Observability and observer‐based control of hybrid systems

Abstract

Addressing significant application domains is important to gain further understanding of the implications of hybrid modeling on control algorithms and to evaluate whether using hybrid formalism can be of substantial help in solving complex, real-life, control problems. In many application domains, hybrid controller synthesis problems are addressed by assuming full hybrid state information, although in many realistic situations state measurements are not available. Hence, to make hybrid controller synthesis relevant, the design of hybrid state observers is of fundamental importance. Moreover in many cases, for example, in communication systems or in fault detection, algorithms for state estimation are important by themselves. Observability has been extensively studied both in the continuous and in the discrete state domains. More recently, various researchers investigated observability of hybrid systems. The definitions of observability and the criteria for assessing this property varied depending on the class of systems under consideration and on the knowledge that is assumed at the output. We believe that this Special Issue, dedicated to observability for hybrid systems, is a useful contribution for the control community as it examines a topic that is in continuous evolution and that offers great opportunities for novel research. Six papers have been selected to reflect the most current research activities. The first two papers address discrete-time linear systems that may switch in an unknown and unpredictable way among different modes taken from a finite set (switching systems). The paper by M. Baglietto, G. Battistelli and L. Scardovi tackles the problem of mode observability with unknown but bounded noises affecting both the system and measurement equations. The paper by P. Caravani and E. De Santis presents conditions of stabilizability for switching systems in terms of a new definition of control invariance. These conditions are based on the realization of a discrete observer that allows the reconstruction of the discrete state in certain intervals of the time basis. The paper by M.D. Di Benedetto, S. Di Gennaro and A. D'Innocenzo proposes a novel definition of observability motivated by safety critical applications given with respect to a subset of critical discrete states. These states model unsafe or unallowed behaviors. This definition is adapted for the class of discrete event systems, for the class of switching systems, and for the class of hidden Markov models. A safety control problem for discrete time block-triangular order preserving hybrid automata with imperfect continuous state information is solved in the paper by D. Del Vecchio. A dynamic feedback law is constructed to guarantee that the continuous state is always outside a bad set. The proposed algorithms, which have linear complexity in the number of variables, are applied to a collision avoidance problem arising in the context of intelligent transportation. The last two papers deal with stochastic systems. The paper by S. Battilotti presents an observer design for a class of single-output nonlinear systems with Markov jumps. The Markov jump process interferes with a deterministic nonlinear dynamics at random times and retains its state for a certain amount of time (dwell time). The novelty is given by the reset mechanism adopted for the estimation process itself, depending on the reset values of the state process. In the paper by A. Germani, C. Manes and P. Palumbo, the state estimation problem is investigated for a class of stochastic linear switching-output systems. In these systems, the output matrix switches in a finite set of possible values according to a not directly measured discrete Markov sequence. The proposed real-time algorithm is based on the optimal polynomial filtering approach, which achieves simultaneous estimation of both the continuous system state and the switching parameter. We conclude by thanking the authors for the quality of their contributions and for the effort in completing their papers on time with the tight deadlines of this Special Issue. We are grateful to Prof. Mike J. Grimble, the Managing Editor of this Journal, for his help and support and to the reviewers who contributed greatly to make this issue a significant contribution to Observability and Observer Design for Hybrid Systems.