E. De Santis

On observability and detectability of continuous-time linear switching systems

The notion of observability and detectability for a particular class of hybrid systems, linear continuous-time switching systems, is investigated. We compare some of the definitions of observability previously offered and we analyze their drawbacks. A novel definition of observability is proposed corresponding to the possibility of reconstructing the state of the system from the knowledge of the discrete and continuous outputs and inputs. The notion of detectability is also introduced. Sufficient and necessary conditions for these properties to hold for switching systems are presented.

Computation of maximal safe sets for switching systems

The problem of determining maximal safe sets and hybrid controllers is computationally intractable because of the mathematical generality of hybrid system models. Given the practical and theoretical relevance of the problem, finding implementable procedures that could at least approximate the maximal safe set is important. To this end, we begin by restricting our attention to a special class of hybrid systems: switching systems. We exploit the structural properties of the graph describing the discrete part of a switching system to develop an efficient procedure for the computation of the safe set. This procedure requires the computation of a maximal controlled invariant set. We then restrict our attention to linear discrete-time systems for which there is a wealth of results available in the literature for the determination of maximal controlled invariant sets. However, even for this class of systems, the computation may not converge in a finite number of steps. We then propose to compute inner approximations that are controlled invariant and for which a procedure that terminates in a finite number of steps can be obtained. A tight bound on the error can be given by comparing the inner approximation with the classical outer approximation of the maximal controlled invariant set. Our procedure is applied to the idle-speed regulation problem in engine control to demonstrate its efficiency.

Constrained control of switching systems: a positive invariant approach

This paper presents conditions for the stabilization of switching discrete-time linear systems with constrained control by using a positive invariance approach. A state feedback control law is used to construct the stabilizing controller. A numerical example is presented to illustrate the technique.

Controlled invariance and feedback laws

We consider controlled invariance for cones, translated cones, polyhedra and various special polyhedral structures. For any polyhedron, if controlled invariance occurs, then all and nothing but the admissible controls can be obtained by an inequative feedback controller. For each special polyhedral structure we compare this feedback controller against piecewise affine, linear and affine feedback controllers.

Design of Luenberger-Like Observers for Detectable Switching Systems

We address the problem of reconstructing the state of a switching system from partial knowledge of the dynamics of the plant. Under the assumption that the discrete state of the switching system is known, we show that if the conditions ensuring detectability are satisfied, a suitable Luenberger-like observer solves the problem of estimating the continuous component of the stateWe address the problem of reconstructing the state of a switching system from partial knowledge of the dynamics of the plant. Under the assumption that the discrete state of the switching system is known, we show that if the conditions ensuring detectability are satisfied, a suitable Luenberger-like observer solves the problem of estimating the continuous component of the state

Observability of Internal Variables in Interconnected Switching Systems

Given a pair of linear switching systems, we study (conditions that ensure the observability of internal variables arising in the input-output interconnection of the switching systems on the basis of the knowledge of input and hybrid output of the interconnected system. Some sufficient and necessary conditions are provided for the reconstruction of hybrid states and of (hybrid) latent variables associated to the interconnected system

Digital control of continuous-time switching systems with safety constraints

Relations between safe sets for a continuous-time switching system S and an appropriate discrete-time switching system associated with S are established. Exploiting these relations, the maximal controlled invariant set for the continuous-time switching system can be arbitrarily closely approximated by an appropriate controlled invariant set for the discretized switching system. Moreover, the safety control problem in a continuous time domain is reduced to a safety problem in a discrete time domain.

OBSERVABILITY OF DISCRETE TIME LINEAR SWITCHING SYSTEMS1

Abstract In previous work, we defined observability (detectability) for continuous-time linear switching systems, as the possibility of reconstructing (respectively, asymptotically reconstructing) the hybrid state of the system from the knowledge of the hybrid outputs for a suitable choice of the control input. In this paper, we address observability for discrete-time linear switching systems. While in continuous-time linear switching systems detection is possible in an arbitrarily small amount of time, in the discrete time case it requires a finite, non-zero number of instants. We derive a sufficient testable condition for the reconstruction of the hybrid state.

A structural approach to the control of switching systems with an application to automotive engines

We consider switching dynamical systems, a particular class of hybrid systems, where transitions between different configurations are determined by external uncontrollable events. The control problem solved consists of maintaining the output of the systems in a given subset of the output space, independently of the transitions and of the presence of disturbances. A structural approach is presented, which allows to simplify considerably the procedure for the determination of the solution available in literature. The theory is applied to idle-speed regulation.

Positiveness of dynamic systems with nonpositive coefficient matrices

This note gives a dynamical solution for linear dynamical feasibility problems, and applies this result to a generalization of the concept of positive systems. More specifically, we require the existence of a positive state and/or output evolution for any initial state in a given set. We give conditions for various sets of initial states. Finally, by means of an example, we illustrate our results. >