The time response of a control system consists of two parts: the transient response and the steady-state response.
After applying input \(r(t)\) to the control system, the output \(c(t)\) takes certain time to reach steady state. So, the output will be in transient state till it goes to a steady state.
Transient response describes the system’s behavior as it transitions from its initial state to its final state, immediately following a change or disturbance.
Steady-state response characterizes how the system output behaves as time approaches infinity, once the transient effects have faded away and the system has reached a stable condition.
Thus, the system response \(c(t)\) may be written as
\(\boxed{\displaystyle c(t) = c_{tr}(t) + c_{ss}(t)}\)
where \(c_{tr}\) is the transient response, and \(c_{ss}\) is the steady-state response.
See: Complete Response of a Circuit
Transient Response Specifications
Delay Time
The delay time \((t_d)\) is the time required for the response to reach half the final value the very first time.
Rise Time
The rise time \((t_r)\) is the time required for the response to rise from \(10\%\) to \(90\%\), \(5\%\) to \(95\%\), or \(0\%\) to \(100\%\) of its final value.
Peak Time
The peak time \((t_p)\) is the time required for the response to reach the first peak of the overshoot.
Overshoot
The overshoot is when a signal exceeds its target. It is often associated with ringing (oscillation of a signal). The undershoot is the same phenomenon in the opposite direction.
The maximum overshoot \((M_{p})\) is the amount that the waveform overshoots the steady-state (final) value, or value at the peak time \((t_p)\).
The amount of the maximum (percent) overshoot directly indicates the relative stability of the system.
Settling Time
The settling time \((t_s)\) is the time required for the response curve to reach and stay within a range about the final value of size specified by absolute percentage of the final value (usually \(2\%\) or \(5\%\) percentage error criterion).
The settling time \((t_s)\) is related to the largest time constant \((\tau)\) of the control system.