The work presented in this paper is concerned with the identification of switched linear systems from input-output data. The main challenge with this problem is that the data are available only as a mixture of observations generated by a finite set of different interacting linear subsystems so that one does not know a priori which subsystem has generated which data. To overcome this difficulty, we present here a sparse optimization approach inspired by very recent developments from the community of compressed sensing. We formally pose the problem of identifying each submodel as a combinatorial ℓ0 optimization problem. This is indeed an NP-hard problem which can interestingly, as shown by recent literature, be relaxed into a (convex) ℓ1-norm minimization problem. We present sufficient conditions for this relaxation to be exact. The whole identification procedure allows us to extract the parameter vectors (associated with the different subsystems) one after another without any prior clustering of the data according to their respective generating-submodels. Some simulation results are included to support the potentialities of the proposed method.

The paper presents realization theory of discrete-time linear switched systems (abbreviated by DTLSSs). We present necessary and sufficient conditions for an input-output map to admit a discrete-time linear switched state-space realization. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observability. Further, we prove that minimal realizations are unique up to isomorphism. We also discuss algorithms for converting a linear switched system to a minimal one and for constructing a state-space representation from input-output data. The paper uses the theory of rational formal power series in non-commutative variables.

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