Networked control system with asynchronous samplings and quantizations in both transmission and receiving channels

Abstract

This study addresses a problem of the controlling networked control systems (NCSs) which is consisted of the continuous-time plant and controller. In both transmission and receiving channels, asynchronous sampling and different logarithmic quantization effects are considered. By categorizing three cases of asynchronous sampling and using two properties of quantizer which are sector bounded and convex combination, sufficient conditions of the existence of desired controllers for each asynchronous case are presented in the form of linear matrix inequalities (LMIs). Simulation results are given to illustrate the validity of the proposed methods.

Introduction

During the past two decades, since the technologies for high-speed communication networks have been rapidly developed, the NCSs have widely studied and attracted much attention by many researchers [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. The NCSs are the systems linked the components such as the controller, sensor, actuator through communication networks. The communication networks are used for sharing data such as control, reference, and plant output signals with components. The NCSs give rise to many advantages such as low cost, reduced weight, simple installation, easy maintenance, and flexible system structure. The great potential of the NCSs in applications has been found in wide-ranging research areas including factory automation, communication-based distributed mobile, unmanned vehicles, aircrafts, spacecrafts, and so on. As these appropriateness of the studying NCSs, various control schemes are employed to achieve control purposes [11], [12], [13], [14]. In [11], an NCS with random delay was formulated as a kind of Markovian jump system, and then the output feedback networked-predictive-controller which compensates mixed random network-induced delays was designed for guaranteeing the stability of the system. In [12], the feasibility problem of fuzzy logic control method for an NCS was investigated based on implementation results of servo motor control using a Profibus-DP network and the system performance was compared with the conventional proportional-integral-derivative controllers. In [14], ${\mathcal{H}}_{\infty }$ predictive controller was designed for an NCS with data dropouts and time-varying delay in both forward and backward channels by using the switched Lyapunov function technique, in which the closed-loop system not only guarantees asymptotically stable, but also achieves desired control performance.

The quantization and sampling can be found in many real systems and many researchers have concentrated their effort to deal with it [15], [16], [17], [18], [19], [20], [21], [22], [23]. Especially, the NCSs have the higher possibility to occur the problems caused by quantizations and samplings than other systems. In practice, it is clear that the data exchange between components of the NCSs through networks arise by network devices (the transmitter and receiver). As well known, the network devices are digital device, so it has not only their own sampling instant, but also quantization levels. These nature may make it difficult to analyze the NCSs, especially the NCSs with continuous-time plant. In this regard, much attempt has been made by many researchers in order to solve the problems caused by the sampling and quantization effect of the NCSs [24], [25], [26], [27], [28], [29]. In [24], the problem of a reset state observer-based control for asymptotic stability of linear systems was investigated by using logarithmic quantized measurements and the reset technique. Both a state feedback controller and an observer-based output feedback controller for networked systems with discrete and distributed delays subject to quantization and packet dropout were designed in [25]. Kim et al. [26] focused on analyzing the time relation of an NCS, so by the packet analyzer, the robust ${\mathcal{H}}_{\infty }$ stabilization of an uncertain NCSs with network-induced delays and packet dropout was investigated.

In existing works stated above, many factors which can be occurred in the real NCSs such as the packet dropout, time-delay, and saturation have been considered. However, the results have still neglected some problems raised by network devices, so we tender the following two challenges:

• Most of the previous works on the NSC have considered just a quantizer even though there exist at least four network devices which are the source of the quantization effect in both transmission and receiving channels. It is very difficult to deal with a chain of quantizers which can be expressed by a quantizer function whose input is also a quantizer function, i.e. $q_1\left(q_2\right(a\left)\right)$ where $q_i(·)$ (i=1,2) is a quantizer. Of course, there are a few papers on consecutive quantizers [30], [31]. In [30], both state and input signal quantization were addressed, but two quantization effects can be handled separately because one quantizer is effected to plant states, x(t), and the other is imposed to output signals of the sliding mode controller, v(t). In other words, two quantizers, $q_1\left(x\right(t\left)\right)$ and $q_2\left(v\right(t\left)\right)$, are individually appeared in mathematics. Unlike this, if a normal feedback controller is employed to the NCSs, the term of the function of function, i.e. $q_2\left({\mathit{Kq}}_1\right(x\left(t\right)\left)\right)$, is arisen. Hu and Yue [31] also dealt with the problem of two consecutive quantization effects by constraining control gain matrix, but their supposed condition for control gain matrix is too restrict. In this regard, this paper takes two consecutive quantizers into account and gives a new technique to solve this problem.

• Moreover, the structure of the NCSs in most researches is the discrete-time plant with the discrete-time controller (DPDC) or the continuous-time plant with the discrete-time controller (CPDC). The most favorite system formulation for analyzing the NCSs is the DPDC case because all existing signals in the DPDC case are discrete-time signal and it is easier to deal with output signals of each network device than other cases. So, the DPDC case holds a majority in the published papers on the NCSs. Several researchers have paid their attention to the CPDC case [27], [28], [29], but they had transformed the CPDC case to the DPDC case by discretizing the continuous-time plant with sampling periods. These cases are still lacking to describe the practical NCSs. As an example, if we have only the solution for the CPDC case, then the continuous-time controller should be also changed to the discrete-time controller when the wired connection of the continuous-time closed-loop system is substituted to the wireless network, even though it takes additional cost and effort. Besides, the case of the continuous-time plant with the continuous-time controller can be easily founded in real world. To send continuous-time signals through a network, the signal must be sampled and encoded as a digital format by network devices. If the sampling of these network devices is not synchronized, the time relation between output signals of each network device is very complicate. In this respect, we figure out the time relation between signals of the NCSs with the continuous-time plant and controller.

Motivated by the above observation, this study primarily addresses an issue in systematical perspective that has been overlooked by other works. In this paper, an NCS with the continuous-time plant and controller is considered, and the signals of the system go after the following data flow: in transmission channel, the output signals of continuous-time plant are sampled and quantized by a transmitter, and are sent to a receiver. The receiver also outs the sampled and quantized signals. These discrete-time signals are transformed as continuous-time signals by a zero-order-hold (ZOH) in the controller side, then the continuous-time controller exports the continuous-time control signals. As the same data flow in transmission channel, these control signals are applied to the continuous-time plant through a transmitter, receiver, and ZOH in receiving channel with sampled and quantized effect. This process is significantly different from the commonly considered NCS, and generates two main problems, one is caused by two successive quantizers, the other is the time relation between output signals of each network device. In order to treat this problem, this paper supposes that: (a) not only sector bound property of quantizer but also convex combination technique are employed for dealing with two successive quantizers, (b) a novel NCS model is formulated within framework of time relation between output signals of each network device. To this end, three cases of asynchronous sampling are considered. By taking logarithmic quantizer and using discontinuous Lyapunov functional approach which fully uses the information of the sampling pattern, the solvability of the designing problems of a controller is formulated in terms of LMIs. Finally, three numerical examples are included to show the effectiveness of the proposed methods.

Notations: ${\mathbb{R}}^n$ is the n-dimensional Euclidean space, $X>0$ (respectively, $X\ge$ 0) means that the matrix X is a real symmetric positive definite matrix (respectively, positive semi-definite). I denotes the identity matrix. $\mathit{diag}\{\cdots \}$ denotes block diagonal matrix. $\star$ in a matrix represents the elements below the main diagonal of a symmetric matrix. $\mathit{Sym}\left\{X\right\}$ indicates $X+X^T$. $\mathit{quot}(a,b)$ indicates the quotient function, and its output is $\frac{a-\mathrm{mod}(a,b)}{b}$. $\mathrm{mod}(a,b)$ denotes the remainder of the Euclidean division of a by b. $X_{\left[f\right(t\left)\right]}\in {\mathbb{R}}^{m\times n}$ means that the elements of the matrix X include the values of f(t). The space of functions $\theta :[a,b]\to {\mathbb{R}}^n$, which are absolutely continuous on $[a,b)$, have a finite ${\lim }_{\varphi \to b^{-}}\theta \left(\varphi \right)$ and have square integrable first order derivatives denoted by $W_n[a,b)$ with the norm $\parallel \theta {\parallel }_{W_n[a,b)}={\max }_{\varphi \in [a,b]}\left|\theta \right(\varphi \left)\right|+{\left[{\int }_a^b\left|\dot{\theta }\right(s){|}^2\mathit{ds}\right]}^{\frac{1}{2}}$.