Solving a differential equation in matrix form but adding a constantx1′&nbsp;=2x1+3x2,x2′&nbsp;=x1−x2.

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yopopolin10d

Answered question

2022-04-20

Solving a differential equation in matrix form but adding a constant
$\begin{aligned} x_{1}^{'} & = 2 x_{1} + 3 x_{2}, \\ x_{2}^{'} & = x_{1} - x_{2} . \end{aligned}$

Answer & Explanation

smachttenbem

Beginner2022-04-21Added 18 answers

Step 1
This is a common form in control theory:
$\dot{x} = A x + B u$
where x and u are state and input vectors, while A and B are system matrices of appropriate dimension.
You can solve this in Laplace domain:
$s X (s) - x (0) = A X (s) + B U (s)$
from which it follows:
$X (s) = (s I - A)^{- 1} B U (s) + (s I - A)^{- 1} x (0)$
Step 2
Matrices and vectors for your example are:
$x = [\begin{matrix} x_{1} \\ x_{2} \end{matrix}], A = [\begin{matrix} 2 & 3 \\ 1 & - 1 \end{matrix}], B = [\begin{matrix} 5 \\ - 4 \end{matrix}]$
and $u (t) = μ (t)$ (step-wise function) with transformation $U (s) = \frac{1}{s}$ .

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