Measurement dynamic feedback output regulation in hybrid linear systems with state jumps

Abstract

Summary This work investigates the problem of designing a hybrid dynamic feedback regulator that forces the output of a hybrid linear system to asymptotically converge to the reference generated by a hybrid exogenous system, asymptotically rejects the exogenous disturbance, and attains global asymptotic stability of the compensated hybrid linear dynamics. The class of hybrid linear systems addressed exhibits a continuous‐time linear behavior except at isolated points of the time axis, where the state is subject to discontinuities that are caused by a jump behavior. In the presence of possibly unequally spaced state jumps, under the only constraint that the minimum time between any two consecutive jumps is no smaller than a given positive real constant, both implicit and explicit sufficient conditions for the existence of a solution to the stated problem are shown. The explicit condition is constructive, in the sense that it outlines the algorithmic procedure for the synthesis of the hybrid feedback regulator, provided that a certain output‐nulling hybrid controlled invariant subspace be known. Then, a necessary and sufficient condition for the existence of such subspace is proven, so that a computational means to derive it, if any exists, is given. Finally, the devised approach is applied to a numerical example borrowed from the literature, with the twofold aim of illustrating its implementation and making a comparison with the available method.

On the Zero-Dynamics of a Class of Hybrid LTI Systems: A Geometric Approach

“…We assume that the time domain is not known in the sense that measures (or estimates) of the jumping instants are not available. We underline that the class of system (1) under study is also referred to as impulsive systems (e.g., see [16]).…”

“…In this paper, we address the problem of defining the zero-dynamics for linear hybrid systems with aperiodic timedriven jumps in the geometric framework developed in [13], [14] for general results and recently extended to this hybrid context in, for example, [15], [16]. In doing so, no knowledge of the jumping instants is assumed.…”

Disturbance decoupling by state feedback for uncertain impulsive linear systems

“…For this reason, impulsive linear systems have recently attracted the interest of many researchers in relation to a number of control problems. By extending to the framework of impulsive linear systems approaches and techniques previously developed for linear systems, solvability conditions have been obtained for stabilization problems, [4][5][6] linear quadratic control problems, 7,8 disturbance decoupling problems, [9][10][11][12] output regulation problems, [13][14][15][16][17][18][19][20][21] and observation problems. [22][23][24][25] Structural geometric methods obtained by extending the classical geometric approach of Basile and Marro 26 and of Wonham 27 have been shown to be particularly effective to deal with the mentioned problems for this class of systems.…”

State estimation for linear hybrid systems with periodic jumps and unknown inputs

“…The disturbance decoupling problem is studied for the class of linear hybrid systems with periodic jumps in Reference , here the conditions for global asymptotic stability for a special subclass of systems are provided. More recently, in the same vein, implicit and explicit sufficient conditions are provided in Reference for the existence of a hybrid dynamic feedback regulator. In particular, in Reference , an observer design is proposed for detectable linear hybrid systems.…”