# A New Modified Non-Approximative Method for Dynamic Systems Direct Calculation
## 3 Problem Solution and Simulation Example
### Direct Non-Approximative Method of Calculation  
By successive derivation of Equations (8) and (24a), we gradually obtain:  
\[\mathbf{x}_{k}=(\mathbf{F}_{\Delta})^{k}\mathbf{x}_{0}+\sum_{l=0}^{k-1}\mathbf{F}_{\Delta}{}^ {k-l-1}\mathbf{G}_{\Delta}\{\mathbf{u}_{l}\}. \tag{25}\]  
where the \(\mathbf{F}_{\Delta}\) and \(\mathbf{G}_{\Delta}\) matrices can be determined by Equation (24b). Thus,  
**Proof**:  
\[ \begin{array}{c}\mathbf{x}_{1}=\mathbf{F}_{\Delta}{}^{1}\mathbf{x}_{0}+\mathbf{F}_{\Delta}{} ^{0}\mathbf{G}_{\Delta}\mathbf{u}_{0}\\ \mathbf{x}_{2}=\mathbf{F}_{\Delta}{}^{2}\mathbf{x}_{0}+\mathbf{F}_{\Delta}{}^{1}\mathbf{G}_{ \Delta}\mathbf{u}_{0}+\mathbf{F}_{\Delta}{}^{0}\mathbf{G}_{\Delta}\mathbf{u}_{1}\\ \vdots\\ \vdots\\ \vdots\\ \mathbf{x}_{k}=\mathbf{F}_{\Delta}{}^{k}\mathbf{x}_{0}+\mathbf{F}_{\Delta}{}^{k-1}\mathbf{G}_{ \Delta}\mathbf{u}_{0}+\mathbf{F}_{\Delta}{}^{k-2}\mathbf{G}_{\Delta}\mathbf{u}_{1}+\\ +\mathbf{F}_{\Delta}{}^{k-(k-1)}\mathbf{G}_{\Delta}\mathbf{u}_{k-2}+\mathbf{F}_{\Delta}{}^{k- k}\mathbf{G}_{\Delta}\mathbf{u}_{k-1}\end{array}  \tag{26a}\]  
Therefore, we have to know the exciting vector \(\mathbf{u}\) as a function of \(k\) or \(l\). Since the functions of \(\mathbf{u}\) vector are analytics, it is sufficient to replace continuous times with discrete ones. If not, then a lookup table must be used. The relation (25) can be also used for \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{n}\) & \(\mathbf{0}\) & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\ldots\) & \(\mathbf{\hat{\mathbf{\mathbf{\gamma}}}}\) \\ \hline \(x_{1}(n.t)\) & \(0\) & \(42.6718\) & \(50.6871\) & \(48.2844\) & \(\ldots\) & \(\ldots\) & \(10\) \\ \(x_{2}(n.t)\) & \(0\) & \(40.4274\) & \(66.0035\) & \(81.5198\) & \(\ldots\) & \(\ldots\) & \(100\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of variables \(x_{1}(t)\) and \(x_{2}(t)\) at discrete time instant \(n.\tau\), \(u_{1}(t)=U\). non-linear systems when all non-stationary elements of \(F_{\Delta\text{non}}\) and \(G_{\Delta\text{non}}\) are transferred into the vectors of the input exciting functions \(\mathbf{u}_{\Delta fict}\):  
\[\mathbf{x}_{k}=\Big{(}\mathbf{F}_{\Delta fict}\Big{)}^{k}\mathbf{x}_{0}+\sum_{l=1}^{k}\mathbf{F }_{\Delta fict}^{k-l}\mathbf{G}_{\Delta fict}\Big{\{}\mathbf{u}_{\Delta fict,\ 1}\Big{\}} \tag{26b}\]  
where matrices \(F_{\Delta fict}\), \(G_{\Delta fict}\) are stationary ones, and there is no need to count them in each calculation step. \(\square\)  
If the excitation function is fictitiously (\(\tau\)) or truly periodically (\(T/2\)) repeating, then we can use \"the long jumps\" approach. Outgoing from Equation (23), we substitute the fundamental and transition matrices with the small integration step (\(l\)) by matrices with periodically repeating a long jump, for example, \(\tau\) or \(T/2\) and index \(k\) by \(n\), where discrete time is equal to \(n\). \(\tau\) or \(n\). \(T/2\), respectively. \[\mathbf{F}_{\tau}=\left(\mathbf{F}_{\Delta}\right)^{k}\ ;\ \ \ \mathbf{G}_{\tau}=\sum_{l=0}^{k-1}\mathbf{F }_{\Delta}^{k-l-1}\mathbf{G}_{\Delta} \tag{27}\]  
The relations (27) for \(\mathbf{F}_{\tau}\), \(\mathbf{G}_{\tau}\) for \(k=\frac{\tau}{h}\). are also valid for \(\mathbf{F}_{T/2}\), \(\mathbf{G}_{T/2}\) for \(k=\frac{T/2}{h}\). Figure 3 shows the courses of total system energies represented by the instantaneous energy of the state variables \(x_{1}\) and \(x_{2}\)[10; 11] as a function of \(k\), Equation (3a,b), calculated by the direct DNM method in equidistance instants _k.h._ Thus,  
\[E(\tau)=\frac{1}{2}\Big{[}C_{1}x_{1}^{2}(\tau)+C_{2}x_{2}^{2}(\tau)\Big{]}. \tag{28}\]  
where \(C_{1}\) and \(C_{2}\) are electrical constants (inductance \(L\) and capacitance \(C\)). Therefore, Equation (28) expresses a direct relationship between the total energy \(E(\tau)\) and the matrices \(\mathbf{F}_{\tau}\) and \(\mathbf{G}_{\tau}\) because of  
\[ \begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix} _{\tau}=\mathbf{F}_{\tau} \begin{bmatrix}x_{1}\\ x_{2}\end{bmatrix} _{0}\] (28a) for Figure 3a, and \[