## 3 Problem Formulation and \(H_{2}\) Controller Design
### Closed-Form Solution of the Nonlinear Time-Varying Riccati-Like Equation  
To consider the solution \(J_{i2}(e,t)\) of the nonlinear Riccati-like equation, Equation (12) can be shown in a more distinct form, with the state transformation in Equation (11) as:  
\[J_{i2}(e,t)=\Gamma_{i2}^{T}\left[ \begin{array}{cc}M_{i2}(r_{i})&0_{3\times 3 }\\ 0_{3\times 3}&Y_{i2}\end{array} \right]\Gamma_{i2} \tag{13}\]  
where \(\Gamma_{i2}\) and \(Y_{i2}\) are some designed positive distinct symmetric constant matrices and will be worked out from a pair of algebraic Riccati-like equations later. Studying the second and third terms on the left-hand side of the time-varying Riccati-like Equation (12), and applying the tracking error dynamic system in Equation (5) and solution \(J_{I2}(e,t)\) in Equation (13), the following formulation can be described as:  
\[J_{I2}(e,t)W_{I2}(e,t)+W_{I2}^{T}(e,t)I_{I2}(e,t)=\left[ \begin{array}{cc}0_{3 \times 3}&Y_{I2}\\ Y_{I2}&0_{3\times 3}\end{array} \right]+\Gamma_{I2}^{T}\left[ \begin{array}{cc}- \mathbb{M}_{I2}(r_{i})&0_{3\times 3}\\ 0_{3\times 3}&0_{3\times 3}\end{array} \right]\Gamma_{I2} \tag{14}\]  
The above equation can be verified easily as below:  
\[V_{I2}^{T}(e,t)J_{I2}(e,t)=(\Gamma_{I2}^{-1}VM_{I2}^{-1}(r))^{T}\Gamma_{I2}^{T }\left[ \begin{array}{cc}M_{I2}(r)&0_{3\times 3}\\ 0_{3\times 3}&Y_{I2}\end{array} \right]\Gamma_{I2}=V_{I2}^{T}\Gamma_{I2} \tag{15}\]  
According to results in Equations (14) and (15), it is easy to confirm that Equation (12) can be represented as an algebraic Riccati-like equation:  
\[\left[ \begin{array}{cc}0_{3\times 3}&Y_{I2}\\ Y_{I2}&0_{3\times 3}\end{array} \right]+Q_{I2}-\Gamma_{I2}^{T}VR_{I2}^{-1}iV^{T} \Gamma_{I2}=0 \tag{16}\]  
To assume one solution \(\Gamma_{I2}\) can be obtained mathematically and from Equation (15), the optimal robust control laws \(u_{I2}^{*}(e,t)\) in Equation (8) can be described as:  
\[u_{I2}^{*}(e,t)=-R_{I2}^{-1}V^{T}\Gamma_{I2}e_{I2}(t) \tag{17}\]  
By selecting  
\[R_{I2}=\beta_{I2}^{2}I_{6\times 6} \tag{18}\]  
where \(\beta_{I2}>0\). To assume the weighted matrix \(Q_{I2}\) in Equation (7), it can be factorized by the Cholesky factorization, shown below:  
\[Q_{I2}=\left[ \begin{array}{cc}Q_{112}^{T}Q_{112}&Q_{122}\\ Q_{212}^{T}&Q_{222}^{T}Q_{222}\end{array} \right] \tag{19}\]  
Applying the definitions of \(V\) and \(\Gamma_{I2}\) in Equations (10) and (11) together with assumption Equations (18) and (19), the Riccati-like Equation (16) can be separated into the following equations:  
\[Q_{112}^{T}Q_{112}-\frac{1}{\beta_{I2}^{2}}\Gamma_{112}^{T}\Gamma_{112}=0 \tag{20}\]  
\[Y_{I2}+Q_{122}-\frac{1}{\beta_{I2}^{2}}\Gamma_{112}^{T}\Gamma_{122}=0 \tag{21}\]  
\[Y_{I2}+Q_{122}^{T}-\frac{1}{\beta_{I2}^{2}}\Gamma_{122}^{T}\Gamma_{112}=0 \tag{22}\]  
\[Q_{222}^{T}Q_{222}-\frac{1}{\beta_{I2}^{2}}\Gamma_{122}^{T}\Gamma_{122}=0 \tag{23}\]  
Based on Equations (20) and (23), the matrices \(\Gamma_{112}\) and \(\Gamma_{122}\) can be described as:  
\[\Gamma_{112}=\beta_{I2}Q_{112} \tag{24}\]  
and  
\[\Gamma_{122}=\beta_{I2}Q_{222} \tag{25}\]From \(\Gamma_{112}\) and \(\Gamma_{122}\) in Equations (24) and (25), the solution \(\Gamma_{i2}\) can be represented as:  
\[\Gamma_{i2}=\left[ \begin{array}{cc}\beta_{i2}Q_{112}&\beta_{i2}Q_{222}\\ 0_{3\times 3}&I_{3\times 3}\end{array} \right] \tag{26}\]  
For fulfilling by \(\Gamma_{112}=\psi_{i2}I_{3\times 3}\) in Equation (11), the weighted matrix \(Q_{112}\) in Equation (19) must be chosen as the below diagonal type:  
\[Q_{112}=d_{112}I_{3\times 3} \tag{27}\]  
By selecting a positive scale, \(d_{112}\) and \(\psi_{i2}\), they then have the relationship:  
\[\psi_{i2}=\beta_{i2}d_{112} \tag{28}\]  
By the same token, substituting Equations (26) into (21) and (22), we can obtain the following equation:  
\[Q_{i2}=\left[ \begin{array}{cc}Q_{112}^{T}Q_{112}&Q_{122}\\ Q_{122}^{T}&Q_{222}^{T}Q_{222}\end{array}