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This paper proposes to use Nesterov's fast gradient method for the solution of linear quadratic model predictive control (MPC) problems with input constraints. The main focus is on the method's a priori computational complexity certification which consists of deriving lower iteration bounds such that a solution of pre-specified suboptimality is obtained for any possible state of the system. We investigate cold-and warm-starting strategies and provide an easily computable lower iteration bound for cold-starting and an asymptotic characterization of the bounds for warm-starting. Moreover, we characterize the set of MPC problems for which small iteration bounds and thus short solution times are expected. The theoretical findings and the practical relevance of the obtained lower iteration bounds are underpinned by various numerical examples and compared to certification results for a primal-dual interior point method.

Faster, cheaper, and more power efficient optimization solvers than those currently possible using general-purpose techniques are required for extending the use of model predictive control (MPC) to resource-constrained embedded platforms. We propose several custom computational architectures for different first-order optimization methods that can handle linear-quadratic MPC problems with input, input-rate, and soft state constraints. We provide analysis ensuring the reliable operation of the resulting controller under reduced precision fixed-point arithmetic. Implementation of the proposed architectures in FPGAs shows that satisfactory control performance at a sample rate beyond 1 MHz is achievable even on low-end devices, opening up new possibilities for the application of MPC on embedded systems.Index Terms-Embedded systems, optimization algorithms, predictive control of linear systems.

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Abstract-Linear quadratic model predictive control (MPC) with input constraints leads to an optimization problem that has to be solved at every instant in time. Although there exists computational complexity analysis for current online optimization methods dedicated to MPC, the worst case complexity bound is either hard to compute or far off from the practically observed bound. In this paper we introduce fast gradient methods that allow one to compute a priori the worst case bound required to find a solution with pre-specified accuracy. Both warmand cold-starting techniques are analyzed and an illustrative example confirms that small, practical bounds can be obtained that together with the algorithmic and numerical simplicity of fast gradient methods allow online optimization at high rates.