During the past several years, there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems. This paper aims to briefly survey recent results in this field. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability. After a brief review of the stability analysis under restricted switching and the multiple Lyapunov function theory, the switching stabilization problem is studied, and a variety of switching stabilization methods found in the literature are outlined. Then the switching stabilizability problem is investigated, that is under what condition it is possible to stabilize a switched system by properly designing switching control laws. Note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature. A necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here.

Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results

This paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete-time domain. The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. The first one is classical while the second is new and uses a slack variable, which makes it useful for design problems. These two conditions are proved to be equivalent for stability analysis. Investigating the static output feedback control problem, we show that the second condition is, in this case, less conservative. The reduction of the conservatism is illustrated by a numerical evaluation.

Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach

Networked control systems (NCSs) have, in recent years, brought many innovative impacts to control systems. However, great challenges are also met due to the network-induced imperfections. Such network-induced imperfections are handled as various constraints, which should appropriately be considered in the analysis and design of NCSs. In this paper, the main methodologies suggested in the literature to cope with typical network-induced constraints, namely time delays, packet losses and disorder, time-varying transmission intervals, competition of multiple nodes accessing networks, and data quantization are surveyed; the constraints suggested in the literature on the first two types of constraints are updated in different categorizing ways; and those on the latter three types of constraints are extended.

Network-Induced Constraints in Networked Control Systems—A Survey

In this paper, the stability and stabilization problems for a class of switched linear systems with mode-dependent average dwell time (MDADT) are investigated in both continuous-time and discrete-time contexts. The proposed switching law is more applicable in practice than the average dwell time (ADT) switching in which each mode in the underlying system has its own ADT. The stability criteria for switched systems with MDADT in nonlinear setting are firstly derived, by which the conditions for stability and stabilization for linear systems are also presented. A numerical example is given to show the validity and potential of the developed techniques.

Stability and Stabilization of Switched Linear Systems With Mode-Dependent Average Dwell Time

There are many communication imperfections in networked control systems (NCS) such as varying transmission delays, varying sampling/transmission intervals, packet loss, communication constraints and quantization effects. Most of the available literature on NCS focuses on only some of these aspects, while ignoring the others. In this paper we present a general framework that incorporates communication constraints, varying transmission intervals and varying delays. Based on a newly developed NCS model including all these network phenomena, we will provide an explicit construction of a continuum of Lyapunov functions. Based on this continuum of Lyapunov functions we will derive bounds on the maximally allowable transmission interval (MATI) and the maximally allowable delay (MAD) that guarantee stability of the NCS in the presence of communication constraints. The developed theory includes recently improved results for delay-free NCS as a special case. After considering stability, we also study semi-global practical stability (under weaker conditions) and performance of the NCS in terms of Lp gains from disturbance inputs to controlled outputs. The developed results lead to tradeoff curves between MATI, MAD and performance gains that depend on the used protocol. These tradeoff curves provide quantitative information that supports the network designer when selecting appropriate networks and protocols guaranteeing stability and a desirable level of performance, while being robust to specified variations in delays and transmission intervals. The complete design procedure will be illustrated using a benchmark example.

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Networked Control Systems With Communication Constraints: Tradeoffs Between Transmission Intervals, Delays and Performance

Stabilizing Switched Control Design and Pole Placement

The main objective of this paper is to give an interpretation of using non-convex and disconnected level sets Lyapunov functions in the stability analysis of discrete time systems obtained by the discretization of a continuous time Lur'e system. For simplicity reasons, Euler discretization scheme is used to illustrate the features of the proposed method. The main result of this paper shows that it is possible to build, for the original continuous time system, a sequence of bounded connected sets that converges to the origin using this type of Lyapunov functions. To this end, sufficient LMI conditions ensuring the stability of the discrete-time model and an upper bound on the error between the sampled state and the continuous trajectory are used to prove the proposed results. An example will be considered to illustrate this questioning.

On using disconnected level sets Lyapunov functions in the context of sampled-data systems

We investigate the scenario where a plant, modeled as a linear deterministic discrete-time system, is controlled through a wireless communication network. The controller is designed by emulation, meaning that we construct it to stabilize the origin of the plant while ignoring communication constraints and then implement the control law over the network. The transmissions over the wireless channel are time-varying and uncertain, in particular, the probability of successful communication (i.e., no packet is dropped) between the plant and the controller depends on the resources utilized, such as the allocated bandwidth or the transmission signal power. As a result, we provide conditions on the varying inter-transmission times to ensure the mean square stability of the closed-loop system. The novelty is that stability properties are characterized by not only the length of the inter-transmission interval (often called the MATI in the deterministic literature) but also by the probability of successful communication before and after this interval, referred to as the stochastic allowable transmission interval (SATI). These conditions could then be utilized by radio engineers/researchers to design energy-efficient communication strategies while ensuring the mean square stability of the control system.

/pdf/stochastic-maximum-allowable-transmission-intervals-for-the-4t2eqht8yy.pdf

Stochastic maximum allowable transmission intervals for the stability of linear wireless networked control systems

This chapter presents the different aspects of the design methodology that has been used to cope with the RCAM design problem. A quadratic approach involving robust pole placement in a disk and the solution of parameter-dependent Riccati equations is proposed. Robustness as well as performance is accounted for by robust disk pole location that ensures robust stability and provides a good transient behaviour together with a specific selection of the weighting matrices involved in the Riccati equations.

The Lyapunov approach

We consider two problems for discrete-time switched systems with autonomous, general nonlinear modes. The first is optimal control of the switches so as to minimize the discounted infinite-horizon sum of the costs. The second problem occurs when switches are a disturbance, and the worst-case cost under any sequence of switches is sought. We use an optimistic planning (OP) algorithm that can solve general optimal control with discrete inputs such as switches. We extend the analysis of OP to provide sequences of switches with certification (upper and lower) bounds on the optimal and worst-case costs, and to characterize the convergence rate of the gap between these bounds. Since a minimum dwell time between switches must often be ensured, we introduce a new optimistic planning variant that can handle this case, and analyze its convergence rate. Simulations for linear and nonlinear modes illustrate that the approach works in practice.

http://busoniu.net/files/papers/cdc15.pdf

Jamal Daafouz

Papers

Stabilizing Switched Control Design and Pole Placement

On using disconnected level sets Lyapunov functions in the context of sampled-data systems

Stochastic maximum allowable transmission intervals for the stability of linear wireless networked control systems

The Lyapunov approach

Planning methods for the optimal control and performance certification of general nonlinear switched systems