## 3 Optimal Tracking and Disturbance-Rejection
### The Higher-Order Filter  
To achieve an even faster closed-loop tracking response than that with the PD Controller (25), a higher-order filter can be applied. Here, the calculation of the filter parameters is based on the process model. However, this does not mean that the main advantage of the proposed tuning method is lost (note that the process model is not required for the calculation of the controller parameters). Instead, the process model will be estimated from the characteristic areas, where the only required parameter from the user is the estimated process time-delay, which can be easily acquired in practice from the process time-response. The chosen process model is a second-order process with zero and the estimated pure time-delay  
\[G_{M}(s)=\frac{K_{M}(1+b_{1m}s)e^{-T_{dm}s}}{s(1+a_{1m}s+a_{2m}s^{2})}, \tag{32}\]  
where \(T_{dm}\) and \(K_{M}\) represent the process time-delay and steady-state gain, respectively. Note that the model parameters (except for the estimated time-delay, \(T_{dm}\)) can be calculated directly from the equations for the characteristic areas \(A_{0}\) to \(A_{3}\) (8):  
\[ \begin{gathered} K_{M}=A_{0},\\ a_{1m}=\frac{2A_{1}\left(A_{2}-\frac{A_{0}T_{dm}^{2}}{2}\right)-2A_{0 }\left(A_{3}-\frac{A_{0}T_{dm}^{3}}{6}\right)+\left(A_{1}-A_{0}T_{dm}\right) \left(A_{0}T_{dm}^{2}-2A_{1}T_{dm}\right)}{A_{0}^{2}T_{dm}^{2}-2A_{0}A_{1}T_{ dm}-2A_{0}A_{2}+2A_{1}^{2}},\\ a_{2m}=\frac{\left(A_{2}-\frac{A_{0}T_{dm}^{2}}{2}\right)\left(2A_{2}- A_{0}T_{dm}^{2}\right)-2\left(A_{1}-A_{0}T_{dm}\right)\left(A_{3}-\frac{A_{0}T_{dm}^{ 3}}{6}\right)+\left(A_{1}-A_{0}T_{dm}\right)\left(A_{1}T_{dm}^{2}-2A_{2}T_{ dm}\right)}{A_{0}^{2}T_{dm}^{2}-2A_{0}A_{1}T_{dm}-2A_{0}A_{2}+2A_{1}^{2}},\\ b_{1m}=\frac{2A_{1}\left(A_{2}-\frac{A_{0}T_{dm}^{2}}{2}\right)-2A_{0 }\left(A_{3}-\frac{A_{0}T_{dm}^{3}}{6}\right)-\left(\frac{A_{1}}{A_{0}T_{dm}} \right)\left(A_{1}^{2}-A_{0}A_{2}\right)}{A_{0}^{2}T_{dm}^{2}-2A_{0}A_{1}T_{ dm}-2A_{0}A_{2}+2A_{1}^{2}}.\end{gathered}  \tag{33}\]For minimum-phase processes, the model transfer function can be simplified by setting \(b_{1m}=0\), which in turn simplifies the identification procedure. Namely, all the process model parameters, including the process time-delay \(T_{dm}\), can be calculated analytically [126] from Expression (8):  
\[ \begin{array}{c}K_{M}=A_{0},\\ \frac{T_{dm}^{3}}{\theta}-\frac{T_{dm}^{2}A_{1}}{2A_{0}}+T_{dm}\Big{(}\frac{A _{1}^{2}}{A_{0}^{2}}-\frac{A_{2}}{A_{0}}\Big{)}-\frac{A_{3}^{3}}{A_{0}^{3}}+ \frac{2A_{1}A_{2}}{A_{0}^{2}}+\frac{A_{3}}{A_{0}}=0,\\ a_{1m}=\frac{A_{1}}{A_{0}}-T_{dm},\\ a_{2m}=-\frac{A_{2}+A_{1}a_{1m}}{A_{0}}+\frac{T_{dm}^{2}}{2},\\ b_{1m}=0,\end{array}  \tag{34}\]  
where \(T_{dm}\) is calculated from the above third-order equation. The result (among three solutions) is the lowest positive real number. The closed-loop transfer function from the reference to the process output, when process \(G_{P}\) is replaced by the process model \(G_{M}\), is the following:  
\[G_{CL}(s)=\frac{G_{F}K_{M}\Big{(}K_{D}s^{2}+K_{P}s+K_{I}\Big{)}(1+b_{1m}s)e^{ -T_{dm}s}}{(1+a_{1m}s+a_{2m}s^{2})s^{2}+K_{M}(K_{D}s^{2}+K_{P}s+K_{I})(1+b_{1m }s)e^{-T_{dm}s}}. \tag{35}\]  
We can now define the desired closed-loop transfer function as follows:  
\[G_{CLD}(s)=\frac{e^{-T_{dm}s}}{\Big{(}1+\frac{T_{CL}}{n}s\Big{)}^{n}}, \tag{36}\]  
where  
\[ \begin{array}{c}n=2\quad T_{dm}>0\\ n=3\quad T_{dm}=0\end{array} . \tag{37}\]  
\(T_{CL}\) is the desired closed-loop time constant. Note that parameter \(T_{CL}\) is related to the speed of the response of the PD controller, so it will be automatically calculated, and no user input is required. Let us define the closed-loop time constant \(T_{CL0}\) equal to the integral of the control error at the unit stepwise reference change:  
\[T_{CLO}=\int_{t=0}^{t=\infty}e(t)dt=\frac{1}{A_{0}K_{P1}}, \tag{38}\]  
where \(K_{P1}\) is the proportional gain of the PD controller.