Delay Scheduling of a LQR and PID Controlled Pendubot Using CTCR Method

Delays are a common physic effect that is present in a huge quantity of industrial systems with feedback control. Sometimes the impact of the delay presence in a system is neglected without any difference in the performance of the controller. But in some cases, the Delay quantity reaches levels that increase the overshoot significantly or destabilizes the system. For this system the CTCR method can be used to design a “delayed scheduled” controller able to reject the delays effect. To be able to apply this method it is necessary for both the system and the controller to be Linear or linearized. In this article the study case is an articulated inverted pendulum or also called “Pendubot”. This configuration of pendulum was selected because, in spite of being a simple pendulum, it has four equilibrium points, where two equilibrium points were selected to be controlled, the most unstable which has the maximum potential energy for each link and the most stable with the lowest potential energy. The purpose of work with those equilibrium points is compare the difference between the delay rejection in each point with similar setting times and overshoots from controllers. In order to get an accurate model, the state of the current is added along with the viscous friction terms for each joint in the pendulum. To tune the linear and non-linear model an experimental validation was carry on the physical prototype from the Universidad Industrial de Santander (VIE-5373 UIS). The control laws applied were a classical PID and LQR control. Due the controller sample frequency is extremely high in comparison to the states response, all the controllers and linear models were implemented in continuous space. In the first tuning process it was observed that the PID control gets a significantly lowest performance than the LQR control in the unstable equilibrium point, for that reason the comparison between points only was carried on with the LQR equation of feedback system, this can be done in different ways, but for the PID control the Mason theorems was applied and for LQR control only with matrix operation the equation can be obtained. After applying the “Delay scheduling” it was observed that the tuned LQR get a highest delay rejection that PID. An observer fact was that although the controllers for each point have a similar performance the delay pockets have completely different values due the final poles locations for each point. Also was observed that the system only gets one stability pocket, this could because only one delay in the actuator was induced.