# Pitch Channel Control of a REMUS AUV with Input Saturation and Coupling Disturbances
## 3 Control Strategy
### Augment \(L_{1}\) Adaptive Control with Riccati-Based Anti-Windup Compensator  
In this part, a modern anti-windup scheme is extended into the L1 AC controller. To strengthen the feasibility of controller implemented in practice, the problem of actuator saturation is considered. Recently, several modern AW compensators have been used to tackle input saturation [22; 25]. However, most of the solutions to Linear Matrix Inequality (LMI) are prone to numerical errors due to bad initialization [22]. Moreover, the robustness of AW compensator controller is less concerned. The compensator selected is what can modify the performance of the controller when the saturation takes place. Riccati-based AWs is proposed to reduce the computational burden of solving LMI problem [22; 33]. Block diagram of the proposed controller is described in Figure 2.  
As shown in the diagram, the compensator produces two different outputs when the saturation of actuator is activated: one is fed back to control law, and the other is utilized to modify the control signal. The problem of AW compensator is transformed into a state feedback form, and is given by:  
\[ \begin{array}{l}\dot{x}_{aw}=(A+BF)x_{aw}(t)+B\tilde{u}\\ u_{aw}=Fx_{aw}(t)\\ y_{aw}=(C+DF)x_{aw}(t)+D\tilde{u}\end{array}  \tag{16}\]where \(x_{uw}\) and \(u_{uw}\) are two outputs of the AW compensator; \(y_{aw}\) is compensator measured output; and \(F\) is a free parameter. \(A+BF\) is necessary to be \(Hurwitz\). Therefore, the problem of designing a full-order AW compensator becomes that of choosing an appropriate state feedback gain matrix, \(F\). The term \(\tilde{u}\) is computed in Equation (17):  
\[\tilde{u}=u_{c}(t)-u(t) \tag{17}\]  
Considering the saturation of actuator input, the nonlinear function of actuator is  
\[u(t)=sat(u_{c}(t),u_{\max}) \tag{18}\]  
In Equations (17) and (18), \(u_{c}(t)\) is the control signal output of L1 AC controller, \(u(t)\) is the actuator or thrust output signal of compensator, and \(u_{\max}\) is the maximum applicable control signal. Specific form of actuator's output signal is expressed as  
\[u(t)=\left\{ \begin{array}{rl}u_{c}(t),&\text{if }|u_{c}(t)|\leq u_{\max}\\ u_{\max}sign(u_{c}(t)),&\text{if }|u_{c}(t)|>u_{\max}\end{array} \right. \tag{19}\]  
Feedback gain vector \(F\) obtained by computing the following Riccati equation:  
\[ \begin{array}{l}A^{T}P_{AW}+P_{AW}A-P_{AW}BRB^{T}P_{AW}+C^{T}C=0\\ F=-(\rho+\frac{W^{-1}}{\varepsilon})B^{T}P_{AW}\end{array}  \tag{20}\]  
where \(\rho\), \(W\) and \(\varepsilon\) are the AW compensator parameters. The robust analysis of these parameters has been investigated [22]. In this paper, the Riccati-based AW compensator is augmented into L1 AC architecture to modify control signal. The equation for the control law is granted as the following. \[ \begin{array}{l}u_{c}(s)=-kD(s)(\hat{u}(s)-k_{g}r(s)-u_{aw}(t))\\ \hat{u}(s)=\ell\left\{\hat{\mu}\right\}=\ell\left\{\hat{\omega}(t)u(t)+\hat{ \theta}(t)\|x(t)+x_{aw}(t)\|_{\text{L}\infty}+\phi(t)\right\}\end{array}  \tag{21}\]  
where \(u(t)\) is the input of actuator, and \(D(s)\) is often defined as \(1/s\). The L1 AC controller is required to satisfy \(L_{1}\) norm condition to ensure the stability of the system, and, when proper feedback gain \(k\) and filter \(D(s)\) can be chosen, the following inequality holds. \[\left\|G(s)_{L_{1}}\right\|L<1 \tag{22}\]  
where  
\[ \begin{array}{l}G(s)\triangleq H(s)(1-C(s))\\ H(s)\triangleq(sI-A_{m})^{-1}b\\ L\triangleq\max\left\|\theta\right\|_{1},\theta\in\Theta\\ C(s)\triangleq\omega\kappa D(s)/(1+\omega\kappa D(s))\end{array}  \tag{23}\]  
In Equation (23), \(C(s)\) is a strictly stable closed-loop filter, and the steady state value of the gain is \(C(0)=1\).