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Stability of discrete-time stochastic nonlinear systems with event-triggered state-feedback control
Journal Pre-proof Stability of discrete-time stochastic nonlinear systems with event-triggered state-feedback control Wenjuan Xie, Quanxin Zhu
PII: DOI: Reference:
S0378-4371(19)32126-0 https://doi.org/10.1016/j.physa.2019.123823 PHYSA 123823
To appear in:
Physica A
Received date : 27 August 2019 Revised date : 27 November 2019 Please cite this article as: W. Xie and Q. Zhu, Stability of discrete-time stochastic nonlinear systems with event-triggered state-feedback control, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123823. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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Highlights
1. We are concerned with the event-triggered state-feedback control of discrete-time stochastic nonlinear systems, which is more challenging than most of works only considering deterministic systems.
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2. Different from the input and output stability described by operators, the input-to-state stability takes into account the influence of the initial state of the system on the systems future time motion trajectory. 3. Works on discrete-time stochastic nonlinear systems with external disturbances are rare. Our work considers such an interesting question.
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4. Under some suitable conditions, the asymptotic stability in mean square in the case of the state-dependent external disturbance and the input-to-state stability in mean square in the case of the state-independent external disturbance are obtained.
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Stability of discrete-time stochastic nonlinear systems with event-triggered state-feedback control ∗ b
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Wenjuan Xiea and Quanxin Zhua,b
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing 210023, China; a
MOE-LCSM, School of Mathematics and Statistics,
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Hunan Normal University, Changsha, Hunan 410081, China
Abstract
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This paper investigates a class of discrete-time stochastic nonlinear systems with external disturbances. To obtain the stability of the suggested system, we develop an event-triggered mechanism which is more general than those given in previous literatures. On this basis, many techniques such as stochastic analysis theory, classified discussion, comparison principle and reduction to absurdity are successfully applied in this paper. Under some suitable conditions, the asymptotic stability in mean square in the case of the state-dependent external disturbance and the input-to-state stability in mean square in the case of the state-independent external disturbance are obtained. Finally, some discussions and remarks are provided to show the significance of our results.
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Key Words: Discrete-time stochastic system; event-triggered state-feedback control; asymptotic stability; input-to-state stability; external disturbance.
Introduction
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1
In the digital control system, the control task is usually performed periodically, because this facilitates the systematic analysis and synthesis using the sampling data sys∗
This work was jointly supported by the National Natural Science Foundation of China (61773217), Hunan Provincial Science and Technology Project Foundation (2019RS1033), the Scientific Research Fund of Hunan Provincial Education Department (18A013), Hunan Normal University National Outstanding Youth Cultivation Project (XP1180101) and the Construct Program of the Key Discipline in Hunan Province. † The corresponding author is Professor Quanxin Zhu with e-mail: [email protected].
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tem theory. However, in the control system based on digital communication network, this method has the problem of wasting communication and computing resources. In order to alleviate this problem, literatures [1]-[2] proposed the event-triggered control method of alternative the periodic sampling control (or time-triggered control) method. In the event-triggered control, the execution of the control task is determined by the welldesigned event-triggered mechanism, and the control task is executed after the triggering mechanism emits the event. The works in [3]-[7] have shown that the event-triggered control can effectively reduce the number of control tasks while maintaining good control performance. Moreover, event-triggered control based on state-feedback is favored by researchers due to its easy analysis and implementation characteristics, and has achieved abundant results [8]-[16].
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In the theoretical research, if the system accuracy requirements are relatively low, we can use the deterministic system to describe it. However, in real life, the description of practical problems is more and more accurate. And the influence of stochastic disturbance and other factors on the system cannot be ignored. While stochastic systems can operate normally, stability is an indispensable prerequisite. In fact, the stability of stochastic systems has been paid much attention since the 1990s [17]-[27]. It has been widely used in physics, chemistry, mechanics, economics and finance system, aerospace engineering system and control theory. In particular, some stable deterministic systems become unstable under the interference of stochastic noise. At this point, the structure of the system may also change substantially. At the same time, the theoretical study and analysis of the stability problem will become more difficult after the stochastic noise interference is considered in the system.
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In 1989, Sontag published a historic article [28] in which he formally proposed the concept of input-to-state stability. Different from the input and output stability described by operators, the input-to-state stability takes into account the influence of the initial state of the system on the system’s future time motion trajectory. In addition, it adopts a description method fully integrated with Lyapunov stability theory. Therefore, inputto-state stability has received extensive attention in the field of control systems [29][37]. These conclusions lay an important position for the study of input-to-state stability theory in nonlinear control system. Compared with the input-to-state stability research in continuous-time systems [38]-[39], the study of input-to-state stability in stochastic discrete-time systems is still insufficient. The basic description of event-triggered control for stochastic linear discrete-time systems was given in [40]. However, works on discretetime stochastic nonlinear systems with external disturbances are rare. So, there are also many problems worth further research in discrete-time stochastic nonlinear systems. Motivated by the above discussion, a class of discrete-time stochastic nonlinear systems with external disturbances are studied in this paper. By using the stochastic analysis theory, comparison principle, reduction to absurdity and classified discussion techniques, we obtain the asymptotic stability and input-to-state stability in mean square of the suggested system with the help of an event-triggered mechanism. The main contributions of this paper are as follows: 2
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1) We are concerned with the event-triggered state-feedback control of discrete-time stochastic nonlinear systems, which is more challenging than most of works only considering deterministic systems. 2) Different from the input and output stability described by operators, the input-to-state stability takes into account the influence of the initial state of the system on the systems future time motion trajectory. 3) Works on discrete-time stochastic nonlinear systems with external disturbances are rare. Our work considers such an interesting question. 4) Under some suitable conditions, the asymptotic stability in mean square in the case of the state-dependent external disturbance and the input-to-state stability in mean square in the case of the state-independent external disturbance are obtained. 5) The time intervals of the event-triggered mechanism designed in this paper are not less than 2 moments. Therefore, the situation of continuous triggering can be directly avoided.
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The rest sections of this paper are arranged as follows. The system, some definitions, assumptions and preliminary lemmas are presented in Section 2. In Section 3, a specific event-triggered mechanism is designed to guarantee the stability in mean square with respect to a state-dependent disturbance and a state-independent disturbance, respectively. Finally, we close this paper with some discussions and provide directions for future research in Section 4.
Notation and problem formulation
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We present the system, some definitions, necessary assumptions and preliminary lemmas in this section.
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Notations: Let N denote the set of all nonnegative integers. Id represents the identity function. A function γ : R+ → R+ is of class K if γ(·) is a strictly increasing function and γ(0) = 0. Moreover, γ ∈ K∞ if γ ∈ K that satisfies γ(s) → ∞ as s → ∞. We denote by γ ∈ CK and γ ∈ VK if γ ∈ K and γ is concave and convex, respectively. Let KL denote the class of functions β(x, t) from R+ × R+ to R+ with β(·, t) ∈ K for each fixed t and decreasing function β(x, t) → 0 as t → ∞ for each fixed x.
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A discrete-time stochastic nonlinear system with the state-feedback control and an unknown external disturbance is considered as follows: x(k + 1) − x(k) = f (k, x(k), u(k), d(k)) + g(k, x(k), u(k), d(k))ω(k),
(2.1)
where k ∈ N, x(k) ∈ Rn represents the state. u(k) ∈ Rm is the control input. d(k) ∈ Rnd is an external disturbance. f : N×Rn ×Rm ×Rnd → Rn and g : N×Rn ×Rm ×Rnd → Rn×r are the drift and diffusion scaling factors of the dynamics, respectively. ω(k) is a scalar Wiener process on a probability space (Ω, F, P) with E[ω(k)] = 0, E[ω 2 (k)] = 1 and ˆ = 0 (∀ k ̸= k). ˆ E[ω(k)ω(k)] 3
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Let {ki }i∈S denote a sequence of triggering times, which is defined as follows: ki+1 = min{k ≥ ki | ϖ(x(k), x(ki )) > 0} + 1,
k0 = 0,
(2.2)
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where S = {0, 1, 2, . . .} ⊆ N and ϖ : Rn × Rn → R is the event function, to be designed in next section. The state-feedback control u(k) is designed by a continuous function κ : Rn → Rm between actuator triggering times ki and ki+1 : ki ≤ k < ki+1 .
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u(k) = κ(x(ki )),
(2.3)
Define the transmission error e(t) as the difference between states x(ki ) and x(k) in the form of: e(k) = x(ki ) − x(k), ki ≤ k < ki+1 . (2.4)
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Next, we state the definitions of the input-to-state stability and the asymptotic stability in mean square to system (2.1) as follows. Definition 2.1 System (2.1) is called asymptotically stable in mean square (ASMS) if there exist a class KL function β such that for all k ≥ k0 , E(|x(k)|2 ) ≤ β(E(|x(k0 )|2 ), k − k0 ).
(2.5)
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Definition 2.2 System (2.1) is called input-to-state stable in mean square (ISSMS) with respect to external disturbances if there exist functions β ∈ KL and γ ∈ K such that for all k ≥ k0 , E(|x(k)|2 ) ≤ β(E(|x(k0 )|2 ), k − k0 ) + γ(sup E|d(k)|2 ). (2.6) k∈N
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Let xk = x(k), ek = e(k), dk = d(k), ωk = ω(k). Using f (k, x, e, d) and g(k, x, e, d) instead of f (k, x(k), κ(x(k) + e(k)), d(k)) and g(k, x(k), κ(x(k) + e(k)), d(k)), respectively. Then, system (2.1) is equivalent to xk+1 − xk = f (k, x, e, d) + g(k, x, e, d) ωk ,
ki ≤ k < ki+1 .
(2.7)
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Before studying the above two class of stability of system (2.7), we first give some necessary assumptions to show the existence and uniqueness of the solution of the suggested system. Assumption 2.3 There exist nonnegative constants ψ1 , ψ2 , ψ3 such that ˆ 2 ∨ |g(k, x, e, d) − g(k, xˆ, eˆ, d)| ˆ 2 |f (k, x, e, d) − f (k, xˆ, eˆ, d)| ˆ 2, ≤ ψ1 |x − xˆ|2 + ψ2 |e − eˆ|2 + ψ3 |d − d|
for any x, xˆ, e, eˆ ∈ Rn and d, dˆ ∈ Rnd . 4
(2.8)
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Assumption 2.4 Functions f (k, ·, ·, ·) and g(k, ·, ·, ·) admit an equilibrium point, i.e., f (k, 0, 0, 0) ≡ 0,
g(k, 0, 0, 0) ≡ 0.
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Remark 2.5 It is known that Assumption 2.3 is the uniform Lipschitz condition. Under Assumptions 2.3 and 2.4, there exists a unique global solution x(k; x(k0 )) to system (2.7) for any initial data x(k0 ) ∈ L2Fk (Ω; Rn ). Moreover, Assumption 2.4 shows that system 0 (2.7) admits a trivial solution x(k; 0) = 0 when the initial value x(k0 ) ≡ 0. To obtain the stability of our suggested system in next section, we first present the comparison principle for discrete-time stochastic systems as follows: Lemma 2.6 ([41], Lemma 2.1) For any continuous function h ∈ VK with Id − h ≥ 0, if there exist a KL-function β(s, k) satisfying with 0 ≤ Ey(k) < ∞. Then,
y(k0 ) = y0 ,
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E[y(k + 1) − y(k)] ≤ −Eh(y(k)), Ey(k) ≤ β(Ey0 , k),
k ∈ N.
Lemma 2.7 ([42], Theorem 1) Assume that there exist K∞ -function v, v¯, α ˜ , a K-function ¯ n λ and a continuous Lyapunov function V : R → R+ satisfying the following conditions:
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v(|xk |2 ) ≤ V (xk ) ≤ v¯(|xk |2 ), ¯ E[V (xk+1 ) − V (xk )] ≤ −˜ α(E|xk |2 ) + λ(sup E|dk |2 ),
(2.9) (2.10)
k∈N
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Then, system (2.7) is ISSMS.
Event-triggered feedback control
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As shown in Figure 1, we will develop a specific event-triggered control approach for stochastic nonlinear system (2.7) with external disturbances in this section. The function of event-triggering mechanism (2.2) is designed as: ϖ(r1 , r2 ) = −πx (|r1 |2 ) + πe (|r2 − r1 |2 ),
where r1 , r1 ∈ Rn , πe = (2 + 3ψ2 )Id and πx = ρ ◦ (Id + ψ1 , ψ2 , ψ3 given in Assumption 2.3 and ρ ∈ CK∞ .
(3.1)
1 ρ)−1 − 3(ψ1 Id + ψ3 ϱ) with 3ψ1
Moreover, we show that the event-triggered stochastic system is ASMS and ISSMS with respect to a state-independent disturbance. 5
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e(k+1)
u(k+1) = u(ki+1)
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u(k+1) = u(ki)
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x(k+1)−x(k) = f (k, x(k), e(k), d(k))+g(k, x(k), e(k), d(k)) ω(k)
if ̟(x(k), x(ki)) > 0
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Figure 1: The event-triggered stochastic system with external disturbances
State-dependent disturbance
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3.1
In the case of the state-dependent disturbance, we assume that the second order of the external disturbance dk is upper bounded by a function ϱ ∈ K∞ of xk as follows: |dk |2 ≤ ϱ(|xk |2 ),
k ∈ N.
(3.2)
Theorem 3.1 Suppose that Assumptions 2.3 and 2.4 hold. If there exist functions v ∈ ¯ VK∞ , v¯ ∈ CK∞ , α ∈ K∞ , ϕ, φ ∈ K and V : Rn → R+ satisfying
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v(|xk |2 ) ≤ V (xk ) ≤ v¯(|xk |2 ), ¯ EV (xk+1 ) − EV (xk ) ≤ −Eα(|xk |2 ) + ϕ(E|ek |2 ) ∨ φ(E|dk |2 ),
(3.3) (3.4)
where the external disturbance dk is state-dependent and satisfies (3.2) and following inequality
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[ϕ◦ρ(·)] ∨ [φ◦ϱ(·)] < (Id − θ)◦α(·)
(3.5)
with ρ ∈ CK∞ , ϱ ∈ K∞ , θ ∈ K∞ , θ ◦α ∈ VK∞ and satisfies Id−θ ∈ K∞ , θ ◦α(·) < v¯(·). Then stochastic system (2.7) is asymptotically stable in mean square under the event-triggered control.
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Proof. In the event-triggered control (2.4), the state xk and the error ek determine whether the next time k + 1 be a triggering time. Here, we consider the following two cases: Case I: For any specific k ∈ N, ϖ(xk , xki ) ≤ 0.
By the definition of function ϖ in (3.1), we have 0 ≥ ϖ(xk , xki ) = −πx (|xk |2 ) + πe (|xki − xk |2 ) ] [ 1 = 3(ψ1 Id + ψ3 ϱ) − ρ◦(Id + ρ)−1 (|xk |2 ) + (2 + 3ψ2 )Id(|ek |2 ). 3ψ1 6
(3.6)
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Moreover, ki ≤ k < k + 1 < ki+1 , xk+1 is not transmitted in this case. It yields ek+1 = xki − xk+1 . Then, from Assumptions 2.3, 2.4 and condition (3.2), E|ek+1 |2 has the following property:
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= = ≤ ≤ ≤
E|ek+1 |2 E|xki − xk + xk − xk+1 |2 E|ek − f (k, x, e, d) − g(k, x, e, d)ωk |2 2E|ek |2 + 2E|f (k, x, e, d)|2 + E|g(k, x, e, d)|2 2E|ek |2 + 3ψ1 E|xk |2 + 3ψ2 E|ek |2 + 3ψ3 E|dk |2 3E(ψ1 Id + ψ3 ϱ)(|xk |2 ) + (2 + 3ψ2 )E|ek |2 .
By using (3.7) and (3.6), we have
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E|ek+1 |2 ≤ 3E(ψ1 Id + ψ3 ϱ)(|xk |2 ) + (2 + 3ψ2 )E|ek |2 1 ≤ Eρ◦(Id + ρ)−1 (|xk |2 ). 3ψ1
(3.7)
(3.8)
It follows from (3.6) again that
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|xk |2 − |xk+1 |2 ψ3 2 + 3ψ2 ≤ |xk |2 + ϱ(|xk |2 ) + |ek |2 ψ1 3ψ1 1 1 ≤ ρ◦(Id + ρ)−1 (|xk |2 ). 3ψ1 3ψ1
(3.9)
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Then, by using (3.9), we get |xk+1 |2
1 1 ρ◦(Id + ρ)−1 ](|xk |2 ) 3ψ1 3ψ1 1 1 1 1 = [(Id + ρ)◦(Id + ρ)−1 − ρ◦(Id + ρ)−1 ](|xk |2 ) 3ψ1 3ψ1 3ψ1 3ψ1 1 = (Id + ρ)−1 (|xk |2 ). 3ψ1
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≥ [Id −
Substituting (3.10) into (3.8) that E|ek+1 |2 ≤ Eρ(|xk+1 |2 ),
in the case of ϖ(xk , xki ) ≤ 0. 7
(3.10)
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Case II: For any specific k ∈ N, ϖ(xk , xki ) > 0.
It follows from (2.2), state x transmits at time k + 1, which means ki+1 = k + 1. Thus, ek+1 = xki+1 − xk+1 = 0. And then, E|ek+1 |2 ≤ Eρ(|xk+1 |2 ) in this case.
E|ek |2 ≤ Eρ(|xk |2 ).
EV (xk+1 ) − EV (xk ) −α(E|xk |2 ) + max{ϕ(E|ek |2 ), φ(E|dk |2 )} −α(E|xk |2 ) + max{ϕ◦ρ(E|xk |2 ), φ◦ϱ(E|dk |2 )} −θ◦α(E|xk |2 ) −θ◦α◦¯ v −1 (EV (xk )).
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≤ ≤ ≤ ≤
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By using conditions (3.2)-(3.5) and (3.11), we get
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Moreover, 0 = E|ek0 |2 ≤ Eρ(|xk0 |2 ). Thus, the above discussions yield that for all k ∈ N,
(3.11)
(3.12)
Adding from k0 to k − 1 on the both sides of inequality (3.12), we obtain EV (xk ) − EV (xk0 ) ≤ −
k−1 ∑
θ◦α◦¯ v −1 (EV (xs )),
s=k0
which means 0 ≤ EV (xk ) ≤ EV (xk0 ) < ∞. Then, it follows from Lemma 2.6 that there exists a function β˜ ∈ KL such that
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˜ EV (xk ) ≤ β(EV (xk0 ), k),
since θ◦α ∈ VK, v¯ ∈ CK, θ◦α(·) ≤ v¯(·).
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Moreover, v¯ ∈ CK also yields that
EV (xk ) ≤ ≤ ≤
˜ β(EV (xk0 ), k) ˜ β(E¯ v (|xk0 |2 ), k) ˜ v (E|xk |2 ), k). β(¯ 0
(3.13)
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On the other hand, by inequality (3.3) and v ∈ VK, we have v(E|xk |2 ) ≤ Ev(|xk |2 ) ≤ EV (xk ).
(3.14)
It follows from (3.13) and (3.14) that ˜ v (E|xk |2 ), k), E|xk |2 ≤ v −1 ◦ β(¯ 0 ˜ v (x), k) in Definition 2.1. So, system (2.7) is ASMS with β(x, k) = v −1 ◦ β(¯ 8
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Remark 3.2 For any specific k ∈ N, E|ek |2 ≤ ρ(E|xk |2 ) can not directly obtained from E|ek+1 |2 ≤ ρ(E|xk+1 |2 ), since xk+1 is determined by xk , ek and dk . Motivated by our discussion in this paper, we designed an event-triggering function ϖ(xk , xki ) in the form of (3.1) to solve this problem. Moreover, composite function πx in (3.1) need not to be positive definite, but only satisfies condition (3.5). Thus, the event-triggered mechanism in this paper is more general than those given in the previous literature.
3.2
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Remark 3.3 Zhang et al. in [43] focused on the discrete-time nonlinear systems with event-triggered controller. On this basis, we generalize their results to the stochastic systems. To obtain the globally asymptotic stabilization, the authors in [43] used the fact that |a − b| ≤ θ ◦ (Id + ρ)−1 (|a|) ⇒ |a − b| ≤ ρ(|b|) for any a, b ∈ R and ρ ∈ K, which was proved in [44]. However, this conclusion is no longer applicable in the proof of Theorem 3.1 in our paper. Since we need to show that inequality |a − b|2 ≤ ρ(|b|2 ) holds, which is equal to (3.11). To overcome this difficulty, we analyze the relationship between E|xk |2 and E|xk+1 |2 separately. And then, we get the desired results. Furthermore, we show that our stochastic system is ASMS with respect to a state-dependent external disturbance. Therefore, we generalize and improve those results obtained in [43].
State-independent disturbance
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Here, we consider the stochastic nonlinear system with event-triggered control, which respects to a state-independent external disturbance dk . This means property (3.2) may be not satisfied. And then, Theorem 3.1 is not valid for this case. In what follows, we focus on the ISSMS of the event-triggered stochastic system with a state-independent disturbance.
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Theorem 3.4 Suppose that conditions of Theorem 3.1 hold without the property (3.2) for the external disturbance dk . Moreover, α in (3.4) satisfies α ∈ VK∞ . Then stochastic system (2.7) is ISSMS under the event-triggered control. Proof. We consider the following two cases: Case I: For any specific k ∈ N, E|ek |2 ≤ ρ(E|xk |2 ).
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It follows from inequalities (3.4) and (3.5) that
EV (xk+1 ) − EV (xk ) ≤ −α(E|xk |2 ) + max{ϕ ◦ ρ(E|xk |2 ), φ(E|dk |2 )} ≤ −α(E|xk |2 ) + max{(Id − θ) ◦ α(E|xk |2 ), φ(E|dk |2 )} ≤ −θ ◦ α(E|xk |2 ) + φ(E|dk |2 ).
Case II: For any specific k ∈ N, E|ek |2 > ρ(E|xk |2 ). 9
(3.15)
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In this case, the fact |ek | = 0 can not satisfied. Thus, k ≥ k0 + 1 and k is not the triggering time. Moreover, there exists i ∈ S such that ki ≤ k − 1 < k < ki+1 .
We now show that |ek−1 |2 > ρ(|xk−1 |2 ) does not hold. In fact, if |ek−1 |2 > ρ(|xk−1 |2 ), and then we have
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ϖ(xk−1 , xki ) [ ] 1 = 3(ψ1 Id + ψ3 ϱ) − ρ◦(Id + ρ)−1 (|xk−1 |2 ) + (2 + 3ψ2 )Id(|ek−1 |2 ) 3ψ1 [ ] 1 = 3(ψ1 Id + ψ2 ρ + ψ3 ϱ) + 2ρ − ρ◦(Id + ρ)−1 (|xk−1 |2 ) 3ψ1 > 0.
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From the event-triggered mechanism (2.4), we obtain that time k is the triggering time, and then ki = k. Then, we have |ek | = 0, which contradicts with the case |ek |2 > ρ(|xk |2 ). Thus, assumption |ek−1 |2 > ρ(|xk−1 |2 ) does not hold. This yields E|ek−1 |2 ≤ Eρ(|xk−1 |2 ) ≤ ρ(E|xk−1 |2 ).
(3.16)
On this basis, we can prove that for any k > k0 ,
E|dk−1 |2 > ϱ(E|xk−1 |2 ).
(3.17)
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In fact, if equality (3.17) does not hold, we have E|ek |2 ≤ ρ(E|xk |2 ) by the proof of Theorem 3.1. This also contradicts with the case E|ek |2 > ρ(E|xk |2 ). Thus, equality (3.17) holds. Combine (3.16) and (3.17), we get
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E|ek |2 = E|(xk − xk−1 ) + (xk−1 − xki )|2 E|f (k − 1, x, e, d) + g(k − 1, x, e, d)ωk−1 − ek−1 |2 3ψ1 E|xk−1 |2 + 3ψ2 E|ek−1 |2 + 3ψ3 E|dk−1 |2 + 2E|ek−1 |2 [3ψ1 Id + (2 + 3ψ2 )ρ](E|xk−1 |2 ) + 3ψ3 E|dk−1 |2 σE|dk−1 |2 ,
= ≤ ≤ ≤
(3.18)
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where σ = [3ψ1 Id + (2 + 3ψ2 )ρ] ◦ ϱ−1 + 3ψ3 Id.
Without losing generality, letting d(−1) = 0. And then, inequality (3.18) also holds in k = k0 . By condition (3.4), we have EV (xk+1 ) − EV (xk ) ≤ −α(E|xk |2 ) + max{ϕ ◦ σ(E|dk−1 |2 ), φ(E|dk |2 )},
for any k ∈ N. 10
(3.19)
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Combine inequalities (3.15) and (3.19) in above two cases, we obtain EV (xk+1 ) − EV (xk ) ≤ − min{α(E|xk |2 ), θ ◦ α(E|xk |2 )} + max{ϕ ◦ σ(E|dk−1 |2 ), φ(E|dk |2 )} ≤ − min{α(E|xk |2 ), θ ◦ α(E|xk |2 )} + max{ϕ ◦ σ(sup E|dk |2 ), φ(sup E|dk |2 )}. k∈N
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Thus, from Lemma 2.7, the system (2.7) is ISSMS under the event-triggered mechanism with α ˜ and λ in (2.10) are taken as α ˜ (·) = min{α(·), θ ◦ α(·)} and λ(·) = max{ϕ ◦ σ(·), φ(·)}, respectively.
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Remark 3.5 In CaseII of the proof of Theorem 3.4, we have ki ≤ k−1 < k < k+1 ≤ ki+1 , which means ki+1 − ki ≥ 2. In reality, this fact always holds only if πx is positive definite. And ϖ(xki , xki ) = −πx (|xki |2 ) ≤ 0 for each i ∈ S. So, ki+1 = min{k ≥ ki | ϖ(xk , xki ) > 0} + 1 ≥ ki + 2 since the “ + 1” term. Obviously, if feedback controller triggered at time k, then k − 1 and k + 1 are both not the triggering times. Therefore, the situation of continuous triggering can be directly avoided. However, if πx is negative definite, it is easy to obtain ϖ(xk , xki ) > 0. And then, ki+1 = k + 1 (i ∈ N) holds for all k ∈ N.
Concluding remarks
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Remark 3.6 Obviously, the system suggested in [44] is a class of nonlinear continuous systems. Different from that, we consider a class of discrete-time stochastic nonlinear systems in this paper. In the proof of our results, there are lots of techniques has been used, such as comparison principle, classified discussion, reduction to absurdity and so on. Moreover, with the help of the event-triggered mechanism, we realize the asymptotic stability in mean square of the suggested stochastic system when the external disturbance dk is state-dependent. While, even if the external disturbance dk is state-independent, we can also obtain the suggested stochastic system is input-to-state stable in mean square.
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In this paper, a class of discrete-time stochastic nonlinear systems have been investigated. More precisely, we consider the external disturbance and the event-triggered feedback control into coefficients. By using comparison principle, classified discussion, reduction to absurdity and many other technique, we realize the stability in mean square of the suggested stochastic system. Especially, in the case of state-dependent external disturbance, we assume that disturbance is upper bounded by a function of state. And then, the suggested stochastic system is ASMS. On the other hand, in the case of stateindependent external disturbance, the suggested stochastic system is ISSMS. Finally, some discussions and remarks are given to illustrate that our results are meaningful. In future, we will extend our results to some more complex event-triggered stochastic systems, such as discrete-time stochastic delayed systems, continuous-time stochastic nonlinear systems with Markovian switching, continuous-time stochastic delayed systems and 11
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so on. Moreover, time-triggered control and self-triggered control for stochastic systems will also be considered.
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[2] K.J. ˚ Astr¨om and B. Bernhardsson, “Comparison of periodic and event based sampling for first-order stochastic systems”, IFAC Proc. Volumes, 32(2), 5006-5011, 1999. [3] T. Henningsson, E. Johannesson and A. Cervin, “Sporadic event-based control of first-order linear stochastic systems”, Automatica, 44, 2890-2895, 2008.
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[4] W.P.M.H. Heemels, K.H. Johansson and P. Tabuada, “An introduction to eventtriggered and self-triggered control”, IEEE Conf. Decis. Control, 3270-3285, 2012. [5] D.V. Dimarogonas, E. Frazzoli and K.H. Johansson, “Distributed event-triggered control for multi-agent systems”, IEEE Trans. Autom. Control, 57(5), 1291-1297, 2012. [6] X.F. Wang and M.D. Lemmon, “Event-triggering in distributed networked control systems”, IEEE Trans. Autom. Control, 56(3), 586-601, 2011.
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[10] D.R. Ding, Z.D. Wang and B. Shen, “Event-triggered consensus control for discretetime stochastic multi-agent systems: The input-to-state stability in probability”, Automatica, 62, 284-291, 2015. [11] H.J. Wang, P. Shi and R.K. Agarwal, “Network-based event-triggered filtering for Markovian jump systems”, Int. J. Control, 89(6), 1096-1110, 2016. [12] D. Liuzza, D.V. Dimarogonas, M. Bernardo and K.H. Johansson, “Distributed model based event-triggered control for synchronization of multi-agent systems”, Automatica, 73, 1-7, 2016.
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[16] K. Liu and E. Fridman, “Wirtingers inequality and Lyapunov-based sampled-data stabilization”, Automatica, 48, 102-108, 2012. [17] I. Karafyllis and M. Krstic, “Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold”, IEEE Trans. Automat. Control, 57, 1141-1154, 2012.
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[18] F. Mazenc, M. Malisoff and T.H. Dinh, “Robustness of nonlinear systems with respect to delay and sampling of the controls”, Automatica, 49, 1925-1931, 2013. [19] P. Pepe, “Stabilization in the sample-and-hold sense of nonlinear retarded systems”, SIAM J. Control Optim., 52, 3053-3077, 2014. [20] L. Karafyllis and M. Krstic, “Sampled-data stabilization of nonlinear delay systems with a compact absorbing set”, SIAM J. Control Optim., 54, 790-818, 2016.
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[21] H. Dong, Z. Wang and B. Shen, “Variance-constrained H-infinity control for a class of nonlinear stochastic discrete time-varying systems: The event-triggered design”, Automatica, 72, 28-36, 2016.
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[22] Y. Tang, H. Gao and J. Kurths, “Robust H-infinity self-triggered control of networked systems under packet dropouts”, IEEE Trans. Cybern., 46, 3294-3305, 2016. [23] Z. Hu and F. Deng, “Robust H-infinity control for networked systems with transmission delays and successive packet dropouts under stochastic sampling”, Int. J. Robost Nonlinear Control, 27, 84-107, 2017.
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[24] M.G. Hua, L. Zhang, F.Q. Yao, J.J. Ni, W.L. Dai, Y.L. Cheng, “Robust H∞ filtering for continuous-time nonhomogeneous Markov jump nonlinear systems with randomly occurring uncertainties”, Signal Processing, 148, 250-259, 2018. [25] M.G. Hua, D.D. Zheng, F.Q. Deng, J.T. Fei, P. Cheng, X.S. Dai, “H∞ filtering for nonhomogeneous Markovian jump repeated scalar nonlinear systems with multiplicative noises and partially mode-dependent characterization”, IEEE Trans. Syst. Man Cybern. Syst., DOI: 10.1109/TSMC.2019.2919146. [26] E.D. Sontag, “Smooth stabilization implies coprime factorization”, IEEE Trans. Autom. Control, 34(4), 435-443, 1989. 13
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[27] X. Liu and H.J. Marquez, “Preservation of input-to-state stability under sampling and emulation: Multirate case”, Int. J. Control, 80(12), 1944-1953, 2007. [28] P. Tabuada, “Event-triggered real-time scheduling of stabilizing control tasks”, IEEE Trans. Autom. Control, 52(9), 1680-1685, 2007.
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[30] H. Huang, D. Li and Y. Xi, “Design and input-to-state practically stable analysis of the mixed H-2/H-infinity feedback robust model predictive control”, IET Control Theory Appl., 6(4), 498-505, 2012. [31] C.K. Ahn, “Input-to-state stable nonlinear filtering for a class of continuous-time delayed nonlinear systems”, Int. J. Control, 86(6), 1179-1185, 2013.
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[32] C. Cai and A.R. Teel, “Robust input-to-state stability for hybrid systems”, SIAM J. Control Optim., 51, 1651-1678, 2013. [33] A. Girard, “Dynamic triggering mechanisms for event-triggered control”, IEEE Trans. Autom. Control, 60, 1992-1997, 2015. [34] X.T. Wu, Y. Tang and W.B. Zhang, “Input-to-state stability of impulsive stochastic delayed systems under linear assumptions”, Automatica, 66, 195-204, 2016.
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[35] X.T. Wu, P. Shi, Y. Tang and W.B. Zhang, “Input-to-state stability of nonlinear stochastic time-varying systems with impulsive effects”, Int. J. Robust Nonlinear Control, 27(10), 1792-1809, 2017.
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[38] T. Henningsson, “Event-based control and estimation with stochastic disturbances”, Lund University Press, ISSN 0280-5316, 2008. [39] P. Zhao, Y. Zhao and R.W. Guo, “Input-to-state stability for discrete-time stochastic nonlinear systems”, Proc. 34th Chinese Control Conf., 2015. [40] D.R. Ding, Z.D. Wang, B. Shen and G.L. Wei, “Event-triggered consensus control for discrete-time stochastic multi-agent systems: The input-to-state stability in probability”, Automatica, 62, 284-291, 2015.
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[41] P.P. Zhang, T.F. Liu and Z.P. Jiang, “Input-to-state stabilization of nonlinear discrete-time systems with event-triggered controllers”, Syst. Control Lett., 103, 1622, 2017.
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[42] T.F. Liu and Z.P. Jiang, “A small-gain approach to robust event-triggered control of nonlinear systems”, IEEE Trans. Autom. Control, 60(8), 2072-2083, 2015.
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Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
PII: DOI: Reference:
S0378-4371(19)32126-0 https://doi.org/10.1016/j.physa.2019.123823 PHYSA 123823
To appear in:
Physica A
Received date : 27 August 2019 Revised date : 27 November 2019 Please cite this article as: W. Xie and Q. Zhu, Stability of discrete-time stochastic nonlinear systems with event-triggered state-feedback control, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123823. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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Highlights
1. We are concerned with the event-triggered state-feedback control of discrete-time stochastic nonlinear systems, which is more challenging than most of works only considering deterministic systems.
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2. Different from the input and output stability described by operators, the input-to-state stability takes into account the influence of the initial state of the system on the systems future time motion trajectory. 3. Works on discrete-time stochastic nonlinear systems with external disturbances are rare. Our work considers such an interesting question.
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4. Under some suitable conditions, the asymptotic stability in mean square in the case of the state-dependent external disturbance and the input-to-state stability in mean square in the case of the state-independent external disturbance are obtained.
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Stability of discrete-time stochastic nonlinear systems with event-triggered state-feedback control ∗ b
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Wenjuan Xiea and Quanxin Zhua,b
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing 210023, China; a
MOE-LCSM, School of Mathematics and Statistics,
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Hunan Normal University, Changsha, Hunan 410081, China
Abstract
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This paper investigates a class of discrete-time stochastic nonlinear systems with external disturbances. To obtain the stability of the suggested system, we develop an event-triggered mechanism which is more general than those given in previous literatures. On this basis, many techniques such as stochastic analysis theory, classified discussion, comparison principle and reduction to absurdity are successfully applied in this paper. Under some suitable conditions, the asymptotic stability in mean square in the case of the state-dependent external disturbance and the input-to-state stability in mean square in the case of the state-independent external disturbance are obtained. Finally, some discussions and remarks are provided to show the significance of our results.
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Key Words: Discrete-time stochastic system; event-triggered state-feedback control; asymptotic stability; input-to-state stability; external disturbance.
Introduction
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1
In the digital control system, the control task is usually performed periodically, because this facilitates the systematic analysis and synthesis using the sampling data sys∗
This work was jointly supported by the National Natural Science Foundation of China (61773217), Hunan Provincial Science and Technology Project Foundation (2019RS1033), the Scientific Research Fund of Hunan Provincial Education Department (18A013), Hunan Normal University National Outstanding Youth Cultivation Project (XP1180101) and the Construct Program of the Key Discipline in Hunan Province. † The corresponding author is Professor Quanxin Zhu with e-mail: [email protected].
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tem theory. However, in the control system based on digital communication network, this method has the problem of wasting communication and computing resources. In order to alleviate this problem, literatures [1]-[2] proposed the event-triggered control method of alternative the periodic sampling control (or time-triggered control) method. In the event-triggered control, the execution of the control task is determined by the welldesigned event-triggered mechanism, and the control task is executed after the triggering mechanism emits the event. The works in [3]-[7] have shown that the event-triggered control can effectively reduce the number of control tasks while maintaining good control performance. Moreover, event-triggered control based on state-feedback is favored by researchers due to its easy analysis and implementation characteristics, and has achieved abundant results [8]-[16].
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In the theoretical research, if the system accuracy requirements are relatively low, we can use the deterministic system to describe it. However, in real life, the description of practical problems is more and more accurate. And the influence of stochastic disturbance and other factors on the system cannot be ignored. While stochastic systems can operate normally, stability is an indispensable prerequisite. In fact, the stability of stochastic systems has been paid much attention since the 1990s [17]-[27]. It has been widely used in physics, chemistry, mechanics, economics and finance system, aerospace engineering system and control theory. In particular, some stable deterministic systems become unstable under the interference of stochastic noise. At this point, the structure of the system may also change substantially. At the same time, the theoretical study and analysis of the stability problem will become more difficult after the stochastic noise interference is considered in the system.
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In 1989, Sontag published a historic article [28] in which he formally proposed the concept of input-to-state stability. Different from the input and output stability described by operators, the input-to-state stability takes into account the influence of the initial state of the system on the system’s future time motion trajectory. In addition, it adopts a description method fully integrated with Lyapunov stability theory. Therefore, inputto-state stability has received extensive attention in the field of control systems [29][37]. These conclusions lay an important position for the study of input-to-state stability theory in nonlinear control system. Compared with the input-to-state stability research in continuous-time systems [38]-[39], the study of input-to-state stability in stochastic discrete-time systems is still insufficient. The basic description of event-triggered control for stochastic linear discrete-time systems was given in [40]. However, works on discretetime stochastic nonlinear systems with external disturbances are rare. So, there are also many problems worth further research in discrete-time stochastic nonlinear systems. Motivated by the above discussion, a class of discrete-time stochastic nonlinear systems with external disturbances are studied in this paper. By using the stochastic analysis theory, comparison principle, reduction to absurdity and classified discussion techniques, we obtain the asymptotic stability and input-to-state stability in mean square of the suggested system with the help of an event-triggered mechanism. The main contributions of this paper are as follows: 2
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1) We are concerned with the event-triggered state-feedback control of discrete-time stochastic nonlinear systems, which is more challenging than most of works only considering deterministic systems. 2) Different from the input and output stability described by operators, the input-to-state stability takes into account the influence of the initial state of the system on the systems future time motion trajectory. 3) Works on discrete-time stochastic nonlinear systems with external disturbances are rare. Our work considers such an interesting question. 4) Under some suitable conditions, the asymptotic stability in mean square in the case of the state-dependent external disturbance and the input-to-state stability in mean square in the case of the state-independent external disturbance are obtained. 5) The time intervals of the event-triggered mechanism designed in this paper are not less than 2 moments. Therefore, the situation of continuous triggering can be directly avoided.
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The rest sections of this paper are arranged as follows. The system, some definitions, assumptions and preliminary lemmas are presented in Section 2. In Section 3, a specific event-triggered mechanism is designed to guarantee the stability in mean square with respect to a state-dependent disturbance and a state-independent disturbance, respectively. Finally, we close this paper with some discussions and provide directions for future research in Section 4.
Notation and problem formulation
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We present the system, some definitions, necessary assumptions and preliminary lemmas in this section.
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Notations: Let N denote the set of all nonnegative integers. Id represents the identity function. A function γ : R+ → R+ is of class K if γ(·) is a strictly increasing function and γ(0) = 0. Moreover, γ ∈ K∞ if γ ∈ K that satisfies γ(s) → ∞ as s → ∞. We denote by γ ∈ CK and γ ∈ VK if γ ∈ K and γ is concave and convex, respectively. Let KL denote the class of functions β(x, t) from R+ × R+ to R+ with β(·, t) ∈ K for each fixed t and decreasing function β(x, t) → 0 as t → ∞ for each fixed x.
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A discrete-time stochastic nonlinear system with the state-feedback control and an unknown external disturbance is considered as follows: x(k + 1) − x(k) = f (k, x(k), u(k), d(k)) + g(k, x(k), u(k), d(k))ω(k),
(2.1)
where k ∈ N, x(k) ∈ Rn represents the state. u(k) ∈ Rm is the control input. d(k) ∈ Rnd is an external disturbance. f : N×Rn ×Rm ×Rnd → Rn and g : N×Rn ×Rm ×Rnd → Rn×r are the drift and diffusion scaling factors of the dynamics, respectively. ω(k) is a scalar Wiener process on a probability space (Ω, F, P) with E[ω(k)] = 0, E[ω 2 (k)] = 1 and ˆ = 0 (∀ k ̸= k). ˆ E[ω(k)ω(k)] 3
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Let {ki }i∈S denote a sequence of triggering times, which is defined as follows: ki+1 = min{k ≥ ki | ϖ(x(k), x(ki )) > 0} + 1,
k0 = 0,
(2.2)
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where S = {0, 1, 2, . . .} ⊆ N and ϖ : Rn × Rn → R is the event function, to be designed in next section. The state-feedback control u(k) is designed by a continuous function κ : Rn → Rm between actuator triggering times ki and ki+1 : ki ≤ k < ki+1 .
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u(k) = κ(x(ki )),
(2.3)
Define the transmission error e(t) as the difference between states x(ki ) and x(k) in the form of: e(k) = x(ki ) − x(k), ki ≤ k < ki+1 . (2.4)
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Next, we state the definitions of the input-to-state stability and the asymptotic stability in mean square to system (2.1) as follows. Definition 2.1 System (2.1) is called asymptotically stable in mean square (ASMS) if there exist a class KL function β such that for all k ≥ k0 , E(|x(k)|2 ) ≤ β(E(|x(k0 )|2 ), k − k0 ).
(2.5)
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Definition 2.2 System (2.1) is called input-to-state stable in mean square (ISSMS) with respect to external disturbances if there exist functions β ∈ KL and γ ∈ K such that for all k ≥ k0 , E(|x(k)|2 ) ≤ β(E(|x(k0 )|2 ), k − k0 ) + γ(sup E|d(k)|2 ). (2.6) k∈N
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Let xk = x(k), ek = e(k), dk = d(k), ωk = ω(k). Using f (k, x, e, d) and g(k, x, e, d) instead of f (k, x(k), κ(x(k) + e(k)), d(k)) and g(k, x(k), κ(x(k) + e(k)), d(k)), respectively. Then, system (2.1) is equivalent to xk+1 − xk = f (k, x, e, d) + g(k, x, e, d) ωk ,
ki ≤ k < ki+1 .
(2.7)
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Before studying the above two class of stability of system (2.7), we first give some necessary assumptions to show the existence and uniqueness of the solution of the suggested system. Assumption 2.3 There exist nonnegative constants ψ1 , ψ2 , ψ3 such that ˆ 2 ∨ |g(k, x, e, d) − g(k, xˆ, eˆ, d)| ˆ 2 |f (k, x, e, d) − f (k, xˆ, eˆ, d)| ˆ 2, ≤ ψ1 |x − xˆ|2 + ψ2 |e − eˆ|2 + ψ3 |d − d|
for any x, xˆ, e, eˆ ∈ Rn and d, dˆ ∈ Rnd . 4
(2.8)
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Assumption 2.4 Functions f (k, ·, ·, ·) and g(k, ·, ·, ·) admit an equilibrium point, i.e., f (k, 0, 0, 0) ≡ 0,
g(k, 0, 0, 0) ≡ 0.
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Remark 2.5 It is known that Assumption 2.3 is the uniform Lipschitz condition. Under Assumptions 2.3 and 2.4, there exists a unique global solution x(k; x(k0 )) to system (2.7) for any initial data x(k0 ) ∈ L2Fk (Ω; Rn ). Moreover, Assumption 2.4 shows that system 0 (2.7) admits a trivial solution x(k; 0) = 0 when the initial value x(k0 ) ≡ 0. To obtain the stability of our suggested system in next section, we first present the comparison principle for discrete-time stochastic systems as follows: Lemma 2.6 ([41], Lemma 2.1) For any continuous function h ∈ VK with Id − h ≥ 0, if there exist a KL-function β(s, k) satisfying with 0 ≤ Ey(k) < ∞. Then,
y(k0 ) = y0 ,
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E[y(k + 1) − y(k)] ≤ −Eh(y(k)), Ey(k) ≤ β(Ey0 , k),
k ∈ N.
Lemma 2.7 ([42], Theorem 1) Assume that there exist K∞ -function v, v¯, α ˜ , a K-function ¯ n λ and a continuous Lyapunov function V : R → R+ satisfying the following conditions:
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v(|xk |2 ) ≤ V (xk ) ≤ v¯(|xk |2 ), ¯ E[V (xk+1 ) − V (xk )] ≤ −˜ α(E|xk |2 ) + λ(sup E|dk |2 ),
(2.9) (2.10)
k∈N
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Then, system (2.7) is ISSMS.
Event-triggered feedback control
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As shown in Figure 1, we will develop a specific event-triggered control approach for stochastic nonlinear system (2.7) with external disturbances in this section. The function of event-triggering mechanism (2.2) is designed as: ϖ(r1 , r2 ) = −πx (|r1 |2 ) + πe (|r2 − r1 |2 ),
where r1 , r1 ∈ Rn , πe = (2 + 3ψ2 )Id and πx = ρ ◦ (Id + ψ1 , ψ2 , ψ3 given in Assumption 2.3 and ρ ∈ CK∞ .
(3.1)
1 ρ)−1 − 3(ψ1 Id + ψ3 ϱ) with 3ψ1
Moreover, we show that the event-triggered stochastic system is ASMS and ISSMS with respect to a state-independent disturbance. 5
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e(k+1)
u(k+1) = u(ki+1)
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u(k+1) = u(ki)
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x(k+1)−x(k) = f (k, x(k), e(k), d(k))+g(k, x(k), e(k), d(k)) ω(k)
if ̟(x(k), x(ki)) > 0
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Figure 1: The event-triggered stochastic system with external disturbances
State-dependent disturbance
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3.1
In the case of the state-dependent disturbance, we assume that the second order of the external disturbance dk is upper bounded by a function ϱ ∈ K∞ of xk as follows: |dk |2 ≤ ϱ(|xk |2 ),
k ∈ N.
(3.2)
Theorem 3.1 Suppose that Assumptions 2.3 and 2.4 hold. If there exist functions v ∈ ¯ VK∞ , v¯ ∈ CK∞ , α ∈ K∞ , ϕ, φ ∈ K and V : Rn → R+ satisfying
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v(|xk |2 ) ≤ V (xk ) ≤ v¯(|xk |2 ), ¯ EV (xk+1 ) − EV (xk ) ≤ −Eα(|xk |2 ) + ϕ(E|ek |2 ) ∨ φ(E|dk |2 ),
(3.3) (3.4)
where the external disturbance dk is state-dependent and satisfies (3.2) and following inequality
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[ϕ◦ρ(·)] ∨ [φ◦ϱ(·)] < (Id − θ)◦α(·)
(3.5)
with ρ ∈ CK∞ , ϱ ∈ K∞ , θ ∈ K∞ , θ ◦α ∈ VK∞ and satisfies Id−θ ∈ K∞ , θ ◦α(·) < v¯(·). Then stochastic system (2.7) is asymptotically stable in mean square under the event-triggered control.
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Proof. In the event-triggered control (2.4), the state xk and the error ek determine whether the next time k + 1 be a triggering time. Here, we consider the following two cases: Case I: For any specific k ∈ N, ϖ(xk , xki ) ≤ 0.
By the definition of function ϖ in (3.1), we have 0 ≥ ϖ(xk , xki ) = −πx (|xk |2 ) + πe (|xki − xk |2 ) ] [ 1 = 3(ψ1 Id + ψ3 ϱ) − ρ◦(Id + ρ)−1 (|xk |2 ) + (2 + 3ψ2 )Id(|ek |2 ). 3ψ1 6
(3.6)
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Moreover, ki ≤ k < k + 1 < ki+1 , xk+1 is not transmitted in this case. It yields ek+1 = xki − xk+1 . Then, from Assumptions 2.3, 2.4 and condition (3.2), E|ek+1 |2 has the following property:
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= = ≤ ≤ ≤
E|ek+1 |2 E|xki − xk + xk − xk+1 |2 E|ek − f (k, x, e, d) − g(k, x, e, d)ωk |2 2E|ek |2 + 2E|f (k, x, e, d)|2 + E|g(k, x, e, d)|2 2E|ek |2 + 3ψ1 E|xk |2 + 3ψ2 E|ek |2 + 3ψ3 E|dk |2 3E(ψ1 Id + ψ3 ϱ)(|xk |2 ) + (2 + 3ψ2 )E|ek |2 .
By using (3.7) and (3.6), we have
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E|ek+1 |2 ≤ 3E(ψ1 Id + ψ3 ϱ)(|xk |2 ) + (2 + 3ψ2 )E|ek |2 1 ≤ Eρ◦(Id + ρ)−1 (|xk |2 ). 3ψ1
(3.7)
(3.8)
It follows from (3.6) again that
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|xk |2 − |xk+1 |2 ψ3 2 + 3ψ2 ≤ |xk |2 + ϱ(|xk |2 ) + |ek |2 ψ1 3ψ1 1 1 ≤ ρ◦(Id + ρ)−1 (|xk |2 ). 3ψ1 3ψ1
(3.9)
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Then, by using (3.9), we get |xk+1 |2
1 1 ρ◦(Id + ρ)−1 ](|xk |2 ) 3ψ1 3ψ1 1 1 1 1 = [(Id + ρ)◦(Id + ρ)−1 − ρ◦(Id + ρ)−1 ](|xk |2 ) 3ψ1 3ψ1 3ψ1 3ψ1 1 = (Id + ρ)−1 (|xk |2 ). 3ψ1
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≥ [Id −
Substituting (3.10) into (3.8) that E|ek+1 |2 ≤ Eρ(|xk+1 |2 ),
in the case of ϖ(xk , xki ) ≤ 0. 7
(3.10)
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Case II: For any specific k ∈ N, ϖ(xk , xki ) > 0.
It follows from (2.2), state x transmits at time k + 1, which means ki+1 = k + 1. Thus, ek+1 = xki+1 − xk+1 = 0. And then, E|ek+1 |2 ≤ Eρ(|xk+1 |2 ) in this case.
E|ek |2 ≤ Eρ(|xk |2 ).
EV (xk+1 ) − EV (xk ) −α(E|xk |2 ) + max{ϕ(E|ek |2 ), φ(E|dk |2 )} −α(E|xk |2 ) + max{ϕ◦ρ(E|xk |2 ), φ◦ϱ(E|dk |2 )} −θ◦α(E|xk |2 ) −θ◦α◦¯ v −1 (EV (xk )).
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≤ ≤ ≤ ≤
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By using conditions (3.2)-(3.5) and (3.11), we get
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Moreover, 0 = E|ek0 |2 ≤ Eρ(|xk0 |2 ). Thus, the above discussions yield that for all k ∈ N,
(3.11)
(3.12)
Adding from k0 to k − 1 on the both sides of inequality (3.12), we obtain EV (xk ) − EV (xk0 ) ≤ −
k−1 ∑
θ◦α◦¯ v −1 (EV (xs )),
s=k0
which means 0 ≤ EV (xk ) ≤ EV (xk0 ) < ∞. Then, it follows from Lemma 2.6 that there exists a function β˜ ∈ KL such that
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˜ EV (xk ) ≤ β(EV (xk0 ), k),
since θ◦α ∈ VK, v¯ ∈ CK, θ◦α(·) ≤ v¯(·).
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Moreover, v¯ ∈ CK also yields that
EV (xk ) ≤ ≤ ≤
˜ β(EV (xk0 ), k) ˜ β(E¯ v (|xk0 |2 ), k) ˜ v (E|xk |2 ), k). β(¯ 0
(3.13)
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On the other hand, by inequality (3.3) and v ∈ VK, we have v(E|xk |2 ) ≤ Ev(|xk |2 ) ≤ EV (xk ).
(3.14)
It follows from (3.13) and (3.14) that ˜ v (E|xk |2 ), k), E|xk |2 ≤ v −1 ◦ β(¯ 0 ˜ v (x), k) in Definition 2.1. So, system (2.7) is ASMS with β(x, k) = v −1 ◦ β(¯ 8
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Remark 3.2 For any specific k ∈ N, E|ek |2 ≤ ρ(E|xk |2 ) can not directly obtained from E|ek+1 |2 ≤ ρ(E|xk+1 |2 ), since xk+1 is determined by xk , ek and dk . Motivated by our discussion in this paper, we designed an event-triggering function ϖ(xk , xki ) in the form of (3.1) to solve this problem. Moreover, composite function πx in (3.1) need not to be positive definite, but only satisfies condition (3.5). Thus, the event-triggered mechanism in this paper is more general than those given in the previous literature.
3.2
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Remark 3.3 Zhang et al. in [43] focused on the discrete-time nonlinear systems with event-triggered controller. On this basis, we generalize their results to the stochastic systems. To obtain the globally asymptotic stabilization, the authors in [43] used the fact that |a − b| ≤ θ ◦ (Id + ρ)−1 (|a|) ⇒ |a − b| ≤ ρ(|b|) for any a, b ∈ R and ρ ∈ K, which was proved in [44]. However, this conclusion is no longer applicable in the proof of Theorem 3.1 in our paper. Since we need to show that inequality |a − b|2 ≤ ρ(|b|2 ) holds, which is equal to (3.11). To overcome this difficulty, we analyze the relationship between E|xk |2 and E|xk+1 |2 separately. And then, we get the desired results. Furthermore, we show that our stochastic system is ASMS with respect to a state-dependent external disturbance. Therefore, we generalize and improve those results obtained in [43].
State-independent disturbance
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Here, we consider the stochastic nonlinear system with event-triggered control, which respects to a state-independent external disturbance dk . This means property (3.2) may be not satisfied. And then, Theorem 3.1 is not valid for this case. In what follows, we focus on the ISSMS of the event-triggered stochastic system with a state-independent disturbance.
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Theorem 3.4 Suppose that conditions of Theorem 3.1 hold without the property (3.2) for the external disturbance dk . Moreover, α in (3.4) satisfies α ∈ VK∞ . Then stochastic system (2.7) is ISSMS under the event-triggered control. Proof. We consider the following two cases: Case I: For any specific k ∈ N, E|ek |2 ≤ ρ(E|xk |2 ).
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It follows from inequalities (3.4) and (3.5) that
EV (xk+1 ) − EV (xk ) ≤ −α(E|xk |2 ) + max{ϕ ◦ ρ(E|xk |2 ), φ(E|dk |2 )} ≤ −α(E|xk |2 ) + max{(Id − θ) ◦ α(E|xk |2 ), φ(E|dk |2 )} ≤ −θ ◦ α(E|xk |2 ) + φ(E|dk |2 ).
Case II: For any specific k ∈ N, E|ek |2 > ρ(E|xk |2 ). 9
(3.15)
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In this case, the fact |ek | = 0 can not satisfied. Thus, k ≥ k0 + 1 and k is not the triggering time. Moreover, there exists i ∈ S such that ki ≤ k − 1 < k < ki+1 .
We now show that |ek−1 |2 > ρ(|xk−1 |2 ) does not hold. In fact, if |ek−1 |2 > ρ(|xk−1 |2 ), and then we have
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ϖ(xk−1 , xki ) [ ] 1 = 3(ψ1 Id + ψ3 ϱ) − ρ◦(Id + ρ)−1 (|xk−1 |2 ) + (2 + 3ψ2 )Id(|ek−1 |2 ) 3ψ1 [ ] 1 = 3(ψ1 Id + ψ2 ρ + ψ3 ϱ) + 2ρ − ρ◦(Id + ρ)−1 (|xk−1 |2 ) 3ψ1 > 0.
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From the event-triggered mechanism (2.4), we obtain that time k is the triggering time, and then ki = k. Then, we have |ek | = 0, which contradicts with the case |ek |2 > ρ(|xk |2 ). Thus, assumption |ek−1 |2 > ρ(|xk−1 |2 ) does not hold. This yields E|ek−1 |2 ≤ Eρ(|xk−1 |2 ) ≤ ρ(E|xk−1 |2 ).
(3.16)
On this basis, we can prove that for any k > k0 ,
E|dk−1 |2 > ϱ(E|xk−1 |2 ).
(3.17)
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In fact, if equality (3.17) does not hold, we have E|ek |2 ≤ ρ(E|xk |2 ) by the proof of Theorem 3.1. This also contradicts with the case E|ek |2 > ρ(E|xk |2 ). Thus, equality (3.17) holds. Combine (3.16) and (3.17), we get
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E|ek |2 = E|(xk − xk−1 ) + (xk−1 − xki )|2 E|f (k − 1, x, e, d) + g(k − 1, x, e, d)ωk−1 − ek−1 |2 3ψ1 E|xk−1 |2 + 3ψ2 E|ek−1 |2 + 3ψ3 E|dk−1 |2 + 2E|ek−1 |2 [3ψ1 Id + (2 + 3ψ2 )ρ](E|xk−1 |2 ) + 3ψ3 E|dk−1 |2 σE|dk−1 |2 ,
= ≤ ≤ ≤
(3.18)
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where σ = [3ψ1 Id + (2 + 3ψ2 )ρ] ◦ ϱ−1 + 3ψ3 Id.
Without losing generality, letting d(−1) = 0. And then, inequality (3.18) also holds in k = k0 . By condition (3.4), we have EV (xk+1 ) − EV (xk ) ≤ −α(E|xk |2 ) + max{ϕ ◦ σ(E|dk−1 |2 ), φ(E|dk |2 )},
for any k ∈ N. 10
(3.19)
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Combine inequalities (3.15) and (3.19) in above two cases, we obtain EV (xk+1 ) − EV (xk ) ≤ − min{α(E|xk |2 ), θ ◦ α(E|xk |2 )} + max{ϕ ◦ σ(E|dk−1 |2 ), φ(E|dk |2 )} ≤ − min{α(E|xk |2 ), θ ◦ α(E|xk |2 )} + max{ϕ ◦ σ(sup E|dk |2 ), φ(sup E|dk |2 )}. k∈N
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k∈N
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Thus, from Lemma 2.7, the system (2.7) is ISSMS under the event-triggered mechanism with α ˜ and λ in (2.10) are taken as α ˜ (·) = min{α(·), θ ◦ α(·)} and λ(·) = max{ϕ ◦ σ(·), φ(·)}, respectively.
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Remark 3.5 In CaseII of the proof of Theorem 3.4, we have ki ≤ k−1 < k < k+1 ≤ ki+1 , which means ki+1 − ki ≥ 2. In reality, this fact always holds only if πx is positive definite. And ϖ(xki , xki ) = −πx (|xki |2 ) ≤ 0 for each i ∈ S. So, ki+1 = min{k ≥ ki | ϖ(xk , xki ) > 0} + 1 ≥ ki + 2 since the “ + 1” term. Obviously, if feedback controller triggered at time k, then k − 1 and k + 1 are both not the triggering times. Therefore, the situation of continuous triggering can be directly avoided. However, if πx is negative definite, it is easy to obtain ϖ(xk , xki ) > 0. And then, ki+1 = k + 1 (i ∈ N) holds for all k ∈ N.
Concluding remarks
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4
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Remark 3.6 Obviously, the system suggested in [44] is a class of nonlinear continuous systems. Different from that, we consider a class of discrete-time stochastic nonlinear systems in this paper. In the proof of our results, there are lots of techniques has been used, such as comparison principle, classified discussion, reduction to absurdity and so on. Moreover, with the help of the event-triggered mechanism, we realize the asymptotic stability in mean square of the suggested stochastic system when the external disturbance dk is state-dependent. While, even if the external disturbance dk is state-independent, we can also obtain the suggested stochastic system is input-to-state stable in mean square.
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In this paper, a class of discrete-time stochastic nonlinear systems have been investigated. More precisely, we consider the external disturbance and the event-triggered feedback control into coefficients. By using comparison principle, classified discussion, reduction to absurdity and many other technique, we realize the stability in mean square of the suggested stochastic system. Especially, in the case of state-dependent external disturbance, we assume that disturbance is upper bounded by a function of state. And then, the suggested stochastic system is ASMS. On the other hand, in the case of stateindependent external disturbance, the suggested stochastic system is ISSMS. Finally, some discussions and remarks are given to illustrate that our results are meaningful. In future, we will extend our results to some more complex event-triggered stochastic systems, such as discrete-time stochastic delayed systems, continuous-time stochastic nonlinear systems with Markovian switching, continuous-time stochastic delayed systems and 11
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so on. Moreover, time-triggered control and self-triggered control for stochastic systems will also be considered.
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Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.