## 4 Results
### Conditional Optimization under the Influence of Nonzero Initial Conditions
#### 4.2.1 System Description  
**Plant**  
The most general \(SISO\) plant's linear time-invariant discrete-time input-output equation is analyzed,  
\[a_{\
u}py(k+\
u)+a_{(\
u-1)}py(k+\
u-1)+\cdots+a_{1}py(k+1)+a_{0 }py(k)=b_{0}pu_{\
u}(k)+\cdots+b_{(\mu-1)}u_{\
u}(k+\mu-1)+b_{\mu}pu_{\
u}(k+\mu), \tag{21}\] \[a_{\
u}=1,\
u\in\mathcal{N},\mu\in\mathcal{N}_{0},\mu\leq\
u,\]  
where \(k\in\mathcal{N}_{0},y(k+j)\in\mathcal{R}\) is the plant output at time \(k+j\), \(\forall j=0,1,2,\cdots,\
u,u_{\
u}(k+j)\in\mathcal{R}\) is the plant input at time \(k+j\), \(\forall j=0,1,2,\cdots,\mu\), \(\mu\leq\
u\); \(a_{jP}\in\mathcal{R}\), \(\forall j=0,1,2,\cdots,\
u\), and \(b_{jP}\in\mathcal{R}\), \(\forall j=0,1,2,\cdots,\mu\), are real numbers. In our case, it is Equation (11), such that  
Figure 1: Membership functions. \(a_{1P}=1\), \(a_{0P}=-0.9398\), \(b_{0P}=0.1023\), \(\omega(k)=y(k)\), \(u(k)=u_{P}(k)\). The compact form of Equation (11) is as follows, [43,45]:  
\[\mathbf{A}_{P}^{(1)}\mathbf{y}^{1}=\mathbf{B}_{P}^{(0)}\mathbf{u}_{P}^{0}, \mathbf{A}_{P}^{(1)}=\left[ \begin{array}{cc}-0.9398&1\end{array} \right], \mathbf{B}_{P}^{(0)}=\left[ \begin{array}{cc}0.1023\end{array} \right], \tag{22}\]  
\[\mathbf{y}^{1}=\left[ \begin{array}{cc}y(k)&y(k+1)\end{array} \right]^{T}, \mathbf{u}_{P}^{0}=\left[ \begin{array}{cc}u_{P}(k)\end{array} \right], \tag{23}\]  
where \(\mathbf{A}_{P}^{(1)}\), \(\mathbf{B}_{P}^{(0)}\) are extended plant matrices and \(\mathbf{y}^{1}\), \(\mathbf{u}_{P}^{0}\) are the extended output and input vectors of the plant. **Controller**  
The first-order PS controller general form is  
\[\frac{\Delta u(k)}{T}+u(k)=K\varepsilon(k)+K_{S}T\sum_{i=0}^{i=k-1},\quad \bigg{/}\Delta \tag{24}\]  
\[\frac{\Delta u(k+1)-\Delta u(k)}{T}+\Delta u(k)=K\Delta\varepsilon(k)+K_{S}T \varepsilon(k),\quad\bigg{/}\cdot T \tag{25}\]  
\[u(k+2)-2u(k+1)+u(k)+Tu(k+1)-Tu(k)=KT\varepsilon(k+1)-KT\varepsilon(k)+K_{S}T^ {2}\varepsilon(k), \tag{26}\]  
\[u(k+2)+(T-2)u(k+1)+(1-T)u(k)=KT\varepsilon(k+1)+\Big{(}K_{S}T^{2}-KT\Big{)} \varepsilon(k). \tag{27}\]  
Compact form of this first order \(PS\) controller is:  
\[\mathbf{A}_{C}^{(2)}\mathbf{u}^{2}=\mathbf{B}_{C}^{(1)}\varepsilon^{1}, \tag{28}\]  
where \(\mathbf{A}_{C}^{(2)}\), \(\mathbf{B}_{C}^{(1)}\) are extended controller matrices and \(\mathbf{u}^{2}\) and \(\varepsilon^{1}(k)\) are extended output and input vectors of the controller, as follows,  
\[\mathbf{A}_{C}^{(2)}=\left[ \begin{array}{cc}1-T,&T-2,&1\end{array} \right], \mathbf{B}_{C}^{(1)}=\left[ \begin{array}{cc}K_{S}T^{2}-KT,&KT\end{array}  \right],T=0.002[\mathrm{s}], \tag{29}\]  
\[\mathbf{u}^{2}=\left[ \begin{array}{cc}u(k)&u(k+1)&u(k+2)\end{array} \right]^{ T},\varepsilon^{1}=\left[ \begin{array}{cc}\varepsilon(k)&\varepsilon(k+1)\end{array}  \right]^{T}. \tag{30}\]  
**The system**  
In the compact form the closed loop system is defined by:  
\[\mathbf{A}^{(3)}(\alpha,\beta)\mathbf{y}^{3}=\mathbf{B}^{(2)}(\alpha,\beta) \mathbf{i}^{2},\forall k\in\mathcal{N}_{0},\alpha=K,\beta=K_{S}, \tag{31}\]  
where \(\mathbf{A}^{(3)}\), \(\mathbf{B}^{(2)}\) are extended system matrices, \(\mathbf{y}^{3}\), \(\mathbf{i}^{2}\) are extended output and input vectors of the system as follows,  
\[\mathbf{A}^{(3)}(\alpha,\beta)=\left[ \begin{array}{cc}a_{0}(\alpha,\beta)&a _{1}(\alpha,\beta)&a_{2}(\alpha,\beta)&a_{3}(\alpha,\beta)\end{array} \right]= \left[ \begin{array}{cc}-2.046\cdot 10^{-4}\alpha+4.1\cdot 10^{-7}\beta-0.93794 \\ 2.046\cdot 10^{-4}\alpha+0\beta+2.8758\\ 0\alpha+0\beta-2.9378\\ 0\alpha+0\beta+1\end{array} \right]^{T}, \tag{32}\]  
\[\mathbf{B}^{(2)}(\alpha,\beta)=\left[ \begin{array}{cc}\mathbf{B}_{0}(\alpha, \beta)&\mathbf{B}_{1}(\alpha,\beta)&\mathbf{B}_{2}(\alpha,\beta)\end{array} \right], \tag{33}\]  
\[\mathbf{B}^{(2)}(\alpha,\beta)=\left[ \begin{array}{cc}-2.046\cdot 10^{-4}\alpha+4.092 \cdot 10^{-7}\beta,&0.1021\end{array} \right]\left[ \begin{array}{cc}2.046\cdot 10^{- 4}\alpha,&-0.2044\end{array} \right]\left[ \begin{array}{cc}0&0.1023\end{array}  \right]\right], \tag{34}\]  
\[\mathbf{y}^{3}=\left[