Mishchenko M. Model predictive control in linear discrete systems

Mishchenko M. D. Model predictive control in linear discree systems. - Qualifying scientific work. Manuscript. Thesis for a Doctor of Philosophy degree in specialty 01.05.04 “System analysis and theory of optimal solutions” (124 — System analysis). — National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 2024. The aim of the dissertation research is to develop a terminal control process intended for system stabilization. The research is dedicated to development of alternative model predictive control (MPC) based stabilization algorithms for discrete-time linear systems. The core idea of the MPC is to generate control signals by choosing a control sequence which corresponds to the best (by some criterion) trajectory prediction on a limited horizon. In practice, it is done by solving an optimization problem, whose objective function depends on future state’s prediction. Usage of the mathematical optimization apparatus instead of the Z-transform allows to avoid ad-hoc controller tuning. It also allows to generate fast stabilization trajectories by using the classic linear system’s evolutionary equation as a future state predictor and constraints on controls as optimization problem’s constraints. A new kind of algorithms capable of terminal stabilization control emerged from results of this research. These algorithms are able to bring system’s state to zero (or at least into its neighborhood in nondeterministic case) in finite time and keep it there indefinitely. They can stabilize not only strictly stable systems, but also semi-stable and unstable ones while respecting control resource constraints. They are capable of it even if there are random perturbations affecting the system. These methods are applicable to controlling technical and any other systems describable in linear discrete-time form. During research it became apparent 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.