Control for discrete singular hybrid systems

The problems of stability, state feedback control and static output feedback control for a class of discrete-time singular hybrid systems are investigated in this paper. A new sufficient and necessary condition for a class of discrete-time singular hybrid systems to be regular, causal and stochastically stable is proposed in terms of a set of coupled strict linear matrix inequalities (LMIs). Sufficient conditions are proposed for the existence of state feedback controller and static output feedback controller in terms of a set of coupled strict LMIs, respectively. Finally, two illustrative examples are provided to demonstrate the applicability of the proposed approach.     

An eigenstructure assignment approach for constrained linear continuous-time singular systems

This paper establishes necessary and sufficient algebraic conditions for positive invariance of convex polyhedra with respect to some linear continuous-time singular systems. They can be considered as an extension for linear singular systems of the classical positive invariance relations for regular linear systems. For a stabilizable and impulse controllable singular system with constrained inputs, a stabilizing state feedback control guaranteeing the closed-loop positive invariance of some polyhedral sets determined from the feedback matrix is studied. An analysis of the closed-loop positive invariance relations is thus presented in terms of eigenstructure and stability properties. An eigenstructure assignment technique is proposed depending on the number of stable finite poles of the singular system.     

A discrete delay decomposition approach to stability of linear retarded and neutral systems

This paper is concerned with stability of linear time-delay systems of both retarded and neutral types by using some new simple quadratic Lyapunov-Krasovskii functionals. These Lyapunov-Krasovskii functionals consist of two parts. One part comes from some existing Lyapunov-Krasovskii functionals employed in [Han, Q.-L. (2005a). Absolute stability of time-delay systems with sector-bounded nonlinearity. Automatica, 41, 2171-2176; Han, Q.-L. (2005b). A new delay-dependent stability criterion for linear neutral systems with norm-bounded uncertainties in all system matrices. International Journal of Systems Science, 36, 469-475]. The other part is constructed by uniformly dividing the discrete delay interval into multiple segments and choosing proper functionals with different weighted matrices corresponding to different segments. Then using these new simple quadratic Lyapunov-Krasovskii functionals, some new discrete delay-dependent stability criteria are derived for both retarded systems and neutral systems. It is shown that these criteria for retarded systems and neutral systems are always less conservative than the ones in Han (2005a) and Han (2005b) cited above, respectively. Numerical examples also show that the results obtained in this paper significantly improve the estimate of the discrete delay limit for stability over some other existing results.     

A Lyapunov approach to analysis of discrete singular systems

In this paper, a new type generalized Lyapunov equation for discrete singular systems is proposed. Then it is applied to study problems such as pole clustering, controllability and observability for discrete singular systems. First, some necessary and sufficient conditions for pole clustering are derived via the solution of this new type Lyapunov equation. Further, the relationship between the solution of the Lyapunov equation and structure properties of discrete singular systems will be investigated based on these results. Finally, a type of generalized Riccati equation is proposed and its solution is used to design state feedback law for discrete singular systems such that all the finite poles of the closed-loop systems are clustered into a specified disk.     

Design of switched controllers for discrete singular bilinear systems

In this paper, switched controllers are designed for a class of nonlinear discrete singular systems and a class of discrete singular bilinear systems. An invariant principle is presented for such switched nonlinear singular systems.The invariant principle and the switched controllers are used to globally stabilize a class of singular bilinear systems thatare not open-loop stable.     

Robustness of dynamic output feedback control for singular perturbation systems

The problem of the robustness of dynamic output feedback control for singular perturbation systems is investigated. The solution to this problem is reduced to the simultaneous design of static output feedback controllers for the fast subsystem and the so-called auxiliary system of the slow subsystem. Some conditions are proposed to ensure the robustness of the actual closed-loop system. The exact upper bound of the parasitic parameter for the controlled system is also determined. Finally, an actual model which failed in dynamic output feedback control in [4] is reexamined successfully here.     

Stability and stabilization of continuous-time singular hybrid systems

In this paper, the problems of stability and state feedback stabilization for a class of continuous-time singular hybrid systems are investigated. A new sufficient and necessary condition for a continuous-time singular hybrid system to be regular, impulse-free and stochastically stable is proposed in terms of a set of coupled strict linear matrix inequalities (LMIs). Moreover, a new sufficient and necessary condition is presented for the existence of the state feedback controller in terms of a set of coupled strict matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the obtained theoretical results.     

Jump modes analysis and observer design for discrete singular systems

This paper is concerned with the reconstructability and detectability of the jump modes of discrete singular systems. The notion of jump modes can be considered as the discrete counterpart of the impulsive modes of continuous singular systems which has been extensively studied. The effect of jump modes on the existence of reduced order normal Luenberger observers for discrete singular systems is investigated. Necessary and sufficient conditions for the existence of normal Luenberger observers and the corresponding observer design procedure are obtained.     

An LMI approach to robust H-infinity control for uncertain singular time-delay systems

The problem of robust H-infinity control for a class of uncertain singular time-delay systems is studied in this paper. A new approach is proposed to describe the relationship between slow and fast subsystems of singular time- delay systems, based on which, a sufficient condition is presented for a singular time-delay system to be regular, impulse free and stable with an H-infinity performance. The robust H-infinity control problem is solved and an explicit expression of the desired state-feedback control law is also given. The obtained results are formulated in terms of strict linear matrix inequalities （LMIs） involving no decomposition of system matrices. A numerical example is given to show the effectiveness of the proposed method.     

On stability of linear all-pass systems

In this paper we analyze the class of all-pass, linear and stable systems. We give a condition on the zero distribution of the transfer function of such a system to guarantee stability and we show that stability is not preserved under cascade decomposition.     

Analysis of input-to-state stability for discrete time nonlinear systems via dynamic programming

The input-to-state stability (ISS) property for systems with disturbances has received considerable attention over the past decade or so, with many applications and characterizations reported in the literature. The main purpose of this paper is to present analysis results for ISS that utilize dynamic programming techniques to characterize minimal ISS gains and transient bounds. These characterizations naturally lead to computable necessary and sufficient conditions for ISS. Our results make a connection between ISS and optimization problems in nonlinear dissipative systems theory (including L₂-gain analysis and nonlinear H_∞ theory). As such, the results presented address an obvious gap in the literature.     

On robust stability of neutral systems with time-varying discrete delay and norm-bounded uncertainty

This paper investigates the robust stability of uncertain linear neutral systems with time-varying discrete delay. A delay-dependent stability criterion is obtained and formulated in the form of a linear matrix inequality. Two numerical examples are given to indicate significant improvements over some existing results.     

Stochastic stability of Positive Markov Jump Linear Systems

This paper investigates on the stability properties of Positive Markov Jump Linear Systems (PMJLS’s), i.e. Markov Jump Linear Systems with nonnegative state variables. Specific features of these systems are highlighted. In particular, a new notion of stability (Exponential Mean stability) is introduced and is shown to be equivalent to the standard notion of 1-moment stability. Moreover, various sufficient conditions for Exponential Almost-Sure stability are worked out, with different levels of conservatism. The implications among the different stability notions are discussed. It is remarkable that, thanks to the positivity assumption, some conditions can be checked by solving Linear Programming feasibility problems.     

Solvability of static output-feedback stabilization for LTI positive systems

This paper addresses the stabilization problem of positive linear systems which have nonnegative states whenever the initial conditions are nonnegative. The synthesis of static output-feedback controllers that ensure the positivity and asymptotic stability of the closed-loop system is investigated. It is shown that this important problem is completely solved for single-input and single-output positive systems. The proposed approach can be applied to multi-input positive systems with controllers having one rank gains. All the provided conditions are necessary and sufficient and can be solved in terms of Linear Programming.     

Characterization and computation for the bound ε in linear time-invariant singularly perturbed systems

This paper gives a complete solution to the ε-bound problem in a linear time-invariant singularly perturbed system. A criterion is presented to determine the existence of the bound.     

A stability result for linear parameter-varying systems

A negative-real-part eigenvalue assumption on a linear parameter-varying system implies exponential stability under a condition that includes either slowly varying parameters, or slowly accelerating parameters.     

On the stability of slowly time-varying linear systems

New conditions are given in both deterministic and stochastic settings for the stability of the system x=A(t)x when A(t) is slowly varying. Roughly speaking, the eigenvalues of A(t) are allowed to wander into the right half-plane as long as on average they are strictly in the left half-plane.This work was funded by the NSF under Grant ECS-8806063, and was completed while the author was with the Department of Electrical and Computer Engineering, Johns Hopkins University, Baltimore, MD 21218, U.S.A.     

Excitability, stability, and sign of equilibria in positive linear systems

The relationship between sign and stability of equilibria in positive linear systems is investigated in this paper. In particular it is proved that the well-known result “stability and irreducibility imply strict positivity” can be better specified by introducing the notion of excitability. In fact stability, excitability and strict positivity of the equilibria are three perfectly balanced properties since any one of them is implied by the two others.     

Simple stability criteria for nonlinear time-varying discrete systems

We present two sufficient conditions for asymptotic stability of nonlinear time-varying discrete systems. Our main results generalize Mori's result for linear discrete time-invariant systems to nonlinear time-varying systems.     

Improved results on stability of continuous-time switched positive linear systems

In this paper, the problems of stability for switched positive linear systems (SPLSs) under arbitrary switching are investigated in a continuous-time context. The so-called “copositive polynomial Lyapunov function” (CPLF) giving a generalization of copositive types of Lyapunov function is first proposed, which is formulated in a higher order form of the positive states of the underlying systems. It is illustrated in this paper that some classical types of Lyapunov functions can be seen as special cases of the proposed CPLF. Then, new stability conditions are developed by the new Lyapunov function approach. It is also proved that the conservativeness of the obtained criteria can be further reduced as the degree of the Lyapunov function increases. A numerical example is given to demonstrate the effectiveness and less conservativeness of the developed techniques.