Closed-form preconditioner design for linear predictive control

Abstract

Model Predictive Control (MPC) with linear models and constraints is extensivelybeing utilized in many applications, many of which have low power requirements and limitedcomputational resources. In these resource-constrained environments, many designers chooseto utilize simple iterative first-order optimization solvers, such as the Fast Gradient Method.Unfortunately, the convergence rate of these solvers is affected by the conditioning of the problemdata, with ill-conditioned problems requiring a large number of iterations to solve. In order toreduce the number of solver iterations required, we present a simple closed-form method forcomputing an optimal preconditioning matrix for the Hessian of the condensed primal problem.To accomplish this, we also derive spectral bounds for the Hessian in terms of the transferfunction of the predicted system. This preconditioner is based on the Toeplitz structure of theHessian and has equivalent performance to a state-of-the-art optimal preconditioner, withouthaving to solve a semidefinite program during the design phase.