## 3 Explicit MPC Controller Design
### Construction of Explicit MPC Model  
For a specific flight stage, the state space model is established at different low pressure shaft speeds, by linearizing the integrated model of the turbofan engine and jet noise. The state space model and equilibrium parameters corresponding to other \(N_{l}\) values are obtained by using \(N_{l}\) values as interpolation variables, so as to establish the variable parameter linear model within the whole range of \(N_{l}\) variation in the flight state. Under a certain value of \(N_{l}\), the discrete state space model of the system can be written as:  
\[ \begin{array}{c}\Delta x(k+1)=A\Delta x(k)+B\Delta u(k)\\ \Delta y(k)=C\Delta x(k)+D\Delta u(k)\end{array}  \tag{16}\]  
Among them, state \(x^{T}=[N_{h},N_{l},SPL]\), and input \(u^{T}=\left[w_{f},\alpha_{fan},A_{microjet}\right]\). \(\Delta\) represents the difference relative to the corresponding value of the balance point at the speed, that is, if the balance point corresponding to a certain \(N_{l}\) is \(\{x_{0},u_{0},y_{0}\}\), then \(\Delta x=x-x_{0}\). The discrete state space model is used as the prediction model to calculate the output of the system corresponding to the input at multiple times in the future. Assuming a finite control horizon \(N\) and prediction horizon \(M\), the control increment sequence is \(\Delta U=\left[\Delta u(k),\Delta u(k+1)\cdots\Delta u(k+N-1)\right]^{T}\), and the state increment sequence is \(\Delta X=\left[\Delta x(k+1),\Delta x(k+2)\cdots\Delta x(k+N-1)\cdots\Delta x (k+M)\right]^{T}\). Each term in the state increment sequence can be expanded with Equation (16), as follows:  
\[ \begin{array}{c}\Delta x(k+1)=A\Delta x(k)+B\Delta u(k)\\ \Delta x(k+2)=A^{2}\Delta x(k)+AB\Delta u(k)\\ +B\Delta u(k+1)\\ \vdots\\ \Delta x(k+N)=A^{N}\Delta x(k)+A^{N-1}B\Delta u(k)\\ +\cdots+B\Delta u(k+N-1)\\ \Delta x(k+N+1)=A^{N+1}\Delta x(t)+A^{N}B\Delta u(k)\\ +\cdots+(AB+B)\Delta u(k+N-1)\\ \vdots\\ \Delta x(k+M)=A^{M}\Delta x(k)+A^{M-1}B\Delta u(k)+\cdots\\ +\left(A^{M-N}B+\cdots+B\right)\Delta u(k+N-1)\end{array}  \tag{17}\]  
Figure 5: Active jet noise reduction controller structure based on explicit model prediction method. The output increment sequence is \(\Delta Y=[\Delta y(k),\Delta y(k+1)\cdots\Delta y(k+N-1)\cdots\Delta y(k+M-1)]^{T}\). With matrix multiplication, \(\Delta Y\) can be expressed as follows:  
\[\Delta Y=S_{x}\Delta x(k)+S_{u}\Delta U \tag{18}\]  
where  
\[\Delta U=\left[ \begin{array}{c}\Delta u(k)\\ \vdots\\ \Delta u(k+N-1)\end{array} \right]_{(N\times 1)}\]  
\[S_{x}=\left[ \begin{array}{c}C\\ CA\\ CA^{2}\\ \vdots\\ CA^{M-1}\end{array} \right]_{(M\times 1)} \tag{19}\]  
\[S_{u}=\left[ \begin{array}{cccc}D&0&0&\cdots&0\\ CB&D&0&\cdots&0\\ CAB&CB&D&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ CA^{M-2}B&CA^{M-3}B&CA^{M-4}B&\cdots&D+\sum\limits_{i=0}^{M-N-1}CA^{i}B\end{array}  \right]_{(M\times N)}\]  
The track performance of the controller is determined by the output errors. The smaller their absolute value, the better the track performance. In addition, the variation of control sequence must be limited. At the time \(k+i\), the track performance cost \(J(k+i)\) is defined as:  
\[ \begin{split} J(k+i)=(\Delta r(k+i)-\Delta y(k+i))^{T}Q(\Delta r (k+i)-\Delta y(k+i))\\ +\Delta u(k+i)^{T}R\Delta u(k+i)\end{split}  \tag{20}\]  
where \(Q\), \(R\) are the weight matrices, and \(\Delta r(k+i)\) is the target command. The total performance cost \(J(x(k))\) is the sum of \(J\) in a finite horizon. It can be calculated as follows:  
\[J(x(k))=(\Delta R-\Delta Y)^{T}W_{Q}(\Delta R-\Delta Y)+\Delta U^{T}W_{R} \Delta U \tag{21}\]  
where  
\(\Delta R=[\Delta r(k),\Delta r(k+1)\cdots\Delta r(k+N-1)\cdots\Delta r(k+M-1) ]^{T}\);  
\(W_{Q}=\text{diag}(Q,Q,\cdots Q)\)  
\(W_{R}=\text{diag}(R,R,\cdots R)\). Considering Equation (18), Equation (21) can be simplified into the following form:  
\[