied, the sampler will be triggered to sample the system state immediately. Then the sampled data is transmitted to the subcontroller for calculating the control input which will be further used by the subsystem.

Remark 2.3 It should be pointed out that the parameter μ has great influence on the event-trigger instants, i.e. different values of μ correspond to different event-trigger frequencies. The less μ is selected, the shorter the event-trigger period is. Hence, μ should be selected in accordance with the specific control requirement and control capacity.

Let { r q } q ∈ N be a given time sequence satisfying r 1 < r 2 < ⋯ < r p , where r q is the switching instant. Meanwhile, define the event-triggered time instants as t 0 < t 1 < t 2 ⋯ < t k < ⋯ .

Assumption 2.1 There is a number τ s > 0 such that any two switches are separated by at least τ s to evade zeno phenomena, which means r q + 1 − r q ≥ τ s for any q > 0 [

Substituting the state-feedback controllers (4) into (1) (2), the event-triggered switched stochastic closed-loop system is obtained for t ∈ [ t k , t k + 1 ) as follows:

d x ( t ) = [ A ¯ σ ( t ) x ( t ) + B σ ( t ) K σ ( t ) e t k ( t ) + D 1 σ ( t ) v ( t ) ] d t + [ A ¯ 1 σ ( t ) x ( t ) + B σ ( t ) K σ ( t ) e t k ( t ) + D 2 σ ( t ) v ( t ) ] d w ( t ) (7)

z ( t ) = C ¯ σ ( t ) x ( t ) + D σ ( t ) K σ ( t ) e t k ( t ) (8)

where A ¯ σ ( t ) = A σ ( t ) + B σ ( t ) K σ ( t ) , A ¯ 1 σ ( t ) = A 1 σ ( t ) + B 1 σ ( t ) K σ ( t ) , C ¯ σ ( t ) = C σ ( t ) + D σ ( t ) K σ ( t ) .

We introduce the following definitions and lemmas, which will be useful in the succeeding discussion.

Definition 2.1 (Average dwell time [

Definition 2.2 (Finite-time stochastic stabilizable [

Ξ [ x T ( 0 ) R x ( 0 ) ] < c 1 ⇒ Ξ [ x T ( t ) R x ( t ) ] < c 2 , ∀ t ∈ [ 0 , T ] . (9)

Definition 2.3 (Finite-time H ∞ stochastic stabilization [

1) Switched system (7) (8) with control input (4) is finite-time stochastic stabilizable.

2) Under the zero initial condition, there is

Ξ ∫ 0 T z T ( t ) z ( t ) d t ≤ γ 2 Ξ ∫ 0 T v T ( t ) v ( t ) d t , (10)

where the prescribed value γ is the attenuation level.

Lemma 2.1 [

X T Y + Y T X ≤ ε X T X + ε − 1 Y T Y . (11)

Lemma 2.2 [

λ min ( W ) x T x ≤ x T W x ≤ λ max ( W ) x T x . (12)

Lemma 2.3 (Schur complement [

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 .

In this section, we focus on the finite-time stabilization of the switched stochastic system (7) (8) with event-triggered control input (4), and some sufficient conditions which can ensure the switched stochastic system (7) (8) is finite-time H ∞ stochastic stabilizable are given by the following theorem.

Theorem 3.1 For any σ ( t ) = i ∈ S = { 1 , 2 , ⋯ , p } , a given positive definite matrix R, if there exist positive constants ε , γ , η > 1 α > 0 and symmetric positive definite matrices P i , i ∈ S satisfying

P i = R 1 2 Q i R 1 2 = R 1 2 H i T H i R 1 2 (13)

P i ≤ η P j (14)

such that the following LMIs hold

Θ < 0 (15)

where

Θ = [ μ 2 Ω − α P i 0 0 Π 1 P i Π 2 Π 3 0 − Ω 0 0 0 0 0 * * − γ 2 I 0 0 0 0 * * * − ε − 1 I 0 0 0 * * * * − ε I 0 0 * * * * * − I 0 * * * * * * − I ] (16)

Π 1 = [ A i + B i K i B i K i D 1 i 0 0 0 0 0 0 ] , Π 2 = [ C i + D i K i D i K i 0 ] , Π 3 = [ H i ( A 1 i + B 1 i K i ) H i B 1 i K i H i D 2 i ] .

Then, under the event-triggering strategy (5), the event-triggered state-feedback controllers (4) and any switching signal with the average dwell time satisfying

τ a ≥ T ln η ln inf i ∈ S [ λ min ( Q i ) ] c 2 sup i ∈ S [ λ max ( Q i ) ] c 1 + γ 2 d − α T (17)

the switched closed-loop stochastic system (7) (8) is finite-time H ∞ stochastic stable (FTSS) with respect to .

Proof Assume that subsystem σ ( t ) = i ∈ S = { 1 , 2 , ⋯ , p } is active on the interval [ r q , r q + 1 ) . When t ∈ [ r q , r q + 1 ) , define the following stochastic Laypunov-Krasovskii functional candidate:

V i ( x ( t ) ) = x T ( t ) P i x ( t ) (18)

By Itô formula, define a weak infinitesimal operator L, then, the stochastic derivative of V i ( x ( t ) ) = x T ( t ) P i x ( t ) is given by

d V i ( x ( t ) ) = L V i ( x ( t ) ) d t + 2 x T ( t ) P i [ A ¯ 1 i x ( t ) + B 1 i K 1 i e t k ( t ) + D ¯ 1 i v ( t ) ] d w ( t ) (19)

We have the weak infinitesimal operator of L V i ( x ( t ) ) as follows:

L V i ( x ( t ) ) = x T ( t ) P i [ A ¯ i x ( t ) + B K i i e t k ( t ) + D ¯ 1 i v ( t ) ] + [ A ¯ i x ( t ) + B K i i e t k ( t ) + D ¯ 1 i v ( t ) ] T P i x ( t ) + [ A ¯ 1 i x ( t ) + B K 1 i i e t k ( t ) + D ¯ 2 i v ( t ) ] T P i [ A ¯ 1 i x ( t ) + B K 1 i i e t k ( t ) + D ¯ 2 i v ( t ) ] (20)

The relationship between the switching instants and event-triggered instants will be discussed as the following two cases.

Case 1 Suppose that there is no sampling in [ r q , r q + 1 ) , i.e. t k ≤ r q < r q + 1 ≤ t k + 1

In view of event condition (5), together with Lemma 2.3, the following inequality can be deduced:

L V i ( x ( t ) ) − α V i ( x ( t ) ) + μ 2 x T ( t ) Ω x ( t ) − e t k T Ω e t k + z T ( t ) z ( t ) − γ 2 v T ( t ) v ( t ) ≤ ξ T ( t ) Θ i ξ ( t ) (21)

where ξ ( t ) = [ x T ( t ) e t k T v T ( t ) ] T ,

Θ i = [ Θ 11 P i B i K i + ( A 1 i + B 1 i K i ) T P i B 1 i K i + ( C i + D i K i ) T D i K i P i D 1 i + ( A 1 i + B 1 i K i ) T P i D 2 i * ( B 1 i K i ) T P i ( B 1 i K i ) + ( D i K i ) T D i K i − Ω ( B 1 i K i ) T P i D 2 i * * D 2 i T P i D 2 i − γ 2 I ] (22)

Θ 11 = P i ( A i + B i K i ) + ( A i + B i K i ) T P i + ( A 1 i + B 1 i K i ) T P i ( A 1 i + B 1 i K i ) + ( C i + D i K i ) T ( C i + D i K i ) + μ 2 Ω − α P i

From (15) and (21), there is

L V i ( x ( t ) ) ≤ α V i ( x ( t ) ) + γ 2 v T ( t ) v ( t ) − z T ( t ) z ( t ) (23)

Integrate both sides of the inequality (23) from r q to t, and obtain

V i ( x ( t ) ) ≤ e α ( t − r q ) V i ( x ( t ) ) + ∫ r q t e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s , t ∈ [ r q , r q + 1 ) (24)

Case 2 For any t ∈ [ r q , r q + 1 ) , there are n sampling, i.e. t k ≤ r q < t k + 1 < t k + 2 < ⋯ < t k + n < r q + 1 , ∀ n ∈ N , where t k + 1 , t k + 2 , ⋯ , t k + n is the updating sequence of the event-triggered controller.

On the intervals [ r q , t k + 1 ) , [ t k + 1 , t k + 2 ) , ⋯ , [ t k + n , r q + 1 ) , (23) (24) can also be similarly received respectively. Then, the following inequalities can be established.

V i ( x ( t ) ) ≤ { e α ( t − r q ) V i ( x ( r q ) ) + ∫ r q t e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s , t ∈ [ r q , t k + 1 ) e α ( t − t k + 1 ) V i ( x ( t k + 1 ) ) + ∫ t k + 1 t e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s , t ∈ [ t k + 1 , t k + 2 ) ⋮ e α ( t − t k + n ) V i ( x ( t k + n ) ) + ∫ t k + n t e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s , t ∈ [ t k + n , r q + 1 ) (25)

On the other hand, it can be derived from (14) that

V σ ( r q ) ( x ( r q ) ) ≤ η V σ ( r q − ) ( x ( r q − ) ) (26)

Suppose that 0 = r 1 < r 2 < ⋯ < r q < T , where r 1 , r 2 , ⋯ , r q are the switching sequence. Correspondingly, from (25) (26) and definition 2.1, we have

V i ( x ( t ) ) ≤ η e α ( t − r q ) V i ( x ( r q − ) ) + ∫ r q t e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s ≤ η e α ( t − r q ) { e α ( r q − r q − 1 ) V i ( x ( r q − 1 ) ) + ∫ r q − 1 r q e α ( r q − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s } + ∫ r q t e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s = η e α ( t − r q − 1 ) V i ( x ( r q − 1 ) ) + η ∫ r q − 1 r q e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s + ∫ r q t e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s

≤ η 2 e α ( t − r q − 1 ) V i ( x ( r q − 1 − ) ) + η ∫ r q − 1 r q e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s + ∫ r q t e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s ≤ η 2 e α ( t − r q − 1 ) V i ( x ( r q − 2 ) ) + η 2 ∫ r q − 2 r q − 1 e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s + η ∫ r q − 1 r q e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s + ∫ r q t e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s

≤ ⋯ ≤ η N σ ( 0 , T ) e a ( t − r 1 ) V i ( x ( 0 ) ) + ∫ r 1 t η N σ ( s , t ) e α ( t − s ) [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s = e α t + N σ ( 0 , T ) ln η V i ( x ( 0 ) ) + ∫ 0 t e α ( t − s ) + N σ ( s , t ) ln η [ γ 2 v T ( s ) v ( s ) − z T ( s ) z ( s ) ] d s ≤ e α T + ln η τ a T V i ( x ( 0 ) ) + ∫ 0 T e ( α + ln η ) T γ 2 v T ( s ) v ( s ) d s (27)

Furthermore, by using lemma 2.2, it follows from (3) (12) (27) that:

Ξ [ V i ( x ( t ) ) ] ≤ e ( α + ln η τ a ) T [ sup i ∈ S λ max ( Q i ) ] x 0 T R x 0 + e ( α + ln η τ a ) T γ 2 d (28)

and

Ξ [ V i ( x ( t ) ) ] ≥ inf i ∈ S [ λ min ( Q i ) ] x T ( t ) R x ( t ) (29)

Taking (27) (28) (29) and (17) into account, the following conclusion is obtained

Ξ [ x T ( t ) R x ( t ) ] ≤ e ( α + ln η τ a ) T { [ sup i ∈ S λ max ( Q i ) ] c 1 + γ 2 d } inf i ∈ S [ λ min ( Q i ) ] < c 2 (30)

Therefore, the H ∞ control performance is obtained from Definition 2.3. This completes the proof.

Theorem 3.2 For any σ ( t ) = i ∈ S = { 1 , 2 , ⋯ , p } , given positive definite matrix R, and positive constants ε , γ , η > 1 , α > 0 consider the switched closed-loop stochastic system (7) (8) with the event-triggering strategy (5), the event-triggered state-feedback controllers (4) can be obtained, if there exist symmetric positive definite matrices X i , Y i , i ∈ S with appropriate dimensions satisfying

Θ ˜ = [ μ 2 X i T Ω X i − α X i T Y i T Y i X i T 0 0 Π ˜ 1 Y i T Y Π ˜ 2 Π ˜ 3 0 − Ω 0 0 0 0 0 * * − γ 2 I 0 0 0 0 * * * − ε − 1 I 0 0 0 * * * * − ε I 0 0 * * * * * − I 0 * * * * * * − I ] (31)

where Π ˜ 1 = [ X i T A i X i + X i T B i Y i X i T B i Y X i X i T D 1 i X i 0 0 0 0 0 0 ] , Π ˜ 2 = [ X i T C i X i + X i T D i Y i D i Y i 0 ] ,

Π ˜ 3 = [ X i T Y i R − 1 2 ( A 1 i X i + B 1 i Y i ) X i T Y i R − 1 2 B 1 i Y i X i T Y i R − 1 2 D 2 i X i ]

then the corresponding controller gains of the event-triggered H ∞ controllers (5) can be obtained as

K i = Y i X i − 1 (32)

Proof Let P i 1 2 = Y i , pre- and post-multiplying both sides of the inequality (15) by diag { X i T , I , ⋯ , I } and diag { X i , I , ⋯ , I } respectively. By Schur complement, the proof can be completed.

The event-triggered finite-time H ∞ control problem has been investigated for switched stochastic system with exogenous disturbance. For the proposed event-triggering schemes, the prescribed H ∞ performance level of the switched stochastic system has been guaranteed by adopting Lyapunov-Krasovski function method and average dwell time method in a given finite-time interval. In order to avoid the Zeno behavior, a lower bound on the triggered inter-event intervals has been estimated. Furthermore, sufficient conditions for H ∞ control performance analysis and control design have been provided in terms of LMIs technique, respectively.

The author declares no conflicts of interest regarding the publication of this paper.

Zhang, A.Q. (2020) Event-Triggered Finite-Time H_{∞} Control for Switched Stochastic Systems. Journal of Applied Mathematics and Physics, 8, 2103-2114. https://doi.org/10.4236/jamp.2020.810157