Define transient response system and steady-state response of a system. In addition, mention the significance of these responses of the system.
Derive the expression for a unit-impulse response and unit-ramp response of a standard second-order system using the unit-step response of the system.
The Correct Answer and Explanation is :
Transient Response and Steady-State Response
Transient Response:
The transient response of a system is the part of the response that lasts temporarily after a disturbance or input is applied. It occurs during the period when the system is moving from its initial state to a steady-state condition. The transient response is influenced by the system’s dynamics (e.g., poles and zeros) and includes components like oscillations, overshoot, rise time, and settling time. It represents the non-steady portion of the response and is typically significant when analyzing system stability and performance.
Steady-State Response:
The steady-state response is the system’s output after all transients have died out. It describes the behavior of the system in response to a persistent input as time approaches infinity. This response depends primarily on the system’s structure and the type of input signal. It is critical for determining the accuracy and performance of the system under normal operating conditions.
Significance:
- Transient Response: Determines how quickly and smoothly the system reacts to changes. It is critical in applications where overshoot, oscillations, or time delays may impact performance.
- Steady-State Response: Ensures the system delivers the desired output over time. It is significant for accuracy and reliability in steady operating conditions.
Derivation of Unit-Impulse and Unit-Ramp Responses
The standard second-order system is represented as:
[
H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}
]
where ( \omega_n ) is the natural frequency, and ( \zeta ) is the damping ratio.
1. Unit-Step Response ((y_{step}(t))):
The time-domain unit-step response is:
[
y_{step}(t) = 1 – e^{-\zeta\omega_n t} \left[ \cos(\omega_d t) + \frac{\zeta}{\sqrt{1-\zeta^2}} \sin(\omega_d t) \right]
]
where ( \omega_d = \omega_n \sqrt{1-\zeta^2} ).
2. Unit-Impulse Response ((y_{impulse}(t))):
The unit-impulse response is the derivative of the unit-step response:
[
y_{impulse}(t) = \frac{d}{dt} y_{step}(t).
]
Differentiating ( y_{step}(t) ):
[
y_{impulse}(t) = \omega_n^2 e^{-\zeta\omega_n t} \left[ \sin(\omega_d t) + \frac{\omega_d}{\omega_n} \cos(\omega_d t) \right].
]
3. Unit-Ramp Response ((y_{ramp}(t))):
The unit-ramp response is the convolution of the unit-step response with a ramp input:
[
y_{ramp}(t) = \int_0^t y_{step}(\tau) d\tau.
]
Substituting ( y_{step}(t) ), solve for ( y_{ramp}(t) ) using integration.
These responses describe how a second-order system reacts to standard inputs, aiding in understanding system dynamics and performance.