TIME DOMAIN SPECIFICATIONS
The performance of a control system is usually measured in terms of its response to a step input. The step input is used because it is easy to generate and gives the system a nonzero steady-state condition, which can be measured.
Most commonly used time domain performance measures refer to a second-order system with the transfer function:
where ωn is the undamped natural frequency of the system and ζ is the damping ratio of the system.
When a second-order system is excited with a unit step input, the typical output response is as shown in Figure 7.3. Based on this figure, the following performance parameters are usually defined: maximum overshoot; peak time; rise time; settling time; and steady-state error.
The maximum overshoot, Mp , is the peak value of the response curve measured from unity. This parameter is usually quoted as a percentage. The amount of overshoot depends on the damping ratio and directly indicates the relative stability of the system.
The peak time, Tp , is defined as the time required for the response to reach the first peak of the overshoot. The system is more responsive when the peak time is smaller, but this gives rise to a higher overshoot.
The rise time, Tr , is the time required for the response to go from 0 % to 100 % of its final value. It is a measure of the responsiveness of a system, and smaller rise times make the system more responsive.
The settling time, Ts , is the time required for the response curve to reach and stay within a range about the final value. A value of 2–5 % is usually used in performance specifications.
The steady-state error, Ess , is the error between the system response and the reference input value (unity) when the system reaches its steady-state value. A small steady-tate error is a requirement in most control systems. In some control systems, such as position control, it is one of the requirements to have no steady-state error.
Having introduced the parameters, we are now in a position to give formulae for them (readers who are interested in the derivation of these formulae should refer to books on control theory). The maximum overshoot occurs at at peak time (t = Tp ) and is given by
i.e. overshoot is directly related to the system damping ratio – the lower the damping ratio, the higher the overshoot. Figure 7.4 shows the variation of the overshoot (expressed as a percentage) with the damping ratio.
The peak time is obtained by differentiating the output response with respect to time, letting this equal zero. It is given by
and the steady-state error when a unit step input is applied can be found from
