Synthesis of Third-Order Time-Optimal Control System for Plants with Extremum Time Response

Abstract

We derive an algorithm of a time optimal (maximum speed) controller for a third-order dynamical system. A model with an extreme second-order transient response with delay was adopted as the object of research; the constant speed electric actuator is represented by an integrator. The synthesis is based on the Pontryagin maximum principle and the description of the system dynamics in the state space via canonical variables. The verification of the correctness of the obtained result is carried out according to Feldbaum’s theorem on the number of switchings of the direction of motion of the regulating body on the control interval. To calculate the canonical state variables, it is proposed to use the position of the regulator, the controlled parameter, and the derivative calculated based on its values measured on real plants. The transition from the measured physical parameters to the canonical variables is performed using the similarity transformation formulas derived in the paper. A solution is given regarding the calculation of the specified values of the state variables, and an algorithm is presented for their prediction in order to compensate for the net delay in the plant model.