Model predictive control (MPC), in general, requires solving numerically a constrained optimization problem repeatedly on-line. A question that often process control engineers face is for what class of processes, model predictive control admits an analytical solution, which requires significantly less computational time to obtain.

The question motivated the study presented in [1]. Soroush and Muske [1] studied several special cases of MPC that take analytical forms. In the particular case of shortest prediction horizon, they showed that the model predictive control problem has an analytical closed-form solution. Seron et al. [2] characterized regions of the state-space wherein for a general single-input single-output (SISO) linear system, constrained finite-horizon linear model predictive control and unconstrained finite-horizon linear model predictive control with clipping (saturation) provide identical solutions. Marjanovic et al. [3] showed that for a general SISO system with input constraints and certain conditions imposed, saturated infinite horizon linear quadratic regulator (SIHLQR) control provides the same control sequence as the constrained infinite horizon linear quadratic regulator (CIHLQR) control. They also showed that saturated LQR (SIHLQR) is equivalent to the CIHLQR in the case of first-order SISO systems, subject to both state and control constraints.

This paper presents a study of the role that the directionality of a process [4,5] plays in analytical model predictive control of the process. In other words, it characterizes the class of processes whose model predictive control has an analytical solution. It is shown theoretically that for processes with no directionality, one can design model predictive controllers with a very wide range of prediction horizons, that have analytical solutions. Several process examples are simulated numerically to demonstrate the theoretical results. This study has been built upon the results presented in [6].

[1] Soroush, M., and K. Muske, "Analytical Model Predictive Control," in Nonlinear Model Predictive Control, F. Allgower and A. Zheng (eds.), Progress in Systems and Control Theory series, Vol. 26, Birkhauser-Verlag, Birkhauser Verlag, Basel, 163-179 (2000)

[2] Seron, M., J. De Dona, and G. Goodwin, “Global analytical model predictive control with input constraints,” Proceedings of the 39th Conference on Decision and Control, Sydney, Australia (2000)

[3] Marjanovic, O., B. Lennox, P. Goulding, and D. Sandoz: “Minimising conservatism in infinite horizon LQR control,” Systems and Control Letters, 46 (4), 271-279 (2002)

[4] Soroush, M., and N. Mehranbod, “Optimal Compensation for Directionality in Processes with a Saturating Actuator,” Comput. & Chem. Engin., 26(11), 1633 (2002)

[5] Soroush, M., and S. Valluri, “Optimal Directionality Compensation in Processes with Input Saturation Nonlinearities,” International J. of Contr., 72(17), 1555 (1999)

[6] Soroush, M., and Y. Dimitratos, “Control System Selection: A Measure of Control Quality Loss in Analytical Control,” DYCOPS 7, Boston, MA (2004)