# First order dynamic system

I have a system that is defined as follows:

$$\dot{x}(t) = Ax(t) + Bu(t),\quad\quad x(0)=x_0.$$

It has a well known solution:

$$x(t) = e^{tA}x_{0} + \int_0^t \! e^{(t-\tau)A}Bu(\tau) \, \mathrm{d}\tau$$

I know that one of the tasks during exam will be to draw the the graph of the solution given that A and B are numbers. If B is zero then this is no problem, but if it's not I don't really know how to calculate that integral. Could anyone please explain the steps that I need to perform to solve this?

PS> I've already found some examples but none of them showed the intermediate steps, so I can't figure out anything :/

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What is $u(t)$? – Hans Lundmark Feb 1 '11 at 17:47
X(t) is the systems state while u(t) is the input. If you have eg. a RC (resistor-capacitor) system then the voltage on the capacitor would be x(t) (the systems current state) and the source voltage would be u(t). – kubal5003 Feb 1 '11 at 17:50
But shouldn't the solution depend on $u$ then? – Hans Lundmark Feb 1 '11 at 18:09
Yes, there is a missing $u(\tau)$ in the integral.  In general, without further information about the input $u(t)$, you can't really simplify that integral further. – cch Feb 1 '11 at 18:58
You're right. I've investigated the problem further and probably the only possible options during the exam are that u(t) is constant , which is easy to solve. This is great, because I don't have to seek a "generic" solution to this problem. Hurray! :D PS> I am truly sorry for wasting your and anyone else's time. It took me some time to dig through all the materials and discover this. – kubal5003 Feb 1 '11 at 19:01

## 1 Answer

That integral is almost never computed. This is generally a Laplace transform problem where $u$ is typically a Heaviside(step), Dirac (delta) or some other Laplace transformable from the tables type of function, hence you don't need to assume $u$ a constant function. If one takes the Laplace transform of the state equation:

$$\begin{align} sX(s) - x(0) &= AX(s)+BU(s)\\ X(s) &= (sI-A)^{-1}x(0) + (sI-A)^{-1}BU(s) \end{align}$$ Possibly applying the Partial Fractional Expansion, via the inverse Laplace transform one gets $x(t)$ explicitly.

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Answering the unanswered Movement! :) – user13838 Nov 27 '11 at 22:23
upvoting the unupvoted. – Gerry Myerson Feb 26 '12 at 2:07
@GerryMyerson Hahaha, thanks! – user13838 Feb 26 '12 at 8:27
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