PRESAS: Block‐structured preconditioning of iterative solvers within a primal active‐set method for fast model predictive control

Abstract

Summary Model predictive control (MPC) for linear dynamical systems requires solving an optimal control structured quadratic program (QP) at each sampling instant. This article proposes a primal active‐set strategy, called PRESAS, for the efficient solution of such block‐sparse QPs, based on a preconditioned iterative solver to compute the search direction in each iteration. Rank‐one factorization updates of the preconditioner result in a per‐iteration computational complexity of 𝒪(Nm2), where m denotes the number of state and control variables and N the number of control intervals. Three different block‐structured preconditioning techniques are presented and their numerical properties are studied further. In addition, an augmented Lagrangian based implementation is proposed to avoid a costly initialization procedure to find a primal feasible starting point. Based on a standalone C code implementation, we illustrate the computational performance of PRESAS against current state of the art QP solvers for multiple linear and nonlinear MPC case studies. We also show that the solver is real‐time feasible on a dSPACE MicroAutoBox‐II rapid prototyping unit for vehicle control applications, and numerical reliability is illustrated based on experimental results from a testbench of small‐scale autonomous vehicles.

A Dual Active-Set Solver for Embedded Quadratic Programming Using Recursive LDL$^{T}$ Updates

“…Remark 9 (Exploiting sparsity): We want to stress that our focus here is on solving dense QPs from the condensed MPC problem formulation. When solving large problems, other solvers that use sparse formulations [6], [11], [24], [28], [29] might be more efficient. If the ideas herein can be modified to exploit sparsity is a topic for future research.…”

Sequential Quadratic Programming Algorithm for Real-Time Mixed-Integer Nonlinear MPC

“…where z includes all optimization variables and the index set I denotes the integer variables. Next, we summarize the main ingredients of the BB-ASIPM solver [13] that uses a B&B Fig. 5: Branch-and-bound (B&B) method as a binary search tree.…”

“…method with reliability branching and warm starting [16], block-sparse presolve techniques [13], early termination and infeasibility detection [17] within a fast convex QP solver [18].…”

“…Recent work [13] indicates that, by exploiting the particular structure of the MIOCPs, real-time solvers can achieve performance comparable to commercial tools, e.g., GUROBI [14] and MOSEK [15], especially for small to medium-scale problems. Therefore, we use the tailored BB-ASIPM solver [13], using a branch-and-bound (B&B) method with reliability branching and warm starting [16], block-sparse presolve techniques [13], early termination and infeasibility detection [17] within a fast convex quadratic programming (QP) solver based on an active-set interior point method (ASIPM) [18].…”

A combined first‐ and second‐order approach for model predictive control

“…By using warm start and the property of the parametric nature of MPC, the so-called "online active-set method" 6 performs better compared with a conventional QP solver. Methods [7][8][9] that exploit the banded structure of the Karush-Kuhn-Tucker (KKT) matrix of the underlying QP result in Riccati-recursion computations, which lead to computational complexities that are linear in the prediction horizon. It is shown in Reference 10 that the banded structure is a special case of a chordal structure, and the Riccati recursion can be summarized in the so-called message passing algorithm 11 for chordal problems.…”