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Publication# Approximate explicit MPC using bilevel optimization

Abstract

A linear quadratic model predictive controller (MPC) can be written as a parametric quadratic optimization problem whose solution is a piecewise affine (PWA) map from the state to the optimal input. While this `explicit solution' can offer several orders of magnitude reduction in online evaluation time in some cases, the primary limitation is that the complexity can grow quickly with problem size. In this paper we introduce a new method based on bilevel optimization that allows the direct approximation of the non-convex receding horizon control law. The ability to approximate the control law directly, rather than first approximating a convex cost function leads to simple control laws and tighter approximation errors than previous approaches. Furthermore, stability conditions also based on bilevel optimization are given that are substantially less conservative than existing statements.

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Ontological neighbourhood

Model predictive control

Model predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints. It has been in use in the process industries in chemical plants and oil refineries since the 1980s. In recent years it has also been used in power system balancing models and in power electronics. Model predictive controllers rely on dynamic models of the process, most often linear empirical models obtained by system identification.

Linear–quadratic regulator

The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem. One of the main results in the theory is that the solution is provided by the linear–quadratic regulator (LQR), a feedback controller whose equations are given below. LQR controllers possess inherent robustness with guaranteed gain and phase margin, and they also are part of the solution to the LQG (linear–quadratic–Gaussian) problem.

Optimal control

Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure.

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