Stability and -gain analysis of Networked Control Systems under Round-Robin scheduling: A time-delay approach
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 -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 -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 -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 ‘’ stands for matrix transposition, denotes the dimensional Euclidean space with vector norm , is the set of all real matrices, and the notation , for means that is symmetric and positive definite. The symmetric elements of the symmetric matrix will be denoted by . The space of functions , which are absolutely continuous on , have a finite and have square integrable first order derivatives is denoted by with the norm denotes the set {0,1,2,3,…}.
Section snippets
Problem formulation and the switched system model
Consider the following system controlled through a network (see Fig. 1): where is the state vector, is the control input, is the disturbance, is controlled output, and are system matrices with appropriate dimensions. These matrices can be uncertain with polytopic type uncertainty. The system has several nodes (distributed sensors, a controller node and an actuator node) which are connected via two
Useful lemmas
We will apply the following lemmas: Lemma 1 Let there exist positive numbers and a functional such thatLet the function be continuous from the right for satisfying (8), absolutely continuous for and satisfies If along(8) with and then(8) with and is asymptotically stable. If along(8) with and for some
Example 1: batch reactor
We illustrate the efficiency of the given conditions on the benchmark example of a batch reactor under the dynamic output feedback with [29], [15], [17], where As in [15], the controlled output is chosen to be equal to the measured output . Thus,
We start with the stability analysis in the disturbance-free case, where
Conclusions
In this paper, a time-delay approach has been introduced for the exponential stability and -gain analysis of NCS with Round-Robin scheduling, variable communication delay and variable sampling intervals. The closed-loop system is modeled as a switched system with multiple and ordered time-varying delays. By developing appropriate Lyapunov–Krasovskii-based methods, sufficient conditions are derived in terms of LMIs. The batch reactor example illustrates the advantages of the new method over
Acknowledgments
This work was partially supported by Israel Science Foundation (grant No 754/10) and by China Scholarship Council.
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