# A New Modified Non-Approximative Method for Dynamic Systems Direct Calculation
## 3 Problem Solution and Simulation Example
### Direct Non-Approximative Method of Calculation
By comparing it with Table 5, we find that the differences in the values of the variables are negligible, but the calculation time is reduced against \[\mathbf{x}_{n}=\left(\mathbf{F}_{\tau}\right)^{n}\!\!\mathbf{x}_{0}+\sum_{m=0}^{n-1}\mathbf{F}_{ \tau}^{\,n-m-1}\mathbf{G}_{\tau}\{\mathbf{u}_{m}\}. \tag{29}\]5  
Time circumstances: computation time \(t_{\mathrm{comp}}\) (s) is shown in Table 7 for the Laplace-Carson method (first line), implicit Euler method (\(\mathrm{Eul}^{-1}\)--second line), and direct non-approximative method DNM (third line). We can also obtain--still for \(u_{1}(t)=U\)--a steady-state solution by inverse transform and using the finite value theorem (if needed). The lemma of finite value can be used to compute steady-state values of the \(x_{n}\):  
\[\mathbf{x}_{n}=\left(\mathbf{F}_{\tau}\right)^{n}\!\!\mathbf{x}_{0}+\sum_{m=0}^{n-1}\mathbf{F}_{ \tau}^{\,n-m-1}\mathbf{G}_{\tau}\{\mathbf{u}_{m}\}. \tag{29}\]6 (35a) or \[\mathbf{x}_{n}=\left(\mathbf{F}_{\tau}\right)^{n}\!\!\mathbf{x}_{0}+\sum_{m=0}^{n-1}\mathbf{F}_{ \tau}^{\,n-m-1}\mathbf{G}_{\tau}\{\mathbf{u}_{m}\}. \tag{29}\]7  
where \(z_{j}\) are the roots of the denominator of the fraction z-function. _Case of \(u_{1}(t)=U\mathrm{sign}\bigl{(}\sin\frac{2\pi}{\tau}t\bigr{)},\ u_{2}(t)=0\)_  
Applying the z-transform (Equation (31b)) yields:  
\[\mathbf{x}_{n}=\left(\mathbf{F}_{\tau}\right)^{n}\!\!\mathbf{x}_{0}+\sum_{m=0}^{n-1}\mathbf{F}_{ \tau}^{\,n-m-1}\mathbf{G}_{\tau}\{\mathbf{u}_{m}\}. \tag{29}\]8  
Applying the inverse z-transformation using the lemma of residua gives:  
\[\mathbf{x}_{n}=\left(\mathbf{F}_{\tau}\right)^{n}\!\!\mathbf{x}_{0}+\sum_{m=0}^{n-1}\mathbf{F}_{ \tau}^{\,n-m-1}\mathbf{G}_{\tau}\{\mathbf{u}_{m}\}. \tag{29}\]9  
where \(z_{j}\) is the \(j\)-th root of the denominator of Equation (30b), i.e., \(z_{1}=F_{T/2}\), \(z_{2}=-E\). Then,  
\[\mathbf{x}_{n+k}=\mathbf{F}_{k}(\mathbf{F}_{\tau})^{n}\mathbf{x}_{0}+\mathbf{F}_{k}\sum_{m=0}^{n-1} \mathbf{F}_{\tau}^{\,n-m-1}\mathbf{G}_{\tau}\{\mathbf{u}_{m}\}.\]0  
In this case, the matrices \(\mathbf{F}_{T/2},\ \mathbf{G}_{T/2}\) can be calculated using Equations (35) or (25) for step \(k=\frac{T/2}{h}\). The results of this case are shown in \[\mathbf{x}_{n+k}=\mathbf{F}_{k}(\mathbf{F}_{\tau})^{n}\mathbf{x}_{0}+\mathbf{F}_{k}\sum_{m=0}^{n-1} \mathbf{F}_{\tau}^{\,n-m-1}\mathbf{G}_{\tau}\{\mathbf{u}_{m}\}.\]1 \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(k\) & **0** & **1 \(\tau\)/\(h\)** & **2 \(\tau\)/\(h\)** & **3 \(\tau\)/\(h\)** & \(\dots\) & \(\mathbf{\dot{\tau}}\) \\ \hline \(x_{1}(k.h)\) & 0 & 42.6808 & 52.4828 & 48.4414 & \(\dots\) & \(\dots\) & 10 \\ \(x_{2}(k.h)\) & 0 & 40.4278 & 66.0040 & 81.5202 & \(\dots\) & \(\dots\) & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Values of variables \(x_{1}(t)\) and \(x_{2}(t)\) at discrete time instant \(t=n.t\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & **0** & **3** & **33** & **70** & \(\dots\) & \(\dots\) & **100** \\ \hline L-CT (s) & 0.0070 & 0.3034 & 2.7754 & 6.2100 & \(\dots\) & \(\dots\) & 10.0811 \\ \(\mathrm{Eul}^{-1}\) (s) & 0 & 0.3031 & 2.9460 & 6.2905 & \(\dots\) & \(\dots\) & 9.0285 \\ DNM (s) & 0 & 2.2914 & 2.3089 & 2.2983 & \(\dots\) & \(\dots\) & 2.4048 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Computation time at discrete time instant \(t=n.t\) for \(u_{0}(t)=U=\mathrm{const}\).