## 3 Observer and Control Law Design
### Control Law Design  
According to Section 2, in order to design a feedback controller and feedforward compensator, the existence conditions of the controllers should be determined first. The following theorem gives the existing conditions of the controller. **Theorem 1**. : _The problem has a solution if the system is SS and matrices exist, \(G_{i}\in\mathbb{R}^{n\times n}\), and \(H_{i}\in\mathbb{R}^{m\times n_{2}}\), satisfying the following equations:_  
\[\left\{ \begin{aligned} \dot{G}_{i}&=A_{i}G_{i}+B_{i}H_{i}-G_{i}A_{m},\\ & 0=C_{i}G_{i}-C_{m}.\end{aligned} \right. \tag{23}\]  
**Proof of Theorem 1. ** Let  
\[\left\{ \begin{array}{l}\delta\hat{x}(t)=\hat{x}(t)-G_{i}x_{m}(t),\\ \delta u(t)=u(t)-H_{i}x_{m}(t),\\ \delta y(t)=y(t)-y_{m}(t).\end{array} \right. \tag{24}\]  
After, we take the derivatives on both sides of the first expression, and we obtain the following equation:  
\[\delta\hat{x}(t)=\hat{x}(t)-G_{i}\hat{x}_{m}(t)-\hat{G}_{i}x_{m}(t)\]  
We substitute \(\hat{x}\)(t) to the original formula:  
\[=\bar{R}_{i1}\hat{x}_{1}^{(1)}(t)+\bar{R}_{i2}\hat{y}(t)-G_{i}A_{m}x_{m}(t)- \hat{G}_{i}x_{m}(t)\]  
We substitute \(\hat{x}_{1}^{(1)}(t)\) by (22) in the above equation:  
\[=R_{i1}(\hat{z}(t)+\bar{R}_{i21}\hat{y}(t))+\bar{R}_{i2}\hat{y}(t)-G_{i}A_{m}x_ {m}(t)-\hat{G}_{i}x_{m}(t)\]  
Similarly, we substitute \(\hat{z}(t)\) from the above equation:  
\[= R_{i1}\Big{(}\bar{A}_{i1}^{(2)}z(t)+\bar{B}_{i1}^{(2)}u(t)+\Big{(} \bar{A}_{i12}^{(2)}+\bar{A}_{i11}^{(2)}H_{i21}\Big{)}y(t)+\bar{H}_{i21}\hat{y} (t)\Big{)}+R_{i2}\hat{y}(t)-G_{i}A_{m}x_{m}(t)\] \[-\hat{G}_{i}x_{m}(t)\] \[= \bar{R}_{i1}A_{i1}^{(2)}z(t)+\Big{(}\bar{R}_{i1}\bar{A}_{i12}^{( 2)}+\bar{R}_{i1}\bar{A}_{i11}^{(2)}\bar{H}_{i21}\Big{)}y(t)+\bar{R}_{i1}\bar{ B}_{i1}^{(2)}(u(t)-H_{i}x_{m}(t))\] \[+\bar{R}_{i1}\bar{H}_{i21}\hat{y}(t)+\bar{R}_{i2}\hat{y}(t)-G_{i}A _{m}x_{m}(t)-\hat{G}_{i}x_{m}(t)+\bar{R}_{i1}\bar{B}_{i1}^{(2)}H_{i}x_{m}(t)\]  
By (22), we can conclude that \(z(t)=\hat{x}_{1}^{(1)}(t)-\bar{H}_{i21}y(t)\), which means equation above is:  
\[= \bar{R}_{i1}\bar{A}_{i11}^{(2)}\Big{(}\hat{x}_{1}^{(1)}(t)-\bar{H }_{i21}y(t)\Big{)}+\Big{(}\bar{R}_{i1}\bar{A}_{i12}^{(2)}+\bar{R}_{i1}\bar{A}_ {i11}^{(2)}\bar{H}_{i21}\Big{)}y(t)\] \[+(\bar{R}_{i2}+\bar{R}_{i1}\bar{H}_{i21})\hat{y}(t)-\big{(}G_{i}A _{m}+\hat{G}_{i}+B_{i}H_{i}\big{)}x_{m}(t)+\bar{R}_{i1}\bar{B}_{i1}^{(2)}(u(t) -H_{i}x_{m}(t))\]  
By (4) and (5) in Lemma 1 and Equation (23), and the equation above, we have:  
\[=\bar{R}_{i11}\bar{A}_{i11}^{(2)}\hat{x}_{1}^{(1)}(t)+\bar{R}_{i1}\bar{A}_{i 12}^{(2)}y(t)+B_{i}\delta u(t)-\big{(}G_{i}A_{m}+\hat{G}_{i}\big{)}x_{m}(t)\]  
By (1) of Lemma 1, \(\bar{R}_{i1}\bar{A}_{i11}^{(2)}=A_{i}\bar{R}_{i1}-\bar{R}_{i2}\bar{A}_{i21}^{( 2)}\). Combined with (2) in Lemma 1, \(\bar{R}_{i1}\bar{A}_{i11}^{(2)}=A_{i}\bar{R}_{i1}+\bar{R}_{i2}\bar{R}_{i3}\), we plug into the above equation  
\[=\bar{R}_{i1}\hat{x}_{1}^{(1)}(t)+\bar{R}_{i2}\hat{y}(t)-G_{i}A_{m}x_{m}(t)- \hat{G}_{i}x_{m}(t)\]0 \[=\bar{R}_{i1}\hat{x}_{1}^{(1)}(t)+\bar{R}_{i2}\hat{y}(t)-G_{i}A_{m}x_{m}(t)- \hat{G}_{i}x_{m}(t)\]1 \[=\bar{R}_{i1}\hat{x}_{1}^{(1)}(t)+\bar{R}_{i2}\hat{y}(t)-G_{i}A_{m}x_{m}(t)- \hat{G}_{i}x_{m}(t)\]2 \[=\bar{R}_{i1}\hat{x}_{1}^{(1)}(t)+\bar{R}_{i2}\hat{y}(t)-G_{i}A_{m}x_{m}(t)- \hat{G}_{i}x_{m}(t)\]3 \[=\bar{R}_{i1}\hat{x}_{1}^{(1)}(t)+\bar{R}_{i2}\hat{y}(t)-G_{i}A_{m}x_{m}(t)- \hat{G}_{i}x_{m}(t)\]4  
By Theorem 1 and Equations (3) and (6) in Lemma 1, we are able to show that:  
\[\left\{ \begin{array}{l}\delta\hat{x}(t)=A_{i}\delta x(t)+B_{i}\delta u(t),\\ \delta y(t)=C_{i}\delta\hat{x}(t)+(C_{i}G_{i}-C_{m})x_{m}(t)=C_{i}\delta\hat{x}(t).\end{array} \right. \tag{25}\]Clearly, system (23) and system (1) have the same structures and, hence, the state feedback control law \(u(t)=K_{i}x(t)\), which can stabilize system (1), and the state feedback \(\delta u(t)=K_{i}\delta x(t)\) can also stabilize system (21), which is to say that the system  
\[\left\{ \begin{array}{l}\delta\dot{x}(t)=[A_{i}+B_{i}K_{i}]\delta x(t),\\ y(t)=C_{i}\delta x(t).\end{array} \right.