## 8 Fuzzy Incremental-N State Model  
An iterative algorithm has been proposed to obtain an incremental-N state model that introduces N integrators and can achieve zero steady-state error in response to polynomial type inputs and disturbances of order N-1. To do this, the extended state matrices are initialized by assigning them to the traditional discrete-state matrices:  
\[ \begin{split} A_{a^{N}}&=A\\ B_{a^{N}}&=B\\ C_{a^{N}}&=C\end{split}  \tag{44}\]  
Using a loop, the extended state matrices are recalculated N times:  
\[ \begin{split} for\;\;i&=1:N\\ A_{a^{N}}&=\left[\begin{array}{cc}I&C_{a^{N}}A_{a^{N}} \\ 0&A_{a^{N}}\end{array}\right]\\ B_{a^{N}}&=\left[\begin{array}{cc}C_{a^{N}}B_{a^{N}}\\ B_{a^{N}}\end{array}\right]\\ C_{a^{N}}&=\left[\begin{array}{cc}I&0\end{array}\right]\\ end\end{split}  \tag{45}\]  
In this way, the incremental-N state model would be defined:  
\[ \begin{split}\left[\begin{array}{c}y(k+1)\\ \Delta x_{a^{N-1}}(k+1)\end{array}\right]&=\left[\begin{array}{cc}I&C_{a^{N- 1}}A_{a^{N-1}}\\ 0&A_{a^{N-1}}\end{array}\right]\left[\begin{array}{c}y(k)\\ \Delta x_{a^{N-1}}(k)\end{array}\right]+\left[\begin{array}{cc}C_{a^{N-1}}B_ {a^{N-1}}\\ B_{a^{N-1}}\end{array}\right]\Delta^{N}u(k)\\ y(k)&=\left[\begin{array}{cc}I&0\end{array}\right]\left[\begin{array}{c} y(k)\\ \Delta x_{a^{N-1}}(k)\end{array}\right]\end{split}  \tag{46}\]  
In matrix form,  
\[ \begin{split} x_{a^{N}}(k+1)&=A_{a^{N}}x_{a^{N}}(k)+B_{a^{N}} \Delta^{N}u(k)\\ y(k)&=C_{a^{N}}x_{a^{N}}(k)\end{split}  \tag{47}\]  
where \(A_{a^{N}}\in\Re^{((Nq+n)\times(Nq+n))}\), \(B_{a^{N}}\in\Re^{((Nq+n)\times p)}\), and \(C_{a^{N}}\in\Re^{(q\times(Nq+n))}\), thus obtaining the new extended vector of incremental-N state, \(x_{a^{N}}(k)\in\Re^{(Nq+n)}\). This would be  
\[x_{a^{N}}(k)=\left[ \begin{array}{c}y(k)\\ \Delta y(k)\\ \Delta^{2}y(k)\\ \vdots\\ \Delta^{(N-1)}y(k)\\ \Delta^{N}x(k)\end{array} \right] \tag{48}\]  
and the reference state is  
\[x_{a^{N}r}=\left[ \begin{array}{c}y_{r}\\ \Delta y_{r}\\ \Delta^{2}y_{r}\\ \vdots\\ \Delta^{(N-1)}y_{r}\\ 0\end{array} \right] \tag{49}\]The controller and control action for the incremental-N would be defined:  
\[ \begin{split}\Delta^{N}u(k)&=K(x_{a^{N}r}-x_{a^{N}}(k) )\\ \Delta^{(N-1)}u(k)&=\Delta^{(N-1)}u(k-1)+\Delta^{N}u(k) \\ \Delta^{(N-2)}u(k)&=\Delta^{(N-2)}u(k-1)+\Delta^{(N- 1)}u(k)\\ &\quad\vdots\\ \Delta u(k)&=\Delta u(k-1)+\Delta^{2}u(k)\\ u(k)&=u(k-1)+\Delta u(k)\end{split}  \tag{50}\]  
where the incremental-N state feedback matrix K can be obtained, for example, using the discrete LQR method, whose calculation will use the state matrices \(A^{N}_{u}\) and \(B^{N}_{a}\) obtained through the recursive algorithm of Equation (45), along with the matrices \(Q\in\Re^{((Nq+n)\times(Nq+n))}\) and \(R\in\Re^{(p\times p)}\). For the incremental-N model, as with the previous incremental models, a state observer is required since \(\Delta^{N}x(k)\) is not accessible. Following the approach of Equations (27) and (41), we can estimate the incremental-N state as follows:  
\[ \begin{split}\Delta^{N}x_{e}(k)&=\Delta^{(N-1)}x_{e }(k)-\Delta^{(N-1)}x_{e}(k-1)\end{split}  \tag{51}\]  
Following the approach of the Equations (42) and (51), the observer can be described as  
\[ \begin{split}\Delta^{N}x_{e}(k+1)&=A\Delta^{N}x_{e }(k)+B\Delta^{N}u(k)\\ &+H\Big{(}\Delta^{(N-1)}y(k)-\Delta^{(N-1)}y(k-1)-C\Delta^{N}x_{e }(k)\Big{)}\end{split}  \tag{52}\]  
where \(H\in\Re^{(n\times q)}\) is calculated in the same way as in the Equation (31). Thus, the observer of the extended incremental-N state vector is defined:  
\[ \begin{split} x_{a^{N}e}(k)&=\left[\begin{array}[] {c}y(k)\\ \Delta y(k)\\ \Delta^{2}y(k)\\ \vdots\\ \Delta^{(N-1)}y(k)\\ \Delta^{N}x_{e}(k)\end{array}\right]\end{split}  \tag{53}\]  
Since the goal is to achieve fuzzy control using the incremental-N state model, the method to be used is explained below. First, after completing the linear identification and T-S identification, the nonlinear system is modeled using the incremental-N T-S state model. Starting from Equation (46), we obtain  
\[