Network-based feedback control for systems with mixed delays based on quantization and dropout compensation

doi:10.1016/j.automatica.2011.09.007

Automatica

Volume 47, Issue 12, December 2011, Pages 2805-2809

https://doi.org/10.1016/j.automatica.2011.09.007 Get rights and content

Abstract

This paper deals with the problem of feedback control for networked systems with discrete and distributed delays subject to quantization and packet dropout. Both a state feedback controller and an observer-based output feedback controller are designed. The infinite distributed delay is introduced in the discrete networked domain for the first time. Also, it is assumed that system state or output signal is quantized before being communicated. Moreover, a compensation scheme is proposed to deal with the effect of random packet dropout through communication network. Sufficient conditions for the existence of an admissible controller are established to ensure the asymptotical stability of the resulting closed-loop system. Finally, a numerical example is given to illustrate the proposed design method in this paper.

Introduction

In modern industrial systems, sensors, controllers and plants are often connected over a network medium so-called networked control systems. Due to appealing advantages in NCSs, together with fruitful applications in a broad range of areas, considerable attention has been devoted to the stability and control of NCSs; see for example, Jiang, Mao, and Shi (2010), Seiler and Sengupta (2005), Walsh, Ye, and Bushnell (2002), Yang, Xia, and Shi (2011), Yin, Yu, and Zhang (2010) and Zhao, Liu, and Rees (2009) and the references therein. Nevertheless, it is worth mentioning that the insertion of communication networks in control loops leads to some inevitable phenomena including random delay, packet dropout, quantization errors and so on, which may result in system performance deterioration and have been primarily highlighted in the literature.

Time delays commonly exist in practical NCSs (Liu and Yang, 2011, Shi et al., 2006, Wang et al., 2009, Wu et al., 2011), which are of discrete nature. Another important delay, namely, distributed time delay, which has recently drawn much research interest when modeling a realistic complex system (Xie, Fridman, & Shaked, 2001), has been proposed in the NCSs for the first time in this paper. Much attention has been focused on how to deal with the distributed delays in discrete NCSs, including the construction of novel Lyapunov functions and the use of new inequality techniques. Due to the promising background in NCSs, it is meaningful to propose such a discrete term for NCSs and finally aim to apply a new term of controller, $u (k) = \sum_{m} μ_{m} x (k - m)$ , in practical NCSs.

Quantization always exists in computer-based control systems and quantization errors have adverse effects on the NCSs’ performance. In early 1990s, quantized state feedback was employed to stabilize an unstable linear system by Delchamps (1990). Inspiringly, there is a new trend of research on the quantization effect on NCSs where a quantizer is regarded as an information coder. Consequently, it is necessary to conduct an analysis on the quantizers and understand how much effect the quantization makes on the overall systems.

On the other hand, due to the limited transmission capacity of the network, one of the challenging issues that has inevitably emerged is data loss (Liu, 2010, Shen et al., 2010, Zhao et al., 2009).

Recently, there have been three main methods to deal with control input data loss for real-time NCSs, that is to use zero control input, keep the previous one, or use the predictive control sequence (Liu, 2010). In this paper, the distributed time delays $\sum_{m} μ_{m} x (k - m)$ have been introduced, which include all the cases mentioned above. Although in the current form the coefficients $μ_{m}$ are constants, there is still potential in future that the distributed time delays can be an estimation of the output measurements that have been lost, based on the available measurements. Consequently, it is not a simple system with delays but an attempt to apply a new controller form in NCS.

In response to the above discussion, the networked-based feedback control problem for systems with discrete and distributed delays involving quantization and dropout is investigated in this paper. In the NCSs, it is assumed that the measurement signals are quantized before being communicated. The main contributions of this paper are summarized as follows. (1) The introduction of the distributed delays in the system states for NCSs and will be finally applied in controller design in future work. (2) A compensation scheme is proposed to deal with the effect of random packet dropout through communication network. (3) The practical observer-based output feedback controller is designed.

Notation: The notation used throughout the paper is fairly standard. The superscript “ $T$ ” stands for matrix transposition. $R^{n}$ denotes the $n$ -dimensional Euclidean space and the notation $P > 0 (\geq 0)$ means that $P$ is real symmetric and positive definite (semi-definite). $E {\cdot}$ denotes the expectation.

Section snippets

Problem formulation

Consider the following networked control system: $(£) : x (k + 1) = A x (k) + B x (k - d (k)) + C \sum_{i = 1}^{+ \infty} μ_{i} x (k - i) + D u (k),$ $y (k) = E x (k),$ $x (j) = ϕ (j), j \leq 0,$ where $x (k) \in R^{n}$ is the state vector, $u (k) \in R^{q}$ is the control input, $y (k) \in R^{m}$ is the measured system output, and $A, B, C, D$ and $E$ are known real matrices with appropriate dimensions. The schematic diagram for system (1) is shown in Fig. 1.

In system $(£)$ , the positive integer $d (k)$ denotes the time-varying delay satisfying $d_{1} \leq d (k) \leq d_{2}, k \in N^{+}$ , where $d_{1}$ and $d_{2}$ are known positive

State feedback control

Assume that the state of system $(£)$ is measurable and will be quantized before it is transmitted to the controller through communication network while the data packet dropout happens. To realize the state feedback control for system $(£)$ subject to quantization and packet dropout, the following compensation scheme is constructed at the side of the controller in order to cope with the effect of data loss: $\hat{x} (k) = (1 - θ_{k}) q (x (k)) + θ_{k} \hat{x} (k - 1),$ where $q (\cdot)$ is the logarithmic quantizer defined in (3). Then,

Observer based output feedback control

Suppose that the system output is available and will be quantized before it is transmitted to the controller through network. Similarly, the following compensator is constructed to deal with the packet dropout: $y_{c} (k) = (1 - θ_{k}) q (y (k)) + θ_{k} y_{c} (k - 1),$ where $q (\cdot)$ is the logarithmic quantizer defined in (3). Then, the following output feedback controller is constructed: $\tilde{x} (k + 1) = A \tilde{x} (k) + D u (k) + L (y_{c} (k) - E \tilde{x} (k)),$ $u (k) = K \tilde{x} (k),$ where the gain matrices $L$ and $K$ will be designed later.

As mentioned above, the

Numerical example

In this section, a numerical example is presented to illustrate the effectiveness of the design method obtained in the previous section. Consider system $(£)$ with the following parameters: $A = [\begin{matrix} 0.6 & - 0.1 & 0 \\ 0 & - 0.8 & 0.5 \\ 0.2 & 0 & - 0.7 \end{matrix}], B = [\begin{matrix} 0.2 & - 0.1 & 0 \\ 0.1 & - 0.1 & 0 \\ 0 & - 0.2 & - 0.1 \end{matrix}],$ $C = [\begin{matrix} - 0.2 & 0 & 0.1 \\ - 0.2 & - 0.1 & 0.1 \\ 0 & 0.2 & - 0.1 \end{matrix}], D^{T} = [\begin{matrix} 0.1 & 0.2 & - 2 \end{matrix}],$ $E = [\begin{matrix} 1 & 0.2 & 0.6 \end{matrix}], d (k) = 2 + \frac{1 + {(- 1)}^{k}}{2}, μ_{i} = 2^{- (i + 1)} .$ Assume that the random variable $θ_{k}$ satisfy $\bar{θ} = E {θ_{k}} = 0.8, σ_{\bar{θ}} = E {{(θ_{k} - \bar{θ})}^{2}} = 0.16$ . It is easy to verify that $d_{1} = 2, d_{2} = 3$ , $\bar{μ} = 1 / 2$ . Suppose that the above system states are

Conclusions

In this paper, the feedback control problem for networked systems with discrete and infinite distributed delays subject to quantization and packet dropout has been studied. Specially, the networked system under investigation includes the term of the infinite distributed time delay. Also, it is assumed that the system state or output signal is quantized before being communicated. Moreover, a compensation scheme is proposed to cope with the effect of random packet dropout in the communication

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^☆: This work was supported in part by the National Science Foundation of China (60825303, 61028010, 60834003), 973 Project (2009CB320600), Foundation for the Author of National Excellent Doctoral Dissertation of China (2007B4) and the Key Laboratory of Integrated Automation for the Process Industry (Northeastern University). This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Hideaki Ishii under the direction of Editor André L. Tits.

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Technical communiqueNetwork-based feedback control for systems with mixed delays based on quantization and dropout compensation☆

Abstract

Introduction

Section snippets

Problem formulation

State feedback control

Observer based output feedback control

Numerical example

Conclusions

Automatica

Signal Processing

Automatica

Stabilizing a linear system with quantized state feedback

IEEE Transactions on Automatic Control

H∞ filter design for a class of networked control systems via T-S fuzzy model approach

IEEE Transactions on Fuzzy Systems

Predictive controller design of networked systems with communication delays and data loss

IEEE Transactions on Circuits and Systems II

Quantized static output feedback stabilization of discrete-time networked control systems

International Journal of Innovative Computing Information and Control

Technical communique
Network-based feedback control for systems with mixed delays based on quantization and dropout compensation☆

$H_{\infty}$ filter design for a class of networked control systems via T-S fuzzy model approach