Fabio Danilo Torrisi

HYSDEL-a tool for generating computational hybrid models for analysis and synthesis problems

This paper presents a computational framework for modeling hybrid systems in discrete-time. We introduce the class of discrete hybrid automata (DHA) and show its relation with several other existing model paradigms: piecewise affine systems, mixed logical dynamical systems, (extended) linear complementarity systems, min-max-plus-scaling systems. We present HYSDEL (hybrid systems description language), a high-level modeling language for DHA, and a set of tools for translating DHA into any of the former hybrid models. Such a multimodeling capability of HYSDEL is particularly appealing for exploiting a large number of available analysis and synthesis techniques, each one developed for a particular class of hybrid models. An automotive example shows the modeling capabilities of HYSDEL and how the different models allow to use several computational tools.

Convexity recognition of the union of polyhedra

In this paper we consider the following basic problem in polyhedral computation: Given two polyhedra in R^d, P and Q, decide whether their union is convex, and, if so, compute it. We consider the three natural specializations of the problem: (1) when the polyhedra are given by halfspaces (H-polyhedra), (2) when they are given by vertices and extreme rays (V-polyhedra), and (3) when both H- and V-polyhedral representations are available. Both the bounded (polytopes) and the unbounded case are considered. We show that the first two problems are polynomially solvable, and that the third problem is strongly-polynomially solvable.

Optimal complexity reduction of polyhedral piecewise affine systems

This paper focuses on the NP-hard problem of reducing the complexity of piecewise polyhedral systems (e.g. polyhedral piecewise affine (PWA) systems). The results are fourfold. Firstly, the paper presents two computationally attractive algorithms for optimal complexity reduction that, under the assumption that the system is defined over the cells of a hyperplane arrangement, derive an equivalent polyhedral piecewise system that is minimal in the number of polyhedra. The algorithms are based on the cells and the markings of the hyperplane arrangement. In particular, the first algorithm yields a set of disjoint (non overlapping) merged polyhedra by executing a branch and bound search on the markings of the cells. The second approach leads to non-disjoint (overlapping) polyhedra by formulating and solving an equivalent (and well-studied) logic minimization problem. Secondly, the results are extended to systems defined on general polyhedral partitions (and not on cells of hyperplane arrangements). Thirdly, the paper proposes a technique to further reduce the complexity of piecewise polyhedral systems if the introduction of an adjustable degree of error is acceptable. Fourthly, the paper shows that based on the notion of the hyperplane arrangement PWA state feedback control laws can be implemented efficiently. Three examples, including a challenging industrial problem, illustrate the algorithms and show their computational effectiveness in reducing the complexity by up to one order of magnitude.

Inner and outer approximations of polytopes using boxes

This paper deals with the problem of approximating a convex polytope in any finite dimension by a collection of (hyper)boxes. More exactly, given a polytope P by a system of linear inequalities, we look for two collections I and E of boxes with non-overlapping interiors such that the union of all boxes in I is contained in P and the union of all boxes in E contains P. We propose and test several techniques to construct I and E aimed at getting a good balance between two contrasting objectives: minimize the volume error and minimize the total number of generated boxes. We suggest how to modify the proposed techniques in order to approximate the projection of P onto a given subspace without computing the projection explicitly.

Computation of the constrained infinite time linear quadratic regulator

This paper presents an efficient algorithm for computing the solution to the constrained infinite time linear quadratic regulator (CLQR) problem for discrete time systems. The algorithm combines multi-parametric quadratic programming with reachability analysis to obtain the optimal piecewise affine (PWA) feedback law. The algorithm reduces the time necessary to compute the PWA solution for the CLQR when compared to other approaches. It also determines the minimal finite horizon N/sub S/, such that the constrained finite horizon LQR problem equals the CLQR problem for a compact set of states S. The on-line computational effort for the implementation of the CLQR can be significantly reduced as well, either by evaluating the PWA solution or by solving the finite dimensional quadratic program associated with the CLQR for a horizon of N=N/sub S/.

Discrete-time hybrid modeling and verification

For hybrid systems described by interconnections of linear dynamical systems and logic devices, we recently (A. Bemporad et al., 2000, 2001) proposed mixed logical-dynamical (MLD) systems and the language HYSDEL (HYbrid System DEscription Language) as a modeling tool. For MLD models, we developed a reachability analysis algorithm which combines forward reach-set computation and feasibility analysis of trajectories by linear and mixed-integer linear programming. In this paper, the versatility of the overall analysis tool is illustrated in the verification of an automotive cruise control system for a car with a robotized manual gear shift.

Performance driven reachability analysis for optimal scheduling and control of hybrid systems

We deal with the optimal control problem for piecewise linear and hybrid systems by using a computational approach based on performance-driven reachability analysis. The idea consists of coupling a reach-set exploration algorithm, essentially based on a repetitive use of linear programming, to a quadratic programming solver which selectively drives the exploration. In particular, an upper bound on the optimal cost is continually updated during the procedure, and used as a criterion to discern non-optimal evolutions and to prevent their exploration. The result is an efficient strategy of branch-and-bound nature, which is especially attractive for solving long-horizon hybrid optimal control and scheduling problems.

Efficient mode enumeration of compositional hybrid systems

A hyperplane arrangement is a polyhedral cell complex where the relative position of each cell of the arrangement and the composing hyperplanes are summarized by a sign vector computable in polynomial time. This tool from computational geometry enables the development of a fast and efficient algorithm that translates the composition of hybrid systems into a piecewise affine model. The tool provides also information on the real combinatorial degree of the system which can be used to reduce the size of the search tree and the computation time of the optimization algorithms underlying optimal and model predictive control.

Discrete-time hybrid modeling and verification of the batch evaporator process benchmark

For hybrid systems described by interconnections of linear discrete-time dynamical systems, automata, and propositional logic rules, we recently proposed the Mixed Logical Dynamical (MLD) systems formalism and the language HYSDEL (Hybrid System Descrip- tion Language) as a modeling tool. For MLD models, we developed a reachability analysis algorithm which combines forward reach set computation and feasibility analysis of trajectories by linear and mixed-integer linear programming. In this paper the versatility of the overall analysis tool is illustrated on the batch evaporator benchmark process.

SCHEDULING OF HYBRID SYSTEMS: MULTI PRODUCT BATCH PLANT

Abstract The paper proposes a solution to a class of scheduling problems where the goal is to minimize the schedule (production) time. The algorithm, which takes into account a model of a hybrid system described as MLD ( mixed logical dynamical ) system, is based on performance driven reachability analysis. The algorithm abstracts the behavior of the hybrid system by building a tree of evolution. Nodes of the tree represent reachable states of a process, and the branches connect two nodes if a transition exists between the corresponding states. To each node a cost function value is associated and based on this value, the tree exploration is driven. As soon as the tree is explored, the global solution to the scheduling problem is obtained.