Design of Piecewise Linear LQ Control for Linear Systems with Rate Saturations Using LMI Optimization

A model of the servomechanism used for high-powered actuators in mechanical systems consists of a position feedback loop around the cascade connection of a memoryless saturation function with an integrator with a large time constant. The saturation function in the servomechanism has a linear high gain characteristics for a small input and so the equivalent time constant of the actuator servomechanism becomes small. As the input to the saturation function becomes larger than the linear range for a drastic control demand, the output of the saturation function becomes constant irrespective of the input magnitude and the actuator response has a rate saturation determined by the large time constant of the integrator and so the time lag of the actuator response behind the demand results in the actual plant input much different from the demand and the plant may exhibit an undesirable behavior of the system. Therefore in this paper we consider a control method for a system with rate saturations in the actuator servomechanisms to keep stable by switching the controller gain according to the input magnitudes to the saturation functions so that the inputs to the saturation functions are controlled within the permissible maximum absolute values which are decided according to each level of LQ controller gain groups determined beforehand to ensure the local absolute stability of the total system whose conditions are expressed as a linear matrix inequalities optimization problem by introducing a Lure-type Lyapunov function. In a piecewise linear control the switching function selects a controller gain group by on-line monitoring the inputs to the saturation functions in the actuator servomechanisms. The effectiveness of the design method is illustrated with a practical example of dynamic positioning(DP) system.