## 4 Control System Design
### Operator Theory  
In this section, right coprime factorization and control system design based on operator theory utilizing this factorization are discussed. The operators \(A:Y\to U\) and \(B:U\to U\) are designed to satisfy the Bezout identity, as shown in Equation (13). Both \(A\) and \(B\) are stable, and \(B\) is invertible. When the Bezout identity is satisfied, the operators \(N\) and \(D\) are referred to as the robust right coprime factorization of the plant \(P\). Additionally, \(M:W\to U\) is a unimodular operator [30]. \[AN+BD=M \tag{13}\]  
Figure 10 illustrates the control system based on operator theory. The operators \(N\), \(D\), \(A\), and \(B\) need to be designed to satisfy the Bezout identity given by Equation (13). In Figure 10, \(r\) represents the reference value, \(u\) denotes the plant input, \(\omega\) is the quasi-state, \(y\) is the plant output, \(b\) indicates the feedback signal, and \(e\) represents the error. The design of the control system based on operator theory is undertaken. Plant \(P(u)(t)\) is represented by Equation (14). The variables \(x_{1}\) and \(x_{2}\) correspond to \(i_{LL}\) and \(V_{out}\) in Equation (10), respectively. Additionally, the duty cycle is considered as the control input \(u(t)\). Figure 9: Conceptual diagram of filtered right factorization. \[P: \begin{cases}\frac{dx_{1}(t)}{dt}=-\frac{1}{L}x_{2}(t)+\frac{u(t)}{L}V_{in}\\ \frac{dx_{2}(t)}{dt}=\frac{1}{C_{out}}x_{1}(t)-\bigg{(}\frac{u(t)T_{SW}}{2L_{out} }+\frac{1}{R_{load}C_{out}}\bigg{)}x_{2}(t)+\frac{u(t)T_{SW}}{2L_{out}}V_{in}\\ y(t)=x_{2}(t)\end{cases}  \tag{14}\]  
From Equation (14), we factorized the plant \(P\) into \(\tilde{D}^{-1}\) and \(\tilde{N}\) on the right side. The designed \(\tilde{N}\) and \(\tilde{D}^{-1}\) are shown in Equations (15) and (16), respectively. \[\tilde{N}:y(t)=V_{in}\hat{\omega}(t) \tag{15}\] \[\tilde{D}^{-1}: \begin{cases}\frac{dx_{d1}(t)}{dt}=-\frac{1}{L}x_ {d2}(t)+\frac{u(t)}{L}V_{in}\\ \frac{dx_{d2}(t)}{dt}=\frac{1}{C_{out}}x_{d1}(t)-\bigg{(}\frac{u(t)T_{SW}}{2L_ {out}}+\frac{1}{R_{load}C_{out}}\bigg{)}x_{d2}(t)+\frac{u(t)T_{SW}}{2L_{out}}V_ {in}\\ \tilde{\omega}(t)=\frac{1}{V_{in}}x_{d2}(t)\end{cases}  \tag{16}\]  
Furthermore, from Equation (16), \(\tilde{D}\) is expressed as follows:  
\[\tilde{D}: \begin{cases}x_{d2}(t)=V_{in}\hat{\omega}(t)\\ \frac{dx_{d1}(t)}{dt}=-\frac{1}{L}x_{d2}(t)+\frac{u(t)}{L}V_{in}\\ u(t)=\frac{2L}{T_{SW}(V_{in}-x_{d2})}\bigg{(}C_{out}\frac{dx_{d2}(t)}{dt}-x_{d1 }(t)+\frac{x_{d2}(t)}{R_{load}}\bigg{)}\end{cases}  \tag{17}\]  
The operator \(Q\) is designed to function as a low-pass filter. Here, \(\tau\) represents the time constant of the low-pass filter. The operator \(Q\) is shown in Equation (18):  
\[Q:\frac{d\tilde{\omega}(t)}{dt}=\frac{1}{\tau}(-\hat{\omega}(t)+\omega(t)) \tag{18}\]  
Furthermore, the operators \(N\) and \(D\) are designed as operators passed through the low-pass filter \(Q\), as shown in Equations (19) and (20), respectively. Figure 10: Control system based on operator theory. \[N: \begin{cases}\dfrac{d\tilde{\omega}_{n}(t)}{dt}=\dfrac{1}{\tau}(- \tilde{\omega}_{n}(t)+\omega(t))\\ y(t)=V_{in}\tilde{\omega}_{n}(t)\end{cases}  \tag{19}\] \[D: \begin{cases}\dfrac{d\tilde{\omega}_{d}(t)}{dt}=\dfrac{1}{\tau}(- \tilde{\omega}_{d}(t)+\omega(t))\\ x_{d2}(t)=V_{in}\tilde{\omega}_{d}(t)\\ \dfrac{dx_{d1}(t)}{dt}=-\dfrac{1}{L}x_{d2}(t)+\dfrac{u(t)}{L}V_{in}\\ u(t)=\dfrac{2L}{T_{SW}(V_{in}-x_{d2})}\bigg{(}\text{C}_{out}\dfrac{dx_{d2}(t)} {dt}-x_{d1}(t)+\dfrac{x_{d2}(t)}{R_{load}}\bigg{)}\end{cases}  \tag{20}\]  
The unimodular operator \(M\) is designed as shown in Equation (21):  
\[M:r(t)=V_{in}\omega(t) \tag{21}\]  
By transforming Equation (13), the following equation is obtained:  
\[B=(M-AN)D^{-1} \tag{22}\]  
The operator \(B\), designed based on Equation (22), is presented in Equation (23). Here, the operator \(A\) is assumed to be the identity operator. \[B: