Viability of Hybrid Systems

This introductory chapter provides an overview of the problems addressed in this book, and a summary of the book and its contributions.

In this chapter, we review existing literature on hybrid systems that is related to this work. This is carried out by first considering three specific approaches to viability of hybrid systems, these being due to Nerode and colleagues, Aubin and colleagues and Deshpande–Varaiya. This review is carried out in Sections 2.1–Section 2.3. In Section 2.4, literature related specifically to Chapters 3–8 is then reviewed. Some concluding remarks are made in Section 2.5. Given that hybrid systems is a field that encompasses a variety of problem domains and disciplines, there exists a vast body of related work. Although we do provide some general overview of the field, we will focus on approaches to hybrid systems, and in particular their control, that have the most direct impact on this work.

A hybrid model is adopted based on the simple hybrid system (SHS) model in which a finite control automaton interacts with a continuous–time plant at sample times, these two components being coupled by analog–to–digital and digital–to–analog converters. Three forms of uncertainty are introduced into this model, with particular interest given to transition dynamics that describes the system dynamics over some time subinterval of the sampling interval. The set of possible solutions generated by this instance of a simple hybrid system is characterized in two related ways to define the notion of hybrid trajectory used in this work. Using this notion of hybrid trajectory, an ordering is given to compare both hybrid solution segments as well as overall hybrid trajectories. This definition requires the characterization of the class of functions that are considered admissible in defining hybrid trajectories. The treatment of solutions given in this work provides a unified method of treating the constructive, approximate, and continuous properties of hybrid solutions generated by this simple hybrid system model. The relationships between the continuity of an operator defined relative to the qualitative property being designed for (e.g., viability), the existence of a fixed point of this operator, and the existence of a hybrid trajectory satisfying the desired qualitative property (corresponding to three control problems) are examined. The three–tank problem is introduced using this modeling formalism and modifications of the basic problem are made in order to illustrate various controlled and uncontrolled hybrid behaviour, in particular, various forms of transition dynamics models.

The work presented in this chapter is aimed at providing the computational framework for the fixed point approach given in (Nerode et al., 1995). The approach taken is based on continuous–time viability specifically the work of (Frankowska and Quincampoix, 1991) on the viability kernel of a differential inclusion. The main objective of the work here is to provide a control automaton that can handle sampling explicitly. The governing continuous–time dynamics are represented by a collection of differential inclusions that allows one to capture the effect of dynamic uncertainty in a hybrid system.

In this chapter, the Viability Control Problem is considered for hybrid systems under time–independent state constraints under the three forms of uncertainty transition dynamics, structural uncertainty, and parametric uncertainty. Ensuring that viability remains satisfied under uncertainty will be referred to as the robust viability problem. The nominal dynamics are taken as the collection of constituent control systems having single–valued nominal dynamics with no uncertainty. Two approaches are used for considering uncertainty relative to viability. In the first approach the effect of uncertainty on the nominal design is examined. In the second approach an uncertainty operator is used to determine the effect of uncertainty on the nominal design. To ensure that viability remains satisfied under uncertainty, two possibilities are considered. The uncertainty can be either taken into account at each iteration of the Controllability Operator Fixed Point Approximation Algorithm or compensated by an appropriate nominal design of the control automaton. In the treatment of the former case, we require that the admissible control law class is the same for the nominal and the uncertain case. In the latter case, we allow for either a larger subset of control law classes in the uncertain case or a different set.

Two simulation applications of viability are considered in this chapter. The first is an Active Magnetic Bearing system considered in Section 6.1 in which viability is satisfied by computation of the reachable set for a differential inclusion. The second application considered in Section 6.2 is that of a batch polymerization process in which viability is satisfied by cascade control of a viable controller with an existing PID controller. In both cases, satisfaction of viability is demonstrated through simulation. Some conclusions are made in Section 6.3.

In this chapter, the controllability operator approach to viability of hybrid systems is extended to consider the problems of attainability and viable attainability. In each case, a relation is defined that captures the hybrid system’s behaviour over sampling intervals. Corresponding operators, the attainability operator and the time-independent viable attainability operator, are defined over the entire time interval of existence. This provides a unified approach to three constraint problems of hybrid systems. The development for attainability and viable attainability is examined through a three fluid filled tank example.

In this chapter, we collect facts and properties of the controllability operator. Firstly, in Section 8.1 we show that the ε _n (x ₀)–balls which are removed as part of the satisfaction of viability are continuous functions. This leads to establishing continuity of the controllability operator. Secondly, in Section 8.2 we consider the lattice properties of the control laws. Two orderings of the control law classes are defined, one weak and one strong ordering. Having this, it is established that the set of control law classes with the order relation and over set intersection and union form a lattice. Next conditions for satisfying the order relations are derived. Thirdly, in Section 8.3 homotopies are defined to consider the variation in the value of the controllability operator relative to the base admissible control law class PWC^Δ which corresponds to the collection of piecewise continuous functions over the sampling interval Δ. A conclusion is made in Section 8.4.

In this work, hybrid systems wwere examined by developing a hybrid model and a formalism for ensuring the property of viability be satisfied. Viability is a significant qualitative property to be satisfied by dynamic systems.