Date of final exam: 27/04/2001

E-mail: berardi@eecs.berkeley.edu

Tutor: Prof.  A. Isidori, Università di Roma ‘La Sapienza’

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Control of Hybrid Systems under Safety Specifications   ___________________________________________________________________________________________________________

Advisor:

Prof.  A. Isidori, Università di Roma ‘La Sapienza’, Prof.  M.D. Di Benedetto, Università de L'Aquila

Summary of the thesis

The most recent advances in the field of control systems have been driven by two apparently opposite kind of issues: first, the ever increasing complexity of large-scale distributed control systems -like we encounter in flight control, automated manufacturing systems, transportation systems, etc.- continuously requires new powerful tools in order to handle the most challenging problems faced by the designers. Second, there is concurrently a great deal of interest towards the embedded systems area: embedded systems are quite pervasive in our world today, covering a too much broad range of applications to be mentioned here (from house-hold appliances to communication systems, and more). In spite of their huge flexibility and adaptability they can be pretty much well described in terms of digital devices (i.e. adopting a boolean logic) interacting with different kinds of physical systems, in such a way as to gather information from them, processing data, and eventually controlling the behaviour of their physical counter-part.

In both cases, we see that there is in general a non trivial interaction between some parts of the system which can be modeled by continuous dynamics and other parts exhibiting discrete-event modes of operation. Therefore, when considering either complex large-scale systems or embedded microcontrollers, the dynamics of the overall system can be captured by the paradigm of hybrid systems. Hybrid control systems, in fact, include both continuous and discrete dynamics. In this regard they extend the classical scheme of supervisory control, but, unlike standard supervised control systems, the discrete and continuous parts of a general hybrid system possess a strictly interacting behaviour, making it impossible to separate the two different aspects during the control design process.

Loosely speaking hybrid systems are discrete event systems containing differential (or difference) equations in each discrete mode. Transitions among discrete locations are determined not only by discrete events, but also by the evolution of the underlying continuous dynamics.

Although the notion of a hybrid system as an interacting combination of discrete-event and continuous dynamics is well accepted by the research community, modeling such systems is sometimes a matter of taste. Given the twofold behaviour they possess, the hybrid systems research area includes people from both computer science and control theory. As they generally face distinct issues using different notations and methods, they also have a different perspective on what a hybrid system model should be. Usually computer scientists prefer to deal with hybrid systems having a complex discrete dynamics and a simple (sometimes trivial) continuous counterpart. Control theorists, instead, prefer to focus on hybrid models with a complicate continuous dynamics, leaving the discrete structure at a low level of complexity. This is the main reason for the absence of a unified modeling framework for hybrid systems, that can be accepted universally by the research community.

We introduce a model which is general enough to capture all the features both computer scientists and control theorists want to be included in hybrid systems. However, as we shall see, the complexity of this model is such that it becomes unsuitable when applied to the solution of actual control problems. Thus we have to introduce some restrictive assumptions, depending on the kind of applications we are interested in.

Hybrid systems have been the subject of intensive study in the past few years. In particular, emphasis has been placed on solving problems with safety specifications, which are described by giving a set of good states within which the controlled hybrid system should evolve. The set of all initial states guaranteeing that the evolution of the system remains in the good (or safe) set is the maximal controlled invariant set contained in the set of good hybrid states. Such a set is called maximal safe-set and the collection of all control strategies that make this set invariant is the maximal controller.

In this thesis, we first tackle this problem for a restricted class of hybrid systems, called switching systems, with the goal of obtaining computationally efficient procedures. Switching systems are characterized by a finite state machine (FSM) and a set of dynamical systems, each corresponding to a state of the FSM. The transitions between two different states of the FSM are determined by external uncontrollable events which act as discrete disturbances. This model can be used to represent a number of control problems of practical interest. In our case, the motivation to study this formulation comes from the engine control problem in automotive applications. Then we consider the case of general hybrid systems, deriving a structural procedure similar to the one used to solve the switching systems case. However, dealing with general hybrid systems leads to the loss of some amenable computational features, like the convexity of the involved safe-sets.

Nevertheless, our methods possess important advantages over the procedure already known in the literature, since we can exploit the structure of the FSMs. The proposed approach decomposes the original problem of finding the maximal safe set into a number of different sub-problems, basically each consisting of finding a maximal (robustly) controlled invariant set in a constraining set for a continuous dynamical system. Since no computationally appealing results are available for the determination of maximal controlled invariant sets for general dynamical systems, we propose to linearize and find a discrete-time representation of nonlinear dynamical systems as an important step towards a computationally efficient approach. Indeed, for discrete-time linear systems, several results for the computation of maximal controlled invariant sets have been reported in the literature. Even in this case, the procedure for the computation of the maximal controlled invariant set may not converge in a finite number of steps. We then propose a procedure for approximating such a set and show how to obtain an accurate bound of the error by combining inner and outer approximations. We then proceed to show how these approximations can be computed. A relation is established between a maximal controlled invariant set associated with a general continuous-time dynamical system and that of the corresponding discrete-time system. In the last section of the thesis we show how the techniques introduced previously can be used to solve the idle-speed control problems for automotive engines.
 

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