# Deterministic and Stochastic Approaches to Supervisory Control Design for Networked Systems with Time-Varying Communication Delays

###### Abstract

This paper proposes a supervisory control structure for networked systems with time-varying delays. The control structure, in which a supervisor triggers the most appropriate controller from a multi-controller unit, aims at improving the closed-loop performance relative to what can be obtained using a single robust controller. Our analysis considers average dwell-time switching and is based on a novel multiple Lyapunov-Krasovskii functional. We develop stability conditions that can be verified by semi-definite programming, and show that the associated state feedback synthesis problem also can be solved using convex optimization tools. Extensions of the analysis and synthesis procedures to the case when the evolution of the delay mode is described by a Markov chain are also developed. Simulations on small and large-scale networked control systems are used to illustrate the effectiveness of our approach.

###### keywords:

switched systems, time-delay systems, stochastic switched systems, linear matrix inequality^{†}

^{†}journal: Nonlinear Analysis: Hybrid Systems

url]www.ee.kth.se/ demirel url]www.briat.info url]www.s3.kth.se/ mikaelj

## 1 Introduction

Networked control systems are distributed systems that use communication networks to exchange information between sensors, controllers and actuators ZBP:01; WYB:02. The networked control system architecture promises advantages in terms of increased flexibility, reduced wiring and lower maintenance costs, and is finding its way into a wide range of applications, from automobiles and transportation to process control and power systems, see e.g., WYB:02 – NSS+:05.

The use of a shared communication medium introduces time-varying delays and information losses which may deteriorate the system’s performance, even to the point where the closed-loop system becomes unstable. A conservative approach is to design a robust controller that considers the worst-case delay. However, this might cause poor performance if the actual delay is only rarely close to its upper bound. Therefore, there is currently a renewed interest in adapting the control law to the delay evolution (e.g., CHB:06 – KJF+:12). Inspired by the communication delays that we have experienced in applications, see Figure 1, we design a supervisory control scheme in the sense of Mor:96. This control architecture consists of a finite number of controllers, each designed for a bounded delay variation (corresponding, e.g., to low, medium and high network load) and a supervisor which orchestrates the switching among them.

The analysis of switched systems with fixed time-delays is challenging and has attracted significant attention in the literature, e.g., HCB:06; XiW:05; SZH:06; YaO:08. Only recently, however, attempts to analyze switched systems with time-varying delays have begun to appear. Distinctively, JFK+:09 constructed multiple Lyapunov-Krasovskii functionals that guarantee closed-loop stability under a minimum dwell-time condition for interval time-varying delays. An alternative approach to deal with time-varying delays is to assume that they evolve according to a Markov chain and develop conditions that ensure (mean-square) stability, see e.g., Nil:98 – BeB:98. The work in Nil:98 assumed that the time delay never exceeds a sampling interval, modeled its evolution as a Markov process, and derived the associated LQG-optimal controller. However, this formulation is not able to deal with longer time-delays. The work in XHH:00 proposed a discrete-time Markovian jump linear system formulation, which allows longer (but bounded) time delays, and posed the design of a mode-dependent controller as a non-convex optimization problem. Complementary to these discrete-time formulations, BLL:10 – BeB:98 have investigated the mean-square stability of continuous-time linear systems with random time delays using stochastic Lyapunov-Krasovskii functionals. The papers CHB:08; CHB:08b have applied these techniques to networked control systems with random communication delays and synthesized mode-dependent controllers.

In this paper, we analyze our proposed supervisory control structure by combining a novel multiple Lyapunov-Krasovskii functional with the assumption of average dwell-time switching. The average dwell-time concept, introduced in HeM:99, is a natural deterministic abstraction of load changes in communication networks, where minimal or maximal durations for certain traffic conditions are hard to guarantee. We demonstrate that the existence of a multiple Lyapunov-Krasovskii functional that ensures closed-loop stability under average dwell-time switching can be verified by solving a set of linear matrix inequalities. In addition, we show that the state feedback synthesis problem for the proposed supervisory control structure can also be solved via semi-definite programming. A similar analysis for Markovian time-delays is developed based on a slightly less powerful Lyapunov-Krasovskii functional than the one underpinning our deterministic analysis. Also in this case, we manage to design mode-dependent state feedback controllers using convex optimization.

The organization of the paper is as follows. Section 2.1 presents the supervisory control structure and formalizes the relevant analysis and synthesis problems in a deterministic setting. In Section 2.2, multiple Lyapunov-Krasovskii functionals are constructed for establishing exponential stability of supervisory control systems under average dwell-time switchings. Additionally, LMI conditions that verify the existence of such a multiple Lyapunov functional are derived. State-feedback synthesis conditions are also given in Section 2.3. Section 3 develops a similar analysis framework for stochastic delays. Section 3.1 formulates switched control system problem introduced in Section 2 as a Markovian jump linear system. Section 3.2 develops stochastic exponential mean-square stability conditions for the supervisory control system under stochastic delays. The corresponding state-feedback synthesis conditions are proposed in Section 3.3. Numerical examples are used to demonstrate the effectiveness of the proposed techniques in Section 4. Finally, Section 5 concludes the paper.

Notation: Throughout this paper, denotes the n–dimensional Euclidean space, is the set of all real matrices, and denotes the cone of real symmetric positive definite matrices of dimension . For a real square matrix we define where is its transpose. Additionally, ’’ represents symmetric terms in symmetric matrices and in quadratic forms, denotes the Kronecker product, and is the set of nonnegative (positive) real numbers. Lastly, is the column vector with components .

## 2 Deterministic Switched Systems

### 2.1 System Modeling

We consider the supervisory control system in Figure 2. Here, is the plant to be controlled, described by

(1) |

where and . The network is modelled as a time-varying delay where with is the mode (operating condition) of the network. We assume that the delay in each mode is bounded,

The multi-controller unit uses the mode signal to select and apply the corresponding mode-dependent feedback law

(2) |

In this way, the closed-loop system is described by the following switched linear system with time-varying delay Σ1:˙x(t)=Ax(t)+Aσ(t)x(t-τσ(t)(t)),∀t∈R≥0x(t)=φ(t),∀t∈[-hM+1,0]