Reliable control for event-triggered singular Markov jump systems with partly unknown transition probabilities and actuator faults

doi:10.1016/j.jfranklin.2019.01.008

Journal of the Franklin Institute

Volume 356, Issue 4, March 2019, Pages 1828-1855

https://doi.org/10.1016/j.jfranklin.2019.01.008 Get rights and content

Abstract

In this paper, the problem of reliable controller design for event-triggered singular Markov jump systems with partly known transition probabilities, nonlinear perturbations and actuator faults is studied. To mitigate the burden of data transmissions over network, two event-triggered schemes with different triggering conditions are introduced. The switch law between the two event-triggered schemes is governed by a random variable with Bernoulli distribution. Taking nonlinear perturbations and actuator faults into account, the resulting closed-loop system is converted into a time-delay singular Markov jump system with partly known transition probabilities. Sufficient conditions of stochastically admissible for the resulting closed-loop system are obtained in terms of a group of linear matrix inequalities. The co-design of desirable reliable controller and weighting matrices of event-triggered schemes is presented. Finally, two numerical examples are given to show the effectiveness of the developed results.

Introduction

Different from traditional control systems, networked control systems (NCSs) are a class of control systems wherein sensors, controllers and actuators are linked via a shared and bandwidth limited communication network [1]. On one hand, due to the advantages of low installation and maintenance cost, high reliability, flexible structures, NCSs have been widely used in numerous practical engineering applications, such as remote surgeries [2], intelligent transportation systems [3], mobile robots and so on. On the other hand, the insertion of communication network inevitably leads to a series of issues, such as network-induced delays (transmission delays) [4], packet dropouts (packet loss) and quantization problems[5], which continuously give rise to great interests for research scholars all over the world and numerous outstanding results have been reported in the last two decades [6], [7], [8], [9], [10]. However, the majority of the results are based on the time-triggered scheme(TTS), which transmits the sampled data into the network with constant time intervals. Consequently, it may transmit substantial unnecessary data into the network, especially when the system gradually reaches its equilibrium state. Obviously, the TTS has low utilization rate of network resources. Thus, to alleviate the signal transmission burden of the communication network, it is of great significance to develop a novel approach to overcome the shortages of TTS in theoretical analysis and practical applications. And the event-triggered scheme (ETS) comes into being [11], [12], [13], [14].

The ETS is a type of new control approach that great efforts have been made on it by numerous scholars in the past decade [15], [16], [17], [18], [19], [20], [21]. Instead of updating the control signal periodically, the ETS updates control signals only when the triggering condition holds. Therefore, the mechanism of ETS is that the control task is executed only when the event-triggered error violations the prescribed threshold, instead of the elaspe of fixed time interval as in the TTS. It is obvious to see that the time intervals between two successive events are time-varying and the ETS results in aperiodic data transmissions over network. Actually, it has been proved by the theoretical analysis and experiments ([22]) that the ETS can greatly reduce the total amount of data transmission over network while guaranteeing the desirable system performance. Recently, a large amount of triggering conditions have been developed, which can be coarsely classified into three types: the absolute, relative and mixed triggering conditions [19]. For example, in [14], based on a well-designed absolute ETS, the problem of state feedback control of linear system was discussed, and the performance of the closed-loop system can approximate the continuous time state feedback system with arbitrary precision. Then in [23], the problem of stabilization of systems in the presence of quantization and time-varying network-induced delays was investigated. Taking into account the model uncertainty and network-induced delays, a relative ETS was designed, and model based NCSs and the event-triggered control were combined to mitigate the burden of data transmissions over networks. And in [15], the problem of stability and L_∞ performance for event-triggered systems with dynamic output feedback was studied. By extending early developed ETSs, a mixed decentralized ETS was proposed, which can balance the performance of the system and communication load for nodes. In [24], a new mixed ETS was developed for continuous time and discrete time linear system with output feedback control. Different from Donkers and Heemels [15], a new event triggering error with the form of exponential attenuation and mixed time-dependent triggering parameters were redefined. Recently, the adaptive ETS and hybrid driven schemes have been investigated. In [25], [26], two different adaptive ETSs were constructed, both of which can be obtained online. However, the difference between them is that the former one adjusted the weighting matrix online, and the later one only tuned the coefficient of the triggering condition online. Concerning the hybrid-driven scheme, the switch law between the TTS and ETS was modelled as a random variable subject to Bernoulli distribution in [27] and [28]. Moreover, Liu et al. [27] focused on the stabilization of linear NCSs with network-induced delays, Zha et al. [28] investigated the reliable control for T-S fuzzy systems in the presence of actuator faults and nonlinear perturbations.

Singular systems, are also named as descriptor systems, differential-algebraic systems, degenerate system, or generalized state-space systems, have been widely investigated in the past two decades for the reason that these systems have been extensively employed to model circuit systems, power systems, mechanical engineering systems, economic systems, etc. [29]. Recently, numerous outstanding research results on singular systems have been published. For example, stability analysis and stabilization [30], filtering issues [31], observer designs [32] and so on. However, up to date, there are very few available research results on singular systems under the ETS. In [33], the problem of event-triggered control for linear time-invariant singular system was addressed. The co-design of controller and event-triggered condition subject to admissibility was developed. And then, in [34], the problem of event-triggered H_∞ control for singular systems over networks was discussed. A novel ETS was constructed to monitor the system state in discrete instants. And the problem of dissipative non-fragile controller design for singular systems over the network under the ETS was studied in [35]. Sufficient conditions of regular, impulsive free, asymptotically stable and strictly dissipative for the resulting closed-loop system were obtained. Then in [36], the problem of H_∞ control for singular systems over network with quantizations in both the measured states and inputs under the ETS was addressed. Considering the effect of quantization and ETS, the closed-loop system was transformed into a time-delay singular system. The co-design of the quantized state feedback controller and the ETS was presented. Nevertheless, to the best of authors’ knowledge, no report is addressed on the problem of reliable control for event-triggered singular Markov jump systems with partly known transition probabilities.

Inspired by the discussions presented above, in this paper, we focus on the problem of reliable control for singular Markov jump systems with partly known transition probabilities and actuator faults under the ETS. The main contributions are given as follows: (1) The problem of reliable control for continuous singular Markov jump systems with partly known transition probabilities is discussed. (2) The occurrence of nonlinear perturbations is modeled as a random variable subject to Bernoulli distribution; the random failures of actuators are modeled as certain distributed random variables; two ETSs with different triggering conditions are constructed, and the switch law between them is modeled as a random variable governed by Bernoulli distribution. (3) The co-design of the desired reliable controller and weighting matrices of ETSs for the resulting closed-loop system subject to stochastically admissible is developed.

The rest of the paper is outlined as follows. Section 2 describes the problem under study. In Section 3, sufficient conditions of stochastically admissible and co-design for the reliable controller and weighting matrices of ETSs are presented. Two numerical examples are given in Section 4 to show the effectiveness of the developed method. The paper is concluded in Section 5.

Notation: The notations of this paper are rather standard. $R^{n}$ stands for n-dimensional Euclidean space, $R^{n \times m}$ is the sets of all n × m real matrices. And the notion P > 0 means that P is real symmetric and positive definite. Pr{x} denotes the occurrence probability of x, $E {\cdot}$ stands for the mathematical expectation operator. l₂[0, ∞) represents the space of all square-summable vectors functions over [0, ∞), and ∥x∥ is the standard l₂ norm of x, i.e., $∥ x ∥ = {(x^{T} x)}^{\frac{1}{2}}$ . In symmetric block matrices, (∗) represents a term that is induced by symmetry, and diag{⋅⋅⋅} denotes a block-diagonal matrix.

Section snippets

System description

It is assumed that {r_t, t ≥ 0} is a finite-state Markov jumping process taking N values in a state space $S = {1, 2, \dots, N}$ . The transition probability of the Markov process r_t is given as $Π = {π_{i j}}_{N \times N} (i, j = 1, 2, \dots, N),$ $\begin{matrix} P r {r_{t + h} = j | r_{t} = i} = {\begin{matrix} π_{i j} h + o (h), & if i \neq j, \\ 1 + π_{i i} + o (h), & if i = j, \end{matrix} \end{matrix}$ where π_ij > 0 if i ≠ j; h > 0 and $\lim_{h \to 0} o (h) / h = 0,$ $π_{i i} = - \sum_{j = 1, j \neq i}^{N} π_{i j}$ for each mode i. For $\forall i \in S,$ define $S = S_{k}^{i} ⋃ S_{u k}^{i}$ with $S_{k}^{i} \dot{=} {j, π_{i j} i s k n o w n}, S_{u k}^{i} \dot{=} {j, π_{i j} i s u n k n o w n} .$ And we also define $π_{k}^{i} = \sum_{j \in S_{k}^{i}} π_{i j}$ .

Fix the probability space $(Ω, F, P),$ and consider a

Main result

In this section, a method is addressed for obtaining stochastic admissible conditions and co-design for the resulting closed-loop system (17). Firstly, based on Lyaunpov functional approach, the existence conditions of stochastic admissible for the closed-loop system (17) are presented in Theorem 1.

Theorem 1

For given scalars $\bar{α},$ $\bar{β},$ ${\bar{τ}}_{1}, {\bar{τ}}_{2},$ w_s, δ_a, δ_b, δ_ws ( $s = 1, \dots, m$ ), σ₁ ∈ [0, 1), σ₂ ∈ [0, 1), and matrix $K_{1 i}, K_{2 i} (i = 1, \dots, N),$ the closed-loop system (17) with partly known transition probabilities (2) is

Numerical examples

In this section, two numerical examples are presented to illustrate the effectiveness of the developed results in this paper.

Example 1

Consider the singular Markov jump system (3) with randomly occurring nonlinear perturbations h(x) and g(x), and it is assumed that the nonlinear bounds are $Φ_{1} = d i a g {- 0.06, 0, 0.02}$ and $Φ_{2} = d i a g {0.03, 0.04, 0.06},$ respectively. Other parameters of Eq. (3) are given as follows $\begin{matrix} E & = & [\begin{matrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}], A_{1} = [\begin{matrix} 0 & 1 & 0 \\ - 1 & 0 & - 1 \\ 2 & 1 & 0 \end{matrix}], A_{2} = [\begin{matrix} 0 & 1 & 0 \\ - 0.8 & 0 & - 1 \\ 2 & 1 & 0 \end{matrix}], A_{3} = [\begin{matrix} 0 & 1 & 0 \\ - 0.5 & 0 & - 1 \\ 2 & 1 & 0 \end{matrix}], \\ B_{1} & = & B_{2} = B_{3} = [\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix}], H_{1} = [\begin{matrix} 0.1 & 0 & 0 \\ 0 & 0.1 & 0 \\ 0 & 0 & 0.1 \end{matrix}], \end{matrix}$

Conclusion

To solve the reliable control problem for singular Markov jump systems with partly known transition probabilities, probabilistic actuator faults and randomly occurring nonlinear perturbations under the ETS, we construct different triggering conditions for two ETSs with Bernoulli distribution, which can be employed to determine when the sampled data are released into the network. Based on Lyapunov functional and linear matrix inequality technique, sufficient conditions of stochastically

Acknowledgment

This work was partly supported by the National Natural Science Foundation of China under Grant 61403236.

References (40)

YueD. et al.
Network-based robust h_∞ control of systems with uncertainty
Automatica
(2005)
J. Lunze et al.
A state-feedback approach to event-based control
Automatica
(2010)
M.S. Mahmoud et al.
Aperiodic triggering mechanisms for networked control systems
Inf. Sci.
(2015)
D. Lehmann et al.
Extension and experimental evaluation of an event-based state-feedback approach
Control Eng. Pract.
(2011)
FanQ. et al.
Sampled-data output feedback control based on a new event-triggered control scheme
Inf. Sci.
(2017)
ZhaL. et al.
Reliable control for hybrid-driven TS fuzzy systems with actuator faults and probabilistic nonlinear perturbations
J. Frankl. Inst.
(2017)
FengZ. et al.
Two equivalent sets: application to singular systems
Automatica
(2017)
ZhengG. et al.
Interval observer for a class of uncertain nonlinear singular systems
Automatica
(2016)
WangH. et al.
Network-based h_∞ control for singular systems with event-triggered sampling scheme
Inf. Sci.
(2016)
R. Sakthivel et al.
Dissipative analysis for network-based singular systems with non-fragile controller and event-triggered sampling scheme
J. Frankl. Inst.
(2017)

TianE. et al.

Delay-dependent robust h_∞ control for t-s fuzzy system with interval time-varying delay

Fuzzy Sets Syst.

(2009)

ZhangW. et al.

Stability of networked control systems

IEEE Control Syst. Mag.

(2001)

MengC. et al.

Remote surgery case: robot-assisted teleneurosurgery

Proceedings of the IEEE Conference on Robotics and Automation

(2004)

R.T. Jackeline et al.

Automated and cooperative vehicle merging at highway on-ramps

IEEE Trans. Intell. Transp. Syst.

(2017)

GaoH. et al.

h_∞ estimation for uncertain systems with limited communication capacity

IEEE Trans. Autom. Control

(2007)

W.P.M.H. Heemels et al.

Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance

IEEE Trans. Autom. Control

(2010)

SongG. et al.

Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time observations of state and mode

IET Control Theory Appl.

(2016)

SongG. et al.

Robust quantised control of hybrid stochastic systems based on discrete-time state and mode observations

Int. J. Control

(2017)

XuS. et al.

Improved delay-dependent stability criteria for time-delay systems

IEEE Trans. Autom. Control

(2005)

XiaoN. et al.

Feedback stabilization of discrete-time networked systems over fading channels

IEEE Trans. Autom. Control

(2012)

Cited by (12)

Dissipativity-based asynchronous fault detection for T-S fuzzy nonlinear Markov jump systems subject to uncertain probabilities
2023, Journal of the Franklin Institute
Citation Excerpt :
Moreover, the authors addressed the problem of event-triggered FD for fuzzy stochastic systems in [1], for nonhomogeneous MJSs in [6], and for nonlinear switched systems in [8]. On the other hand, MJSs have received considerable attention from scholars in a broad range of fields including, but not limited to electronics systems, aerospace engineering, chemical production process and robotics systems, and fruitful results have been reported due to their excellent modeling capability [5–7,9,11–15]. However, there are still many challenges worthy of further study.
In this paper, the dissipativity-based asynchronous fault detection problem over unreliable communication is investigated for a class of Takagi-Sugeno fuzzy nonlinear Markov jump systems with time-varying delays and subject to uncertain probabilities. The communication links between the plant and filter are considered to be unreliable, where the stochastic quantization and measurement missing phenomenon are taken into account simultaneously. The asynchronization between the plant and the designed fault detection filter (the random quantizer), is modeled by independent hidden Markov models with a conditional probability matrix, where both the transition probabilities of the plant and the conditional probabilities of the filter are considered to be only partially accessible. By virtue of the Lyapunov stability theory and the slack matrix technique, sufficient conditions guaranteeing the resulting residual system being asymptotically mean square stable and strictly ( $Q$ , $S$ , $R$ )- $α$ -dissipative are deduced. And then, the parameters of the desired fault detection filter are explicitly expressed by solving a set of strict linear matrix inequalities. Finally, two examples are utilized to demonstrate the effectiveness and practicability of the proposed design technique.
Finite-time control for discrete-time nonlinear Markov switching LPV systems with DoS attacks
2023, Applied Mathematics and Computation
Citation Excerpt :
In the past few decades, remarkable results have been obtained about the analysis and synthesis for MSSs with all available transition probabilities (TPs) [23–27]. However, it is worth noting that due to the feasibility, experimental complexity, and cost, many practical control systems are always difficult to guarantee the ideal conditions of all available TPs information, which make some obstacles and difficulties in the application of the proposed theoretical methods [7,28–31]. For the pitch-axis model of autonomous underwater vehicle [32], the random round-trip delay is described by Markov chain, but the transition probability among system modes is not easy to get.
This work studies the finite-time control problem for discrete-time nonlinear Markov switching linear parameter varying (LPV) systems with denial-of-service (DoS) attacks. The aperiodic DoS attacks are introduced into the systems with partially unknown transition probabilities. First, considering the characteristics of aperiodic DoS attacks, an appropriate feedback controller is designed according to that whether it is attacked or not. Furthermore, the finite-time stability criteria of closed-loop system are derived by means of iterative technique and multiple Lyapunov functions. Finally, a turbofan engine model is taken as an example to verify the effectiveness and feasibility of the proposed finite-time control method.
Output feedback l<inf>1</inf> control of positive Markov jump systems: A dynamic event-triggered method
2022, Journal of the Franklin Institute
The output feedback $l_{1}$ control problems of positive Markov jump systems(PMJSs) are studied using event-triggered method in this paper. Firstly, two novel dynamic output event-triggered control strategies(DOETCS) are proposed for continuous-time PMJSs as well as discrete-time PMJSs, which can reduce the event-triggered frequency and save network resources. Then, by constructing multiple linear copositive Lyapunov functions, two sufficient conditions about positivity and stochastic stability with $l_{1}$ -gain performance are presented for the continuous-time closed-loop PMJSs and the discrete-time closed-loop PMJSs. Simultaneously, two desired output feedback $l_{1}$ controllers and two DOETCS are also co-designed to stabilize the PMJSs in continuous and discrete cases, respectively. All the conditions obtained in this paper are in the form of linear programming(LP). Further more, the influence of parameters in the novel DOETCS on the event-triggered frequency is investigated in detail in the section of numerical simulation. Finally, two numerical examples are provided to verify the effectiveness of the proposed design schemes.
Multiple integral techniques for stochastic stability analysis of Markovian jump systems: A unified uncertain transition rates
2022, Journal of the Franklin Institute
This paper studies the stochastic stability problem for Markovian jump systems with unified uncertain transition rates via multiple integral techniques. The considered transition rates unify some existing ones in a framework, which are more general and practical. A multiple-integral-type Lyapunov–Krasovskii functional (MITLKF) is constructed, which contains more ply of integral terms than some existing ones. In order to obtain a tighter bound of the MITLKF, an auxiliary function-based multiple integral inequality (AFMII) is proposed, which encompasses some existing ones as its special cases. Based on these ingredients, a novel stability condition is derived for Markovian jump systems with the unified uncertain transition rates. The effectiveness of the proposed approach is demonstrated by two examples.
HMM-based adaptive attack-resilient control for Markov jump system and application to an aircraft model
2021, Applied Mathematics and Computation
Citation Excerpt :
On the other hand, it is noteworthy that the sensors, actuators and data transmission channels of Markov jump systems may be unreliable with faults, packet dropout and attacks. Considering actuator fault, the reliable control for Markov jump systems are researched through robust control technique [10,11] and dissipative control technique [12]. Specific to unreliable filter with additive gain variations, a resilient filtering strategy is proposed in Chang et al. [13].
This paper deals with adaptive attack-resilient control for Markov jump systems, in which the system mode, sensor and actuator are all attacked. Not only the inputs of system and controller are tampered, but the system mode is also tampered, which may trigger asynchronous switching phenomenon. To describe the asynchronous switching phenomenon, the closed-loop system is modeled as a hidden markov model (HMM). The adaptive control technique is applied to deal with the time-varying attacks on sensor and actuator with reasonable constraints. With the proposed HMM-based adaptive attack-resilient control strategy, the states of Markov system and parameter estimation errors are bounded and the corresponding upper bounds are analyzed. Finally, the proposed strategy is applied to an aircraft model.
Design of Neutral Mixed Delay Multi Stochastic Systems with Event Triggering
2023, Proceedings of 2023 IEEE 12th Data Driven Control and Learning Systems Conference, DDCLS 2023

View all citing articles on Scopus

View full text

Reliable control for event-triggered singular Markov jump systems with partly unknown transition probabilities and actuator faults

Abstract

Introduction

Section snippets

System description

Main result

Numerical examples

Conclusion

Acknowledgment

Automatica

Automatica

Inf. Sci.

Control Eng. Pract.

Inf. Sci.

J. Frankl. Inst.

Automatica

Automatica

Inf. Sci.

J. Frankl. Inst.

Fuzzy Sets Syst.

Stability of networked control systems

IEEE Control Syst. Mag.

Remote surgery case: robot-assisted teleneurosurgery

Proceedings of the IEEE Conference on Robotics and Automation

Automated and cooperative vehicle merging at highway on-ramps

IEEE Trans. Intell. Transp. Syst.

h∞ estimation for uncertain systems with limited communication capacity

IEEE Trans. Autom. Control

Networked control systems with communication constraints: tradeoffs between transmission intervals, delays and performance

IEEE Trans. Autom. Control

Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time observations of state and mode

IET Control Theory Appl.

Robust quantised control of hybrid stochastic systems based on discrete-time state and mode observations

Int. J. Control

Improved delay-dependent stability criteria for time-delay systems

IEEE Trans. Autom. Control

Feedback stabilization of discrete-time networked systems over fading channels

IEEE Trans. Autom. Control

h_∞ estimation for uncertain systems with limited communication capacity