Abstract-We apply an operator splitting technique to a generic linear-convex optimal control problem, which results in an algorithm that alternates between solving a quadratic control problem, for which there are efficient methods, and solving a set of single-period optimization problems, which can be done in parallel, and often have analytical solutions. In many cases, the resulting algorithm is division-free (after some off-line precomputations) and can be implemented in fixed-point arithmetic, for example on a field-programmable gate array (FPGA). We demonstrate the method on several examples from different application areas.Index Terms-Alternating directions method of multipliers (ADMM), convex optimization, embedded control, fixed point algorithms for control, model predictive control (MPC), operator splitting, optimal control.

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A method for certifying exact input trackability for constrained discrete time linear systems is introduced in this paper. A signal is assumed to be drawn from a reference set and the system must track this signal with a linear combination of its inputs. Using methods inspired from robust model predictive control, the proposed approach certifies the ability of a system to track any reference drawn from a polytopic set on a finite time horizon by solving a linear program. Optimization over a parameterization of the set of reference signals is discussed, and particular instances of parameterization of this set that result in a convex program are identified, allowing one to find the largest set of trackable signals of some class. Infinite horizon feasibility of the methods proposed is obtained through use of invariant sets, and an implicit description of such an invariant set is proposed. These results are tailored for the application of power consumption tracking for loads, where the operator of the load needs to certify in advance his ability to fulfill some requirement set by the network operator. An example of a building heating system illustrates the results.

This work presents an algorithmic scheme for solving the infinite-time constrained linear quadratic regulation problem. We employ an accelerated version of a popular proximal gradient scheme, commonly known as the Forward-Backward Splitting (FBS), and prove its convergence to the optimal solution in our infinite-dimensional setting. Each iteration of the algorithm requires only finite memory, is computationally cheap, and makes no use of terminal invariant sets; hence, the algorithm can be applied to systems of very large dimensions. The acceleration brings in 'optimal' convergence rates O(1/k 2 ) for function values and O(1/k) for primal iterates and renders the proposed method a practical alternative to model predictive control schemes for setpoint tracking. In addition, for the case when the true system is subject to disturbances or modelling errors, we propose an efficient warm-starting procedure, which significantly reduces the number of iterations when the algorithm is applied in closed-loop. Numerical examples demonstrate the approach. Index TermsConstrained LQR, Alternating minimization, Operator splitting I. INTRODUCTION An important extension of the famous result of [1] on the closed form solution of the infinitehorizon linear quadratic regulation (LQR) problem is the case where input and state variables are constrained. This problem is computationally significantly more difficult and has been by and large addressed only approximately. A prime example of an approximation scheme is model predictive control (MPC), which approximates the infinite-time constrained problem by a finitetime one. Stability of such MPC controllers is then typically enforced by adding a suitable terminal constraint and a terminal penalty. The inclusion of a terminal constraint limits the feasible region of the MPC controller, and, consequently, the region of attraction of the closedloop system. In practical applications, this problem is typically overcome by simply choosing a "sufficiently" long horizon based on process insight (e.g., dominant time constant). Closed-loop behavior is then analyzed a posteriori, for instance by exhaustive simulation.There have been few results addressing directly the infinite-horizon constrained LQR (CLQR) problem. Among the most well-known efforts are the works

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