Time-response specifications

Specifications for a control system design often involve certain requirements associated with the time response of the closed-loop system. The requirements are specified by the behaviour of the controlled variable or by the control error on well defined test signals. The most important test signal is a unit step on the input of the control system and requirements are placed on the behaviour of the controlled variable , as shown in Figure 7.8. The requirements for a unit step response are expressed in terms of the following standard quantities:

- The
*maximum overshoot*is the magnitude of the overshoot after the first crossing of the steady-state value (100%). This value is normally expressed as a percentage of the steady-state value of the controlled variable. - The
*peak time*is the time required to reach the maximum overshoot. - The
*settling time*is the time for the controlled variable first to reach and thereafter remain within a prescribed percentage of the steady-state value. Common values of are 2%, 3% or 5%. - The
*rise time*is the time required to reach first the steady-state value (100%). It may also be defined as the time to reach the vicinity of the steady-state value particularly for a response with no overshoot, e.g. the time between 10% and 90%. The rise time is defined as the time to first reach 50% of the final value.

Similarly, the behaviour on step disturbances can be characterised as shown in Figure 7.9. Here likewise the terms 'maximum overshoot' and 'settling time' are defined.

These standard quantities are measures of some properties of the control system. and essentially characterise the damping and and the speed, i.e. the dynamics of the control behaviour. The steady-state error as described in section 7.2 is a typical characteristic of the static behaviour.

These quantities describe the deviation of the step response from the ideal case described in section 7.1 and the goal of the design of a control system is to hold them as small as possible. In most cases one can restrict the values of the three quantities , and .