# Design and Implementation of Attitude Stabilization System for Marine Satellite Tracking Antenna
## 4 \(\mathcal{H_{\infty}}\) Loop-Shaping Design  
\(\mathcal{H_{\infty}}\) loop shaping is a popular controller design method that has been successfully applied in many fields [28; 29; 30; 31]. The loop-shaping design procedure described in this section is based on \(H_{\infty}\) robust stabilization combined with classical loop shaping, which was proposed by [PERSON] and [PERSON] [32; 33; 34]. It consists of two steps. First, the open-loop plant is augmented by pre- and post-compensators to give a desired shape to the Bode magnitude plot of the open-loop frequency response. Then, the resulting shaped plant is robustly stabilized with respect to coprime factor uncertainty using \(\mathcal{H_{\infty}}\) optimization [35]. Note that this coprime factor uncertainty does not use the uncertainty information from the plant. The coprime uncertainty description provides a good \"generic\" uncertainty description for cases where we do not use any specific a priori uncertainty information [35]. The plant, expressed in Equation (18), is pre- and post-compensated, respectively, by \(W_{1}\) and \(W_{2}\), as shown below,  
\[G_{\mathrm{s}}=W_{1}\cdot G_{\mathrm{vs}}\cdot W_{2} \tag{19}\]  
where \(W_{1}\) is the pre-compensator, \(W_{2}\) is the post-compensator, and \(G_{\mathrm{s}}\) is the shaped plat. The main function of attitude stabilization system is for disturbance rejection. According to the product performance requirements, the RMS value of pitch tracking error should be less then 0.2 deg while the disturbance from ship is,  
\[D_{\mathrm{s}}=A\cdot sin\left(\frac{2\pi}{T}\cdot t\right) \tag{20}\]  
where \(A=25\) deg and \(T=6\) s.  
In our case, \(W_{1}\) was set to be the inverse of \(G_{\mathrm{vs}}\) to counteract the high open-loop gain of \(G_{\mathrm{vs}}\) at \(w_{n}\). Then, \(W_{2}\) was selected to get acceptable disturbance rejection while keep the system stable. To get good disturbance rejection, the loop gain, \(|G_{\mathrm{s}}(j\omega)|\), should be as large as possible within the bandwidth region. For the existence of unmodeled high-frequency dynamics and limitations on the allowed manipulated inputs, \(|G_{\mathrm{s}}(j\omega)|\) has to drop below one at and above the crossover frequency \(\omega_{c}\). After trial and error, \(W_{2}\) was set to be,  
\[W_{2}=\frac{6.8\cdot(s^{2}+13.3s+361)}{s^{2}\cdot(s^{2}+45s+2025)} \tag{21}\]  
where the phase lead compensator, \(\frac{s^{2}+13.3s+361}{s^{2}+45s+2025}\), are applied for enhancing the responsiveness and stability of the system. The Bode plot of the frequency response of \(G_{\mathrm{s}}\) is shown in Figure 9.  
Next, the procedure of \(\mathcal{H_{\infty}}\) robust stabilization of \(G_{\mathrm{s}}\) is stated. The perturbed plant model of the nominal plant, \(G_{\mathrm{s}}\), is expressed as,  
\[G_{\mathrm{s}p}=(M_{l}+\Delta_{M_{l}})^{-1}(N_{l}+\Delta_{N_{l}}) \tag{22}\]  
where \(M_{l}\) and \(N_{l}\) are stable coprime transfer functions obtained from left coprime factorization of \(G_{\mathrm{s}}\), that is, \(G_{\mathrm{s}}=M_{l}{}^{-1}\cdot N_{l}\). \(\Delta_{M_{l}}\) and \(\Delta_{N_{l}}\) are stable unknown transfer functions which represent the uncertainty in \(G_{\mathrm{s}}\). The family of perturbed plants that should be stabilized are defined as,  
\[G_{\mathrm{s}p} = \left\{(M_{l}+\Delta_{M_{l}})^{-1}(N_{l}+\Delta_{N_{l}}):\right.\] \[\left.\left\|\left[ \begin{array}{cc}\Delta_{N_{l}}&\Delta_{M_{l} }\end{array} \right]_{\infty}<\varepsilon\right\|\right\}\]  
where \(\varepsilon>0\) is the stability margin. The objective of robust stabilization is to maximize \(\varepsilon\). The perturbed feedback system of velocity loop is shown in Figure 10, where \(u\) is the manipulated variable, \(y\) is the rotation velocity of elevation part and is also the measurement of gyroscope, and \(K_{sp}\) is the velocity controller for shaped plant. The stability property is robust if and only if the nominal feedback system is stable and  
\[\gamma\stackrel{{\Delta}}{{=}}\left\|\left[ \begin{array}{c}K_{ sp}\\ I\end{array}