Terminal region characterization and stability analysis of discrete time quasi-infinite horizon nonlinear model predictive control

Abstract

A terminal penalty term and terminal constraints were introduced in the nonlinear MPC (NMPC) formulation for establishing the nominal closed loop stability. A large terminal set gives rise to a large region of attraction for the closed loop system and can also help in reducing the on-line computation cost. In this work, two novel approaches have been developed for characterization of the terminal region for a discrete time quasi-infinite horizon NMPC formulation that employs quadratic stage cost. Similar to the continuous time QIH NMPC formulation by Chen and Allgöwer [1], a stabilizing linear controller developed using linearization at the origin is employed for characterization of the terminal set. The first approach permits use of an arbitrary linear controller while the second approach is based on LQR controller. Using a method of bounding only the higher order nonlinear effects of the system via simulations under linear controller, the proposed approach is further extended to handle terminal region computation for a large dimensional system. Unlike the approaches available in the literature, the proposed approaches provide sufficient degrees of freedom to shape the terminal set. The quadratic control Lyapunov function in the terminal set together with the quadratic stage cost is further used to establish exponential stability of the discrete time QIH NMPC formulation under nominal conditions. Additionally, it is shown that discrete time QIH-NMPC based on the nominal model is input to state stable in the face of bounded disturbances in the state dynamics. Efficacy of the proposed approaches for characterization of the terminal region is demonstrated using a two state example from Chen and Allgöwer [1] and a benchmark CSTR system. Moreover the extension to large dimensional systems is demonstrated using a system consisting of two distillation columns in series. Parametric studies reveal that the choice of sampling interval and the choices of the tuning matrices in both approaches have significant influence on the sizes of terminal sets. Moreover, the simulation results demonstrate that the systems under consideration converge asymptotically to the origin under the nominal conditions and are input to state stable in the face of bounded disturbance.