Estimation

Moving Horizon Estimation of Hybrid Systems

While state estimation techniques for linear systems are very well explored (Luenberger observer, Kalman filter), this is not true for hybrid systems. Since linear filters may exhibit bad performance or even instability when applied to hybrid systems, other methods are needed for hybrid systems. Optimal state estimation is the dual problem to optimal control, similar methods may thus be applied to both.

The system classes considered are, due to their equivalence, Mixed Logical Dynamical (MLD) and Piecewise Affine (PWA) systems. For these systems, an infinite horizon optimization problem cannot be used since the problem is of infinite size. Moving Horizon Estimation (MHE) only considers a finite number of past samples; hence, the optimization problem is solvable. However, convergence of the state estimates to the real value is not guaranteed in general. We developed an algorithm which provides such guarantees. A dynamic programming approach is used and conditions are given for choosing the arrival cost (which is the dual of the cost-to-go in control) and the cost function, such that convergence is guaranteed.

external page G. Ferrari-Trecate, D. Mignone, and M. Morari, "Moving Horizon Estimation for Hybrid Systems". In Proceedings of the American Control Conference, pp. 1684-1688, Chicago, IL, USA, October 2002.

The MHE optimization problem is stated as a mixed-integer quadratic program (MIQP). Such problems can be solved using branch-and-bound (B&B) strategies, where the tree spanned by the binary variables is searched in an appropriate way. We developed an efficient search algorithm which performs well on MIQPs arising from both controller synthesis- and hybrid estimation problems. It both reduces search time for the global minimum and provides good local minima after a short number of iterations.

external page A. Bemporad, D. Mignone, and M. Morari, "An Efficient Branch and Bound Algorithm for State Estimation and Control of Hybrid Systems". In Proceedings of the European Control Conference, Karlsruhe, Germany, 1999.  

Fault Detection

Fault detection is the task of detecting and isolating faults in technical processes from measured input and output data. For our purposes we apply model-based detection algorithms. In model-based detection we use a mathematical model of the dynamical system under investigation. The model incorporates knowledge of the faultless and the faulty system behavior.

We apply moving horizon estimation to detect the fault. As a system model we consider Mixed Logical Dynamical systems. In the MLD framework the occurrence of faults is modeled by unmeasured binary disturbances. The dynamics of the system in the presence of a fault is assumed known. Thus the original MLD system model is extended by including the faults, i.e. binary disturbance, input disturbance, and output disturbance. We denote this extended model as the mixed logical dynamic fault (MLDF) form.

A moving horizon estimator for the MLDF can be formulated to determine the unmeasured variables and thereby determine the fault-state of the system.

D. Mignone, "Control and Estimation of Hybrid Systems with Mathematical Optimization" (Chapters 3.2 and 4.4). PhD Dissertation, ETH Zürich, Switzerland, January 2002.