Optimal control of constrained, piecewise affine systems with bounded disturbances

Abstract

Finite horizon optimal control of piecewise affine systems with a piecewise affine (1-norm or ∞-norm) stage cost and terminal cost is considered. Provided the respective constraint sets are given as the unions of polyhedra, it is shown that the partial value functions and partial optimal control laws are piecewise affine on a polyhedral cover of the set of states that can be steered, by an admissible control policy, to a terminal set of states in a finite number of steps. Existing results only consider the case of systems without disturbances, or systems with disturbances that are independent of the state and input. This paper extends these results to the case where the disturbance is dependent on the state and input.

Inf–sup control of discontinuous piecewise affine systems

“…We recall the problem setup from [11,12], which was treated in detail in [15] for PWA systems with state-and input-dependent disturbances.…”

“…Recently, methods for computing explicit control laws for discrete-time piecewise affine (PWA) systems with constraints have been reported in the control literature [12,15,16,[20][21][22][23]. There are not introduce gaps in the domain of the state update equation, we do not assume that a solution to the optimal control problem exists, and the state update equation is not transformed into a difference inclusion, and thus, the dynamic programming approach is relatively simple from the theoretical point of view.…”

“…The methods [9,10] have served as the foundation for a large number of publications (see e.g. [12][13][14][15][16][17][18][19]) and will also be utilized in this paper.…”

“…As researchers tried to characterize the solution to more difficult optimal control problems (piecewise affine systems, uncertain systems, nonlinear systems etc. ), several variations of the dynamic programming approach originally proposed by Bellman [8] has been employed [9][10][11][12][13][14][15][16][17][18][19]. The dynamic programming approach for inf-sup (or more commonly min-max) control was introduced for linear systems as early as [9] and in a more general framework in [10].…”

Efficient Evaluation of Piecewise Control Laws Defined Over a Large Number of Polyhedra

“…For piecewise affine systems the constrained finite time optimal control (CFTOC) problem can be solved by means of multi-parametric programming [5], [7], [1], [21] and the resulting solution is a time-varying piecewise affine state feedback control law. If the solution to the CFTOC problem is used in a receding horizon control [26], [23] strategy (or model predictive control (MPC)) the time-varying PWA state feedback control law becomes time-invariant and can serve as a control 'look-up table' on-line, thus enabling receding horizon control to be used for fast sampled systems.…”