A Family of High-Performance Solvers for Linear Model Predictive Control

Abstract

In Model Predictive Control (MPC), an optimization problem has to be solved at each sampling time, and this has traditionally limited the use of MPC to systems with slow dynamic. In this paper, we propose an efficient solution strategy for the unconstrained subproblems that give the search-direction in Interior-Point (IP) methods for MPC, and that usually are the computational bottleneck. This strategy combines a Riccati-like solver with the use of high-performance computing techniques: in particular, in this paper we explore the performance boost given by the use of single precision computation, and techniques such as inexact search direction and mixed precision computation. Finally, we test our HPMPC toolbox, a family of high-performance solvers tailored for MPC and implemented using these techniques, that is shown to be several times faster than current state-of-the-art solvers for linear MPC.

Inexact Newton-Type Optimization with Iterated Sensitivities

“…Proof. The third equation in both augmented KKT systems from (14) and (25) at the solution (y , λ , D ) reads as g z D − g w = 0 such that D = g −1 z g w holds. The following equality therefore holds at the solution…”

“…The adjoint-free inexact Newton method with iterated sensitivities (AF-INIS) then uses the same approximate Jacobian matrixJ INIS (ȳ,λ,D) from (15) to solve the augmented set of equations in (25). At each iteration, the corresponding linear system reads as…”

“…Similar to the implementation described in Algorithms 2 and 3, condensing and expansion techniques are used to obtain multiple shooting structured subproblems in each iteration of the SQP algorithm [44]. Tailored convex solvers such as qpOASES [23], qpDUNES [24], and HPMPC [25] can be used to solve these subproblems, especially in the presence of inequality constraints.…”

“…A solution (y , λ , D ) to the alternative augmented system in(25), corresponds to a regular KKT point (y , λ ) for the NLP in (1).…”

“…Proof. At the solution of the adjoint-free augmented KKT system in (25) for the QP formulation in (28), we know that D = A −1 1 A 2 and we use the notation A = A 1 A 2 andà = A −1 1 A. The eigenvalues γ of the iteration matrixJ −1 INIS J AF −1 nINIS are given by the expression det(J AF −(γ +1)J INIS ) = 0, based on the exact and inexact adjoint-free augmented Jacobian matrices…”