The article addresses the problem of efficiently solving linear MPC problems with a terminal quadratic constraint. Typically, linear MPC problems use a polyhedral terminal constraint, leading to a quadratic programming (QP) problem. However, an ellipsoidal terminal constraint may be desirable, resulting in a quadratically-constrained quadratic programming (QCQP) problem, which is generally more computationally demanding to solve.
The authors propose a sparse ADMM-based solver that directly handles the terminal quadratic constraint without reformulating it as a second-order cone (SOC) constraint. The key idea is to modify the ADMM equality constraints in a way that allows for an explicit solution of the projection step related to the ellipsoidal constraint. This retains the simple matrix structures exploited by a previously proposed sparse MPC solver, leading to a computationally efficient approach.
The article provides a detailed description of the proposed solver, including the steps of the ADMM algorithm and the explicit solution for the projection onto the ellipsoid. Two case studies are presented:
Comparison with other state-of-the-art solvers: The proposed solver is compared against several alternatives, including SOC-based approaches and interior-point methods. The results demonstrate the computational advantages of the proposed approach, especially for small-scale systems.
Implementation on an embedded system: The proposed solver is implemented on a Raspberry Pi 4 to control a 12-state, 6-input chemical plant. The results show the suitability of the solver for embedded applications, with low computation times and iteration counts.
The article concludes that the proposed sparse ADMM-based solver is a viable and efficient option for solving linear MPC problems with terminal quadratic constraints, particularly in embedded systems with limited computational resources.
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