Calculation of PID Controller Parameters for Unstable First Order Time Delay Systems

In this paper, a numerical approach for the fractional order proportional-integral-derivative controller (FO-PID) design for the unstable first order time delay system is proposed. The controller design is based on the system time delay. In order to obtain the relation between the controller parameters and the time delay, for several amounts of the plant time delay and the fractional derivative and integral orders, the ranges of stabilizing controller parameters are determined. First, for a typical time delay plant and the fractional order controller, the D-decomposition technique is used to plot the stability region(s). The controller derivative gain has been considered as one. By changing the fractional derivative and integral orders, a small amount in each stage, some ranges of proportional and integral gains are achieved which stabilize the system, independent of the fractional , orders. Therefore a set of different controllers for any specified time delay system is obtained. This trend for several various systems with different values of time delay has been done and the proportional and integral gains of the stabilizing controller have been calculated. Then we have fitted these values to the exponential functions and the proportional and integral gains have been obtained in terms of the system time delay. Using these relations, we can specify some ranges of the proportional and integral gains and obtain a set of stabilizing controllers for any given system with certain time delay. In these relations, fractional derivative and integral orders haven’t part, and therefore can be applied to any fractional order controller design (for 0.1 , 0.9 ). Thus we have reached a numerical approach from the graphical D-decomposition method. In this method, there is freedom in choosing the values of and (they can fall in the range of [0.1, 0.9] ), and there is no need to plot the stability boundaries and check the different regions to determine the stable one. This numerical method does not offer the complete set of the stabilizing controllers. Whenever the system time delay is more, the specified range of proportional and integral gains will be smaller. In other words, the extent of obtained stability region is inversely proportional to the system time delay. Finally, the introduced numerical approach is used for stabilizing an unstable first order time delay system.