Step response of first and second order LTI systems

Frequency response function

Frequency response is a method to quantitative measure of the output spectrum of a system. Frequency response is used to character the dynamics of the system. Also the frequency can measure of magnitude and phase of the output as a function of frequency.For example sine wave is injected into a system at a given frequency, a linear system will respond at that same frequency with a certain magnitude and a certain phase angle relative to the input.

There are two applications of frequency response analysis. For an audio system the objective may be to reproduce the input signal with no distortion. This they require a uniform (flat) magnitude of response up to the bandwidth limitation of the system, with the signal delayed by precisely the same amount of time at all frequencies. That amount of time could be seconds, or weeks or months in the case of recorded media. The other application is the feedback apparatus used to control a dynamic system. And the objective is to give the closed-loop system improved response as compared to the uncompensated system. In gneral the feedback require respond to system dynamics within a very small number of cycles of oscillation (usually less than one full cycle), and with a definite phase angle relative to the commanded control input. For feedback of sufficient amplification, getting the phase angle wrong can lead to instability for an open-loop stable system, or failure to stabilize a system that is open-loop unstable.

Bode plot

A bode plot is a graph of the transfer function of a linear, time-invariant system versus frequency, plotted with a log-frequency axis, to show the system's frequency response. It is usually combine a bode phase magnitude expressing the magnitude of the frequency response gain, and a bode phase plot expressing the frequency responsephase shift.

Youtube video for Bode Diagram tutorial, 2nd video,3rd video, 4th video, 5th video, Last video

Practice Problems
Bode plot practice problems Bode plot examples and problems ( Great wesite)

Question to think about for Frequency response

Extra Resources
Bode plot on Matlab

Second order LTI system

First order System

Frequency response 1

Frequency response 2