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Mathematical Problems in Engineering
Volume 2012, Article ID 142060, 16 pages
Research Article

Toolgraph Design of Optimal and Feasible Control Strategies for Time-Varying Dynamical Systems

Department of Decision Systems (WETI), Gdansk University of Technology, Narutowicza 11/12, 80-952 Gdansk, Poland

Received 30 March 2012; Accepted 21 July 2012

Academic Editor: Zoran Gajic

Copyright © 2012 Z. Kowalczuk and K. E. Olinski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The paper presents a new method for designing optimal and feasible control strategies for time-variant dynamical processes. The key point of the presented idea lies in utilizing a flow graph structure for representing pertinent properties of the autonomous dynamics of a given dynamical process in a time-and-state space, which is composed of certain elementary segments. The structure is referred to as a time-and-state space toolgraph. In the procedure, each segment of the temporary state space is assigned a node of the time-and-state space toolgraph. The flow values are proportional to the cost of driving the operational point of the dynamical process between the centers of adjacent segments. Any of the discrete optimization algorithms can be applied to search for a cheapest path connecting the initial and terminal points of the sought optimal piecewise-linear trajectory of the operational points in the considered time-and-state space. Additional assumptions or restrictions concerning arbitrary forbidden zones for the operational points can be easily taken into account. In such cases the nodes representing the segments partially or entirely belonging to the finite forbidden zones are deposed from the toolgraph structure.