Luc C. G. J. M. Habets

Controlling a Class of Nonlinear Systems on Rectangles

In this paper, we focus on a particular class of nonlinear affine control systems of the form xdot=f(x)+Bu, where the drift f is a multi-affine vector field (i.e., affine in each state component), the control distribution B is constant, and the control u is constrained to a convex set. For such a system, we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant. We then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time. If the control constraints are polyhedral, we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle. This work is motivated by the need to construct discrete abstractions for continuous and hybrid systems, in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs. We show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators

Control of Piecewise-Linear Hybrid Systems on Simplices and Rectangles

A necessary and sufficient condition for the reachability of a piecewise-linear hybrid system is formulated in terms of reachability of a finite-state discrete-event system and of a finite family of affine systems on a polyhedral set. As a subproblem, the reachability of an affine system on a polytope is considered, with the control objective of reaching a particular facet of the polytope. If the polytope is a simplex, necessary and sufficient conditions for the solvability of this problem by affine state feedback are described. If the polytope is a multi-dimensional rectangle, then a solution is obtained using continuous piecewise-affine state feedback.

Constructing decidable hybrid systems with velocity bounds

In this paper, the question of bi-similarity between hybrid systems and their discrete quotients is studied from a new point of view. We consider two classes of hybrid systems: piecewise affine hybrid systems on simplices and piecewise multi-affine systems on multi-dimensional rectangles. Given a fixed partition of the state space, we derive sufficient conditions on the values of the vector fields at the vertices of the polytopes, in order that the constructed hybrid system is bi-similar with its corresponding discrete quotient transition system. The results are based on the fact that affine vector fields on simplices and multi-affine vector fields on rectangles are uniquely determined by their values at the vertices. In this way, an interesting class of decidable hybrid systems is determined. The result is applied to a motion planning problem for planar robots.

Understanding the Bacterial Stringent Response Using Reachability Analysis of Hybrid Systems

In this paper we model coupled genetic and metabolic networks as hybrid systems. The vector fields are multi – affine, i.e., have only product – type nonlinearities to accommodate chemical reactions, and are defined in rectangular invariants, whose facets correspond to changes in the behavior of a gene or enzyme. For such systems, we showed that reachability and safety verification problems can be formulated and solved (conservatively) in an elegant and computationally inexpensive way, based on the fact that multi-affine functions on rectangular regions of the space are determined at the vertices. Using these techniques, we study the stringent response system, which is the transition of bacterial organisms from growth phase to a metabolically suppressed phase when subjected to an environment with limited nutrients.

System equivalence for AR-systems over rings : with an application to delay-differential systems

Abstract. In this paper the notion of autoregressive systems over an integral domain ? is introduced, as a generalization of AR-systems over the rings ℝ[s] and ℝ[s,s−1]. The interpretation of the dynamics represented by a matrix over ? is fixed by the choice of a module ℳ over ?, consisting of all time-trajectories under consideration. In this setup the problem of system equivalence is studied: when do two different AR-representations characterize the same behavior? This problem is solved using a ring extension of ?, that explicitly depends on the choice of the module ℳ of all time-trajectories. In this way the usual divisibility conditions on the system defining matrices can be recovered. The results apply to the class of delay-differential systems with (in)commensurable delays. In this particular application, the ring extension of ? is characterized explicitly.

Abstraction of Biochemical Reaction Systems on Polytopes

Analysis of the dynamic behavior of large-scale biochemical reaction systems can be facilitated by abstraction followed by model checking. A biochemical reaction system can be approximated by a multi-affine system or an affine system on a rectangle. Either of these systems can be abstracted to an automaton. Model checking can then be employed to determine whether the dynamic behavior of the automaton satisfies specific properties. A relation between the system and its abstraction is proved; it is an over-approximation: any discrete state trajectory of the abstraction of the continuous state trajectory is contained in the automaton but the automaton may contain more behavior.

Input-output equations and observability for polynomial delay systems

This paper discusses and strengthens a result by Fliess (1988) about input-output equations for polynomial systems with time delays. The proof given is more detailed and it opens the way for constructive methods for determining the input-output behavior. A method based on Grobners approach is described in detail. Furthermore, some connections with observability are exploited.<<ETX>>

Control of Piecewise-Affine Hybrid Systems - Extended abstract

The character of this paper is an extended abstract of the approach developed by the authors for control of piecewise-affine hybrid systems. Control of hybrid systems is a wide ranging research topic with many theoretical problems. The authors have therefore decided early on to restrict attention to piecewise-affine hybrid systems on polytopes. The approach developed includes for affine systems on polytopes sufficient conditions for the problems (1) leaving the polytope in finite time through a prespecified set of facets; (2) remaining inside a particular polytope forever; and (3) converging to a fixed state inside a particular polytope. For piecewise-affine hybrid system on polytopes a procedure for reachability and control synthesis is proposed consisting of (1) for each discrete state the computation of all possible exit facets and exit events; (2) abstraction of the continuous dynamics to an automaton; and (3) the search of a path in the automaton from an initial discrete state to a target discrete state. Open research issues for control of hybrid systems are stated.

Behavioral controllability of time-delay systems with incommensurable delays

Abstract In this paper it is shown that for time-delay systems in pseudo-state form with (in)commensurable time-delays, the notion of behavioral controllability and spectral controllability are equivalent. This yields a generalized Haut us test for behavioral controllability of time-delay systems. The result is obtained by considering delay systems as convolution systems, and relies on a result of Hormander, that generalizes the Corona Theorem to the Paley-Wiener algebra of holomorphic functions of exponential type that are polynomially bounded on the real axis.