Output regulation by error dynamic feedback in hybrid systems with periodic state jumps

Hybrid systems that exhibit a continuous-time dynamic behavior (flow dynamics) combined with state discontinuities activated by a clock signal (jump dynamics) have been investigated extensively in the last decade owing to their particular effectiveness in representing how some real systems in the fields of science and engineering operate. Examples are provided by biological systems subject to impulsive stimuli; mechanical systems subject to collisions or ruptures; electrical networks with switches; embedded systems combining analog and digital components. Several classes of hybrid systems of this kind can be differentiated according to features of the flow and the jump dynamics, such as being either non-linear or linear, or of the occurrence of the jump behavior, such as being time or state driven. Accordingly, synthesis procedures in control problems that involve hybrid systems raise several typical issues and call for ad-hoc devised methods, depending on the characteristics of the hybrid systems in question. In particular, this study considers hybrid systems with a continuous-time linear flow dynamics subject to periodic time-driven linear state jumps and it addresses a general output regulation problem.

Output regulation has been identified as a basic problem in control theory and it consists in finding a feedback regulator that forces the output of a given plant to asymptotically track the reference generated by a given exogenous system, while maintaining stability of the resulting closed-loop dynamics. Elena Zattoni, at the University of Bologna, and Anna Maria Perdon and Giuseppe Conte, at the Polytechnic University of Marche, in Italy, considered the output regulation problem for multivariable hybrid systems with periodic state jumps from a point of view that differs from those previously adopted in the literature. Their approach provides new insight into the problem and its solvability conditions and their research work is published in the IFAC journal Automatica.

The principal contribution of this study is in stating new necessary and sufficient conditions for the solvability of the regulation problem in purely geometric terms. This result is obtained by employing notions of invariance and of controlled invariance for subspaces of the state space of a hybrid system that generalize the classical ones for linear systems. Such approach has the merit of allowing a neat distinction between structural and stability issues in the considered control problem, so simplifying its study, as well as that of highlighting the way in which the regulated system works, by identifying the subspace of admissible state motions. Although such result cannot be used to provide a direct synthesis procedure in the general case, it serves to state a further sufficient solvability condition that requires the existence of a controlled invariant subspace for the flow dynamics which is also invariant for the jump dynamics. If, on the one side, this condition may appear rather conservative, on the other, it has the advantage of guaranteeing solvability independently of the length of the time period between two consecutive state jumps. Moreover, when it is satisfied, it provides a simple and effective synthesis procedure of the hybrid controller that solves the problem. In the paper, a numerical example is worked out to illustrate how the proposed method is applied.

Professor Conte, the senior author of the article quoted “This paper is part of the research project “Structural properties in control and observation problems for hybrid systems” that aims at studying structural properties of hybrid systems – such as linear switched system and/or linear systems with state jumps – in order to solve specific control and observation problems. The methodology consists in finding suitable notions of controlled and conditioned invariance that generalize those of the classical geometric approach to linear systems. Stabilizability properties of controlled and of conditioned subspaces of the state space are then investigated and characterized. ”

He then added “Results are exploited to provide solvability conditions to non-interacting control problems, regulation problems, model matching problems and observation problems. Updated information and a list of pertinent papers can be found at the authors’ web-pages on Researchgate.net”