Introduction
Once we have obtained a mathematical representation of a system, our next step is to analyze its transient and steady-state response.
Transient and Steady-State Response
The purpose of a closed loop control is to make a system’s output follow the input as cloesly as possible.
Ideally the ooutput would at all times correspond exactly to the input, but this is not possible in real systems which (due to the effects or inertia, inductance, heat transfer, etc.) do not respond instantaneously to changes in the input.
In practice, a system is judged by a number of criteria, the three most important of which are,
- Is the System Absolutely Stable?
- After an input disturbance, the output should settle down to a steady value.
- How Accurate is the System in Steady State?
- The steady-state error should be small.
- How Quickly Does the System Reach Steady State?
- Steady-state should be reached quickly without exessive overshoot or oscillation.
Steady-State Response to Step/Ramp Input
The steady-state response of a system to a step input is very often of interest.
For this, the final value theorem is applied. $$ \lim_{t \to \infty} f(t) = \lim_{s \to 0} sF(s) $$
If the transfer function of the system is denoted $G_{\text{sys}}(s)$, then the final value theorem tells us that for a step* of size $\bar{U}$, $$ u(t) = \bar{U} & \forall t > 0 \newline U(s) = \frac{\bar{U}}{s} \newline \lim_{t \to \infty} u(t) = \lim_{s \to 0} sU(s) = \lim_{s \to 0} s\frac{\bar{U}}{s} = \bar{U} $$
For a ramp input of slope $\bar{U}$, $$ u(t) = \bar{U}t & \forall t > 0 \newline U(s) = \frac{\bar{U}}{s^2} \newline \lim_{t \to \infty} u(t) = \lim_{s \to 0} sU(s) = \lim_{s \to 0} s\frac{\bar{U}}{s^2} = \lim_{s \to 0} \frac{\bar{U}}{s} = \infty $$
Poles and Zeroes
The poles and zeroes of a system are important in determining the system’s response.
Let’s see how we analyze a systems poles and zeroes to determine a system’s response.
Consider, $$ G(s) = \frac{b_{n - 1} s^{n - 1} + b_{n - 2} s^{n - 2} + \ldots + + b_0}{s^n + a_{n - 1} s^{n - 1} + \ldots + + a_0} = \frac{N(s)}{D(s)} $$
Poles of $G(s)$ are the roots of $D(s) = 0$ and zeroes of $G(s)$ are the roots of $N(s) = 0$.
Generally, at poles $G(s) = \infty$ unless the pole is cancelled by a mathcing zero.
At zeroes, $G(s) = 0$ unless the zero is cancelled by a matching pole.
Poles and Zeroes of a First Order System
A system’s output response contains two parts,
- Forced or steady-state response, this is caused by thr poles of the input function, $R(s)$.
- Natural or homogenous response, this is caused by the poles of the transfer function, $G(s)$.