Theory and computation of discrete state space decompositions for hybrid systems

https://doi.org/10.1016/j.ejcon.2012.09.001Get rights and content

Abstract

The problem of verifying whether properties such as reachability and safety hold for a hybrid system can be simplified, for example by using bisimulation. In this paper, we introduce a novel decomposition method to simplify the problem of verifying also other important properties such as stabilizability and detectability. To do so, we characterize general properties of a hybrid system and derive structural decompositions with respect to the characterization, which reduce the verification problem to a set of simpler verification problems on the elements of the decomposition.

Introduction

A rich set of results on stability, safety, observability, and other important properties for hybrid systems is available in the literature (e.g., see [8], [19], [29] for safety, [6], [18], [26] for stability, [2], [3], [5], [7], [11], [12], [26], [30] for observability). However, testing the conditions for these properties to hold is in general computationally intensive.

An effective approach for complexity reduction consists of finding a system that is “equivalent” to the original one with respect to the properties that we want to verify but that is “simpler” to analyze. A powerful tool for complexity reduction is bisimulation equivalence introduced in [21], [24]. Extensions of the notion of bisimulation to continuous and hybrid systems were explored in a number of papers (e.g. [1], [14], [15], [16], [17], [22], [23], [25], [28]). If two systems are bisimilar, then the so-called sequence properties [20], such as reachability and safety, are preserved. Then a procedure that generates “simpler” systems that are bisimilar to a given system under study effectively reduces verification complexity.

In this paper, we address how to reduce complexity for verification problems that involve checking whether a property, which may be more general than a sequence property, e.g. stabilizability, observability or detectability, holds for a hybrid system. The goal is to extract particular subsystems from the original system, so that checking a property on these subsystems is equivalent to checking whether the same property holds for the original system.

The basic idea that inspires our approach can be easily understood with the following simple example (see Fig. 1): consider a hybrid system having three discrete states, say 1, 2 and 3, and let S1, S2 and S3 be the corresponding linear dynamic equations. We suppose that the system evolves as follows. When in mode 1, a switching to mode 2 has to occur within 10 s from the last switching time. Such a commutation is due to a control action, but is subject to temporal constraints. From mode 2, an uncontrolled commutation must occur albeit at an unknown time instant, after 5–10 s from the last switching time. Finally, anytime an uncontrolled commutation may occur from mode 3 to mode 1. The continuous state x before the commutation is reset after switching to a new state that linearly depends on x. We suppose that S2 is controllable and that we want to analyze the stabilizability of the overall hybrid system H.

We claim that stabilizability of the hybrid system is equivalent to stabilizability of the subsystem S3. In fact, if the initial state is mode 3, a transition could never occur. Hence S3 has to be stabilizable. Conversely, if the initial state is mode 1, the controller can force the commutation to mode 2 when it is allowed, and then, since S2 is controllable, the continuous state can be driven to zero within 5 s, which is the dwell time in mode 2. If the transition from 2 to 3 never occurs, then the convergence to zero of the trajectory is ensured. Otherwise two cases are possible: if a transition from 3 to 1 occurs, then we apply the reasoning again with initial state 1. If there is no transition from mode 3, stabilizability of S3 ensures the asymptotic convergence to zero of the trajectory. The same reasoning holds if the initial state is either mode 2 or 3.

The idea is to generalize this analysis to exploit the structure of the FSM of the hybrid system and the characteristics of the subsystems corresponding to the nodes of the FSM to simplify the property verification problem. The question is what are the characteristics of the properties for which a topological simplification of the FSM is feasible.

To answer this question, we introduced a new property classification scheme and a procedure to decompose the system that depends on this classification. To the best of our knowledge, the only contributions dealing with structural decomposition of the discrete state-space of a hybrid system with respect to a given property are reported in [4], where sufficient conditions are given for the generic final-state asymptotic determinability in terms of properties of the cycles of the automaton associated to a hybrid system. Further, in [31], the stability test for a class of hybrid systems, where the continuous dynamics are linear, the mode switching depends on time and on the decision of a supervisor, is reduced to a finite number of stability tests on suitable subsystems.

The paper is organized as follows. In Section 2, background notations and definitions are given. In Section 3, we introduce and classify abstract properties for hybrid systems. In Section 4, we present structural decompositions of the hybrid system discrete state-space with respect to those properties. In Section 5, we illustrate some examples of abstract properties introduced in the previous sections and give some specific results. Concluding remarks are offered in Section 6. The proofs and algorithms that are not necessary to understand the flow of the paper are presented in the Appendix.

Section snippets

Definitions and background

We introduce a class of hybrid systems where the transitions depend either on a disturbance event (switching transitions) or on a discrete input (controlled transitions). For simplicity, we make the reasonable assumption that at each time tR only one discrete disturbance event can affect the system. A discrete disturbance and a discrete input can act simultaneously on the system, and the disturbance has the priority. Systems in this class are called H-systems. Let Q={1,2,,N} be the finite set

Properties for hybrid systems

In this section, we define predicates for a hybrid system, in the sense that we attribute properties to sets of executions of a hybrid system. We will consider predicate logic, so that a property of a set of executions will be either true or false. Our notion of property will be the technical tool used to decompose the given hybrid system into simpler subsystems, so that, if each subsystem satisfies the property, then the given system does too.

Formally we define a property P on subsets of χS as

Discrete structure decompositions

This section presents the main results of the paper and is devoted to decompositions of the given hybrid system S, which affect the discrete state space of S by simplifying its topological structure. Such decompositions are based on the FSM (Q,W,E) and on the functions g and f, but the way in which they can be used to check a property depends on the closure under composition and/or backward extensibility of the property itself. Therefore the results we are going to describe depend on the

Characteristics of some classical properties for hybrid systems

In this section, we consider some well-known properties for hybrid systems, such as asymptotic stabilizability, detectability, reachability and safety, and analyze them with respect to closure under composition and backward extensibility. The decompositions shown in the previous sections can then be applied w.r.t. those particular properties. The properties introduced in 5.5 Finite escape time, 5.6 Norm boundedness are examples of a non-composition closed-backward extendable property, and of a

An example

Consider an LH-system S and let g(e)={tR+:tδm},eE, for some real δm>0. Fig. 3 describes the discrete transitions of S: the dashed links represent uncontrolled transitions and the solid links represent controlled transitions. Assume that a controllable dynamic system is associated with the discrete state q. Then S satisfies the property Ps with respect to the set {q}.

By Proposition 12, S satisfies the property Ps with respect to the set Reach1({q}) (see Fig. 4). Therefore, from (iii) of

Conclusions

We presented abstract properties such as stabilizability and detectability for hybrid systems, and presented results on decomposition of the discrete structure of the hybrid system, which allow a significant reduction of the effort required to test properties such as stabilizability or detectability. Our technique can be used to obtain efficient tests, if combined with state space decompositions defined on the continuous component of the state space.

References (32)

  • A. Bemporad et al.

    Observability and controllability of piecewise affine and hybrid systems

    IEEE Transactions on Automatic Control

    (2000)
  • R.A. De Carlo et al.

    Perspective and results on the stability and stabilizability of hybrid systems

    Proceedings of the IEEE

    (2000)
  • E. De Santis, M.D. Di Benedetto, G. Pola, On observability and detectability of continuous–time linear switching...
  • E. De Santis et al.

    Computation of maximal safe sets for switching systems

    IEEE Transactions on Automatic Control

    (2004)
  • E. De Santis, M.D. Di Benedetto, G. Girasole, Digital idle speed control of automotive engines using hybrid models, in:...
  • E. De Santis, M.D. Di Benedetto (Guest Editors), Special issue on observability and observer-based control of hybrid...
  • Cited by (7)

    • Almost always observable hybrid systems

      2020, Nonlinear Analysis: Hybrid Systems
    • Stabilization and control Lyapunov functions for language constrained discrete-time switched linear systems

      2018, Automatica
      Citation Excerpt :

      The idea of employing regular languages and automata to impose constraints on the switching law has been recently applied to the problem of stability analysis for switched linear systems. The problem of stability of constrained switched linear system is addressed in De Santis and Benedetto (2013), Wang, Roohi, Dullerud, and Viswanathan (2014) and Weiss and Alur (2007) using automata properties while converse Lyapunov theorems, based on the joint spectral radius approach, are provided in Philippe and Jungers (2015). Graph Lyapunov functions and spectral radius are employed in Ahmadi, Jungers, Parrilo, and Roozbehani (2014) and in Lee and Dullerud (2007) directed graphs are used to determine the switching sequences under which the system is stable.

    • Observability Characterization for H-Systems

      2023, Communications and Control Engineering
    View all citing articles on Scopus

    This work was partially supported by the European Union Seventh Framework Programme [FP7/2007-2013] under grant agreement n257462 HYCON2 Network of excellence IST Network of Excellence HyCON2, and by STREP iFLY contract n, TREN/07/FP6AE/S07.71574/037180.

    View full text