Oded Maler

Hybrid Systems: Computation and Control

This class presents recent advances in the theory for control, verification, and simulation of hybrid dynamical systems, and shows the application of the theory to the design of the control architecture for complex, large scale systems. Hybrid dynamical systems are continuous time, continuous variable systems with a phased operation. The phases of operation capture the system’s discrete event or linguistic behavior, while the continuous variable dynamics capture the system’s detailed or “lowerlevel” behavior. The two behaviors influence each other. Hierarchical organization is implicit in hybrid systems, since the discrete event dynamics represent planning which is based on an abstraction of the continuous dynamics. Control systems design based on hybrid models consist of three parts:

Reachability Analysis of Pushdown Automata: Application to Model-Checking

We apply the symbolic analysis principle to pushdown systems. We represent (possibly infinite) sets of configurations of such systems by means of finite-state automata. In order to reason in a uniform way about analysis problems involving both existential and universal path quantification (such as model-checking for branching-time logics), we consider the more general class of alternating pushdown systems and use alternating finite-state automata as a representation structure for sets of their configurations. We give a simple and natural procedure to compute sets of predecessors using this representation structure. We incorporate this procedure into the automata-theoretic approach to model-checking to define new model-checking algorithms for pushdown systems against both linear and branching-time properties. From these results we derive upper bounds for several model-checking problems as well as matching lower bounds.

SpaceEx: scalable verification of hybrid systems

We present a scalable reachability algorithm for hybrid systems with piecewise affine, non-deterministic dynamics. It combines polyhedra and support function representations of continuous sets to compute an over-approximation of the reachable states. The algorithm improves over previous work by using variable time steps to guarantee a given local error bound. In addition, we propose an improved approximation model, which drastically improves the accuracy of the algorithm. The algorithm is implemented as part of SpaceEx, a new verification platform for hybrid systems, available at spaceex.imag.fr. Experimental results of full fixed-point computations with hybrid systems with more than 100 variables illustrate the scalability of the approach.

Kronos: A model-checking tool for real-time systems

KRONOS is a software tool aiming at assisting designers of real-time systems to develop projects meeting the specified requirements. One major objective of KRONOS is to provide a verification engine to be integrated into design environments for real-time systems in a wide range of application areas. Real-time communication protocols, timed asynchronous circuits and hybrid systems are some examples of application domains where KRONOS has already been used. KRONOS has been also used in analysing real-time systems modeled in several other process description formalisms, such as ATP, AORTA, ET-LOTOS, and T-ARGOS. On the other direction, the tool itself provides an interface to untimed formalisms such as labeled-transition systems (LTS) which has been used to exploit untimed verification techniques.

Monitoring Temporal Properties of Continuous Signals

In this paper we introduce a variant of temporal logic tailored for specifying desired properties of continuous signals. The logic is based on a bounded subset of the real-time logic mitl, augmented with a static mapping from continuous domains into propositions. From formulae in this logic we create automatically property monitors that can check whether a given signal of bounded length and finite variability satisfies the property. A prototype implementation of this procedure was used to check properties of simulation traces generated by Matlab/Simulink.

Controller Synthesis for Timed Automata 1

Abstract In this work we tackle the following problem: given a timed automaton, restrict its transition relation in a systematic way so that all the remaining behaviors satisfy certain properties. This is an extension of the problem of controller synthesis for discrete event dynamical systems, where in addition to choosing among actions, the controller have the option of doing nothing and let the time pass. The problem is formulated using the notion of a real-time game, and a winning strategy is constructed as a fixed-point of an operator on the space of states and clock configurations.

Approximate Reachability Analysis of Piecewise-Linear Dynamical Systems

In this paper we describe an experimental system called d/dt for approximating reachable states for hybrid systems whose continuous dynamics is defined by linear differential equations. We use an approximation algorithm whose accumulation of errors during the continuous evolution is much smaller than in previously-used methods. The d/dt system can, so far, treat non-trivial continuous systems, hybrid systems, convex differential inclusions and controller synthesis problems.

Symbolic Controller Synthesis for Discrete and Timed Systems

This paper presents algorithms for the symbolic synthesis of discrete and real-time controllers. At the semantic level the controller is synthesized by finding a winning strategy for certain games defined by automata or by timed-automata. The algorithms for finding such strategies need, this way or another, to search the state-space of the system which grows exponentially with the number of components. Symbolic methods allow such a search to be conducted without necessarily enumerating the state-space. This is achieved by representing sets of states using formulae (syntactic objects) over state variables. Although in the worst case such methods are as bad as enumerative ones, many huge practical problems can be treated by fine-tuned symbolic methods. In this paper the scope of these methods is extended from analysis to synthesis and from purely discrete systems to real-time systems.

Reachability analysis of dynamical systems having piecewise-constant derivatives

Abstract In this paper we consider a class of hybrid systems, namely dynamical systems with piecewise-constant derivatives (PCD systems). Such systems consist of a partition of the Euclidean space into a finite set of polyhedral sets (regions). Within each region the dynamics is defined by a constant vector field, hence discrete transitions occur only on the boundaries between regions where the trajectories change their direction. With respect to such systems we investigate the reachability question: Given an effective description of the systems and of two polyhedral subsets P and Q of the state-space, is there a trajectory starting at some xϵP and reaching some point in Q? Our main results are a decision procedure for two-dimensional systems, and an undecidability result for three or more dimensions.

Reachability Analysis via Face Lifting

In this paper we discuss the problem of calculating the reachable states of a dynamical system defined by ordinary differential equations or inclusions. We present a prototype system for approximating this set and demonstrate some experimental results.