(309c) Nonlinear Model Predictive Control with Explicit Performance Specification

The operation of chemical plants is hindered by numerous challenges such as inherent nonlinearity, stochastic disturbances, complex variable interactions and process constraints. One well established control method that is effective in handling these challenges is model predictive control (MPC), where an open-loop optimal control problem is solved at each sampling instance over a finite time horizon, subject to the dynamic response of the plant and constraints. The abundant studies and various industrial applications of MPC have also contributed towards better understanding of the rigorous stability and performance properties of MPC.

One form of contribution has involved the development of Lyapunov-based MPC, which can explicitly characterize the region from where stability of the closed-loop system is guaranteed in the presence of input and state constraints [1, 2]. This approach is also capable of handling uncertainty through the stochastic characterization of disturbances and robust control formulations [3]. On one hand, robust control approaches counteract the worst-case scenario for the realization of the uncertainty. On the contrary, offset-free MPC augments the state variables with fictitious states that estimate and counteract the uncertainty. The prevailing economic incentive in operating chemical processes has fostered the recent development of economic MPC formulations where the controller determines the set-point internally to satisfy the prescribed economic objective, while a rigorous analysis ensures that stability is preserved.

Regardless of the nature of the predictive controller used, relatively fewer studies have explored the explicit characterization of the desired closed-loop behavior in the MPC formulation. The explicit specification of the control performance has been addressed for linear, single-input single-output (SISO) systems in the frequency domain using internal model control (IMC) [5], where the controller is designed to achieve a pre-specified desired closed-loop transfer function. However, it is not always guaranteed that the transfer function of the IMC-based control law will be consistent with the proportional-integral-derivative (PID) control structure. Moreover, the IMC approach does not generalize readily for multi-input multi-output (MIMO) systems with constraints. In an effort to handle MIMO systems, the funnel control approach [6] is proposed where the time-varying output error feedback controller forces the tracking error to be within a bounded prescribed function. Although the funnel control approach is capable of handling nonlinear MIMO systems, the method does not explicitly consider input constraints. To address this inherent limitation, recent contributions proposed a framework that incorporates explicit performance specification for MIMO systems while cognizant of input constraints [7], which is then implemented in conjunction with offset-free model predictive control. The work considers linear time invariant (LTI) systems that are invertible (i.e. the inputs can be explicitly computed). The proposed explicit performance specification based offset-free MPC, using a linearized plant model, was implemented on a nonlinear chemical process. The previous work, however, does not explicitly handle the issue of lack of full state measurements, constraints on the rate of change of the manipulated input, and process nonlinearity.

Motivated by the above considerations, in this work we generalize the explicit performance specification based control design to handle the output feedback problem, constraints on the rate of change of the manipulated input, and process nonlinearity. The proposed framework consists of a bi-layer optimization scheme. In the first tier, the optimal trajectory is computed for the given performance specification in the form of a prescribed function. Then, in the second layer, a nonlinear model predictive control scheme computes the optimal inputs so as to closely follow the trajectory as given by the first tier. Results are first presented for the output feedback control for a linear systems, then the handling of rate constraints is illustrated for a nonlinear CSTR example, and finally, the approach is implemented on a reactor separator system.

[1] Mahmood, M. and Mhaskar, P., 2014. Constrained control Lyapunov function based model predictive control design. International Journal of Robust and Nonlinear Control, 24(2), pp.374-388.

[2] Mhaskar, P., El-Farra, N.H. and Christofides, P.D., 2006. Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control. Systems & Control Letters, 55(8), pp.650-659.

[3] Mahmood, M. and Mhaskar, P., 2012. Lyapunov-based model predictive control of stochastic nonlinear systems. Automatica, 48(9), pp.2271-2276.

[4] Wallace, M., Das, B., Mhaskar, P., House, J. and Salsbury, T., 2012. Offset-free model predictive control of a vapor compression cycle. Journal of Process Control, 22(7), pp.1374-1386.

[5] Garcia, C.E. and Morari, M., 1982. Internal model control. A unifying review and some new results. Industrial & Engineering Chemistry Process Design and Development, 21(2), pp.308-323.

[6] Ilchmann, A., Ryan, E.P. and Sangwin, C.J., 2002. Tracking with prescribed transient behaviour. ESAIM: Control, Optimisation and Calculus of Variations, 7, pp.471-493.

[7] Wallace, M., Pon Kumar, S.S. and Mhaskar, P., 2016. Offset-Free Model Predictive Control with Explicit Performance Specification. Industrial & Engineering Chemistry Research.