# A New Modified Non-Approximative Method for Dynamic Systems Direct Calculation
## 2 Problem Formulation  
### Linear ODE System in Matrix State-Space Form  
Consider a classical dynamic system equation,  
\[\left.\begin{array}{l}\frac{\mathrm{d}x_{1}(t)}{\mathrm{d}t}-a_{11}x_{1}(t) -\cdots-a_{1n}x_{n}(t)=\\ +b_{11}u_{1}(t)+\cdots+b_{1n}u_{n}(t)\\ \cdots\cdots\cdots\cdots\\ \frac{\mathrm{d}x_{n}(t)}{\mathrm{d}t}-a_{n1}x_{1}(t)-\cdots-a_{m}x_{n}(t)=\\ +b_{n1}u_{1}(t)+\cdots+b_{mn}u_{n}(t)\end{array}\right\} \tag{1}\]  
which can be also presented in matrix state-space form (2):  
\[\frac{\mathrm{d}x(t)}{\mathrm{d}t}=\begin{bmatrix}a_{11}&\ldots&a_{1n}\\ \ldots&\ldots&\ldots\\ a_{n1}&\ldots&a_{nm}\end{bmatrix}\mathbf{x}(t)+\begin{bmatrix}b_{11}&\ldots&b_{1n }\\ \ldots&\ldots&\ldots\\ b_{n1}&\ldots&b_{nm}\end{bmatrix}\mathbf{u}(t) \tag{2}\]  
where  
\[\begin{bmatrix}a_{11}&\ldots&a_{1n}\\ \ldots&\ldots&\ldots\\ a_{n1}&\ldots&a_{nm}\end{bmatrix}=\mathbf{A}, \tag{3a}\] \[\begin{bmatrix}b_{11}&\ldots&b_{1n}\\ \ldots&\ldots&\ldots\\ b_{n1}&\ldots&b_{nm}\end{bmatrix}=\mathbf{B} \tag{3b}\]  
are system and transition matrices, and  
\[\mathbf{x}(t)=\begin{bmatrix}x_{1}(t)\\ \ldots\\ x_{n}(t)\end{bmatrix}\text{ and }\mathbf{u}(t)=\begin{bmatrix}u_{1}(t)\\ \ldots\\ u_{n}(t)\end{bmatrix} \tag{4}\]  
are state variables and exciting vectors, respectively. Thus,  
\[\frac{\mathrm{d}\mathbf{x}(t)}{\mathrm{d}t}=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{u}(t), \tag{5a}\] \[\mathbf{x}(t_{0})=\mathbf{x}_{0} \tag{5b}\]  
Such a system of ODE can be solved analytically; it can also be solved using integral transforms (Laplace or Laplace-Carson [3; 6]) or numerically [1; 2]. For the numerical solution, we consider the one-step implicit [PERSON]'s method, although it is also possible to use any other numerical method (Taylor's expansion, Runge-Kutta, etc.).  
Numerical Solution--Time Discretization  
The one-step explicit [PERSON] method yields:  
\[\mathbf{x}_{k+1}=(\mathbf{E}+h\mathbf{A})\mathbf{x}_{k}+h\mathbf{B}\mathbf{u}_{k} \tag{6}\]where \(h\) is the integration step and \(\mathbf{E}\) is the unity or Eye matrix, respectively. The method is sensitive to the integration step. The stability condition is that \(h\) should be smaller than \(2/\left|\text{Re}\left\{\lambda_{i}\right\}\right|_{\max}\)[2], where \(\lambda_{i}\) is the \(i\)-root of the characteristic polynomial of the system matrix.  
The one-step implicit [PERSON] method yields:  
\[\mathbf{x}_{k+1}=\text{inv}(\mathbf{E}-h\mathbf{A})[\mathbf{x}_{k}+h\mathbf{B}\mathbf{u}_{k}] \tag{7}\]  
where \(\text{inv}(\mathbf{E}-h\mathbf{A})=\mathbf{F}\) is a fundamental matrix of the system. Contrary to above this method, for the negative real part of eigenvalues, it is absolutely stable (A-stabile) for any positive step \(h\)[2]. Another possibility for calculation is to use the well-known Taylor expansion or Runge-Kutta methods. Similar to the [PERSON] implicit method above, these methods are also A-stabile.  
All discrete equations carried out by [PERSON] explicit, implicit, and Taylor expansion methods are easily solvable by numerical computing using Equation (8):  
\[\mathbf{x}_{k+1}=\mathbf{F}\mathbf{x}_{k}+\mathbf{G}\mathbf{u}_{k} \tag{8}\]  
The compact form of this equation,  
\[\mathbf{x}_{k}=f(\mathbf{F},\mathbf{x}_{0},\mathbf{G},\mathbf{u}_{0}) \tag{9}\]  
is often needed and is derived in Section 3. Therefore, the direct calculation of \(\mathbf{x}_{k}\) is our problem for the next solution.