Stability and $L_2$-gain analysis of Networked Control Systems under Round-Robin scheduling: A time-delay approach

Abstract

This paper analyzes the exponential stability and the induced $L_2$-gain of Networked Control Systems (NCS) that are subject to time-varying transmission intervals, time-varying transmission delays and communication constraints. The system sensor nodes are supposed to be distributed over a network. The scheduling of sensor information towards the controller is ruled by the classical Round-Robin protocol. We develop a time-delay approach for this problem by presenting the closed-loop system as a switched system with multiple and ordered time-varying delays. Linear Matrix Inequalities (LMIs) are derived via appropriate Lyapunov–Krasovskii-based methods. Polytopic uncertainties in the system model can be easily included in the analysis. The efficiency of the method is illustrated on the batch reactor and on the cart-pendulum benchmark problems. Our results essentially improve the hybrid system-based ones and, for the first time, allow treating the case of non-small network-induced delay, which can be greater than the sampling interval.

Introduction

Networked Control Systems (NCS) are systems with spatially distributed sensors, actuators and controller nodes which exchange data over a communication data channel. Only one node is allowed to use the communication channel at once. The communication along the data channel is orchestrated by a scheduling rule called protocol. Using such control structures offers several practical advantages: reduced costs, ease of installation and maintenance and increased flexibility. However, from the control theory point of view, it leads to new challenges. Closing the loop over a network introduces undesirable perturbations such as delay, variable sampling intervals, quantization, packet dropouts, scheduling communication constraints, etc. which may affect the system performance and even its stability. It is important in such a configuration to provide a stability certificate that takes into account the network imperfections. For general survey papers we refer to [1], [2], [3]. Recent advancements can be found in [4], [5], [6], [7], [8], [9] for systems with variable sampling intervals, [10] for dealing with the quantization and [11], [12], [13] for control with time delay. Concerning NCS, three main control approaches have been used: discrete-time models (with integration step), input/output time-delay models and impulsive/hybrid models.

In the present paper, we focus on the stability and $L_2$-gain analysis of NCS with communication constraints. We consider a linear (probably, uncertain) system with distributed sensors. The scheduling of sensor information towards the controller is ruled by the classical Round-Robin protocol. The Round-Robin protocol has been considered in [14], [15] (in the framework of hybrid system approach) and in [16], [17] (in the framework of discrete-time systems). In [14], stabilization of the nonlinear system based on the impulsive model is studied. However, delays are not included in the analysis. In [15], the authors provide methods for computing the Maximum Allowable Transmission Interval (MATI — i.e. the maximum sampling jitter) and Maximum Allowable Delay (MAD) for which the stability of a nonlinear system is ensured.

In [16], network-based stabilization of Linear Time-Invariant (LTI) with Round-Robin protocol and without delay have been considered (see also [17] for delays less than the sampling interval). The analysis is based on the discretization and the equivalent polytopic model at the transmission instants. For LTI systems, discretization-based results are usually less conservative than the general hybrid system-based results. However, discrete-time models do not take into account the system behavior between two transmissions and are complicated in the case of uncertain systems. Moreover, it is tedious to include large delays in such models and the stability analysis methods may fail when the interval between two transmissions takes small values.

In the present paper, for the first time, a time-delay approach is developed for the stability and $L_2$-gain analysis of NCS with Round-Robin scheduling. Discrete-time measurements are considered, where the delay may be larger than the sampling interval. We present the closed-loop system as a switched continuous-time system with multiple and ordered time-varying delays. The case of the ordered time-varying delays (where one delay is smaller than another) has not been studied yet in the literature. By developing the appropriate Lyapunov–Krasovskii techniques for this case, we derive LMIs for the exponential stability and for $L_2$-gain analysis. The efficiency and advantages of the presented approach are illustrated by two benchmark examples. Our numerical results essentially improve the hybrid system-based ones [15] and, for the stability analysis, are not far from those obtained via the discrete-time approach [17]. Note that the latter approach is not applicable to the performance analysis. Also, for the first time (under Round-Robin scheduling), the network-induced delay is allowed to be greater than the sampling interval.

Our preliminary results on stability of NCS with constant delay under Round-Robin scheduling have been presented in [18].

Notation: Throughout the paper the superscript ‘$T$’ stands for matrix transposition, ${\mathcal{R}}^n$ denotes the $n$ dimensional Euclidean space with vector norm $\Vert \cdot \Vert$, ${\mathcal{R}}^{n\times m}$ is the set of all $n\times m$ real matrices, and the notation $P>0$, for $P\in {\mathcal{R}}^{n\times n}$ means that $P$ is symmetric and positive definite. The symmetric elements of the symmetric matrix will be denoted by $\ast$. The space of functions $\varphi :[a,b]\to {\mathcal{R}}^n$, which are absolutely continuous on $[a,b)$, have a finite ${lim}_{\theta \to b^{-}}\varphi \left(\theta \right)$ and have square integrable first order derivatives is denoted by $W[a,b)$ with the norm ${\Vert \varphi \Vert }_W={max}_{\theta \in [a,b]}\left|\varphi \left(\theta \right)\right|+{\left[{\int }_a^b{\left|\dot{\varphi }\left(s\right)\right|}^{2}ds\right]}^{\frac{1}{2}}.\mathcal{N}$ denotes the set {0,1,2,3,…}.