# Describing step response in terms of poles and zeros of transfer function?

me and a friend are stuck describing something, we're both coming to terms with using matlab and we're trying to understand the design of a simple control system for controlling the steady-state error of an open loop system. We have a block diagram for such a control system here: simulate this circuit – Schematic created using CircuitLab

As an example, we take the transfer function of the open loop system (G(s)) to be G(s) = 10/(s^2 + 11s + 10)

Then in matlab, we have the following expression used to get the closed-loop transfer function of the system:

[numc,denc] = feedback([0 0 10], [1 11 10], [k2 k1], [0 1]);


We take k2=0 and k1=10 (first order proportional control) This system yields use the following pole-zero map and step response in matlab:  How can I describe the step in response in terms of the poles and zeros of this system? I don't know how to relate the two. Any guidance is greatly appreciated

Thanks

## 3 Answers

The step is a stimulus to the system. The system is defined by the pole zero diagram irrespective of the stimulus and it looks like a 2nd order response so here's the math behind the poles: - You have poles at co-ordinates -5 on the real part of the s-plane and +/-9 on the $j\omega$ of the s-plane. So you can say 5 = $\zeta\omega_n$ and 9 = $\omega_n\sqrt{1-\zeta^2}$. From this you can work out what $\omega_n$ and $\zeta$ are.

Because it looks like a 2nd order system you can use this to define the shape from the above values for $\omega_n$ and $\zeta$: - I've never worked with the feedback function but, reading the documentation I guess that's not the way to define your system.

Maybe the code should be something like:

k1 = 10;
k2 = 0;
G = tf(10, [1 11 10]);% transfer function G(s) = 10/(s^2 + 11s + 10)
K = tf([k2 k1], ); % transfer function K(s) = s k2 + k1
H = feedback(G, K);


Not sure about the Matlab part.

But regarding to your question, the poles an zeros define the transfer function, which defines the impulse response.

With the impulse response, since the system is LTI, you have every other response by convoluting the input with it; for example, convolute a step with the impulse response to get the step response.

You also need to plot the pole zero for the open loop system as well and then you can say that for step response, the P-Z moved from point A to point B. That makes more sense to analyse the system in terms of P-Z.

Also P-Z of a system just indicate that start and the end point of the system, with a trajectory as to how the system moves around as the frequency is changed.