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

http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac11-proceedings/data/html/papers/1135.pdf

Suboptimal Switching State Feedback Control Consistency Analysis for Switched Linear Systems

In this paper, we consider switched linear discrete-time systems. We propose a method to design an observer-based switched control which guarantees that the switched system is asymptotically stable. The main result consists in proving a separation principle for linear discrete-time switched systems. Hence, the design of the switched state feedback control and the switched observer can be carried out independently. Such a design is formulated in terms of Linear Matrix Inequalities (LMI).

/pdf/observer-based-switched-control-design-for-discrete-time-2927msp35k.pdf

Observer-based switched control design for discrete-time switched systems

Trajectory-dependent filter design for discrete-time switched linear systems

The paper focuses on the problem of consensus in the framework of hybrid systems. We consider networks of scalar agents interconnected via directed graphs. Precisely, the network consists of a large number of agents belonging to several clusters. Each cluster is represented by a fixed directed strongly connected graph and almost all the time there is no link between different clusters. In each cluster there exists a specific agent called leader. At specific instants, the leaders interact via a fixed directed strongly connected graph. In this model the agents have continuous dynamics but the states of the leaders are reseted at the instants when they interact between them. In the paper, we first characterize the consensus value of this model and show it depends only on the initial condition and the interaction topologies. Next, we provide sufficient condition in Linear Matrix Inequality (LMI) form for the global uniform exponential stability of the consensus in presence of an almost periodic reset rule. The numerical implementation of this LMI condition requires a polytopic embedding which is provided before the illustrative example.

/pdf/lmi-sufficient-conditions-for-the-consensus-of-linear-agents-5eil122ytr.pdf

LMI sufficient conditions for the consensus of linear agents with nearly-periodic resets

We propose a new LMI approach to the design of optimal switching sequences for polynomial dynamical systems with state constraints. We formulate the switching design problem as an optimal control problem which is then relaxed to a linear programming (LP) problem in the space of occupation measures. This infinite-dimensional LP can be solved numerically and approximately with a hierarchy of convex finite-dimensional LMIs. In contrast with most of the existing work on LMI methods, we have a guarantee of global optimality, in the sense that we obtain an asympotically converging (i.e. with vanishing conservatism) hierarchy of lower bounds on the achievable performance. We also explain how to construct an almost optimal switching sequence.

https://hal.archives-ouvertes.fr/hal-00798196/document

Jamal Daafouz

Papers

Suboptimal Switching State Feedback Control Consistency Analysis for Switched Linear Systems

Observer-based switched control design for discrete-time switched systems

Trajectory-dependent filter design for discrete-time switched linear systems

LMI sufficient conditions for the consensus of linear agents with nearly-periodic resets

Optimal switching control design for polynomial systems: an LMI approach