First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the result into partial fractions, and performing the inverse Laplace transform, the output is obtained. This output starts from zero and asymptotically approaches one. Notably, at a one-time constant (τ), the response reaches 63.2% of its final value, indicating the system's initial response speed.
For a unit-ramp input, the system's response and the error signal are derived similarly. The Laplace transform of the unit-ramp input is substituted into the transfer function, expanded into partial fractions, and then inverse Laplace transformed. The resulting output shows how the system tracks the ramp input. As the time approaches infinity, the error signal stabilizes around the time constant (τ), implying that systems with smaller time constants have reduced steady-state errors, indicating better performance in tracking ramp inputs.
For a unit-impulse input, the system's response is obtained directly using the Laplace transform method. The unit-impulse response function highlights the system's transient behavior, illustrating how the system initially reacts to a sudden input.
Linear time-invariant (LTI) systems exhibit a unique property where the response to the derivative or integral of an input signal can be determined by differentiating or integrating the response to the original signal. This property simplifies the analysis of LTI systems by leveraging known responses. However, this property does not extend to linear time-varying (LTV) or nonlinear systems, where the response to derivatives or integrals of inputs must be computed from first principles.
Understanding the responses of first-order systems to these basic input functions is essential for designing and analyzing more complex systems, as it provides foundational insights into their dynamic behavior and performance metrics.
