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Efficient Sparse ADMM-based Solver for Linear Model Predictive Control with Terminal Quadratic Constraint


Core Concepts
The article presents a sparse ADMM-based solver for linear model predictive control (MPC) problems with a terminal quadratic constraint. The proposed approach directly handles the quadratic constraint without reformulating it as a second-order cone constraint, retaining the simple matrix structures that enable efficient computation.
Abstract
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.
Stats
The article does not provide any explicit numerical data or statistics to support the claims. The performance of the proposed solver is evaluated through comparative case studies and implementation on an embedded system.
Quotes
"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." "The results demonstrate the computational advantages of the proposed approach, especially for small-scale systems." "The results show the suitability of the solver for embedded applications, with low computation times and iteration counts."

Deeper Inquiries

How can the proposed solver be extended to handle multiple ellipsoidal constraints or more general nonlinear constraints

The proposed solver can be extended to handle multiple ellipsoidal constraints or more general nonlinear constraints by modifying the formulation of the optimization problem in the ADMM algorithm. For multiple ellipsoidal constraints, each constraint can be incorporated into the objective function and constraints of the optimization problem. The Lagrangian function in the ADMM algorithm would then include terms corresponding to each ellipsoidal constraint, and the projection steps would need to be modified accordingly. Similarly, for more general nonlinear constraints, the Lagrangian function would need to be adjusted to incorporate these constraints, and the projection steps would need to be adapted to handle the specific form of the constraints. By appropriately formulating the optimization problem and updating the ADMM algorithm, the solver can efficiently handle multiple ellipsoidal constraints or more general nonlinear constraints.

What are the theoretical guarantees (e.g., convergence rate, optimality) of the proposed ADMM-based solver, and how do they compare to other optimization approaches for this class of problems

The theoretical guarantees of the proposed ADMM-based solver include convergence to a solution of the optimization problem and optimality under certain conditions. The ADMM algorithm is known to converge under mild assumptions, such as convexity of the objective function and constraints. The convergence rate of ADMM is typically linear or sublinear, depending on the problem structure and algorithm parameters. In the context of the linear MPC problem subject to terminal quadratic constraints, the proposed solver leverages the structure of the problem to achieve efficient convergence. Compared to other optimization approaches, such as interior-point methods or SQP solvers, the ADMM-based solver may offer advantages in terms of computational efficiency and memory requirements for certain classes of problems. While the theoretical guarantees of the proposed solver are strong, empirical validation and performance comparisons with other solvers are essential to assess its practical effectiveness in real-world applications.

What are the potential applications and benefits of the proposed solver beyond the specific MPC problem considered in the article, and how could it be integrated into a broader range of optimization-based control and decision-making frameworks

The proposed solver has potential applications beyond the specific MPC problem considered in the article. It can be integrated into a broader range of optimization-based control and decision-making frameworks, such as distributed optimization, machine learning, and reinforcement learning. The sparse nature of the solver makes it suitable for embedded systems and real-time applications where computational efficiency is crucial. By extending the solver to handle multiple constraints and nonlinearities, it can be applied to a wide range of optimization problems in various domains, including robotics, finance, energy systems, and healthcare. The benefits of the proposed solver include its simplicity, scalability, and ability to handle complex constraints efficiently. Integrating the solver into optimization frameworks can enhance the performance of control systems, decision-making processes, and resource allocation in diverse applications.
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