The effect of the time-constant of the reference trajectory in predictive control on closed-loop stability: an analytical approach for a particular case

Heuristically, it was claimed in the literature that increasing the time-constant `T<sub>ref</sub>´ of the reference trajectory in predictive control improves the closed-loop stability because the controller will be able to tolerate model mismatch which is difficult for a zero time-constant reference trajectory (a pure set-point) to tolerate. On the other hand, many industrial model predictive control (MPC) technologies like SMC-Idcom, HIECON, PFC, etc... use `T_ref ´ as the main tuning parameter. Smaller time constants demand more aggressive control, while larger time constants result in less agressive action. One may start with `T<sub>ref</sub>´ equal to the open-loop time constant of the control variable, then refine the tuning based on performance/robustness trade-offs. In this paper, the author investigates these issues analytically for linear SISO systems. The analysis done extends the notion of system types in linear control by defining a new type denoted as MGS for which it is possible by using predictive control tools to find analytically a stable equilibrium point for the system and track at the same time a reference trajectory with a gain margin that increases as `T_ref´ increases, improving that way the relative stability of the closed-loop system. MGS systems have wide applications in many areas such as: hybrid systems, power systems, robotic systems, etc