## 2 2 DOF PI-P Controller Design Using FRIT  
Consider the control system of the 2 DOF PI-P controller shown in Figure 1, where the system is subjected to a disturbance. This is the FB-type expression of the 2 DOF PI-P control system [6]. A feedback path exists from \(y(k)\) to \(u(k)\). \(C_{1}(q_{1}\),\(z)\) and \(C_{2}(q_{2})\) are defined as the serial compensator and the feedback compensator in a discrete-time. There are also the step response and the disturbance response. The step response is the response of the controlled variable \(y(k)\) to the set-point variable \(r(k)\), and the disturbance response is the response of the controlled variable \(y(k)\) to the unit step disturbance \(d(k)\). \(G(z)\) is the transfer function of the plant modeled as a discrete time. \(C_{1}(q_{1}\),\(z)\) and \(C_{2}(q_{2})\) are the transfer functions of the controller, where \(q_{1}\) and \(q_{2}\) are the parameter vectors to be tuned in the controller. Figure 1: Closed-loop of the feedback-type (FB)-type 2 DOF (two-degrees-of-freedom) PI-P (proportional-integral/proportional) control system. In addition, \(r(k)\), \(u(k)\), \(e(k)\), \(d(k)\) and \(y(k)\) are the reference signal, the controller output, the error between the reference and the plant output, the disturbance and the plant output, respectively. \(C_{1}(q_{1},z)\) is the controller in the form of a PI controller, and \(C_{2}(q_{2})\) is the controller in the form of the P controller. \[C_{1}\left(q_{1},z\right)=z\left(C_{1}\left(q_{1},s\right)\right) \tag{1}\]  
\[C_{1}\left(q_{1},s\right)=K_{\text{P1}}\left\{1+\frac{K_{I}}{s}\right\} \tag{2}\]  
\[C_{2}\left(q_{2}\right)=K_{\text{P2}} \tag{3}\]  
\[q_{1}=\left[ \begin{array}{cc}K_{\text{P1}}&K_{I}\\ \end{array} \right]^{T} \tag{4}\]  
\[q_{2}=\left[K_{\text{P2}}\right]^{T} \tag{5}\]  
\[q=\left[ \begin{array}{cc}q_{1}&q_{2}\\ \end{array} \right] \tag{6}\]  
where \(z\) denotes the \(z\)-operator and \(z(C_{1}\left(q_{1},s\right))\) denotes \(z\)-transform that converts controller from a continuous-time to a discrete time. \(K_{\text{P1}}\), \(K_{\text{P2}}\) and \(K_{I}\) represent the proportional and integral gains that are to be tuned. In the PI controller, if disturbance response is optimized, the step response tends to have an overshoot. To suppress the overshoot, the P controller is provided in the feedback loop. Basically, Equation (2) can be decoupled in the form of a PI controller:  
\[C_{1}\left(q_{1},s\right)=K_{\text{P1}}+\ \frac{K_{\text{P1}}K_{I}}{s} \tag{7}\]  
\[C_{1}\left(q_{1},s\right)=K_{\text{P1}}+\ \frac{K_{I}^{\prime}}{s} \tag{8}\]  
where \(K_{I}^{\prime}=K_{\text{P1}}K_{I}\). To obtain the fictitious reference signal, the closed loop response to the step input set-point (\(r(k)=1\) and \(d(k)=0\)) is considered. By performing a one-shot experiment to obtain the input/output data \(u_{0}(k)\), \(y_{0}(k)\), where \(k=1,2,3,\ldots,N\), for initial controller parameters \(q_{1}\) and \(q_{2}\) and the reference signal \(r(k)\), the fictitious reference signal can be expressed as:  
\[\widetilde{r}\left(q,k\right)=C_{1}\left(q_{1},z\right)^{-1}u_{0}\left(k \right)+C_{1}\left(q_{1},z\right)^{-1}y_{0}\left(k\right)C_{2}\left(q_{2} \right)+\ y_{0}\left(k\right) \tag{9}\]  
The initial controller parameters \(q_{1}\) and \(q_{2}\) are selected arbitrary, only once and relatively small. After a one-shot experiment is performed in the closed-loop speed control of PMSM to obtain input data \(u_{0}(k)\) and output data \(y_{0}(k)\) as shown in Figure 2, then fictitious reference signal (\(\widetilde{r}\left(q,k\right)\)) is formed as shown in Equation (9). Figure 2: 2 DOF PI-P fictitious reference iterative tuning (FRIT) principle.