An Adjoint-based Numerical Method for Fast Nonlinear Model Predictive Control

Abstract

The application of optimization-based control methods such as nonlinear model predictive control (NMPC) to real-world process models is still a major computational challenge. In this paper, we present a new numerical optimization scheme suited for NMPC. The SQP-type approach uses an inexact constraint Jacobian in its iterations and is based on adjoint derivatives, that can be computed very efficiently. In comparison to a similar real-time algorithm based on directional sensitivities and an exact constraint Jacobian, the computational complexity is significantly reduced. Both algorithms are applied to the model of a thermally coupled distillation column for disturbance rejection. The results provide a proof-of-principle for the proposed adjoint-based optimization approach.

Fast nonlinear model predictive control for teleoperation systems using computationally efficient optimization techniques

“…Its performance, however, strongly depends on the quality of the constraint Jacobians and Hessian approximations. Instead of keeping the linearization fixed for all iterations, an appealing idea is to update the constraint linearization at times [17].…”

“…Here, the Hessian is computed by GuassNewton approximation. The issue of applying the adjoint SQP method in the RTI scheme was proposed in [17]. The adjoint SQP based RTI is initialized with the approximations of Hessian and Jacobian matrices which are kept fixed for all subsequent iterations.…”

Generation of discrete first‐ and second‐order sensitivities for single shooting

“…Different performance results are reported for applications of the adjoint approach to the BDF integration method. A theoretical bound of maximum five times the cost of a system simulation for a discrete first‐order directional gradient has generally been assumed 13, 21. For a different implementation, the computational cost for one directional second‐order derivative has been reported to be between two and four simulation times 6.…”

“…A different implementation for index one linear implicit systems have been reported in Refs 13. and21. A comparison of the adjoint mode for explicit and implicit integration methods can be found in Ref 14…”

A factorization with update procedures for a KKT matrix arising in direct optimal control

“…For a number of optimal control and MPC applications that have been treated recently we refer e.g. to [16,33,43]. The limitations of condensing approaches become noticeable for optimal control problems with long horizons, fine discretizations of the horizon, and for problems with more control parameters than state variables.…”