## 3 Design of Concise Robust Path-Following Control with RRR
### Control Design
#### 3.2.1 Concise Course Control Based on Backstepping  
**Theorem 1.**_Considering the ship course keeping system in Equation (5), the proposed algorithm in Equation (11) based on the Backstepping method is capable of stabilizing the yaw motion and guaranteeing the uniformly asymptotic stability of the closed-loop course system._  
\[u_{c}=-\frac{T}{K}\big{[}f(x_{c2})+(1+k_{c1}k_{c2})e_{c}+(k_{c1}+k_{c2})\dot{e} _{c}\big{]} \tag{11}\]  
_where the course deviation \(e_{c}=\psi-\psi_{d}\), \(k_{c1}\), \(k_{c2}\) are positive design parameters._  
**Proof.** Set \(x_{c1}=\psi\), \(x_{c2}=r\), and then one can get the transformed model of Equation (5).  
\[\left\{\begin{array}{l}\dot{x}_{c1}=x_{c2}\\ \dot{x}_{c2}=f(x_{c2})+b_{c}u_{c}\\ y_{c}=x_{c1}\end{array}\right. \tag{12}\]  
where \(y_{c}\in R\), \(b_{c}=\frac{K}{T}\), \(f(x_{c2})=-\frac{1}{T}x_{c2}\), which is a linear function.  
Define \(z_{c1}=x_{c1}-\psi_{d}\) and \(z_{c2}=x_{c2}-\alpha(z_{c1})\), and then one can get  
\[\dot{z}_{c1}=\dot{x}_{c1}-\dot{\psi}_{d}=x_{c2}-\dot{\psi}_{d}=z_{c2}+\alpha(z _{c1})-\dot{\psi}_{d} \tag{13}\]  
\(\alpha(z_{c1})\) is chosen as an intermediate control function defined as  
\[\alpha(z_{c1})=\dot{\psi}_{d}-k_{c1}z_{c1} \tag{14}\]  
and  
\[\dot{\alpha}(z_{c1})=\ddot{\psi}_{d}-k_{c1}\dot{z}_{c1} \tag{15}\]  
where \(k_{c1}>0\) is one design constant. Substituting Equation (14) into Equation (13), one can get  
\[\dot{z}_{c1}=-k_{c1}z_{c1}+z_{c2} \tag{16}\]  
Considering the following Lyapunov function candidate  
\[V_{c}=\frac{1}{2}z_{c1}^{2}+\frac{1}{2}z_{c2}^{2} \tag{17}\]  
\[\dot{V}_{c}=-k_{c1}z_{c1}^{2}+z_{c2}\big{[}f(x_{c2})-\dot{\alpha}(z_{c1})+z_{ c1}+b_{c}u_{c}\big{]} \tag{18}\]  
To guarantee \(\dot{V}_{c}\leq 0\), one can choose the actual control function as follows  
\[u_{c}=\frac{1}{b_{c}}\big{(}\dot{\alpha}(z_{c1})-f(x_{c2})-z_{c1}-k_{c2}z_{c2} \big{)} \tag{19}\]  
where \(k_{c2}>0\) is another design constant. Substituting Equation (19) into Equation (18), one can get  
\[\dot{V}_{c}=-k_{c1}z_{c1}^{2}-k_{c2}z_{c2}^{2}<0,\ \forall z_{c1}\
eq 0,z_{c2}\
eq 0 \tag{20}\]  
In accordance with [PERSON] stability theory, the course control system can be stabilized by the control law in Equation (19), and all the variables in course control loop are of uniformly asymptotic stability with an equilibrium point \([x_{c1},x_{c2}]=[\psi_{d},\dot{\psi}_{d}]\).  
Setting the reference course as \(\psi_{d}\) and \(\ddot{\psi}_{d}=0\), and substituting Equation (15) into Equation (19), the control law is transformed into Equation (21). It is obvious that the essence of the above Backstepping control design is to compensate and stabilize \(f(x_{\zeta 2})\) with a PD-type control law \(v_{\zeta}\).  
\[\begin{array}{l}u_{\zeta}=-\frac{1}{k_{\zeta}}\Big{[}f(x_{\zeta 2})+(1+k_{\zeta 1}k_{\zeta 2})(x_{\zeta 1}-\dot{\psi}_{d})+(k_{ \zeta 1}+k_{\zeta 2})(\dot{x}_{\zeta 1}-\dot{\psi}_{d})\Big{]}\\ =-\frac{T}{k}\big{[}f(x_{\zeta 2})+(1+k_{\zeta 1}k_{\zeta 2})e_{c}+(k_{\zeta 1}+k_{ \zeta 2})\dot{e}_{c}\big{]}\\ =-\frac{T}{k}\big{[}f(x_{\zeta 2})+v_{c}\big{]}\end{array} \tag{21}\]  
This ends the proof of Theorem 1. \(\square\)