## 3 Design of the Controller
### Theoretical Proof
 \tag{31}\]  
If we set \(h_{1}=b_{L1}+b_{L1}c_{B}b_{u}\)=\(b_{L1}(1+c_{B}b_{u})\), \(b_{1}=(b_{L1}b_{a}+b_{L2}b_{a}+b_{L3}b_{a})\), Formula (31) can be simplified again as follows:  
\[\|\Delta u_{i+1}(k)\|\leq\|I-L_{1}B_{i}\|\|\Delta u_{i}\|+h_{1}\|\Delta q_{i} \|+b_{L2}\|\Delta q_{i+1}\|+b_{L3}\|\Delta q_{i-1}\|+b_{1} \tag{32}\]  
The Formula (32) contains the system output error function and the system state error function; however, we need to eliminate the system state error to obtain the output error function. Substitute the status error (29) into the above system Formula (32) to obtain the following:  
\[\begin{array}{c}\|\Delta u_{i+1}(k)\|\leq\|I-L_{1}B_{i}\|\|\Delta u_{i}\|+h_ {1}\left[\sum\limits_{j=0}^{k-1}\left[h_{2}^{k-1-j}(b_{B}\|\Delta u_{i}\|+b_{ \beta})+h_{2}^{k}  
If we set \(\lambda>\max\{1,h_{2},b_{B}(b_{L2}+b_{L3})\}\), we can obtain the following:  
\[\left[1-\frac{b_{B}(b_{L2}+b_{L3})[1-(h_{2}/\lambda)^{n}]}{\lambda-h_{2}}\right] >0\Rightarrow\left\{ \begin{array}{c}\alpha-h_{2}\
eq 0\\ \alpha>h_{2}+b_{B}(b_{L2}+b_{L3})\\ \alpha-h_{2}>b_{B}(b_{L2}+b_{L3})[1-(h_{2}/\lambda)^{n}]\end{array} \right. \tag{35}\]  
Then, the above inequality (35) can be simplified as follows:  
\[\left\|\Delta u_{i+1}(k)\right\|_{\lambda}\leq A\left\|\Delta u_{i}(k)\right\| _{\lambda}+B \tag{36}\]  
where  
\[A=\frac{\left[\rho+\frac{b_{B}h_{1}}{\lambda-h_{2}}\right]}{\left[1-\frac{b_{ B}(h_{2}+b_{3})[1-(h_{2}/\lambda)^{n}]}{\lambda-h_{2}}\right]},B=\frac{ \left[h_{1}b_{q_{0}}+b_{L2}b_{q_{0}}+b_{L3}b_{q_{0}}+b_{1}+\frac{b_{B}(h_{1}+ b_{2}+b_{L3})[1-(h_{2}/\lambda)^{n}]}{\lambda-h_{2}}\right]}{\left[1-\frac{b_{ B}(h_{2}+b_{L3})[1-(h_{2}/\lambda)^{n}]}{\lambda-h_{2}}\right]} \tag{37}\]  
Iterating the output error from the initial moment to the current moment, the output error at any moment can be obtained:  
\[\left\|\Delta u_{i+1}(k)\right\|_{\lambda}\leq A\left\|\Delta u_{i}(k)\right\| _{\lambda}+B=A^{i}\left\|\Delta u_{1}(k)\right\|_{\lambda}+\frac{B(1-A^{i})}{ 1-A} \tag{38}\]  
If we set \(\lambda\) to be very large and the system parameter \(A\approx\rho<1\), Formula (38) can be simplified as follows:  
\[\underset{i\rightarrow\infty}{\lim}\left\|\Delta u_{i}(k)\right\|_{\lambda} \leq\frac{B}{1-A} \tag{39}\]  
When the output disturbance \(b_{\alpha}=0\), the initial error \(b_{q_{0}}=0\) and the state disturbance signal \(b_{\beta}=0\) are brought into Formulas (37) and (39), the output error of the controller is proved to approach zero asymptotically. Namely, the actual output of the controller is infinitely close to the desired control signal. Step 3: In analyzing the error between the actual trajectory and the expected trajectory, we need to use the state error of the system. According to Formulas (29) and (39), we can find the expression of the state error. To analyze the output error of the system, we also need to multiply the state error by \((\lambda)^{-k}\). Then, the expression can be simplified as follows:  
\[ \begin{array}{l}\left\|\Delta q_{i}\right\|\left(\frac{1}{\lambda^{k}}\right) \leq\sum\limits_{j=0}^{k-1}\frac{h_{2}^{k-1-j}}{\lambda^{k-1-j}}(b_{B}\frac{ \left\|\Delta u_{i}\right\|}{\lambda}+b_{\beta}\frac{1}{\lambda^{j}})+\frac{h_ {2}^{k}}{\lambda^{k}}b_{q_{0}}\\ \leq(b_{B}\left\|\Delta u_{i}\right\|_{\lambda}+b_{\beta})\sum\limits_{j=0}^{k- 1}\frac{h_{2}^{k-1-j}}{\lambda^{k-1-j}}+b_{q_{0}}\\ =b_{B}\frac{[1-(h_{2}/\lambda)^{n}]}{\lambda-h_{2}}\left\|\Delta u_{i}\right\| _{\lambda}+b_{\beta}\frac{[1-(h_{2}/\lambda)^{n}]}{\lambda-h_{2}}+b_{q_{0}} \end{array}  \tag{40}\]  
After infinite iterations, the state error of system can be simplified to the following:  
\[\underset{i\rightarrow\infty}{\lim}\left\|\Delta q_{i}\right\|_{\lambda}\leq b _{B}\frac{[1-(h_{2}/\lambda)^{n}]}{\lambda-h_{2}}\frac{B}{1-A}+b_{\beta} \frac{[1-(h_{2}/\lambda)^{n}]}{\lambda-h_{2}}+b_{q_{0}} \tag{41}\]  
where \(b_{B}\) represents the maximum norm of the control matrix, \(b_{\beta}\) represents the maximum norm of the state disturbance, and \(b_{q_{0}}\) represents the initial error of the system.