In this pap e we prove in a constructive way, the ee ale b e we e pie- a#ne syste and a broad class of hybridsyste de e d by inte line dynamics, automata, and propositional logic. By focusing our inveon the forme class, we show through countethat obse ability and controllability prope rtie cannot be e asilydely from those of the comp tline subsyste Inste we propose practical nume te base onmixe te line programming. Keywords--- Hybrid syste controllability,obse ability, pie line syste pie a#ne syste mixe teline programming I. Introducti In recent yearsb oth control and computer science haveb een attractedb y hybridsystem [1], [2], [23], [25], [26],b ecause they provide a unified framework fordescribgARB( cesses evolving accordingto continuous dynamics, discrete dynamics, and logic rules. The interest is mainly motivatedb y the large variety of practical situations, for instance real-time systems, where physical processes interact with digital controllers. Several modelingformalisms h...

We propose a new technique for the identification of discrete-time hybrid systems in the Piece-Wise Affine (PWA) form. This problem can be formulated as the reconstruction of a possibly discontinuous PWA map with a multi-dimensional domain. In order to achieve our goal, we provide an algorithm that exploits the combined use of clustering, linear identification, and pattern recognition techniques. This allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures. Moreover, the clustering step (used for classifying the datapoints) is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions. Measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure.

There are numerous application examples for hybrid systems verification in recent literature. Most of them were introduced to illustrate a new approach to hybrid systems verification, and are therefore of a limited size. Others are case studies that serve to prove that an approach can be applied to real world problems. Verification of these typically requires a lot of domain experience to obtain a tractable, verifiable model. Verification of a case study yields a singular result that is hard to compare and time-consuming to reproduce.

Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approximating reachable sets using oriented rectangular hulls (ORHs), the orientations of which are determined by singular value decompositions of sample covariance matrices for sets of reachable states. The orientations keep the over-approximation of the reachable sets small in most cases with a complexity of low polynomial order with respect to the dimension of the continuous state space. We show how the use of ORHs can improve the efficiency of reachable set computation significantly for hybrid systems with nonlinear continuous dynamics.

In this paper we propose a procedure for synthesizing piecewise linear optimal controllers for hybrid systems and investigate conditions for closed-loop stability. Hybrid systems are modeled in discrete-time within the mixed logical dynamical (MLD) framework[8], or, equivalently [7], as piecewise affine (PWA) systems. A stabilizing controller is obtained by designing a model predictive controller (MPC), which is based on the minimization of a weighted 1/∞-norm of the tracking error and the input trajectories over a finite horizon. The control law is obtained by solving a mixed-integer linear program (MILP) which depends on the current state. Although efficient branch and bound algorithms exist to solve MILPs, these are known to be NP-hard problems, which may prevent their on-line solution if the sampling-time is too small for the available computation power. Rather than solving the MILP on line, in this paper we propose a different approach where all the computation is moved off line, by solving a multiparametric MILP (mp-MILP). As the resulting control law is piecewise affine, on-line computation is drastically reduced to a simple linear function evaluation. An example of piecewise linear optimal control of the heat exchange system [16] shows the potential of the method.

In this paper, we formulate the problem of characterizing the stability of a piecewise affin (PWA) system as a verification problem. The basic idea is to take the whole R^n as the set of initial conditions, and check that all the trajectories go to the origin. More precisely, we test for semi-global stability by restricting the set of initial conditions to an (arbitrarily large) bounded set X(0), and label as "asymptotically stable in T steps" the trajectories that enter an in variant set around the origin within a finite time T ,or as "unstable in T steps" the trajectories which enter a (very large) set X_inst . Subsets of X (0) leadin ton2W of the two previous cases are labeled as "nv classifiable in T steps". The domain of asymptotical stability in T steps is a subset of the domain of attraction ofan equilibrium poin t, an has the practicalmeanca of collectin inPv)v convW2xvP from which the settlin time of the system is smaller than T . In addition it can be computed algorithmically i...

In this paper we focus on the identification of discrete-time hybrid systems in the Mixed-Logic Dynamical (MLD)/Piece-Wise Affine (PWA) form. This problem can be formulated as the reconstruction of a possibly discontinuous PWA map with a multi-dimensional domain. In order to achieve our goal, we propose an algorithm that exploits the combined use of clustering, linear identification, and classification techniques. This allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid.

Abstract — Bounded model checking (BMC) has recently emerged as a very powerful methodology for the verification of purely discrete systems. Given a horizon of interest, bounded model checking verifies whether all finite-horizon trajectories satisfy a temporal logic formula by first translating the problem to a large satisfiability SAT-problem and then relying on extremely powerful state-of-the art SAT-solvers for a counterexample or a certification of safety. In this paper we consider the problem of bounded model checking for a general class of discrete-time hybrid systems. Critical to our approach is the abstraction of continuous trajectories under discrete observations with a purely discrete system that captures the same discrete sequences. Bounded model checking can then be applied to the purely discrete, abstracted system. The performance of our approach is illustrated by verifying temporal properties of a hybrid model of an electronic height controller. I.

Abstract. We report on how implementing a Model Based Automotive SW Engineering Process in an industrial setting can ensure the correctness of automotive applications when a process based on formal models is used. We show how formal methods, in particular model checking, can be used to ensure consistency of the models and can prove that the models satisfy selected functional and safety requirements. The technique can also be used to automatically generate test vectors from the model. Hence we show how in many ways formal verification techniques can add value to the models used for different purposes in developing automotive applications. 1

This report documents the progress made in the PA work-package during the rst two years. This work-package is the most demanding from a scientic point of view and occupied a large part of the participants time in the second year.