Characterisation Of Receding Horizon Control For Constrained Linear Systems

Abstract

This paper characterises the geometric structure of receding horizon control (RHC) of linear, discrete-time systems, subject to a quadratic performance index and linear constraints. The geometric insights so obtained are exploited to derive a closed-form solution for the case where the total number of constraints is less than or equal to the number of degrees of freedom, represented by the number of control moves. The solution is shown to be a partition of the state space into regions for which an analytic expression is given for the corresponding control law. Both the regions and the control law are characterised in terms of the parameters of the open-loop optimal control problem that underlies RHC and can be computed off line. The solution for the case where the total number of constraints is greater than the number of degrees of freedom is addressed via an algorithm that iteratively uses the off-line solution and avoids on-line optimisation.

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

“…Remark 6.4 It is shown in [4][5][6] that in the case of non-degeneracies, the optimal cost function of a parametric quadratic programming problem represents a strictly convex, piecewise quadratic function (this can also be proven using dynamic programming).…”

“…Recall (see [4][5][6][7][8]) that a parametric linear/quadratic programming problem is defined as follows with respect to…”

“…In control theory, this structure of control laws appeared in the last decade as an approximation of the classical nonlinear control laws with respect to a predefined error [1][2][3]. Then, it is shown that this PWA structure is inherited by the exact optimal solution of a linear model predictive control (MPC) problem with respect to a linear/quadratic cost function [4][5][6][7][8][9].…”

Inf–sup control of discontinuous piecewise affine systems

“…which is to be minimized with respect to u for all values of x ∈ X := [ −5, 7], that is, 7]; clearly the infimum J * (0) = 0 cannot be attained, that is, u * such that f (0, u * ) = J * (0) = 0. In fact, a minimum does not exist for any x ∈ [−5, 2).…”

“…[2,3] and references therein. Parametric programming has had a resurgence of interest recently due to the observation that explicit control laws for some model predictive control problems [4] can be obtained by viewing the initial state as a vector of parameters [5][6][7]. As researchers tried to characterize the solution to more difficult optimal control problems (piecewise affine systems, uncertain systems, nonlinear systems etc.…”