A geometric approach for reachability and observability of linear switched impulsive systems

This paper is concerned with the reachability and observability of linear switched impulsive systems with singular impulse matrices. First some new concepts with respect to the reachability and unobservability are introduced. Especially, span reachability is proposed because the reachable sets of switched impulsive systems do not always constitute subspaces. Then the geometric characterization of the span reachable and unobservable sets is presented. Moreover, the relations between the span reachable set, unobservable set and the invariant subspaces of such systems are discussed. Finally, corresponding criteria applied to linear impulsive systems and linear switched systems are also discussed.     

The distinguishability of linear control systems

When observabilities of hybrid systems such as piecewise-affine hybrid systems are considered, the distinguishability of subsystems takes a very important role. However, properties and even definitions of such distinguishabilities have not been extensively studied in the literature. This paper will give a definition of the distinguishability of two linear control systems and present the necessary and sufficient condition of the distinguishability, giving an answer for the problem posed by Vidal et al.     

Necessary and sufficient conditions for distinguishability of linear control systems

Distinguishability plays a crucial rule in studying observability of hybrid system such as switched system. Recently, for two linear systems, Lou and Si gave a condition not only necessary but also sufficient to the distinguishability of linear systems. However, the condition is not easy enough to verify. This paper will give a new equivalent condition which is relatively easy to verify.     

Controllability and observability of coordinated linear systems

This paper develops concepts of controllability and observability for the class of coordinated linear systems, a special class of linear hierarchical systems with a top-to-bottom information structure. The usual concepts of reachability and indistinguishability for linear systems are refined, taking into account the decentralized nature of coordinated linear systems. The corresponding controllability and observability decompositions are derived, and several concepts of controllability and observability for coordinated linear systems are introduced and discussed. These results are then combined to a definition and characterization of stabilizability via dynamic measurement feedback for this class of systems.     

Quasidifferentiability and observability of linear nonstationary systems

We suggest a method for studying the observability of linear nonstationary ordinary differential systems on the basis of the quasidifferentiability of the output variables with respect to some lower-triangular matrix. This approach permits one to weaken the smoothness requirement on the coefficients in the statement of observability criteria.     

F-controllability and observability of linear retarded systems

The paper gives new results onF-controllability of linear retarded functional differential equations. The concept ofF-controllability, weaker than the approximate controllability in the state space, has been introduced by this author in 1976. In the present paper new conditions are presented which, in the case of systems with one delay, become easily verifiable criteria ofF-controllability. A duality betweenF-controllability and observability is discussed, and several examples are given.This research has been supported in part from NSERC of Canada Grant A-9240 and in part from Grant FCAC 1978/79 and 1979/80 of the Quebec Ministry of Education.     

Balanced truncation for linear switched systems

We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians.     

On observability of singular linear hybrid systems

The linear system of discrete–continuous time equations with constant coefficients is considered under the assumption that it is unsolved with respect to the derivative of a continuous component part of the desired vector function. In terms of input data the necessary and sufficient solvability condition of the initial value problem is obtained. Algebraic criteria of total observability for such a system are also substantiated.     

Unknown-input present-state observability of discrete-time linear systems

A generalized definition of the present-state unknown-input observability is given as well as the corresponding unobservable subspace for discrete-time linear dynamical systems whose initial state is known, unknown, or partially known. It is shown that the time sequence of the unobservable subspaces have interesting properties explicable via generalized reachability subspaces, namely, the properties of conditional monotonicity and those associated with it. Furthermore, an observer is constructed to determine the present state.     

On stability for switched linear positive systems

This paper addresses the stability properties of switched linear positive systems in continuous-time as well as in discrete-time. In the discrete-time case, some sufficient and necessary conditions for asymptotic stability are derived for pairs of second order systems. Similar conditions are also established for a finite number of second order systems. Furthermore, for higher order systems, some results on stability are provided in a similar manner. In particular, in this case, a common linear Lyapunov function guaranteeing the stability of the switched positive systems can be easily located by means of geometry properties. In the continuous-time case, a finite number of second order systems are considered. Some equivalent conditions for stability of such systems are developed.     

On observability and stability in infinite-dimensional linear systems

It seems to be generally known (Ref. 1) that, in the case of finite-dimensional systems, the state stability properties of the system may be inferred from the output stability properties if the system is completely observable.In the case of infinite observable systems, the notion of observability generalizes in two distinct ways, weak and strong notions. The above-mentioned stability properties remain true only for the case of strong observability.This research was partially supported by the Bat-Sheva de Rothschild Fund for Advancement of Science and Technology.     

Periodical stabilization of switched linear systems

Periodical stabilization problems for switched linear systems are investigated in this paper. For autonomous switched systems, if there exists a stable convex combination of the subsystems, then a periodically switching signal can be constructed such that the overall system is asymptotically stable. Based on this fact, for switched control systems, corresponding sufficient conditions are presented under which constant/switching direct/observer-based state feedback controller can be designed such that the corresponding closed-loop systems are asymptotically stable under some periodically switching signal. Some numerical examples are given to illustrate our results.     

On the observability of time-discrete conservative linear systems

We consider various time discretization schemes of abstract conservative evolution equations of the form , where A is a skew-adjoint operator. We analyze the problem of observability through an operator B. More precisely, we assume that the pair (A,B) is exactly observable for the continuous model, and we derive uniform observability inequalities for suitable time-discretization schemes within the class of conveniently filtered initial data. The method we use is mainly based on the resolvent estimate given by Burq and Zworski in [N. Burq, M. Zworski, Geometric control in the presence of a black box, J. Amer. Math. Soc. 17(2) (2004) 443-471 (electronic)]. We present some applications of our results to time-discrete schemes for wave, Schrödinger and KdV equations and fully discrete approximation schemes for wave equations.     

Commuting and stable feedback design for switched linear systems

In the paper, commuting and stable feedback design for switched linear systems is investigated. This problem is formulated as to build up suitable state feedback controller for each subsystem such that the closed-loop systems are not only asymptotically stable but also commuting each other. A new concept, common admissible eigenvector set (CAES), is introduced to establish necessary/sufficient conditions for commuting and stable feedback controllers. For second-order systems, a necessary and sufficient condition is established. Moreover, a parametrization of the CAES is also obtained. The motivation comes from stabilization of switched linear systems which consist of a family of LTI systems and a switching law specifying the switching between them, where if all the subsystems are stable and commuting each other, then the total system is stable under arbitrary switching.     

On uniform controller design for linear switched systems

Traditional method to design a controller for each subsystem of switched systems will increase the complexity of the controller’s realization. Hence this paper gives sufficient conditions for designing a uniform output feedback controller for linear switched systems, and this common controller can be used for all subsystems of the switched systems. Then the output stabilization problem for a particular class of linear switched systems under this uniform output feedback controller has been studied. An illustrative example is given in order to highlight the proposed method.     

Controllability and observability of a class of linear impulsive systems

This paper studies the controllability and observability of a class of linear piecewise constant impulsive systems. Necessary and sufficient criteria for reachability and controllability are established, respectively. It is proved that the reachability is equivalent to the controllability under some mild conditions. Then, necessary and sufficient criteria for observability and determinability of such systems are established, respectively. It is also proved that the observability is equivalent to the determinability under some mild conditions. Our criteria are of geometric type, they can be transformed into algebraic type conveniently. Finally, a numerical example is given to illustrate the utility of our criteria.     

Asymptotic disturbance attenuation properties for uncertain switched linear systems

In this paper, the disturbance attenuation properties in the sense of uniformly ultimate boundedness are investigated for a class of switched linear systems with parametric uncertainties and exterior disturbances. The aim is to characterize the conditions under which the switched system can achieve a finite disturbance attenuation level. First, arbitrary switching signals are considered, and a necessary and sufficient condition is given. Secondly, conditions on how to restrict the switching signals to achieve finite disturbance attenuation levels are investigated. Two cases are considered here that depend on whether all the subsystems are uniformly ultimately bounded or not. Both discrete-time and continuous-time switched systems are considered, and the techniques are based on multiple polyhedral Lyapunov functions and their extensions.     

Controller synthesis for constrained discrete-time switched positive linear systems

This paper investigates the controller synthesis for a class of discrete-time switched linear systems with bounds on the controls and the states. First, the synthesis of state-feedback controllers guaranteeing positivity and stability of the closed-loop system is studied for sign-restricted inputs. Also, the results can be extended to asymmetrically bounded controls and constrained states. All the derived conditions are shown as linear programming framework. In addition, a cost function is proposed to maximize the length of the box constraints on the initial state. Finally, several numerical examples illustrate the validity of the developed results.