Graphical Determination of Optimum Trajectories for Third-Order Systems

Abstract

The present paper explains the use of two phase planes in the graphical determination of optimum trajectories for third-order systems, depending on the sign of a single control function. The control function is defined on these planes by means of different contours. Accordingly, the control signal is known at the different points on these planes. Once the control signal is found, the state trajectory is determined. Most of the arbitrary initial states are treated and, in particular, the cases of separate steps in each of the error and its first as well as second time derivatives. This work also explains the use of the graphical solution in obtaining the maximum error and switching times.