Practical aspects of model predictive control in linear systems and cognitive maps

Abstract

The subject of this paper is peculiarities of model predictive control (MPC) application in linear discrete-time system stablization. While stabilization in linear systems is already a well-studied problem in control theory, the MPC approach gives opportunity to produce faster stabilization trajectories at cost of higher amount of computations required. Significant progress in capabilities of computers since emergence of the control theory makes the MPC approach feasible in modern times. The MPC approach gives an opportunity to achieve significantly better results, but its application requires great care. It is due to many unobvious and undesirable effects it leads to if used incorrectly. These effects are discussed, explained and demonstrated one by one on examples in this paper. Analysis of their causes reveals requirements for MPC-based stabilizing control algorithm which allow resulting controller to operate reliably. It also appears that in most cases an optimal stabilization trajectory is not unique, i.e. it is possible to choose between optimal trajectories to improve some kind of secondary objective. In addition, as an example which is valuable by itself, stabilization in linear cognitive maps is discussed separately. Being an example of discrete-time linear system, linear cognitive maps are susceptible of application of the same control strategies and algorithms to their impulses. But if nature of linear cognitive map is disregarded, their state starts to wander under pressure of external random perturbation (i.e. noise) even though stabilizing controller mitigates their influence on cognitive map’s impulses. Ability of the MPC approach to consider secondary objectives allowed to mitigate this effect at least partially. In particular, it is achieved here by seeking a particular objective cognitive map state as a secondary objective in search for a stabilization trajectory. It is also demonstrated here, that only a certain hyperplane in cognitive map’s state-space is reachable under assumption, that its impulse is zero at the end of trajectory.