## 3 The MPC Controller Design for the Controllable Pitch Propeller
### Optimized Design  
The optimization problem design is based on two main considerations. On the one hand, the controlled output is required to be consistent with the reference input. On the other hand, imposing constraints is necessary. For the first aspect, the objective function is chosen as the form:  
\[J_{1}=\sum\limits_{i=1}^{p}\sum\limits_{j=1}^{n_{c}}\left(\Gamma_{y_{i}}(y_{cj }(k+i|k)-r_{j}(k+i))\right)^{2} \tag{22}\]  
where \(i=1,2,\cdots,p\) is the \(j\) component of the reference input, and \(\Gamma_{y_{i}}\) is the weighting factor for the \(j\) difference between the controlled output and the reference input. During controller design, the performance of the controllable pitch propeller is influenced by the variation between the controlled output and the reference input. When the error requirement is high, the weighting factor \(\Gamma_{y_{i}}\) can be increased. Depending on the needs of a controllable pitch propeller, the weighting factor can be time-varying. \[J_{1}=\sum\limits_{i=1}^{p}\sum\limits_{j=1}^{n_{c}}\left(\Gamma_{y_{j},i}(y_{ cj}(k+i|\ k)-r_{j}(k+i))\right)^{2}=\sum\limits_{i=1}^{p}\big{|}\big{|}\Gamma_{y _{i}}(y_{c}(k+i|k)-r(k+i))\big{|}\big{|}^{2} \tag{23}\]  
A mathematical description of the constraints on the action is needed for proper control action. The difference in the control input \(\Delta u\) can be introduced into the objective function design. The control action can be constrained by weighting. \[J_{2}=\sum\limits_{i=1}^{m}\|\ \Gamma_{u,i}\Delta u(k+i-1)\|^{2} \tag{24}\]  
Ultimately, the optimization problem combining Equations (23) and (24) can be described as \(\min\limits_{\Delta U(k)}J(x(k),\triangle U(k),m,p)\). \[J(x(k),\Delta U(k),m,p)=\sum\limits_{i=1}^{p}\big{|}\big{|}\Gamma_{y,i}(y_{c}( k+i|k)-r(k+i))\big{|}\big{|}^{2}+\sum\limits_{i=1}^{m}\big{|}\Gamma_{u,j}\Delta u (k+i-1)\|^{2} \tag{25}\]  
According to Equation (20), Equation (25) is transformed into the following equation:  
\[J(x(k),\Delta U(k),m,p)=\!\Big{|}\Big{|}\Gamma_{y}(Y_{p}(k+1\big{|}k)-R(k+1)) \Big{|}\Big{|}^{2}\!+\Big{|}\Big{|}\Gamma_{u}\Delta U(k)\Big{|}\Big{|}^{2} \tag{26}\]The weighting matrices are shown as follows:  
\[\Gamma_{y}=diag(\Gamma_{y,1},\Gamma_{y,2},\cdots,\Gamma_{y,p}),\;\Gamma_{u}=diag( \Gamma_{u,1},\Gamma_{u,2},\cdots,\Gamma_{u,m}) \tag{27}\]  
The reference input sequence is:  
\[R(k+1)= \begin{bmatrix}r(k+1)&r(k+2)&\cdots&r(k+p)\end{bmatrix} ^{T} \tag{28}\]  
Equation (26) is a quadratic matrix function, which can be decomposed for ease of solution. \[J(x(k),\Delta U(k),m,p)=\rho^{T}\rho \tag{29}\]  
\[\rho= \begin{bmatrix}\Gamma_{y}(Y_{p}(k+1|k)-R(k+1))\\ \Gamma_{u}\Delta U(k)\end{bmatrix}  \tag{30}\]  
Substituting the predictive Equation (21) with the auxiliary variable Equation (30), we can obtain the following:  
\[ \begin{array}{l}\rho=\begin{bmatrix}\Gamma_{y}(S_{x}\Delta x(k)+ly_{c}(k)+S _{d}\Delta d(k)+S_{u}\Delta U(k)-R(k+1))\\ \Gamma_{u}\Delta U(k)\end{array} \\ = \begin{bmatrix}\Gamma_{y}S_{u}\\ \Gamma_{u}\end{bmatrix} \Delta U(k)- \begin{bmatrix}\Gamma_{y}(R(k+1|-S_{x} \Delta x(k)-ly_{c}(k)-S_{d}\Delta d(k))\\ 0\end{bmatrix} \\ = \begin{bmatrix}\Gamma_{y}S_{u}\\ \Gamma_{u}\end{bmatrix} \Delta U(k)- \begin{bmatrix}\Gamma_{y}(E_{p}(k+1|k))\\ 0\end{bmatrix} \\ =Az-b\end{array} \tag{31}\]  
To facilitate the subsequent calculations, the above formula is simplified. \[z=\Delta U(k),A= \begin{bmatrix}\Gamma_{y}S_{u}\\ \Gamma_{u}\end{bmatrix} ,b= \begin{bmatrix}\Gamma_{y}(E_{p}(k+1|k))\\ 0\end{bmatrix}  \tag{32}\]  
\[E_{p}(k+1|k)=R(k+1)-S_{x}\Delta x(k)-ly_{c}(k)-S_{d}\Delta d(k) \tag{33}\]  
The optimization problem \(\underset{\Delta U(k)}{\min}J(x(k),\triangle U(k),m,p)\) can be transformed into \(\underset{z}{\min}\rho^{T}\rho\). \[\rho=Az-b \tag{34}\]  
\(\rho^{T}\rho=\left(Az-b\right)^{T}\left(Az-b\right)\) is derived to obtain the extreme points.