Date of final exam: 11/06/2004

E-mail: pola@ing.univaq.it

Tutor: Prof.  M.D. Di Benedetto, Università di L'Aquila

Co-Tutor: Prof.  E. De Santis, Università di L'Aquila

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SWITCHING SYSTEMS: ANALYSIS AND CONTROL   ___________________________________________________________________________________________________________

Advisor:

Prof.  M.D. Di Benedetto, Università di L'Aquila

Summary of the thesis

  Hybrid systems have been the subject of intense research over the past few years because of their expressive power that is able to capture various non-smooth phenomena in diverse application areas. However in many situations the resulting hybrid systems are very complex and therefore a general control theory for those systems is very difficult to devise. In the past few years, researchers focused on several subclasses of hybrid systems with the attempt of characterizing their dynamical properties and to synthesize controllers ensuring some prescribed performance (e.g. piecewise affine systems, introduced by Sontag in 1981, mixed logic digital systems, introduced by Bemporad et al. in 1999, switched systems introduced by Wicks in 1998, switching systems introduced by Berardi et al. in 1999 and jump systems introduced by Tugnait in 1982).

  Object of investigation of this thesis is the study of dynamical properties and of hybrid controllers synthesis for the class of continuous-time Switching Linear Systems (SLSs), i.e. hybrid systems where the discrete variables obey a discrete-event system and where the continuous variables obey linear control systems whose dynamical parameters depend on the discrete states.

  In the context of analysis of SLSs our contribution is summarized hereafter:

- Stabilizability. We proposed a definition of stabilizability and we derived a necessary and sufficient condition for its characterization based on a Kalman-like decomposition of the hydrid state space: in this way the stabilizability problem of an SLS can be cast into the stability problem of a suitable autonomous SLS extracted from the original one. Therefore an insight on the problem of stability of autonomous SLSs was addressed and some checkable sufficient conditions were derived (This last issue was studied in collaboration with the University of Twente, The Netherlands, in the group of Dr. J.W. Polderman.)

- Observability and Detectability. We proposed a general definition of observability and detectability, based on the reconstructability of the hybrid state evolution from measures of continuous inputs and of discrete and continuous outputs. A checkable sufficient and necessary condition characterizing observability was derived and a sufficient condition for detectability was found on the basis of a Kalman-like decomposition of the hybrid state space: this result reduces the detectability problem of an SLS to the stability problem of a suitable autonomous SLS extracted from the original one.

- Bimulation Theory. We proposed a general notion of hybrid bisimulation for the class of SLSs. An algebraic characterization of hybrid bisimulation and an algorithmic procedure converging in a finite number of steps to the maximal hybrid bisimulation were derived. Hybrid state reduction was performed by hybrid bisimulation. Moreover, simulation and abstraction characterizations were obtained. Finally, connections between observability, bisimulation-based reduction and simulation-based abstraction were illustrated. This topic was studied in collaboration with the University of Twente, The Netherlands, in the group of Prof. A.J. van der Schaft.

  In the context of controller synthesis for SLSs, the topic of interest was the so-called problem with safety specifications for the class of continuous-time SLSs. Given a set of "good" states within which the controlled hybrid system should evolve, the problem is to find the "maximal safe set", i.e. the set of all hybrid initial states guaranteeing that there exists a hybrid control strategy that maintains the evolution of the system in the good set. Our contributions in this research area consists in (i) a technique for computing a sequence of sets that arbitrarily close approximate the maximal safe-set for continuous-time SLSs from the interior; and in (ii) the design of a digital hybrid controller that solves the safety problem. We also proved that the synthesised hybrid controller solves also the stabilizability problem. The proposed approach was based on the definition of an appropriate discretization of the continuous-time SLS under consideration. With respect to previous works in this area, our procedure does not assume that sampling times and switching times are synchronized and, consequently, that the problem translates into a pure discrete-time problem.

  Finally an automotive control problem was considered for testing our theoretical results. The focus was on the idle speed control problem that deals with the task of maintaining, while in the idle mode, the engine speed into a given range, rejecting torque disturbances due to accessory loads and preventing the engine from stalling. By abstracting this engineering problem a safety problem was found. Therefore our analysis concentrated on computing inner approximations of the maximal safe set and a hybrid control strategy solving the safety problem at hand. That problem was addressed at first in the discrete-time domain and finally in the continuous-time domain.

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  Other research activities:

  In collaboration with the University of Cambridge, UK, in the group of Prof. John Lygeros, we considered some classes of Stochastic Hybrid Models developed in the literature: Piecewise Deterministic Markov Processes, introduced by Davis in 1993, Switching Diffusion Processes, introduced by Ghosh et al. in 1997 and Stochastic Hybrid Systems, introduced by Hu et al. in 2000. The descriptive power of the three classes was formally compared and conditions under which the classes coincide were developed. This theoretical analysis was instrumental for the modelling of Air Traffic Management Systems. In fact on the basis of the theoretical results, we proposed a possible modelling of aircrafts and weather in an air traffic management scenario and an analysis of human operators acting in ATM systems, taking particular attention to human errors which are shown to be the main cause of accidents.  

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