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.

/pdf/networked-control-systems-with-communication-constraints-5ak0mdbgn7.pdf

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

https://hal.archives-ouvertes.fr/hal-00875659/document

Nonlinear event-triggered tracking control of a mobile robot: design, analysis and experimental results

Equivalence between the lyapunov-krasovskii functional approach for discrete delay systems and the stability conditions for switched systems

The main goal of this work is to give a general treatment on observer synthesis for LPV systems in the framework of Linear Matrix Inequalities.A special Parameter Dependent Lyapunov Function, called poly-quadratic Lyapunov function, is considered. It incorporates the parameter variations for LPV systems with polytopic parameter dependence and allows to guarantee a so-called poly-quadratic stability which is sufficient to ensure Global Asymptotic Stability. The concept of polytopic observers is introduced. A LMI-based method for the synthesis of this type of observers is proposed. The case when LPV systems are subjected to disturbances or when the parameter is known with a bounded level of uncertainty is further addressed. Conditions to guarantee performances like Input-to-State Stability (ISS), bounded peak-to-peak gain and L2 gain are given. The design of polytopic Unknown Input Observers both in the deterministic and in the noisy or uncertain cases is also presented. Finally, two illustrative examples dealing with polytopic observers for chaos synchronization and air path management of a turbocharged Spark Ignition engine are detailed.

Polytopic Observers for LPV Discrete-Time Systems

The design of a scheduled observer which allows us to estimate the state of an affine LPV (linear parameter varying) system is investigated. The state and the gain matrices of the observer are scheduled by using an interpolation method which is linear according to each parameter but which is nonlinear according to the parameter vector. The stability of the estimation error is based on the existence of an affine parameter-dependent Lyapunov function. The problem of the observer design and the existence of a such Lyapunov function is interpreted as an LMI feasibility problem with a rank constraint. An example is presented.

State estimation for affine LPV systems

This work is motivated by the fact that many real systems are characterized by two features. The first one is that they are obtained by interconnecting a bunch of simpler subsystems that have to synchronize in order to reach a global goal. The second one is that each subsystem presents dynamics that evolves on different time-scales. Taking into account the two features leads to the problem of synchronization in networks of singularly perturbed systems. In this work we are providing a preliminary study that considers the problem where each subsystem is linear and the network topology is represented by a connected undirected graph that is fixed in time. We show that we can proceed to a time-scale separation of the overall network dynamics and design the controls that synchronize the slow dynamics and the fast ones. Applying the joint control actions to the network of singularly perturbed systems we obtain an approximation of the synchronization behavior imposed for each scale. The methodology requires a variable transformation to overcome the fact that we are dealing with non-standard singularly perturbed systems. One example illustrates the synchronization behavior of linear singularly perturbed systems.

/pdf/synchronization-in-networks-of-linear-singularly-perturbed-e6j598m67w.pdf

Jamal Daafouz

Papers

Nonlinear event-triggered tracking control of a mobile robot: design, analysis and experimental results

Equivalence between the lyapunov-krasovskii functional approach for discrete delay systems and the stability conditions for switched systems

Polytopic Observers for LPV Discrete-Time Systems

State estimation for affine LPV systems

Synchronization in networks of linear singularly perturbed systems