(390d) Proportional State-Feedback Controller Design Using MPC Structure and Carleman Approximation Method

Model Predictive Control (MPC) has gained widespread recognition due to its ability to account directly for constraints. The cost is that all the advanced Model Predictive Controllers need to solve a nonlinear optimization problem at each sampling time. Thus, MPC is usually a computationally expensive control approach and this issue may cause a delay in the input signal which potentially deteriorates the controller's performance and may cause closed-loop system instability. Thus, MPC designs are not always practical, especially for large scale or fast evolving processes. Some papers utilize Carleman approximation method to represent nonlinear dynamic systems in a bilinear form. This approximation not only provides the analytical solution of the system, but also facilitates sensitivity derivations of the objective function with respect to decision variables ^[1-4]. This sensitivity information significantly accelerates the online computations. However, these articles do not discuss the stability of the resulting closed-loop system.

Motivated by the above discussion, this work tunes a proportional state feedback controller employing the nonlinear MPC structure combined with the Carleman approximation method. Therefore, the analytical solution and sensitivity of the objective function with respect to the proportional gain vector are available. This proposed state feedback control in the MPC structure, gives a smoother control law rather than a traditional nonlinear MPC which generates a piecewise constant input. Also, the analytical calculation of the Hessian matrix in addition to the gradient vector, in this work, reduces the online computation efforts even more.

The proposed controller would inherit the same nominal stability properties of the ideal nonlinear MPC, when there exists no Carleman approximation error. However, Carleman approximation method, with a finite dimension, always has a dynamic error which might endanger the stability of the closed-loop system. This work discusses the conditions required to guarantee the input-to-state stability in the presence of the Carleman approximation.

References:

[1] N. Hashemian and A. Armaou, "Simulation, model-reduction and state estimation of a two-component coagulation process," AIChE J., 2016, 62, pp 1557-1567, DOI:10.1002/aic.15146.

[2] N. Hashemian and A. Armaou, "Fast Moving Horizon Estimation of nonlinear processes via Carleman linearization,"Proceedings of the American Control Conference, pp. 3379 - 3385, Chicago, IL, USA, 2015,DOI: 10.1109/ACC.2015.7171854.

[3] Y. Fang and A. Armaou, "Carleman approximation based Quasi-analytic Model Predictive Control for Nonlinear Systems, " AIChE J., 2016, Accepted Author Manuscript, DOI:10.1002/aic.15298.

[4] Y. Fang and A. Armaou, "Nonlinear Model Predictive Control Using a Bilinear Carleman linearization-based Formulation for Chemical Processes," Proceedings of the American Control Conference, pp. 5629 - 5634, Chicago, IL, USA, 2015, DOI: 10.1109/ACC.2015.7172221.