## 3 Design of Composite Control Method
### Optimized Design of Disturbance Observer
#### 3.2.2 Evaluation Function Definition  
1. Evaluation function for a two-degree-of-freedom system:  
According to Equations (19) and (20), the impact of disturbance \(d\) on the output depends on the sensitivity function \(S(s)\), and the effect of sensor noise \(\xi\) on the output depends on the complementary sensitivity function \(T(s)\). Meanwhile, based on the small gain theorem (see Equation (14)), the robust stability conditions of the closed-loop system also depend on \(T(s)\). In a two-degrees-of-freedom control system, we define the performance function of the disturbance observer with order and structure constraints as [26]:  
\[\max\gamma,\min_{Q(s)}\left\|\left[ \begin{array}{l}\gamma W_{S}(s)S(s)\\ W_{T}(s)T(s)\end{array} \right]\right\|_{\infty}= \tag{21}\] \[\max\gamma,\min_{Q(s)}\left\|\left[ \begin{array}{l}\gamma W_{S} (s)(1+P_{\text{n}}(s)K(s))^{-1}(1-Q(s))\\ W_{T}(s)(P_{\text{n}}(s)K(s)+Q(s))(1+P_{\text{n}}(s)K(s))^{-1}\end{array} \right] \right\|_{\infty}<1\]  
In the evaluation function of the two-degrees-of-freedom system, the transfer function has a complex form, especially concerning the complementary sensitivity function related to the robust stability conditions. Due to the robust stability conditions and structural constraints, this \(H_{\infty}\) control problem cannot be directly solved through systematic methods. It is necessary to transform this problem into a form that can be systematically solved. The performance function of the two-degrees-of-freedom system can be converted into the performance function of the inner-loop DOB system for solving. 2. Evaluation function for the inner loop DOB system:  
Figure 12: Feedback control system with DOB. The inner-loop DOB system, as indicated by the red dashed box in Figure 12, has an input-output equation when the controlled object and the model are consistent (\(P_{\text{n}}=P\)):  
\[ \begin{array}{l}y=\frac{P}{1+Q[P-P_{\text{n}}]P_{\text{n}}^{-1}}u-\frac{P(1- Q)}{1+Q[P(s)-P_{\text{n}}]P_{\text{n}}^{-1}}d-\frac{PQ}{P_{\text{n}}+Q[P-P_{ \text{n}}]}\xi\\ =P(s)u-P(s)(1-Q(s))d-Q(s)\end{array} . \tag{22}\]  
In the DOB system, without considering the outer loop, disturbances \(d\) and detection noise \(\xi\) pass through the sensitivity function (\(S_{\text{DOB}}(s)=1-Q(s)\)) and complementary sensitivity function (\(T_{\text{DOB}}(s)=Q(s)\)) of the DOB system, respectively, affecting the system. To ensure that the system was not influenced by external disturbances and measurement noise, the terms involving \(d\) and \(\xi\) in the equation above had to be minimized. This involved making \(1-Q(s)\) and \(Q(s)\) sufficiently small; however, the sum of these two terms is always equal to \(1\), making it impossible for both to be minimized simultaneously. Therefore, a trade-off needed to be considered during the design. Similar to the \(H_{\infty}\) control mixed sensitivity problem, a weighted function was used to address the frequency trade-off issue. The evaluation function for the DOB system is defined as follows:  
\[\max\gamma,\min_{Q(s)\in\Omega_{k}}\left\| \begin{bmatrix}\gamma W_{\text{C}}(s) (1-Q(s))\\ W_{Q}(s)Q(s)\end{bmatrix} \right\|_{\infty}<1. \tag{23}\]  
The evaluation Function (23) involves complex norm conditions and constraints related to orders and relative orders, making it impractical to solve using standard \(H_{\infty}\) control problem-solving methods. Reference [27] transformed the optimization problem with order constraints into a standard \(H_{\infty}\) control problem. Thus, Equation (23) can be further transformed into an unconstrained optimization problem:  
\[\max\gamma,\min_{\widetilde{K}(s)}\left\| \begin{bmatrix}\gamma W_{\text{C}}(s )(1+\widetilde{P}(s)\widetilde{K}(s))^{-1}\\ W_{Q}(s)\widetilde{P}(s)\widetilde{K}(s)(1+\widetilde{P}(s)\widetilde{K}(s)) ^{-1}\end{bmatrix} \right\|_{\infty}<1, \tag{24}\]  
where \(\widetilde{P}(s)\) and \(\widetilde{K}(s)\)) represent the virtual plant and virtual controller of the open-loop system.