Explicit sub-optimal linear quadratic regulation with state and input constraints

Abstract

Optimal feedback solutions to the in nite horizon LQR problem with state and input constraints based on receding horizon real-time quadratic programming are well known. In this paper we develop an explicit solution to the same problem, eliminating the need for realtime optimization. A suboptimal strategy, based on a suboptimal choice of a nite horizon and imposing additional limitations on the allowed switching between active constraint sets on the horizon, is suggested in order to address the computer memory and processing capacity requirements of the explicit solution. It is shown that the resulting feedback controller is piecewise linear, and the piecewise linear structure is explored and exploited for computational analysis of stability and performance of the suboptimal constrained LQR. The piecewise linear structure can also be exploited for e cient real-time implementation of the controller.

Evaluation of piecewise affine control via binary search tree

“…An algorithm to solve the mp-QP is also provided, however, a more efficient algorithm is developed in (Tøndel et al, 2001). An alternative solution strategy is given in (Johansen et al, 2000), where pre-determination of a small set of sampling instants where the active set is allowed to change gives a suboptimal solution. Sub-optimality of mp-QP is also introduced in (Bemporad and Filippi, Conditionally accepted for publication with minor revisions) by adding small slacks to the optimality conditions, and in (Johansen and Grancharova, 2002), by imposing an orthogonal structure to the state space partitioning.…”

Explicit model predictive control of gas–liquid separation plant via orthogonal search tree partitioning

“…It was shown that the solution (the control input) has an explicit representation as a piecewise linear (PWL) state feedback on a polyhedral partition of the state space, see also [2,8,11,12], and they develop an mp-QP algorithm to compute this function.…”

Design of robust explicit model predictive controller via orthogonal search tree partitioning

“…In [1] it was recognized that the constrained linear MPC problem is a multi-parametric quadratic program (mp-QP), when the state is viewed as a parameter to the problem. It was shown that the solution (the control input) has an explicit representation as a piecewise linear (PWL) state feedback on a polyhedral partition of the state space, see also [2,8,14,15], and they develop an mp-QP algorithm to compute a representation of this function. Some of these approaches have been further extended to ensure robustness of the explicit MPC controllers against disturbances [3,11,13].…”