Control for Markov sampled-data systems with event-driven transmitter
- Wenxia Cui^{1}Email author,
- Lu Li^{1} and
- Ruijuan Liu^{1}
https://doi.org/10.1186/s13662-015-0681-6
© Cui et al. 2015
Received: 15 April 2015
Accepted: 30 October 2015
Published: 5 November 2015
Abstract
In this paper, feedback control problem is considered for networked systems with discrete, infinite distributed delays and sampled-data. A Markov chain is used to characterize the random sampled measurement process of the networked control systems. In addition, an event-driven transmitter is introduced to transmit the control signal according to the measurement sampling period. Based on Lyapunov functional and the matrix analysis techniques, several sufficient conditions are given to ensure the asymptotical stability in the mean square of the addressed control systems. Furthermore, a novel output feedback controller is proposed with both sampling and event-driven transmitter-induced delay indexes. Finally, a simulation example is provided to illustrate the effectiveness of the theoretical results and the proposed method.
Keywords
1 Introduction
In the traditional feedback control systems, the connections between system components are established by point-to-point cables. Compared to the traditional point-to-point systems, networked control systems are real-time control systems where sensors, actuators and controllers are interconnected by a shared digital communication network. Networked control systems (NCSs) offer many advantages such as lower cost, simpler installation, easier maintenance, and resource sharing [1]. And NCSs have great applications in aircrafts and spacecrafts control, robotics, process control and vehicles [2]. Therefore, the field of NCSs has been becoming a hot research topic [3–5].
Since digital microprocessors are quickly becoming indispensable in practical applications, control designing problems of systems tend to be implemented on digital platforms [6, 7]. The periodic sampling leads to conservativeness in the usage of computational resource and bandwidth, because the constant sampling period is chosen to guarantee stability in the worst case [8]. For reducing the usage of computational resource and limited bandwidth, the nonuniform sampler was employed in the implementations of NCSs. Recently, several initial attempts have been proposed to study the stability of NCSs with nonuniformly sampled systems [9, 10].
On the other hand, time delays widely exist in practical systems due to the unreliable communication channel [11, 12]. It is well known that time delay makes the analysis and synthesis of NCSs more complex and important. And time delay is also the major cause for NCS performance deterioration and potential system instability [13]. Discrete time delay is common [14], Liou and Ray proposed the synthesis of a stochastic regulator in the presence of randomly varying delays from the controller to actuator [15]. Distributed time delay \(\sum_{r=1}^{+\infty}\mu_{r}x(k-r)\) is another important delay, which has recently drawn much research interest when modeling a realistic complex system [16].
With the very interesting results reported in [17, 18], it is seen that, in some cases, the activities of sensors/actuators are even triggered by events characterizing stochastic processes, e.g., Markov process [19, 20]. Experimental result [21] shows that the event-triggered control scheme can efficiently reduce the number of control task executions so that communication resources can be saved significantly while retaining satisfactory closed-loop performance.
In practical engineering, discrete and distributed time delays always appear simultaneously in the systems, and the measurement, communication and control updates need the nonuniform sampler. It is therefore essential and challenging to investigate the control for Markov sampled-data systems with event-driven transmitter, which has great potential in practical applications. Therefore, for the mixed time-delay NCSs, an interesting problem is to find a co-design method of the event-triggered control scheme in this paper. In consequence of the above discussion, the networked-based feedback control problem with event-driven transmitter is investigated for NCSs. The main contributions of this paper are the following ones: (1) we consider nonuniform sample data, discrete and distributed time-delays, and present criteria for ensuring stochastic stability of the closed-loop networked system; (2) A novel output feedback controller incorporating both Markov-based sampling, \(\sum_{r=1}^{+\infty}\mu_{r}x(k-r)\) and event-driven transmitter-induced delay indexes is proposed.
2 System description
3 Observer-based output feedback control
4 Main results
In the following, the main results of this paper will be presented, which can be used to study the asymptotical stability in the mean square of system (8).
Theorem 1
Proof
Next, we present the results on the solvability of the control problem based on Theorem 1, where the cone complementarity linearization approach is introduced to deal with the constraint. The main result is concluded in the following theorem by using the Schur complement method and letting \(G_{0}=\bar{P}_{0}^{-1}\), \(G_{i+1}=\bar{P}_{i+1}^{-1}\).
Remark 1
In [4], Markov-based sample data was reflected on the networked control systems, in which the sampling-induced delay index was modeled by a Markov chain. Different from [4], our model includes discrete time-delays, distributed time-delays \((\sum_{r=1}^{+\infty}\mu_{r}x(k-r))\) and Markov-based sample data.
Remark 2
Unlike the method in [4], Theorem 1 gives the sufficient and necessary condition for the stability, which helps to reduce the conservatism. It should be pointed out that Theorem 1 can be easily applied to stability analysis by LMIs conditions for systems with time-varying and distributed delays in this paper.
Theorem 2
Remark 3
According to [23], the computational complexity of the controller design by solving LMIs is defined by \(T(F)=O(F^{3})\) [4], where F is the total number of scalar decision variables. From (16) and the minimization problem, we can see that F satisfies \(F = N(4n^{2} +2n)+(2N-1)(nq+nm)\) for Theorem 2. Therefore, the computational complexity of Theorem 2 is \(O(N^{3}n^{6})\) for the system dimension n and the transition matrix dimension N.
5 Numerical examples
In this section, the simulation results are presented to illustrate the theoretical results derived in this paper.
Example 1
6 Conclusions
In this paper, we have presented a theoretical framework to analyze network-based output feedback control for Markov sampled-data systems with mixed delays. At first, the networked control systems model is constructed including nonuniform sample data, discrete and distributed time-delays. Furthermore, the novel output feedback controller incorporating both Markov-based sampling and event-driven transmitter-induced delay indexes is proposed.
Based on the results obtained in this paper, a new form of controller \(u(k)=\sum_{r}k_{r}x(k-r)\) will be developed and most efforts will be made to solve different \(k_{r}\) in our subsequent work. And we will consider other approaches to give the sufficient and necessary conditions to ensure stability of the closed-loop system.
Declarations
Acknowledgements
This research is supported by the Start-up Project of Shanghai University of Engineering Science (E1-0501-15-0101), the Outstanding Young Teacher Training Project of Shanghai (E1-8500-15-01087).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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