On Invertibility of Switched Linear Systems

Abstract

We address the invertibility problem for switched systems, which is the problem of recovering the switching signal and the input uniquely given an output and an initial state. In the context of hybrid systems, this corresponds to recovering the discrete state and the input from partial measurements of the continuous state. In solving the invertibility problem, we introduce the concept of singular pairs for two systems. We give a necessary and sufficient condition for a switched system to be invertible, which says that the individual subsystems should be invertible and there should be no singular pairs. When the individual subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included.

High-order sliding mode observers for nonlinear autonomous switched systems with unknown inputs

“…On the other hand, for the unknown operating mode case, in Bejarano and Pisano [6], based on a property of strong detectability and using a LMI approach, are designed two state observers for some classes of switched linear systems with unknown inputs. Considering that the output and the initial state are available, in Vu and Liberzon [32] necessary and sufficient conditions for a switched system to be invertible are proposed, i.e. condition for recovering the switching signal and the input uniquely.…”

Robust reconstruction of the discrete state for a class of nonlinear uncertain switched systems

“…the problem of reconstructing the input and switching signal uniquely from given output and initial state, was addressed in [12] for the class of switched nonlinear systems affine in the control variables. In general, the problem of invertibility for switched systems, especially linear switched systems, has received considerable attention [13][14][15][16].…”

Failure detection and reconstruction in switched nonlinear systems

“…The new problem about the invertibility of switched linear systems is addressed initially in [6], where the authors give a necessary and sufficient condition for a switched system to be invertible. The authors' work is particularly based on the classical invertibility of linear systems combined by an algorithm for finding switching signal.…”

“…This new approach will show us that the complexity of the invertibility of the switched nonlinear systems does not depend on the complexity of the systems, that is to say that: complex systems complex computation of the invertibility. Note that other approaches based on finding change of variables and diffeomorphism to compute the inverse dynamic, are proposed in [6] and recently in [9] for linear switched systems and nonswitched systems respectively (based principally on the results given in [10][11][12]). …”