Improved non-fragile feedback control for stochastic jump system based on observer and quantized measurement

doi:10.1016/j.jfranklin.2020.09.024

Journal of the Franklin Institute

Volume 357, Issue 17, November 2020, Pages 12433-12453

https://doi.org/10.1016/j.jfranklin.2020.09.024 Get rights and content

Abstract

This paper elaborates the topic of observer-based non-fragile mixed passivity and H_∞ control for stochastic Markovian jump system with mode-dependent time-varying delays and uncertain parameters via quantized measurements. Improved sufficient conditions are acquired via constructing a mode-dependent Lyapunov–Krasovskii (L–K) functional and using the free-weight matrix technique. A suitable observer-based non-fragile feedback controller is provided to guarantee stochastic stability of the closed-loop system. Two examples including an electrical RLC circuit are presented to illustrate the correctness of the proposed approach in this paper.

Introduction

Over the past a few decades, many scholars are attracted to the Markovian jump systems (MJS) in virtue of the superiority of expressing random changes for physical systems suitably [1], [2], [3], [4], [5]. A mass of satisfactory results have been obtained in the intensive study about stability analysis [6], [7], [8] and control synthesis [9], [10], [11], [12] for MJS. In practice, many dynamic systems are often corrupted by noise, therefore, the topics of $I t \hat{o}$ stochastic systems become hot, which can simulate various practical systems affected by certain types of environmental noise, see, [13], [14], [15]. When a $I t \hat{o}$ stochastic system encounters Markovian jump changes, a stochastic Markovian jump system (SMJS) is generated [16], [17].

In practical systems, time delays are often confronted, which tend to reduce the performance and even lead to the instability of the system [18], [19], [20], [21]. The topic of stability and controller design for SMJS with time delays has been extensively studied in recent years [22], [23], [24], [25]. To mention a few, a new criteria was established to show the stability of uncertain MJS with polytopic parameter uncertainties and delays in [23]; the issue of stability analysis for uncertain time-delay MJS was taken into account in [24], the mode-dependent state feedback controller was designed for stochastic Markovian jump delay systems based on LaSalle theorem in [25].

There exist quite small uncertainties that lead to the closed-loop system unstable in controller implementation. Thus, up till now, considerable topics of non-fragile controller have been researched [26], [27]. How to design an insensitive feedback controller has been the principal problem of this control scheme, see, [28], [29], [30], [31], [32], [33]. For instance, both additive and multiplicative controller uncertainties were considered in [28]; the issue of non-fragile controller based on average dwell-time method was investigated in [29].

Noting that it is often difficult to acquire the states of many dynamic systems directly in real fields. Fortunately, the constructed observer can be employed to estimate the system states. The observer-based controller is widely applied in many practical plants [34], [35], [36], [37], [38]. Based on observer, [34] designed a non-fragile controller for stochastic systems, [38] developed mixed passive and H_∞ control for a class of MJS. However, mode-independent delays, $I t \hat{o}$ stochastic effects and Markovian jump changes were not mentioned simultaneously in the above literature. According to our knowledge, the problem of observer-based non-fragile mixed passivity and H_∞ control for SMJS with mode-dependent time-varying delays and uncertain parameters has not been fully investigated yet.

On the other hand, it is worth mentioning that quantization of input signals or measurement output is indispensable in some practical cases, and quantization can cause undesirable effect on system performance even stability [39]. Thus, considering the limited communication capacity of many dynamic systems, how to mitigate quantization effect and realize quantization control effectively has always been a significant research topic [20], [37]. Very recently, many important results of quantized feedback control have been reported, see, [12], [16], [37] and the references therein. Given all that, this paper will address the issue of non-fragile mixed passivity and H_∞ control for SMJS with mode-dependent time-varying delays and uncertain parameters based on observer and quantized measurements.

The main contributions of this paper are summed up as follows:

(1) A non-fragile controller and a non-fragile observer are considered simultaneously, an improved mode-dependent and delay-dependent L-K functional is constructed in this paper.

(2) Time-varying delays $τ_{r_{t}} (t)$ for SMJS are mode-dependent, derivative constraint condition of ${\dot{τ}}_{r_{t} = i} (t) \leq h < 1$ is extended to ${\dot{τ}}_{r_{t} = i} (t) \leq h,$ thus, the constraint condition is more general than the most existed results.

(3) Quantized measurement is utilized to solve feedback control for delayed SMJS based on observer.

Notation: $E (\cdot)$ denotes the expectation operator; | · | represents the Euclidean vector norm; matrix $Q > 0$ ( $Q \geq 0$ ) refers to $Q$ is positive definite (positive semidefinite); $S y m (X)$ is defined as $S y m (X) = X + X^{T}$ ; * expresses a term that is induced by symmetry expressions.

Section snippets

Problem formulation and preliminaries

We consider the following SMJS with time-varying delays: ${\begin{matrix} d x (t) = [(A (r_{t}) + Δ A (t, r_{t})) x (t) + (A_{τ} (r_{t}) + Δ A_{τ} (t, r_{t})) x (t - τ_{r_{t}} (t)) + B_{1} (r_{t}) u (t) \\ + D_{1} (r_{t}) ν (t)] d t + [(E (r_{t}) + Δ E (t, r_{t})) x (t) + (E_{τ} (r_{t}) \\ + Δ E_{τ} (t, r_{t})) x (t - τ_{r_{t}} (t) + B_{2} (r_{t}) u (t)] d ω (t), \\ y (t) = C_{1} (r_{t}) x (t) + C_{1 τ} (r_{t}) x (t - τ_{r_{t}} (t)), \\ z (t) = C_{2} (r_{t}) x (t) + C_{2 τ} (r_{t}) x (t - τ_{r_{t}} (t)) + B_{3} (r_{t}) u (t) + D_{2} (r_{t}) ν (t), \end{matrix}$ where $x (t) \in R^{n}$ is the state; $y (t) \in R^{p}$ is the measured output; $z (t) \in R^{q}$ is the controlled output; $u (t) \in R^{p}$ is the control input; $ν (t) \in R^{m}$ is the disturbance input and belongs to L₂[0, ∞); ω(t) is a

Main results

Theorem 1

The delayed SMJS (13) with $ν (t) = 0$ is said to be stochastically stable, if for all $i \in S,$ there exist matrices $Q > 0,$ $R > 0,$ $Z > 0,$ $J > 0,$ $P_{i} > 0$ and scalars τ > 0, μ > 0, h, δ > 0, such that: $\begin{matrix} Π_{i} = [\begin{matrix} Π_{11 i} & Π_{12 i} & Z \\ * & Π_{22 i} & - Z - J \\ * & * & - Z - R \end{matrix}] < 0 \end{matrix}$ where $\begin{matrix} Π_{11 i} = \sum_{j = 1}^{N} π_{i j} P_{j} + P_{i} {\tilde{A}}_{i} + {\tilde{A}}_{i}^{T} P_{i} + (1 + δ τ) Q - Z + {\tilde{E}}_{i}^{T} (P_{i} + τ Z) {\tilde{E}}_{i} + τ^{2} {\tilde{A}}_{i}^{T} R {\tilde{A}}_{i}, \\ Π_{12 i} = P_{i} {\tilde{A}}_{τ i} + Z + J^{T} + {\tilde{E}}_{i}^{T} (P_{i} + τ Z) {\tilde{E}}_{τ i} + τ^{2} {\tilde{A}}_{i}^{T} R {\tilde{A}}_{τ i}, \\ Π_{22 i} = - (1 - h) Q - Z - J - J^{T} + {\tilde{E}}_{τ i}^{T} (P_{i} + τ Z) {\tilde{E}}_{τ i} + τ^{2} {\tilde{A}}_{τ i}^{T} R {\tilde{A}}_{τ i} . \end{matrix}$

Proof

For the sake of simplicity, we define the drift term of the delayed SMJS (13) as f(t), where $f (t) = {\tilde{A}}_{i} ζ (t) + {\tilde{A}}_{τ i} ζ (t - τ_{i} (t)) + {\tilde{D}}_{i} ν ($

Simulation examples

Example 1 Consider the time-varying delay SMJS (1) with the following parameters: $\begin{matrix} A_{1} & = [\begin{matrix} 0.1 & 0 \\ 0 & 0.1 \end{matrix}], & A_{2} & = [\begin{matrix} 0.2 & 0 \\ 0 & 0.2 \end{matrix}], & A_{τ 1} & = [\begin{matrix} - 1.1 & - 0.2 \\ 0 & - 0.5 \end{matrix}], & A_{τ 2} & = [\begin{matrix} - 1 & - 0.1 \\ 0 & - 0.4 \end{matrix}], \\ E_{1} & = [\begin{matrix} 0.2 & 0 \\ 0 & 0.2 \end{matrix}], & E_{2} & = [\begin{matrix} 0.3 & 0 \\ 0 & 0.3 \end{matrix}], & E_{τ 1} & = [\begin{matrix} 0.2 & 0 \\ 0 & 0.2 \end{matrix}], & E_{τ 2} & = [\begin{matrix} 0.3 & 0 \\ 0 & 0.3 \end{matrix}], \\ C_{11} & = [\begin{matrix} 0.1 & 0.1 \end{matrix}], & C_{12} & = [\begin{matrix} 0.6 & 0.6 \end{matrix}], & C_{1 τ 1} & = [\begin{matrix} 0.5 & 0.5 \end{matrix}], & C_{1 τ 2} & = [\begin{matrix} 0.6 & 0.6 \end{matrix}], \\ C_{21} & = [\begin{matrix} 0 & 0.7 \end{matrix}], & C_{22} & = [\begin{matrix} 0 & 0.8 \end{matrix}], & C_{2 τ 1} & = [\begin{matrix} 0 & 0.7 \end{matrix}], & C_{2 τ 2} & = [\begin{matrix} 0 & 0.8 \end{matrix}], \\ B_{11} & = [\begin{matrix} 0 & 0.5 \end{matrix}]^{T}, & B_{12} & = [\begin{matrix} 0 & 0.6 \end{matrix}]^{T}, & B_{21} & = [\begin{matrix} - 0.1 & 0.1 \end{matrix}]^{T}, & B_{22} & = [\begin{matrix} - 0.2 & 0.2 \end{matrix}]^{T}, \\ M_{P 1} & = [\begin{matrix} - 1.5 & - 0.1 \end{matrix}]^{T}, & M_{P 2} & = [\begin{matrix} - 1.6 & - 0.2 \end{matrix}]^{T}, & M_{L 1} & = [\begin{matrix} - 0.1 & 0.1 \end{matrix}]^{T}, & M_{L 2} & = [\begin{matrix} - 0.2 & 0.2 \end{matrix}]^{T}, \\ N_{A 1} & = [\begin{matrix} 0.1 & 0.1 \end{matrix}], & N_{A_{τ} 1} & = [\begin{matrix} 0.1 & 0.1 \end{matrix}], & N_{E 1} & = [\begin{matrix} 0.1 & 0.1 \end{matrix}], & N_{E_{τ} 1} & = [\begin{matrix} 0.1 & 0.1 \end{matrix}], \\ N_{A 2} & = [\begin{matrix} 0.2 & 0.2 \end{matrix}], & N_{A_{τ} 2} & = [\begin{matrix} 0.2 & 0.2 \end{matrix} \end{matrix}$

Conclusions

This paper has elaborated the topic of observer-based non-fragile mixed passivity and H_∞ control for SMJS with mode-dependent time-varying delays and uncertain parameters via quantized measurements. Improved sufficient conditions have been acquired via constructing a mode-dependent L–K functional and using the free-weight matrix technique. A suitable observer-based non-fragile feedback controller has been provided to guarantee stochastic stability of the closed-loop system. Two examples

Declaration of Competing Interest

The authors declare that they have no conict of interest.

Acknowledgments

The authors would like to thank the anonymous Referees and the Editors for their valuable comments and suggestions, which greatly improved the exposition and quality of the work. This work was partially supported by the National Natural Science Foundation of China under Grants 61773191, 61673169; Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions under Grant 2019KJI010; the Natural Science Foundation of Shandong Province for Outstanding Young Talents in

References (39)

Z. Shu et al.
Robust stabilization of Markovian delay systems with delay-dependent exponential estimates
Automatica
(2006)
R. Sakthivel et al.
Mixed h_∞ and passive control for singular Markovian jump systems with time delays
J. Frankl. Inst.
(2015)
L. Xiong et al.
Stability analysis for neutral Markovian jump systems with partially unknown transition probabilities
J. Frankl. Inst.
(2012)
Y. Shen et al.
h_∞ control of Markov jump time-delay systems under asynchronous controller and quantizer
Automatica
(2019)
Y. Li et al.
Robust delay-dependent h_∞ filtering for uncertain Takagi-Sugeno fuzzy neutral stochastic time-delay systems
J. Frankl. Inst.
(2019)
Y. Kao et al.
h_∞ sliding mode control for uncertain neutral-type stochastic systems with Markovian jumping parameters
Inf. Sci.
(2015)
Y. Tan et al.
Co-design of event generator and quantized fault detection for time–delayed networked systems with sensor saturations
J. Frankl. Inst.
(2017)
X. Zhang et al.
An improved reciprocally convex inequality and an augmented Lyapunov–Krasovskii functional for stability of linear systems with time-varying delay
Automatica
(2017)
H. Li et al.
Parameter-dependent robust stability for uncertain Markovian jump systems with time delay
J. Frankl. Inst.
(2011)
G. Zhuang et al.
State feedback control for stochastic Markovian jump delay systems based on lasalle-type theorem
J. Frankl. Inst.
(2018)

D. Zhao et al.

Non-fragile fault-tolerant control for nonlinear Markovian jump systems with intermittent actuator fault

Nonlinear Anal. Hybrid Syst.

(2019)

R. Kavikumar et al.

Reliable non-fragile memory state feedback controller design for fuzzy Markov jump systems

Nonlinear Anal. Hybrid Syst.

(2020)

S. Xu et al.

Non-fragile positive real control for uncertain linear neutral delay systems

Syst. Control Lett.

(2004)

J. Xia et al.

Non-fragile finite-time extended dissipative control for a class of uncertain discrete time switched linear systems

J. Frankl. Inst.

(2018)

B. Zhang et al.

Output-feedback stabilization of singular LPV systems subject to inexact scheduling parameters

Automatica

(2019)

J. Zhou et al.

Non-fragile observer-based $H_{\infty}$ control for stochastic time-delay systems

Appl. Math. Comput.

(2016)

W. Xia et al.

Observer-based mixed passive and h_∞ control for uncertain Markovian jump systems with time delays using quantized measurements

Nonlinear Anal. Hybrid Syst.

(2019)

N. Xiao et al.

Stabilization of Markov jump linear systems using quantized state feedback

Automatica

(2010)

S. Xu et al.

Delay-dependent h_∞ control and filtering for uncertain Markovian jump systems with time-varying delays

IEEE Trans. Circuits Syst. Regul. Pap.

(2007)

Cited by (3)

Stability of T-S fuzzy system under non-fragile sampled-data H<inf>∞</inf> control using augmented Lyapunov-Krasovskii functional
2023, Journal of the Franklin Institute
This article is concerned with the non-fragile sampled-data control for T-S fuzzy system with parameter uncertainties. Firstly, a novel augmented Lyapunov-Krasovskii functional with sufficient sampled-data information is constructed. And a novel $h (t)$ -depended exponential stability criterion with $H_{\infty}$ performance is gotten by reciprocally convex matrix inequality. Beyond that, compared with the existing methods, the gain matrices for non-fragile sampled-data controller expected are less conservative by linear matrix inequality technique. And numerical examples are provided to support the viability and validity of the results.
Observer-based finite-time asynchronous sliding mode control for Markov jump systems with time-varying delay
2022, Journal of the Franklin Institute
The issue of finite-time sliding mode control (SMC) is studied for a class of Markov jump systems, in which parameter uncertainties, external disturbances and time-varying delay are considered. Firstly, a suitable observer-based SMC law is devised so that state trajectory of the system can reach the designed sliding mode surface in finite-time, the gain of the controller is asynchronous to the mode of original system. Meanwhile, the sufficient conditions of finite-time boundedness in the sliding phase and reaching phase are derived by the time partition strategy. Moreover, the gains of the observer and the observer-based controller will be acquired by using the linear matrix inequalities tool. In fine, emulation products are used to confirm the merits of the SMC strategy.
Non-fragile Sliding Mode Control for Enterprise Knowledge Workers System with Time Delay
2021, Communications in Computer and Information Science

View full text

Improved non-fragile feedback control for stochastic jump system based on observer and quantized measurement

Abstract

Introduction

Section snippets

Problem formulation and preliminaries

Main results

Simulation examples

Conclusions

Declaration of Competing Interest

Acknowledgments

Automatica

J. Frankl. Inst.

J. Frankl. Inst.

Automatica

J. Frankl. Inst.

Inf. Sci.

J. Frankl. Inst.

Automatica

J. Frankl. Inst.

J. Frankl. Inst.

Nonlinear Anal. Hybrid Syst.

Nonlinear Anal. Hybrid Syst.

Syst. Control Lett.

J. Frankl. Inst.

Automatica

Appl. Math. Comput.

Nonlinear Anal. Hybrid Syst.

Automatica

Delay-dependent h∞ control and filtering for uncertain Markovian jump systems with time-varying delays

IEEE Trans. Circuits Syst. Regul. Pap.

Delay-dependent h_∞ control and filtering for uncertain Markovian jump systems with time-varying delays