Elena Zattoni

The output regulation problem with stability for linear switching systems: A geometric approach

Abstract This work presents a solution for the output regulation problem with quadratic stability under arbitrary switching in linear switching systems. The extension to other stability requirements, like asymptotic stability in particular, is also considered and it is shown to affect only few specific features of the proposed solution. The main reason is that the geometric approach, which is at the basis of the developed methodology, establishes a neat separation between the structural aspects and the stability aspects of the problem. For the same reason, continuous-time systems and discrete-time systems are given a unified treatment as far as the structural issues are concerned, while different technicalities characterize the discussion of the stability issues.

Disturbance Decoupling with Stability in Continuous-Time Switched Linear Systems Under Dwell-Time Switching

Abstract This work deals with state feedback compensation of disturbance inputs in continuous-time switched linear systems, with the requirement that the closed-loop systems be exponentially stable under switching signals with a sufficiently large dwell-time. Constructive conditions for the problem to be solvable are shown, on the assumption that the given switched linear system has zero initial state. The effects of nonzero initial states are inspected. The theoretical background consists of both classic and novel ideas of the geometric approach, enhanced with notions specifically oriented to switched linear systems.

Model matching problems for switching linear systems

Abstract This paper investigates the problem of designing a feedback compensator to force the response of a plant modeled by a switching linear system to match that of a prescribed, switching linear model, for any choice of the switching law. The problem is stated by considering both the situation in which the state of the model is measurable and that in which it is not. Accordingly, static compensators or, alternatively, dynamic ones will be sought. The additional requirement of asymptotic stability of the compensated system is introduced by reasonably restricting the class of admissible switching laws. Using geometric methods, that extend classic ones to the framework of switching systems, a complete solution, in terms of necessary and sufficient conditions that are algorithmically checkable, is given for matching without stability and for matching with asymptotic stability for a mildly restricted class of plants.

A Geometric Approach to Output Regulation for Linear Switching Systems

Abstract This paper considers the problem of asymptotic output regulation by output dynamic feedback for continuous-time linear switching systems, with the requirement of asymptotic stability of the regulation loop. Using tools and methods of the geometric approach, necessary and sufficient conditions for the existence of solutions, under suitable assumptions, are found. A synthesis procedure is outlined in case a stronger sufficient condition holds. The case of different stability requirements, in particular that of quadratic stability, is discussed.

A constructive condition for inaccessible signal rejection with quadratic stability in discrete-time linear switching systems

This work deals with rejection of inaccessible signals in discrete-time linear switching systems, with the requirement that the compensated system be quadratically stable under arbitrary switching. A constructive condition is provided for devising a switching state feedback achieving zero output for any admissible inaccessible input sequence and any initial state in a certain subspace, while attaining quadratic stability under arbitrary switching of the closed-loop dynamics. The constructive condition is twofold, since structural and stability issues are considered independently of each others. The methodological bases consist of both classic and novel ideas of the geometric approach, enhanced with the notion of quadratic stability under arbitrary switching. The theoretical results are supported by a complete computational framework, developed in Matlab by using the algorithms of the geometric approach and the LMI solvers.

Output-feedback model matching with strong stability in continuous-time switched linear systems

This work deals with the problem of model matching in switched linear systems. On appropriate assumptions, conditions are shown that allow the synthesis of a switched dynamic output feedback capable of achieving model matching with asymptotic stability under restricted switching of both the switched compensator and the closed-loop switched system.

A geometric approach to output regulation for discrete-time switched linear systems

This paper considers the problem of asymptotic output regulation by dynamic feedback for discrete-time switched linear systems, with the requirement of asymptotic stability of the regulation loop. Using the methods of the geometric approach, sufficient conditions for solvability of the problem, under suitable assumptions, are shown. A synthesis procedure is outlined.

Measurable disturbance rejection with stability in continuous-time switched linear systems under dwell-time switching

This work deals with rejection of disturbance inputs accessible for measurement in continuous-time switched linear systems with dwell-time constraints on the switching signals. The measurable disturbance rejection problem is stated as the problem of finding a dynamic feedforward switched compensator achieving zero output and exponential stability of the compensated switched linear system over a class of switching signals with a sufficiently large dwell-time, in the presence of any admissible measurable disturbance input. The synthesis of the compensator is based on a pair of sufficient conditions which respectively address the structural issue and the stabilizability issue. The former condition is expressed in geometric terms as the inclusion of the image of the disturbance input matrix in the sum of the so-called maximal robust controlled invariant subspace and the image of the control input matrix, for all the modes of the given switched system. The second condition is expressed as the exponential stabilizability under dwell-time switching of the internal switched dynamics of the maximal robust controlled invariant subspace.

Dynamic feedforward compensation of measurable signals in discrete-time linear switching systems

This work deals with rejection of signals accessible for measurement in discrete-time linear switching systems, with the requirement that the compensated system be quadratically stable under arbitrary switching. The synthesis of a feedforward dynamic switching compensator is first presented. A combined switching control scheme, also including measurement dynamic feedback, is then introduced with the aim of extending the proposed methodology to switching plants which are quadratically stabilizable by linear state feedback and by linear output injection. The methodology developed is based on the combined use of geometric approach tools and linear matrix inequalities.

A geometric approach to the general autonomous regulator problem in the time-delay framework

Abstract The aim of this paper is to show the applicability of geometric techniques to a regulation problem for linear, time-delay systems. Given a plant whose dynamics equations include delays and an exosystem that generates a reference signal, the problem we consider consists in finding a feedback regulator which guarantees asymptotic stability of the regulation loop and asymptotic command following of the reference signal, for any initial condition of the overall system in the presence of disturbances. By associating to the time-delay plant a corresponding abstract system with coefficients in a ring, it is possible to place our investigation in a finite dimensional algebraic context, where intuition and results obtained in the classical case, that is without delays, may be exploited. In particular, using tools and methods of the geometric approach to systems with coefficients in a ring, sufficient conditions for the solvability of the considered problem are found and a constructive procedure, which works under specific hypotheses, is given.