+ Enhanced Numerical Methods for MPC

The focus of this research project, funded by the National Science Foundation Award CMMI-1562209, is on developing computationally efficient algorithms and solvers for the real-time implementation of Model Predictive Control (MPC).

1. Inexact Newton-Kantorovich methods for model predictive control

Newton-type methods (e.g. Sequential Quadratic Programming) are known to have strong convergence properties in terms of iteration count. However, the computational cost of the individual iterations may be large thereby leading to unsatisfactory real-time behavior. Inexact Newton methods can be used to reduce the computational cost per iteration by relaxing the tolerance of each sub-problem. Likewise, the Newton-Kantorovich method bypasses the need to recompute the second order terms after each iteration, thus reducing the cost of formulating each sub-problem. In this research, we have established convergence results for inexact Newton-Kantorovich methods applied to the optimal control problem in MPC. Based on these results, we have developed an inexact Newton-Kantorovich formulation of the Sequential Quadratic Programming and the Nonsmooth Newton (NSN) methods. Simulations featuring automotive and aerospace applications illustrate that, although the resulting methods require more iterations to converge, the decrease in cost-per-iteration ensures an overall reduction in the average and maximum computational time. See Figure 1 for comparison of the worst case and average computation time in the spacecraft attitude control example.

Figure 1: Comparison of different methods in terms of average and maximum computation time. The comparison demonstrates advantages of inexact SQP/NSN-Kantorovich implementation.

References:

[1] K. Butts, A. Dontchev, M. Huang, and I. Kolmanovsky, "A perturbed chord (Newton-Kantorovich) method for constrained nonlinear model predictive control," IFAC-PapersOnLine, 49(18), pp. 253-258.

[2] A.L. Dontchev, M. Huang, I. Kolmanovsky, and M. Nicotra, “Inexact Newton-Kantorovich methods for constrained nonlinear model predictive control,” IEEE Transactions on Automatic Control, under review.

2. Dynamically Embedded Model Predictive Control

The Dynamically Embedded MPC [3], [4] is a novel framework that approaches Model Predictive Control from a different perspective. The idea was first introduced in [3] and consists in treating the solver as a dynamic feedback law that evolves in parallel with the plant dynamics. Using a continuous-time formulation and classical nonlinear control theory, the convergence and closed-loop stability properties are demonstrated. The main advantage of the method is that it is computationally inexpensive and can be implemented at high update frequency. This makes the resulting control law more reactive to external disturbances. See Figure 2.

Figure 2: Closed-loop responses of F-16 aircraft pitch angle, pitch rate and angle of attack when controlled using the traditional discrete-time MPC (Subscript 1, in teal) and the dynamically embedded MPC (Subscript 2, in green). The aircraft is affected by an unmeasured disturbance. Faster update rate possible with DEMPC improves disturbance rejection and transient response properties.

Matlab software: Example implementation of dynamically embedded MPC in Matlab/Simulink for F-16 example.

References:

[3] M.M. Nicotra, D. Liao-McPherson, and I.V. Kolmanovsky, “Dynamically embedded model predictive control,” Proceedings of 2018 American Control Conference, Milwaukee, Wisconsin, 2018.

[4] M.M. Nicotra, D. Liao-McPherson, and I. Kolmanovsky, "Embedding constrained Model Predictive Control in a continuous-time dynamic feedback," IEEE Transactions on Automatic Control, 2019.