Reduced order controller for FO-IMC with desired phase margin and gain cross-over frequency

Fractional order (FO) controller provide more number of tuning parameters than integer order (IO) counterpart. In this paper, a fractional order (FO) internal model control (IMC) is considered which provides two tuning parameters $(\lambda$ and $\beta)$ as compared to integer order (IO) counterpart. These two parameters are tuned to get the desired $\varphi_{m}$ and $\omega_{g}$ for higher order systems. Usually, we use IO approximation of FO terms in the controllers. It is observed that 5-15 order IO approximation provide almost accurate behavior for the FO terms. The problem with such approximation is the resulting higher order controller. In this work, a balanced truncation method is used to get a reduced order controller which retains important properties of higher order controller. The viability of reduced order controller is measured in terms of maximum sensitivity $(M_{s})$ and the frequency at which the system is most sensitive $(\omega_{m})$. Further, the results are compared in terms of rise time $(T_{r})$, settling time $(T_{s})$, maximum overshoot (%$M)$, integral square error (ISE), integral absolute error (IAE) and integral of the time weighted absolute error (ITAE). The importance of using reduced order controller is checked using three examples: first, a minimum phase (MP) system, second, a non-minimum phase (NMP) system with right hand plane (RHP) zero and third, a first order plus time delay (FOPTD) system. It is observed that the lower order controller can be used for higher order controller.