Transient and Steady State Analysis of Linear Time Invariant (LTI) Systems
Time Response Analysis
When the energy state of any system is disturbed, and the disturbances occur at input, output or both ends, then it takes some time to change from one state to another state. This time that is required to change from one state to another state is known as transient time and the value of current and voltage during this period is called transient response.
Depending upon the parameters of the system, the transient may have oscillations which may be either sustained or decaying in nature.
Thus time response of a control system is divided into two parts-
Transient State response
It deals with the nature of the response of a system when subjected to an input.
Steady State Analysis
It deals with the estimation of the magnitude of steady-state error between input and output.
Different Type of Standard Test Signals
The various inputs or disturbances affecting the performance of a system are mathematically represented as a standard test signal.
Characteristics of Time-domain Analysis
Transient Time: The time required to change from one state to another is called the transient time.
Transient Response: The value of current and voltage during the time change is called transient response.
So, we can say that the transient response is the part of the response which goes to zero as time increases and the steady-state response is the part of the total response after transient has died. If the steady-state response is the part of the output does not match with the input then the system has a steady state error.
Test input signal for transient analysis
For the analysis of the time response of a control system, the following input signals are used.
A unit step function is denoted by u(t) and is defined as
u(t) = 0 ; t=0 = 1 ;0=t
Step function is also called displacement function. If input is R(S), then R(s) = 1/s
This function starts from the origin and linearly decreases or increases with time as shown in the figure above.
Let r(t) be the ramp function then
r(t) = 0 ; t<0 = Kt ; t>0
Where 'K' is the slope of the line, for a positive value of 'K' the slope is upward, and the slope is downward for the negative value of 'K.'
The value of r(t) is zero when t<0 and is a quadratic function of time when t>0.
Therefore r(t) = 0 ; t<0 = (Kt^2)/2 ; t>0
Where 'K' is constant for unit parabolic function K = 1. The unit parabolic function is defined as
r (t) = 0 ; t<0 = t^2/2 ; t>0
A unit impulse function is defined as
Thus we can say that impulse function has zero value everywhere except at t=0 where the amplitude is infinite.