In this article, we are going to discuss the steady-state response. We will see what is steady state response in Time domain analysis. We will then discuss some of the standard test signals used in finding the response of a response. We also discuss the first-order response for different signals. We will discuss the second-order system and its cases. Along with these, we will also discuss some specifications and errors. Later in the article, we will discuss some applications, advantages, and disadvantages of steady-state response.
Table of Content
What is Steady State Response?
Time domain analysis of a system refers to the analysis of system performance in time. When the output or response of a system varies with respect to time then it is called a time response. Time responses are of two types transient response and steady-state response. Transient response is the part of the time response that goes to zero as time tends to be large. This response is for that time when the system needs to become steady. Steady response is the time response of the system when it has become steady and transient practically vanishes as time goes to infinity. Steady means when all the disturbances vanish.
Mathematically, the time response can be written as
c(t) = ctr (t) + css(t)
Where,
- ctr(t) is the transient response
-
css(t) is the steady-state response
Standard Test Signals Used
These are some signals that are used to know the characteristics and performance of a system. These signals are unit impulse, unit step, ramp, parabolic signals. Any arbitrary input signal can be made using these standard signals.
Unit Impulse Signal
The unit impulse signal is one of the signal used in control systems. It is denoted by δ(t). This signal exists only at t=0 and it has very short duration pulse with an area under the curve equal to 1 or unity.
It is defined as follows,
δ(t)= ∞,t=0 and δ(t)= 0, t ≠0
Unit Step Signal
This signal is denoted by u(t). This signal only exists for positive side of the plot. In simple words, the signal is zero for all values less than zero and unity for all values equal and greater than zero. There is sudden change in the system from 0 to 1 at time t=0.
It is defined as follows,
u(t)= 0, t<0 and u(t) =1, t ≥0
Unit Ramp Signal
This signal is defined as r(t). This signal represents a linear increase or decrease in magnitude with time. Ramp function is zero for all values less than zero and for values greater than or equal to zero it increases linearly. The rate of increase of the function is linear, which means for each unit increase in time the function also increases by same amount.
It is defined as follows,
r(t)= 0, t<0 and r(t) =t, t ≥0
Unit ramp signal can be written in terms of step signal as r(t)=tu(t).
Unit Parabolic Signal
It is denoted as p(t). Unit parabolic signal is zero for all values less than zero and follows a quadratic increase with time for values greater than or equal to zero. The rate of increase is non-linear.
It is defined as follows,
p(t)= t2/2; t ≥0 and p(t)=0; t<0
Unit parabolic signal can be written in terms of step signal as p(t)=(t2/2)u(t).
Type and Order of a System
- Type: The number of open loop poles at origin (s=0) is the type of system.
- Order: Total number of open loop poles is the order of closed loop system.
First Order System Response
System with only one pole is called as first order system.
In the above system we can write the transfer function as,
Where,
Therefore,
This equation gives the speed of response of system for an input. Higher the time constant, slower is the response and vice-versa.
Case 1: Input is Unit Impulse
r(t)=
Applying Laplace transform we get,
R(s)=1
We know,
Substituting the value of R(s)=1 we get,
This can be also written as,
Applying inverse Laplace transform we get,
Case 2: Input is Unit Step
r(t)=u(t)
Applying Laplace transform we get,
We know,
Substituting the value of R(s) we get
By applying first partial fraction and then applying the inverse Laplace transform we get
Here, the transient part is
Case 3: Input is Unit Ramp
r(t)=t
Applying Laplace transform we get,
We know,
Substituting the value of R(s) we get
By applying first partial fraction and then applying the inverse Laplace transform we get
Here, the transient part is
Case 4: Input is parabolic
r(t)=
Applying Laplace transform we get,
We know,
Substituting the value of R(s) we get
By applying first partial fraction and then applying the inverse Laplace transform we get
Here, the transient part is
Second Order System
The systems with two poles are called second order systems. The transfer function for second order system is
Where,
-
: System Natural Frequency. It is the frequency at which the system tends to oscillate in the absence of damping force. -
:System Damping Ratio. It is a dimensionless quantity describing decay of oscillations during transient response. -
System damped frequency(
) is the frequency at which the system tends to oscillate in presence of damping force.
Response of Second Order System
There are four cases of responses depending on the damping effect created by
where,
R(s)=1
Therefore,
-
Overdamped system (
)
In overdamped system the transient in the system exponentially decays to steady state without oscillation.
Taking Inverse Laplace transform on both sides we get,
-
Critically Damped System (
)
In this type of system the transient in the system decay to steady state without any oscillation in shortest possible time.
Taking Inverse Laplace Transform
-
Underdamped System (
)
In this system the transient oscillates with amplitude of oscillation gradually decreasing to zero.
Taking inverse Laplace Transform,
Where,
-
Undamped System (
)
In this the system keeps on oscillating at its natural frequency without any decay in amplitude.
Taking Inverse Laplace we get,
Steady State Specification
Delay Time : It is the time required to reach 50% of final value at first instant.
Rise Time: It is the time required to rise from 10% to 90% of final value for overdamped and 0 to 100% of final for underdamped system at first instant.
Peak Time: It is the time required for response to reach the peak value of the response.
Peak Overshoot: Normalized difference between peak value of response and steady state value.
Settling Time: It is the time required to reach and stay within a specified tolerance band of its final value or steady state value. Usually the tolerance band is 2% or 5%.
For 5% Criteria:
For 2% Criteria:
Steady State Error
When the output of the system is deviated from the desired response during steady state, this deviation is known as steady state error. We use final value theorem for calculating the steady state error.
or,
Any system is called controlled system if output follows input at steady state or steady state error is as low as possible.
When we have input as unit step signal the error is known as position error and
When we have input as unit ramp the error is known as velocity error and
When we have input as unit parabolic the error is acceleration error and
Summary
|
Type of system |
Unit Step Input |
Unit Ramp Input |
Unit Parabolic Input |
|---|---|---|---|
|
Type-0 system |
|
|
|
|
Type-1 system |
|
|
|
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Type-2 system |
|
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Advantages and Disadvantages of Steady State Response
Given below are list of Advantages and Disadvantages of Steady State Response
Advantages of Steady State Response
- Steady state response is advantageous to predict the systems output under constant conditions.
- Analysis is easy in steady state as the transient effect and other disturbances are not present.
- It is important for efficiency assessment of a system.
Disadvantages of Steady State Response
- It neglects transient effect which can be significant while analyzing a system
- It always assumes the system to be linear.
- It is limited to time-variant systems.
Applications of Steady State Response
- It is used in control systems for design of controllers by analyzing the output of the system.
- It is used in mechanical systems like engines ,motors for analyzing their performance.
- Steady state response is important for analyzing the filters and amplifiers in electrical systems and communication systems.
Conclusion
In conclusion, the steady state response is the response of the system when the system is steady and it is free of transient. It can be analyzed using first order and second order system using different test signals. It is very useful for analyzing the output of a system at constant conditions. It also has some constraints, it ignores the transient conditions and it assumes the system to be linear. It is also limited to time varying systems. With these limitations also it is used in control system, communication and electrical networks for analyzing the behavior of the system.
FAQs on Steady State Response
1. In which system peak time and peak overshoot are not defined?
Peak time and peak overshoot are not defined for overdamped and critically damped system.
2. What is the difference between steady state and transient state?
The transient response is the response for the time that is required for the system to get to steady state. Steady state response is the response of the system when all disturbances are removed and the system is steady.
3. How is the stability of the system is determined by the value of
When
then we say that negative damping is present and the system is unstable.