Chapter 5 - Transient and Steady State Response Analyses

This chapter is concerned with system responses to aperiodic signals (such as step, ramp, acceleration, and impulse functions of time).

The Time Response is the response of the signal in the time domain (the continuous time function)

This time response is composed of the Transient response and the steady-state response

The transient response is the transition between the initial state and the steady-state

The steady state is the manner in which the system output behaves as t approaches infinity

Absolute Stability wether the system is stable or unstable

critically stable oscillations of the output continue forever

Relative Stability

Steady-State Error

First Order Systems

This the unit step response.

This is the unit ramp response

This is the unit impulse response

Second Order System

General Equation of a Second order System

$\sigma$ is called the attenuation. Which is the reduction in the strength of a signal

$$\sigma = \omega_n \zeta$$

$\omega_n$ the undamped natural frequency is the frequency that the system would oscillate at if the dampening ratio was decreased to zero.

$\omega_d$ is the observed damped natural frequency which is equal to

$$\omega_n \sqrt{1-\zeta^2}$$

$\zeta$ is the damping ratio of the system. The damping ratio $\zeta$ is the ratio of the actual damping B to the critical damping $B_c = 2\sqrt{JK}$ or

This is a graph showing the graphs of the damped signals as you vary the damping ration $\zeta$

Transient Response Specifications

$c(\infty)$ is the steady state output

1. Delay time $t_d$ time to reach 50 percent of desired output
2. Rise time, $t_r$ time to rise from 0 to 100% $$t_r = \frac{1}{w_d} tan^{-1} (- \frac{\omega_d}{\sigma})$$
3. Peak time, $t_p$ time to rise to the first peak $$\frac{\pi}{\omega_d}$$
4. Maximum percent overshoot, $M_p$

$$M_p = \frac{c(t_p) - c(\infty)}{c(\infty)}$$

1. Settling time, $t_s$ time to stay within a 2% or 5% tolerance of the final value

These are shown on the graph below