## 3 Problem Formulation  
Suppose, an \(n\)th-order SISO continuous interval plant \(G_{n}(s)\) for the closed-loop control system [54] of Figure 1 is expressed as  
\[G_{n}(s)=\frac{A(s)}{B(s)}=\frac{[A_{0}^{-},A_{0}^{+}]+\cdots+[A_{n-1}^{-},A_ {n-1}^{+}]s^{n-1}}{[B_{0}^{-},B_{0}^{+}]+[B_{1}^{-},B_{1}^{+}]s+\cdots+[B_{n}^ {-},B_{n}^{+}]s^{n}} \tag{6}\]  
The interval coefficients of numerator polynomial \(A(s)\) are \([A_{i}^{-},A_{i}^{+}]\) for \((i=0,1,\cdots,n-1)\), with \(A_{i}^{+}\) as upper bounds and \(A_{i}^{-}\) as lower bounds, respectively. Likewise, interval coefficients of denominator polynomial \(B(s)\) are \([B_{i}^{-},B_{i}^{+}]\) for \((i=0,1,\cdots,n)\), with \(B_{i}^{+}\) as upper bounds and \(B_{i}^{-}\) as lower bounds, respectively. The power series [55] form of HOCIP, (6), about \(s=0\) and \(s=\infty\), respectively, are expressed as  
\[G_{n}(s)=\lambda_{0}+\lambda_{1}s+\cdots+\lambda_{k}s^{k}+\cdots \tag{7}\]  
\[G_{n}(s)=\mu_{1}s^{-1}+\mu_{2}s^{-2}+\cdots+\mu_{k}s^{-k}+\cdots \tag{8}\]  
where \(\lambda_{i}=[\lambda_{i}^{-},\lambda_{i}^{+}]\) for \(i=0,1,2,\cdots\) and \(\mu_{i}=[\Figure 2 represents the closed-loop system without an FOPID controller. Figure 3 depicts the reduced-order model \(\widehat{G}_{r}\ (s)\) of HOCIP in place of \(G_{n}(s)\). The desired reduced-order continuous interval plant (ROCIP) \(\widehat{G}_{r}\ (s)\) of \(G_{n}(s)\) is represented as  
\[\widehat{G}_{r}\ (s)=\frac{\widehat{E}\ (s)}{\widehat{F}\ (s)}=\frac{[e_{0}^{-},e_{0}^{+ }]+[e_{1}^{-},e_{1}^{+}]s+\cdots+[e_{r-1}^{-},e_{r-1}^{+}]s^{r-1}}{[f_{0}^{-}, f_{0}^{+}]+[f_{1}^{-},f_{1}^{+}]s+\cdots+[f_{r}^{-},f_{r}^{+}]s^{r}} \tag{9}\]  
where \(r<n\). The interval coefficients of numerator polynomial \(\widehat{E}\ (s)\) are \([e_{i}^{-},e_{i}^{+}]\) for \((i=0,1,\cdots,r-1)\), with \(e_{i}^{+}\) as upper bounds and \(e_{i}^{-}\) as lower bounds, respectively. Similarly, the interval coefficients of denominator polynomials \(\widehat{F}\ (s)\) are \([f_{i}^{-},f_{i}^{+}]\) for \((i=0,1,\cdots,r)\) with \(f_{i}^{+}\) as upper bounds and \(f_{i}^{-}\) as lower bounds, respectively. The model, (9), is expressed as  
\[\widehat{G}_{r}\ (s)= \stackrel{{\wedge}}{{\lambda}}_{\lambda}+\stackrel{{ \wedge}}{{\lambda}}_{1}s+\cdots+\stackrel{{\wedge}}{{ \lambda}}_{k}s^{k}+\cdots \tag{10}\] \[(\text{expansion about}\ \ s=0)\] \[\widehat{G}_{r}\ (s)= \stackrel{{\wedge}}{{\mu}}_{1}s^{-1}+\stackrel{{ \wedge}}{{\mu}}_{2}s^{-2}+\cdots+\stackrel{{\wedge}}{{\mu}}_{k} s^{-k}+\cdots\] (11) \[(\text{expansion about}\ \ s=\infty)\]  
where \(\stackrel{{\wedge}}{{\lambda}}_{i}=\left[\ \stackrel{{\wedge}}{{\lambda}}_{i}\, \stackrel{{\wedge}}{{\lambda}}_{i}^{+}\ \right]\) for \(i=0,1,2,\cdots\) and \(\stackrel{{\wedge}}{{\mu}}_{i}=\left[\ \stackrel{{\wedge}}{{\mu}}_{i}^{-}, \stackrel{{\wedge}}{{\mu}}_{i}^{+}\ \right]\) for \(i=1,2,,3,\cdots\) are TMs and MPs of ROCIP.  
The model \(\widehat{G}_{r}\ (s)\) in Figure 3 is an approximation of system \(G_{n}(s)\) in Figure 2 which is obtained by order reduction technique. In this work, the denominator of \(\widehat{G}_{r}\ (s)\) is obtained from improved Routh  
Figure 1: Block diagram of closed loop control system.  
Figure 3: Closed loop system of ROCIP.  
Figure 2: Closed loop system of HOCIP.  
approximation [56] and numerator of \(\widehat{G_{r}}\ (s)\) is derived by matching of TMs and MPs [29]. Figure 4 shows the closed-loop control system with FOPID controller \(\overset{\wedge}{C}\ (s)\) designed for ROCIP. Once the design for ROCIP is established, the same controller is applied to HOCIP as depicted in Figure 5 in order to ensure the applicability of designed controller for original system.