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Unformatted text preview: Fast Model Predictive Control Using Online Optimization Yang Wang * Stephen Boyd ** * Stanford University, Stanford, CA 94305 (e-mail: email@example.com) ** Stanford University, Stanford, CA 94305 (e-mail: firstname.lastname@example.org) Abstract: A widely recognized shortcoming of model predictive control (MPC) is that it can usually only be used in applications with slow dynamics, where the sample time is measured in seconds or minutes. A well known technique for implementing fast MPC is to compute the entire control law offline, in which case the online controller can be implemented as a lookup table. This method works well for systems with small state and input dimensions (say, no more than 5), and short time horizons. In this paper we describe a collection of methods for improving the speed of MPC, using online optimization. These custom methods, which exploit the particular structure of the MPC problem, can compute the control action on the order of 100 times faster than a method that uses a generic optimizer. As an example, our method computes the control actions for a problem with 12 states, 3 controls, and horizon of 30 time steps (which entails solving a quadratic program with 450 variables and 1260 constraints) in around 5msec, allowing MPC to be carried out at 200Hz. 1. INTRODUCTION In classical model predictive control (MPC), the control action at each time step is obtained by solving an online optimization problem. With a linear model, polyhedral constraints, and a quadratic cost, the resulting optimiza- tion problem is a quadratic program (QP). Solving the QP using general purpose methods can be slow, and this has traditionally limited MPC to applications with slow dynamics, with sample times measured in seconds or minutes. One method for implementing fast MPC is to compute the solution of the QP explicitly as a function of the initial state (Bemporad et al. , ndel et al. ); the control action is then implemented online in the form of a lookup table. The major drawback here is that the number of entries in the table can grow exponentially with the horizon, state, and input dimensions, so that explicit MPC can only be applied reliably to small problems (where the state dimension is no more than around 5). In this paper we describe a collection of methods that can be used to greatly speed up the computation of the control action in MPC, using online optimization. Some of the ideas have already been noted in literature, and here we will demonstrate that when used in combination, they allow MPC to be implemented orders of magnitude faster than with generic optimizers. Our main strategy is to exploit the structure of the QPs that arise in MPC (Boyd and Vandenberghe , Wright ). It has already been noted that with an appropriate variable re-ordering, the interior-point search direction at each step can be found by solving a block tridiagonal sys- tem of linear equations. Exploiting this special structure, a problem with state dimension...
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