Transient Response Analysis
March 2015
Transient response and steady state response
“The time response of a control system consists of two parts: the transient response and the steadystate response. By transient response, we mean that which goes from the initial state to the final state.
By steady-state response, we mean the manner in which the system output behaves as it approaches
infinity.”
What is the overshoot?
It is the part of the response when the system overreact to a certain input, that is when the response
reaches the desired value but doesn’t stop there, it rather crosses it before returning back again. We call
the response from the moment of crossing the desired value until returning back the overshoot.
Absolute stability
This term refers to the overall stability of the system, that is, whether the system is working or is broken.
Relative stability
This term refers to the transient response performance of the system. A system may be stable but have
a very slow response or a very large overshoot making it non useful, in this case we consider the system
relatively unstable.
Steady-state error
This term refers to the difference between the set point (input) and the system response (output) at
steady state, i.e. when the system reaches equilibrium.
System order
The degree of the denominator of the transfer function is called the system order. Note that the order
of the closed loop transfer function in a unity feedback system is the same for the open loop transfer
function of the system. The degree of the numerator cannot be greater than the degree of the
nominator.
( )
( )
( )
Where:
( )
( )
( )
( )
Characteristics equation
If we take the denominator and equal it to zero, the resulting equation is called the characteristics
equation. This equation is very important. We can identify the system response by analyzing this
equation.
Test inputs
In order to study and analyze a system, we have to apply an input to it. There is many types of inputs but
the most common inputs used are step, ramp and impulse signals.
First order systems
First order systems have the following common transfer function
Unit-step response of 1st order systems
In order to find the response of the 1st order system to a step input mathematically, we need to find the
response equation first.
Taking Laplace transfer to find the response c(t)
The response will take the following shape
Please note that T is called the time constant of the system, which represents the time required for the
response to reach 63.2% of its final value (at steady state).
According to the required design specifications, we may assign 3T, 4T or 5T as the time required to reach
steady state for an allowed steady-state error criteria of 95%, 98% or 99% respectively.
Unit-ramp response of 1st order systems
In order to find the response of the 1st order system to a ramp input mathematically, we need to find
the response equation first.
Taking Laplace transfer to find the response c(t)
The response will take the following shape
Here we get a constant steady-state error equal in value to the time constant of the system T. Please
note that the error equation would be
Unit-impulse response of 1st order systems
In order to find the response of the 1st order system to an impulse input mathematically, we need to
find the response equation first.
Taking Laplace transfer to find the response c(t)
The response will take the following shape
Second order systems
The second order systems have the following general transfer function
Here we can see that the characteristics equation is consisting primarily from two parameters, these
parameters are very important
ζ is called the damping ration, it represents the amount of the damping in the system response.
Wn is called the undamped natural frequency, it represents the response frequency when the system
have no damping (ζ =0).
Step response of 2nd order systems
The response can be divided into three categories according to the value of ζ
According to the value of the damping ratio, the system will respond as shown in the following figure
Notice that the response will be very quick for very small values of the damping ratio but will produce
very strong overshoots and oscillations, while very large values would yield no overshoots or oscillations
but on the expense of a very slow and sluggish response.
Impulse response of 2nd order systems
Impulse response would be according to the following equations
The response curves would look like those of the following figure
Ramp response of 2nd order systems
For a ramp input, the second order system will exhibit a steady-state error of
Therefore the response would appear, in relation to the input, as in the following figure
Transient response of higher order systems
Consider the following transfer function of a higher order system
Notice that in a linear system the overall response is the algebraic sum of the responses of each of its
parts. Therefore, the response of a stable higher order system is the sum of a number of exponential
curves and damped sinusoidal curves, depending on the type of each pole in the characteristics
equation.
Transient response specifications
In order to better understand the transient response of control systems, we need to know that the
overall system response is the algebraic sum of the responses of all the poles of the characteristics
equation.
Now let’s consider a second order system, the poles of the characteristics equation would be
If the imaginary part of the pole is equal to zero, it will lay on the real axis of the s-plane, resulting in a
damping ratio of 1 or larger. But if we have imaginary part in the pole, the value of the damping ratio
would be less than 1 and the system would be under-damped.
The following figure shows a graphical representation of a pole (denoted by a circle) in the s-plane
Notice that
,
and
Now, to provide requirements for the design of control systems, the following specifications are used

Rise time
Represents the time required by the response to reach the final value for the first time.

Peak time
Represents the time required by the response to reach the peak of the overshoot (for underdamped response)

Maximum overshoot
It is the highest value the response reaches before returning to the desired value in an underdamped response

Settling time
It is the time required by the response to reach the final value at steady state.
Delay time
It is the time required by the response to reach half (50%) of the final value.
