# Concise Robust Control of the Nonlinear Ship in Course-Keeping Control
## 4 Design of a Nonlinear Robust Controller for Ship Course Control  
### Design of Concise Robust (CROB) Controller  
\(H_{\infty}\) robust control theory solves the problem of robust controller design for MIMO systems in the frequency domain. However, the whole design process is not only based on difficult mathematical theory but also requires many experiments to be carried out to choose the proper weight function to obtain a robust controller with better performance. Moreover,the order of the controller obtained is always too high, which is a time-consuming task compared with the robustness of the controller; in other words, the price is too high. Given the correlation between the mixed sensitivity algorithm and the compensation sensitivity function \(T\) and sensitivity function \(S\) in the robust control theory, \(G\) represents the transfer function for the system and \(K\) is the controller designed; the controller can be determined by constructing the four parameters related to compensation sensitivity function \(T\), such as maximum singular value, bandwidth, closing-door slope, and closed-loop of peak spectrum. Because these four parameters are of engineering significance, the controller is designed in accordance with the closed-loop transfer function of the system. Because of the correlation between \(S\) and \(T\), the shape of \(S\) can be determined indirectly once the shape of T is set up, and the robust performance and robust stability of the system are guaranteed. This method is called the closed-loop gain-forming algorithm, which is also named the concise robust control method. If the \(H_{\infty}\) mixed sensitivity control algorithm is the product of positive thinking, the loop-shaping algorithm can be said to be the product of divergent thinking, and the concise robust control algorithm based on closed-loop gain-shaping is the product of reverse thinking. The typical \(S/T\) Singular Value (SV) curve is shown in Figure 2. To make the system robust and stable, the closed-loop spectrum of the system is required to be low-pass, and the maximum singular value is set to 1 to ensure the tracking of the reference signal without static error; the bandwidth of the system determines the control performance of the system, and the closing-door slope rate of the spectrum determines the sensitivity of the system to the interference outside the invalid frequency band. The larger the slope, the less sensitive the influence on the interference, and the stronger the robustness of the system. However, if the slope of the closing door is too large, the order of the controller is too high, which is not practicable for the realization of the controller. Generally speaking, the slope of door-closing can be set as \(-20\) db/dec, \(-40\) db/dec, and \(-60\) db/dec. Assuming that the door-closing slope of the closed-loop system transfer function spectrum is \(-20\) dB/DEC, the singular value curve of the complementary sensitivity function T is approximately expressed as the spectrum curve of the first-order inertial system with the maximum singular value of 1:  
\[\frac{1}{Ts+1}=\frac{GK}{1+GK} \tag{7}\]  
If the bandwidth of the closed-loop system is \(1/T\), then the controller is  
\[K=\frac{1}{GT\,s} \tag{8}\]  
Figure 2: Typical singular value for \(S/T\). If the door-closing slope of the closed-loop system transfer function spectrum is \(-40\) dB/DEC, the singular value curve of \(T\) is approximately expressed as the spectrum curve of the second-order inertial system with the maximum singular value of \(1\).