Computation of Observability Regions for Piecewise Affine Systems: A Projection-Based Algorithm

Abstract

In this paper we consider the problem of computing sets of observable states for discrete-time, piecewise affine systems. When the maximal set of observable states is fulldimensional, we provide an algorithm for reconstructing it up to a zero measure set. The core of the method is a quantifier elimination procedure that, in view of basic results on piecewise linear algebra, can be performed via the projection of polytopes on subspaces. We also provide a necessary condition on the minimal length of the observability horizon in order to expect a full-dimensional set of observable states. Numerical experiments highlight that the new procedure is considerably faster than the one proposed in [1].

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“…O O T is the union of finitely many not necessarily closed polytopes (Gati and Ferrari-Trecate 2005). This property can be proved by exploiting general results in piecewise linear algebra (Sontag 1981(Sontag , 1982.…”

“…To be more precise, as shown in Gati and Ferrari-Trecate (2005), rankðC i Þ < n implies that all the states in the interior of X i are unobservable but no conclusion can be drawn for the points on the boundary of X i . However, if rankðC i Þ < n, the index i is not added to I and this may result in discarding some observable states in the 498 G. Ferrari-Trecate and M. Gati boundary of X i .…”

“…As detailed in Gati and Ferrari-Trecate (2005), when the sets Q i are closed, a possible way for computing the matrices and the regions in (4) is to resort to the mixed-logical dynamical representation of system (1) and then use an algorithm based on mode enumeration. For details about the conversion of mixed-logic dynamical models into PWA ones (and vice-versa) we defer the reader to Bemporad et al (2000) and Geyer et al (2003).…”

“…By using tools of piecewise linear algebra (Sontag 1981(Sontag , 1982 and the system representation (4), in (Gati and Ferrari-Trecate 2005) we derived the following algorithm for computing a set B T of unobservable states that is equal to B B T up to a set of zero measure.…”

“…Then, we exploit a recently developed algorithm for computing the maximal set of states (up to a set of zero measure) that can be observed on the basis of inputs and outputs collected over a finite time-horizon (Gati and Ferrari-Trecate 2005). In particular, the PWA structure of the model guarantees that this set is a finite collection of polytopes.…”