## 4 Results
### Conditional Optimization under the Influence of Nonzero Initial Conditions
#### 4.2.3 Performance Index  
Because the system steady state error \(\varepsilon_{s}\) is equal to zero and the control algorithm contains the sum (S) component, to calculate the performance index, we use Equation (45),  
\[I=\sum_{k=0}^{\infty}\varepsilon^{2}(k),\quad\varepsilon(k)=r(k)-y(k), \tag{45}\]  
Figure 2: The loci in the \(\alpha\beta\) parameter plane of the constant damping coefficient \(\zeta=0.7\). where error \(\varepsilon(k)\) occurred as a result of all simultaneous actions on the system-nonzero initial conditions and external nonzero input. In practice, it is more reasonable and accurate compared to the error \(\varepsilon(k)\) influenced solely by nonzero external input and for all zero initial conditions that have been mostly used in the literature so far. The full block diagram [43; 45] of the system, in general case, is shown in Figure 3.  
The plant \(\mathcal{Z}\) transfer function is \(G_{P}(z)\). The plant \(\mathcal{Z}\) transfer function matrix with respect to the plant vector of initial conditions \(\mathbf{c}_{0P}\) is \(\mathbf{G}_{0P}(z)\). Likewise, the controller \(\mathcal{Z}\) transfer function is \(G_{\mathrm{C}}(z)\) and the controller \(\mathcal{Z}\) transfer function matrix with respect to the controller vector of initial conditions \(\mathbf{c}_{0C}\) is \(\mathbf{G}_{0C}(z)\). The classical transfer function matrix with respect to the external input, for the short transfer function matrix \(\mathbf{G}(z)\), is only one element of the \(\mathbf{F}(z)\). There is another element of the \(\mathbf{F}(z)\), transfer function matrix \(\mathbf{G}_{0}(z)\) which is related to overall initial conditions. The only appropriate transfer function matrix, without any controversy, for testing Lyapunov stability properties is \(\mathbf{G}_{0}(z)\), which arises from \(\mathbf{F}_{md}(z)\), i.e., \(\mathbf{G}_{\mathrm{o}rnd}(z)\). From the block diagram, it is easy to obtain \(\mathcal{Z}\)-transform \(Y(z)\) of the output \(y(k)\):  
\[Y(z)=\mathbf{F}(z)\mathbf{V}(z)=\left[ \begin{array}{cc}\mathbf{G}(z)&\mathbf{ G}_{0}(z)\end{array} \right]\cdot\left[ \begin{array}{c}\mathbf{I}(z)\\ \mathbf{c}_{0}\end{array} \right]=\left[ \begin{array}{c}\frac{G_{P}(z)G_{ \mathrm{C}}(z)}{1+G_{P}(z)G_{\mathrm{C}}(z)}\\ \frac{G_{P}(z)}{1+G_{P}(z)G_{\mathrm{C}}(z)}\\ \frac{G_{P}(z)}{1+G_{P}(z)G_{\mathrm{C}}(z)}\mathbf{G}_{0C}^{\mathcal{T}}(z)\\ \frac{1}{1+G_{P}(z)G_{\mathrm{C}}(z)}\mathbf{G}_{0P}^{\mathcal{T}}(z)\end{array}  \right]^{T}\cdot\left[ \begin{array}{c}\mathbf{I}(z)\\ \mathbf{c}_{0C}\\ \mathbf{c}_{0P}\end{array} \right],\mathbf{I}(z)=\left[ \begin{array}{c}R(z) \\ D(z)\end{array} \right], \tag{46}\]  
Equivalent full block diagram of the system shown in Figure 3 is displayed in Figure 4.  
It should be noted that in the created TS model of the DC motor, disturbance was not considered. However, the proposed methodology below could be applied in the same way as in the case of presence of the disturbance \(D(z)\), where it should be necessary to determine the transfer function of DC motor in relation to the disturbance, as well as the transfer function matrix in relation to the initial conditions of the disturbance. From the most general block diagram, Figures 3 and 4, all the necessary transfer functions are calculated below for the DC motor considered in this paper, which is not affected  
Figure 4: Equivalent full block diagram of the closed loop system. Figure 3: Full block diagram of the closed loop system. by the disturbance. After applying Z-transform, including all initial conditions on plant Equation (11), the following is obtained:  
\[Y(z)=\left[ \begin{array}{cc}G_{P}(z)&G_{0P}(z)\end{array} \right]\left[ \begin{array} []{c}U(z)\\ c_{0P}\end{array}