G. Ferrari-Trecate, D. Mignone, and M. Morari. Moving horizon estimation for piecewise affine systems. AIChE annual meeting - "Exchanging Ideas for Innovation" , 2000. Los Angeles, California, US, 12-17 November.
Many practical control, estimation and fault detection problems involvehybrid systems, here losely defined as systems involving both continuousand discrete variables. Various approaches have been proposed for modelinghybrid systems. Often engineering systemsinclude ``logic'' components (eg. if-then-else rules, finite state machine, etc.) which are conveniently described via propositional logic. Moreover, in addition to a quantitative system description there might be some available qualitative information about the behavior of the system, for instance in terms of heuristic knowledge.Recently it was shown (Bemporad and Morari, 1999) that expressing logicalpropositions in the form of linear constraints on integer variables leadsto a powerful modeling framework, the so called mixed logic dynamical (MLD)form. It allows to describe a broad number of important classes of systems, likepiecewise linear systems, systems with mixed discrete/continuous inputs andstates, and many others more (Bemporad and Morari, 1999). Theframework permits to include and prioritize constraints, and incorporate heuristic rules inthe description of the model.In this work we first show that the MLD form can be used as a new tool tomodel systems with faults. Next we formulate the moving horizon estimation (MHE)problem for MLD systems, which can be considered dual to (receding horizon) model predictive control. At each time step we solve a least squaresestimation problem over a finite horizon extending backwards from the current time. The resulting optimization problem is a mixed-integer quadratic program (MIQP), for which efficient solvers exist. We propose to use thismoving horizon estimation formulation to estimate states and faults of hybrid systems in the MLD form. Next we propose a state smoothing algorithm based on MHE. We provide sufficient conditions on the time horizon and the additional penalties in the MHE cost function to guarantee asymptotic convergence of the MHE scheme. Moreover we propose an algorithm for the computation of the additional penalties that allows to implement MHE by solving Mixed-Integer Quadratic Programs. Finally we consider some aspects of the mathematical problems involved in the estimation procedure.