# Discrete Time Constrained Optimal Control

Author(s):F. Borrelli |
Conference/Journal:vol. Diss. ETH Nr. 14666 |

Abstract:In this thesis we address the optimal control problem and receding horizon control for discrete-time systems. We extend the theory of the Linear Quadratic Regulator in several important directions. In particular, we study the optimal control problem with linear constraints on states and inputs, consider different norms in the objective function and different classes of dynamical systems with and without uncertainty. We use the following approach to establish the properties of the optimal control law and the value function. We formulate the finite time optimal control problems as mathematical programs where the input sequence is the optimization vector. Depending on the dynamical model of the system, the nature of the constraints, and the cost function used, a different mathematical program is obtained. The current state of the dynamical system enters the cost function and the constraints as a parameter that affects the solution of the mathematical program. We study the structure of the solution of the mathematical program as this parameter changes. We focus on finite time optimal control problems for discrete-time linear and piecewise affine systems. We consider cost functions based on $2$, $1$ and $infty$ norms and we assume polyhedral constraints on inputs and states. We demonstrate that the solution to all these optimal control problems can be expressed as a piecewise affine state feedback law. In particular, we show that for linear systems with polyhedral constraints on inputs and states and any of the aforementioned norms in the cost function, the optimal control law is affine over polyhedral sets and the value function is convex. In the general case of piecewise-affine dynamics, the optimal control law is shown to be affine over non-convex and disconnected sets. Along with the analysis of the solution properties, we also present algorithms that efficiently compute the optimal control law for all the considered cases. For discrete-time linear systems with constraints, complexity reduction and robustness of the optimal control laws are also addressed. Specifically, we propose new algorithms that reduce the storage demands and computational complexity related to the evaluation of the piecewise affine optimal control laws. Moreover, for uncertain linear systems with polyhedral constraints on inputs and states, we develop an approach to compute the state feedback solution to min-max control problems with a linear performance index. Robustness is achieved against additive norm-bounded input disturbances and/or polyhedral parametric uncertainties in the state-space matrices. We show that the robust optimal control law over a finite horizon is a continuous piecewise affine function of the state vector. The applicability of the theoretical results is demonstrated through experiments on two practical case studies, a mechanical laboratory process and a traction control system developed jointly with Ford Motor Company in Michigan. | Year:2002 |

Type of Publication:(03)Ph.D. Thesis | |

Supervisor:M. Morari | |

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