# Unit ste response of a second order system

#### Greeko

Joined Dec 1, 2011
2
Hi i have this following question which i am finding difficulty doing:

Derive an expression for the unit step response of the system;

g(s)=1/(s^2 +2s +17)

Then sketch this response and compare it to the response predicted using MATLAB.

I have found many examples on this type of question however all of them seem to have the numerator and the last number in the denominator to be the same. So the natural freqency(omega) being the square root of the numerator....? from my understanding without the natural frequency i cannot find the damping factor and therefore cannot fill the blanks in the response for the unit step...?

Any help will be great, thanks

#### Georacer

Joined Nov 25, 2009
5,182
Keep in mind that the Laplace transformation is linear. That means that if
$$f(x) \rightarrow ^{\mathfrak{L}} F(x) \leftrightarrow\\ a \cdot f(x) \rightarrow ^{\mathfrak{L}} a \cdot F(x).$$

That means that you can write $$g(s)=\frac{1}{17} \cdot \frac{17}{s^2+2s+17}$$ and solve it as you know.

#### Greeko

Joined Dec 1, 2011
2
Keep in mind that the Laplace transformation is linear. That means that if
$$f(x) \rightarrow ^{\mathfrak{L}} F(x) \leftrightarrow\\ a \cdot f(x) \rightarrow ^{\mathfrak{L}} a \cdot F(x).$$

That means that you can write $$g(s)=\frac{1}{17} \cdot \frac{17}{s^2+2s+17}$$ and solve it as you know.
Thanks for that, that makes sence.

#### Vahe

Joined Mar 3, 2011
75
I think the first thing to do is to determine the roots of $$s^2+2s+17$$. You should be able to show that the roots of this system are at $$a, b = -1 \pm j 4$$.

Given $$G(s)=Y(s)/X(s)$$, $$Y(s)$$ is the output (step response) when $$X(s)$$ is the input (unit step $$u(t)$$ and $$X(s)=1/s$$); therefore,

$$Y(s)=G(s)X(s)=\frac{1}{s} \frac{1}{s^2+2s+17}=\frac{K_0}{s} + \frac{K_1}{s+a}+\frac{K_2}{s+b}$$
with the factors $$K_0, K_1, K_2$$ to be determined by partial fraction expansion.

--Vahe