# Control Strategy of In-Port U-Turn for Ships Based on Arctangent Function Nonlinear Feedback
## 4 Proof of System Stability  
After many comparative experiments, this paper uses the arctangent function \(f(e)=\eta\arctan(\omega e)\) to improve the performance of the controller. The controller has a simple structure and is widely used in various engineering control applications; its parameters are easy to adjust. Therefore, this paper uses the robust controller obtained through the CGSA to control a ship's automatic U-turn in a port. After many simulation experiments, the control effect is satisfied when \(\eta=2,\omega=0.3\). 1. Impact on the stability of the system: it can be seen that the function \(\arctan(\omega e)\) has the n-th order derivative at the origin point; through Taylor expansion, it can be obtained that \(\arctan(\omega e)\approx\omega e\). Therefore, \(f(e)=\eta\arctan(\omega e)\approx\eta\omega e\); after the introduction of nonlinear feedback, the system control law changes from \(K\) to \(\eta\omega K\), which is equivalent to adding a coefficient \(\eta\omega<1\) on the basis of the original controller. Moreover, the structure of the system has not changed and does not affect the closed-loop stability of the system [27]. In order to further prove the stability of the system, the Routh criterion is used in this paper. The expression of the second-order Nomoto model of the ship is shown as Equation (18): \[G(s)=\frac{K_{0}}{s(T_{0}s+1)}\] (18) Write the characteristic function of the system as Equation (19) and make a Routh table: \[D(s)=4T_{0}s^{2}+(T_{0}+4)s+1+4\rho\] (19) According to Equation (18), the Routh table of the characteristic function \(D(s)\) can be written. It can be known that, in the first column of the Routh table, \(4T_{0}\), \(T_{0}s+4\), and \((1+4\rho)(T_{0}s+4)\) are all positive numbers. According to the Routh stability criterion, the numbers in the first column of the Routh table are all positive numbers, which can prove that the system is stable. 2. Influence on the stabilization value of the system: Take the step signal, \(\psi_{r}\), as the input signal; according to the final value theorem of Laplace transform, the heading angle of the steady state output of the system \(\psi\) is shown as Equation (20): \[\psi(\infty) =\lim_{t\rightarrow\infty}\psi(t)=\lim_{s\to 0}sF(s)= s\times\frac{G(s)K(s)\eta\omega}{1+G(s)K(s)\eta\omega}\times\frac{\psi_{r}}{s}\] (20) \[\psi(\infty) =s\times\frac{\left(\frac{K_{0}}{s(T_{0}s+1)}\right)\left(\frac{ 1}{K_{0}T_{c}}+\rho+\frac{T_{c}}{K_{0}T_{0}}\right)\eta\omega}{1+\left(\frac{ K_{0}}{s(T_{0}s+1)}\right)\left(\frac{1}{K_{0}T_{c}}+\rho+\frac{T_{c}}{K_{0}T_{0}} \right)\eta\omega}\times\frac{\psi_{r}}{s}\] \[=\frac{K_{0}\left(\frac{K_{0}T_{c}}{K_{0}T_{c}}+\rho+\frac{T_{c} }{K_{0}T_{0}}\right)\eta\omega}{s(T_{0}s+1)+K_{0}\left(\frac{1}{K_{0}T_{c}}+ \rho+\frac{T_{c}}{K_{0}T_{0}}\right)\eta\omega}\times\psi_{r}\] \[=\psi_{r}\]
3. Impact on the dynamic performance of the system: according to the closed-loop gain-shaping algorithm, the open-loop frequency characteristics (\(G(s)\times K(s)\)) of the system need to meet the requirements of high gain at low-frequency and low gain at high-frequency. Furthermore, the introduction of a nonlinear feedback link has little impact on the dynamic performance of the system. 4. The transfer function of the rudder angle from the input \(\delta\) of the system to the output \(\psi_{r}\) is shown as Equation (21): \[\frac{\delta}{\psi_{r}}=\frac{\eta\omega K}{1+\eta\omega GK}\] (21)It can be seen that the introduction of nonlinear function makes the denominator change more significantly than the numerator, so the introduction of a nonlinear feedback link does not affect the original control effect of the controller, and the same control effect can be achieved with less energy. In this part, we choose arctangent function to realize nonlinear feedback. Some reasons for this choice are presented here:  
1. The arctangent function possesses smooth nonlinear characteristics, which can provide different control gains within different error ranges.