## 3 Design of Composite Control Method
### Optimized Design of Disturbance Observer
#### 3.2.2 Evaluation Function Definition
This problem, without any order constraints and satisfying the assumptions of the standard problem, can be solved systematically using an optimization algorithm to obtain the optimal filter \(Q(s)\). This design approach not only meets all the order requirements of DOB's filter \(Q(s)\), such as order conditions, relative order conditions, and internal model order conditions, but also ensures global optimality and convergence in the systematic solving process. 3. Transformation of evaluation functions:  
In the performance functions of the two-degrees-of-freedom system (see Equation (21)) and the DOB system (see Equation (23)), there is a fundamental difference in terms of robust stability. However, in terms of form, it is possible to transform the performance function of the two-degrees-of-freedom system into that of the DOB system. The specific steps are as follows:  
On one hand, the condition \(\left\|W_{T}(s)T(s)\right\|_{\infty}<1\) in Equation (21) is a sufficient condition for the robust stability of the system. By the sensitivity function \(S(s)\) and the properties of the absolute value of complex numbers, we can derive the inequality:  
\[\left|Q(j\omega)\right|<\left|W_{T}^{-1}(j\omega)(1+L(j\omega))\right|-|L(j \omega)|,\ \forall\omega, \tag{25}\]  
where \(L(s)=P_{\text{n}}(s)C(s)\) is the open-loop transfer function of the feedback system at the nominal state, only if the filter \(Q(s)\) satisfies this inequality, the entire control system will be stable. In other words, Equation (25) is a sufficient condition for the robust stability of the closed-loop system. Choosing a stable weighting function \(W_{TD}(s)\) that satisfies\[\left|W_{TD}^{-1}(j\omega)\right|<\left|W_{T}^{-1}(j\omega)(1+L(j\omega))\right|- \left|L(j\omega)\right|\triangleq E(\omega),\ \forall\omega. \tag{26}\]  
The sufficient condition for robust stability in Equation (25) can be reformulated as  
\[\left\|W_{TD}(s)Q(s)\right\|_{\infty}<1. \tag{27}\]  
On the other hand, the term \(max\ \gamma,\underset{Q(s)}{\text{min}}\|\gamma\cdot W_{S}(s)S(s)\|_{\infty}<1\) in Equation (21) represents the performance evaluation function for disturbance rejection. According to Equation (20) with \(T\)(s), the evaluation function can be expressed as:  
\[\max\gamma,\underset{Q(s)}{\text{min}}\left\|\gamma\cdot W_{s}(s)(1+L(s))^{-1} (1-Q(s))\right\|_{\infty}<1. \tag{28}\]  
Choose a weight function \(W_{SD}\)(s) that satisfies  
\[\left|W_{s}(j\omega)(1+L(j\omega))^{-1}\right|\leq\left|W_{SD}(j\omega)\right|,\ \forall\omega. \tag{29}\]  
Then, the sufficient condition of Formula (28) can be expressed as  
\[\max\gamma,\underset{Q(s)}{\text{min}}\|\gamma\cdot W_{SD}(s)(1-Q(s))\|_{ \infty}<1. \tag{30}\]  
According to Equations (27) and (30), the evaluation function of the two-degrees-of-freedom system can be expressed as  
\[\max\gamma,\underset{Q(s)}{\text{min}}\left\| \begin{bmatrix}\gamma W_{SD}(s) (1-Q(s))\\ W_{TD}(s)Q(s)\end{bmatrix} \right\|_{\infty}<1. \tag{31}\]  
Equation (31) is the same as the evaluation function for the DOB system (see Equation (23)). Therefore, the original design problem for the two-degrees-of-freedom system's filter \(Q\)(s) (see Equation (21)) can be transformed into a standard design problem for the inner-loop DOB system. The optimal filter for Equation (31) can be obtained through the standard \(H_{\infty}\) control framework (see Equation (24)).