Limit Cycle Analysis of Flexible Structures Under Discontinuous Feedback Control

A method for calculating all periodic solutions and their domains of attraction for flexible systems under nonlinear feedback control is presented. The systems considered consist of mechanical systems with many flexible modes and a relay type controller coupled with a PID control law operating in a feedback configuration. The proposed approach includes three steps. First, limit cycle frequencies and periodic fixed points are computed exactly, using a block diagonal state-space modal representation of the plant dynamics. Then the relay switching surface is chosen as the Poincare mapping surface and is discretized using the cell mapping method. Finally, the region of attraction for each limit cycle is computed using the cell mapping algorithm and employing an error based convergence criterion. An example consisting of a system with two modes, a relay with hysteresis and a PD controller is used to demonstrate the proposed approach.

Temperature control in a chemical reactor through variable area of the heat transfer surface. Experimental verification

A possibility has been tested in the paper of the feed back control of temperature of the reaction mixture in a batch reactor with an exothermic reaction through the variable area of the cooling surface. The measurement were carried out in a laboratory reactor with a retractable cooler which was being immersed into the reaction mixture. The speed of motion of the cooler was sufficiently high permitting the process of immersion to be regarded as practically instantaneous. The aim of the control was to stabilize the set point temperature of the reaction mixture by a two-point controler. In dependence on the rate of response of the system to a change of the section variable either the ideal relay or the relay with hysteresis control algorithmus were used. The results of measurements showed that with the aid of a retractable cooler the temperature could be controlled safely even in those cases, in which the control by the variable flow rate of the coolant was unfeasible. The verification was carried out in the open-loop instable operating point of the reactor.

Temperature control in a chemical reactor through variable area of the heat transfer surface. Simulation analysis

An algorithm has been developed and on a mathematical model analyzed to stabilize the reaction temperature of a batch reactor. The reaction has been a zero-order one and the reactor has been operated in a instable operating point. The action variable is the heat exchange surface whose area is increased if the temperature is above, or decreased if the temperature is below the set point. The following two-point regulators have been studied: An ideal relay, a relay with hysteresis and an asymmetric PD relay. The effect has been discussed of the parameters of the regulators on the quality of regulation. Stability analysis has been made of the stationary switching cycles and the domains of applicability have been determined for individual regulators with respect to the rate of change of the area of heat exchange surface.