Notes: Stability Analysis in S-Domain | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

 Page 1


 
 
 
 
 
 
  
STABILITY ANALYSIS IN S-DOMAIN 
Stability is an important concept. In this chapter, let us discuss the stability of system 
and types of systems based on stability. 
What is Stability? 
A system is said to be stable, if its output is under control. Otherwise, it is said to be 
unstable. A stable system produces a bounded output for a given bounded input. 
The following figure shows the response of a stable system. 
 
This is the response of first order control system for unit step input. This response has the 
values between 0 and 1. So, it is bounded output. We know that the unit step signal has the 
value of one for all positive values of t including zero. So, it is bounded input. Therefore, 
the first order control system is stable since both the input and the output are bounded. 
Types of Systems based on Stability 
We can classify the systems based on stability as follows. 
 
 Absolutely stable system 
 Conditionally stable system 
 Marginally stable system 
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STABILITY ANALYSIS IN S-DOMAIN 
Stability is an important concept. In this chapter, let us discuss the stability of system 
and types of systems based on stability. 
What is Stability? 
A system is said to be stable, if its output is under control. Otherwise, it is said to be 
unstable. A stable system produces a bounded output for a given bounded input. 
The following figure shows the response of a stable system. 
 
This is the response of first order control system for unit step input. This response has the 
values between 0 and 1. So, it is bounded output. We know that the unit step signal has the 
value of one for all positive values of t including zero. So, it is bounded input. Therefore, 
the first order control system is stable since both the input and the output are bounded. 
Types of Systems based on Stability 
We can classify the systems based on stability as follows. 
 
 Absolutely stable system 
 Conditionally stable system 
 Marginally stable system 
 
 
 
 
 
Absolutely Stable System 
If the system is stable for all the range of system component values, then it is known as the 
absolutely stable system. The open loop control system is absolutely stable if all the 
poles of the open loop transfer function present in left half of ‘s’ plane. Similarly, the 
closed loop control system is absolutely stable if all the poles of the closed loop transfer 
function present in the left half of the ‘s’ plane. 
Conditionally Stable System 
If the system is stable for a  certain  range of  system component values,  then  it  is known     
as conditionally stable system. 
Marginally Stable System 
If the system is stable by producing an output signal with constant amplitude and constant 
frequency of oscillations for bounded input, then it is known as marginally stable 
system. The open loop control system is marginally stable if any two poles of the open 
loop transfer function is present on the imaginary axis. Similarly, the closed loop control 
system is  marginally stable if any two poles of the closed loop transfer function is present 
on the imaginary axis. 
n this chapter, let us discuss the stability analysis in the ‘s’ domain using the RouthHurwitz 
stability criterion. In this criterion, we require the characteristic equation to find the 
stability of the closed loop control systems. 
Routh-Hurwitz Stability Criterion 
Routh-Hurwitz stability criterion is having one necessary condition and one sufficient 
condition for stability. If any control system doesn’t satisfy the necessary condition, then 
we can say that the control system is unstable. But, if the control system satisfies the 
necessary condition, then it may or may not be stable. So, the sufficient condition is helpful 
for knowing whether the control system is stable or not. 
Necessary Condition for Routh-Hurwitz Stability 
The necessary condition is that the coefficients of the characteristic polynomial should be 
positive. This implies that all the roots of the characteristic equation should have negative 
real parts. 
Consider the characteristic equation of the order ‘n’ is - 
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STABILITY ANALYSIS IN S-DOMAIN 
Stability is an important concept. In this chapter, let us discuss the stability of system 
and types of systems based on stability. 
What is Stability? 
A system is said to be stable, if its output is under control. Otherwise, it is said to be 
unstable. A stable system produces a bounded output for a given bounded input. 
The following figure shows the response of a stable system. 
 
This is the response of first order control system for unit step input. This response has the 
values between 0 and 1. So, it is bounded output. We know that the unit step signal has the 
value of one for all positive values of t including zero. So, it is bounded input. Therefore, 
the first order control system is stable since both the input and the output are bounded. 
Types of Systems based on Stability 
We can classify the systems based on stability as follows. 
 
 Absolutely stable system 
 Conditionally stable system 
 Marginally stable system 
 
 
 
 
 
Absolutely Stable System 
If the system is stable for all the range of system component values, then it is known as the 
absolutely stable system. The open loop control system is absolutely stable if all the 
poles of the open loop transfer function present in left half of ‘s’ plane. Similarly, the 
closed loop control system is absolutely stable if all the poles of the closed loop transfer 
function present in the left half of the ‘s’ plane. 
Conditionally Stable System 
If the system is stable for a  certain  range of  system component values,  then  it  is known     
as conditionally stable system. 
Marginally Stable System 
If the system is stable by producing an output signal with constant amplitude and constant 
frequency of oscillations for bounded input, then it is known as marginally stable 
system. The open loop control system is marginally stable if any two poles of the open 
loop transfer function is present on the imaginary axis. Similarly, the closed loop control 
system is  marginally stable if any two poles of the closed loop transfer function is present 
on the imaginary axis. 
n this chapter, let us discuss the stability analysis in the ‘s’ domain using the RouthHurwitz 
stability criterion. In this criterion, we require the characteristic equation to find the 
stability of the closed loop control systems. 
Routh-Hurwitz Stability Criterion 
Routh-Hurwitz stability criterion is having one necessary condition and one sufficient 
condition for stability. If any control system doesn’t satisfy the necessary condition, then 
we can say that the control system is unstable. But, if the control system satisfies the 
necessary condition, then it may or may not be stable. So, the sufficient condition is helpful 
for knowing whether the control system is stable or not. 
Necessary Condition for Routh-Hurwitz Stability 
The necessary condition is that the coefficients of the characteristic polynomial should be 
positive. This implies that all the roots of the characteristic equation should have negative 
real parts. 
Consider the characteristic equation of the order ‘n’ is - 
 
 
 
 
 
 
 
 
 
Note that, there should not be any term missing in the n
th 
order characteristic equation. 
This means that the n
th 
order characteristic equation should not have any coefficient that 
is of zero value. 
Sufficient Condition for Routh-Hurwitz Stability 
The sufficient condition is that all the elements of the first column of the Routh array 
should have the same sign. This means that all the elements of the first column of the 
Routh array should be either positive or negative. 
Routh Array Method 
If all the roots of the characteristic equation exist to the left half of the ‘s’ plane, then the 
control system is stable. If at least one root of the characteristic equation exists to the right 
half of the ‘s’ plane, then the control system is unstable. So, we have to find the roots of the 
characteristic equation to know whether the control system is stable or unstable. But, it is 
difficult to find the roots of the characteristic equation as order increases. 
So, to overcome this problem there we have the Routh array method. In this method, 
there is no need to calculate the roots of the characteristic equation. First formulate the 
Routh table and find the number of the sign changes in the first column of the Routh table. 
The number of sign changes in the first column of the Routh table gives the number of 
roots of characteristic equation that exist in the right half of the ‘s’ plane and the control 
system is unstable. 
Follow this procedure for forming the Routh table. 
 Fill the first two rows of the Routh array with the coefficients of the characteristic 
polynomial as mentioned in the table below. Start with the coefficient of sn and 
continue up to the coefficient of s0. 
 Fill the remaining rows of the Routh array with the elements as mentioned in the 
table below. Continue this process till you get the first column element of row s0s0 
is an. Here, an is the coefficient of s0 in the characteristic polynomial. 
Note - If any row elements of the Routh table have some common factor, then you can 
divide the row elements with that factor for the simplification will be easy. 
The following table shows the Routh array of the n
th 
order characteristic polynomial. 
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STABILITY ANALYSIS IN S-DOMAIN 
Stability is an important concept. In this chapter, let us discuss the stability of system 
and types of systems based on stability. 
What is Stability? 
A system is said to be stable, if its output is under control. Otherwise, it is said to be 
unstable. A stable system produces a bounded output for a given bounded input. 
The following figure shows the response of a stable system. 
 
This is the response of first order control system for unit step input. This response has the 
values between 0 and 1. So, it is bounded output. We know that the unit step signal has the 
value of one for all positive values of t including zero. So, it is bounded input. Therefore, 
the first order control system is stable since both the input and the output are bounded. 
Types of Systems based on Stability 
We can classify the systems based on stability as follows. 
 
 Absolutely stable system 
 Conditionally stable system 
 Marginally stable system 
 
 
 
 
 
Absolutely Stable System 
If the system is stable for all the range of system component values, then it is known as the 
absolutely stable system. The open loop control system is absolutely stable if all the 
poles of the open loop transfer function present in left half of ‘s’ plane. Similarly, the 
closed loop control system is absolutely stable if all the poles of the closed loop transfer 
function present in the left half of the ‘s’ plane. 
Conditionally Stable System 
If the system is stable for a  certain  range of  system component values,  then  it  is known     
as conditionally stable system. 
Marginally Stable System 
If the system is stable by producing an output signal with constant amplitude and constant 
frequency of oscillations for bounded input, then it is known as marginally stable 
system. The open loop control system is marginally stable if any two poles of the open 
loop transfer function is present on the imaginary axis. Similarly, the closed loop control 
system is  marginally stable if any two poles of the closed loop transfer function is present 
on the imaginary axis. 
n this chapter, let us discuss the stability analysis in the ‘s’ domain using the RouthHurwitz 
stability criterion. In this criterion, we require the characteristic equation to find the 
stability of the closed loop control systems. 
Routh-Hurwitz Stability Criterion 
Routh-Hurwitz stability criterion is having one necessary condition and one sufficient 
condition for stability. If any control system doesn’t satisfy the necessary condition, then 
we can say that the control system is unstable. But, if the control system satisfies the 
necessary condition, then it may or may not be stable. So, the sufficient condition is helpful 
for knowing whether the control system is stable or not. 
Necessary Condition for Routh-Hurwitz Stability 
The necessary condition is that the coefficients of the characteristic polynomial should be 
positive. This implies that all the roots of the characteristic equation should have negative 
real parts. 
Consider the characteristic equation of the order ‘n’ is - 
 
 
 
 
 
 
 
 
 
Note that, there should not be any term missing in the n
th 
order characteristic equation. 
This means that the n
th 
order characteristic equation should not have any coefficient that 
is of zero value. 
Sufficient Condition for Routh-Hurwitz Stability 
The sufficient condition is that all the elements of the first column of the Routh array 
should have the same sign. This means that all the elements of the first column of the 
Routh array should be either positive or negative. 
Routh Array Method 
If all the roots of the characteristic equation exist to the left half of the ‘s’ plane, then the 
control system is stable. If at least one root of the characteristic equation exists to the right 
half of the ‘s’ plane, then the control system is unstable. So, we have to find the roots of the 
characteristic equation to know whether the control system is stable or unstable. But, it is 
difficult to find the roots of the characteristic equation as order increases. 
So, to overcome this problem there we have the Routh array method. In this method, 
there is no need to calculate the roots of the characteristic equation. First formulate the 
Routh table and find the number of the sign changes in the first column of the Routh table. 
The number of sign changes in the first column of the Routh table gives the number of 
roots of characteristic equation that exist in the right half of the ‘s’ plane and the control 
system is unstable. 
Follow this procedure for forming the Routh table. 
 Fill the first two rows of the Routh array with the coefficients of the characteristic 
polynomial as mentioned in the table below. Start with the coefficient of sn and 
continue up to the coefficient of s0. 
 Fill the remaining rows of the Routh array with the elements as mentioned in the 
table below. Continue this process till you get the first column element of row s0s0 
is an. Here, an is the coefficient of s0 in the characteristic polynomial. 
Note - If any row elements of the Routh table have some common factor, then you can 
divide the row elements with that factor for the simplification will be easy. 
The following table shows the Routh array of the n
th 
order characteristic polynomial. 
 
 
 
 
 
 
 
 
Example 
Let us find the stability of the control system having characteristic equation, 
 
 
 
Step 1 - Verify the necessary condition for the Routh-Hurwitz 
stability. All the coefficients of the characteristic polynomial, 
are positive. So, the control system satisfies the necessary 
condition. 
Step 2 - Form the Routh array for the given characteristic polynomial. 
Page 5


 
 
 
 
 
 
  
STABILITY ANALYSIS IN S-DOMAIN 
Stability is an important concept. In this chapter, let us discuss the stability of system 
and types of systems based on stability. 
What is Stability? 
A system is said to be stable, if its output is under control. Otherwise, it is said to be 
unstable. A stable system produces a bounded output for a given bounded input. 
The following figure shows the response of a stable system. 
 
This is the response of first order control system for unit step input. This response has the 
values between 0 and 1. So, it is bounded output. We know that the unit step signal has the 
value of one for all positive values of t including zero. So, it is bounded input. Therefore, 
the first order control system is stable since both the input and the output are bounded. 
Types of Systems based on Stability 
We can classify the systems based on stability as follows. 
 
 Absolutely stable system 
 Conditionally stable system 
 Marginally stable system 
 
 
 
 
 
Absolutely Stable System 
If the system is stable for all the range of system component values, then it is known as the 
absolutely stable system. The open loop control system is absolutely stable if all the 
poles of the open loop transfer function present in left half of ‘s’ plane. Similarly, the 
closed loop control system is absolutely stable if all the poles of the closed loop transfer 
function present in the left half of the ‘s’ plane. 
Conditionally Stable System 
If the system is stable for a  certain  range of  system component values,  then  it  is known     
as conditionally stable system. 
Marginally Stable System 
If the system is stable by producing an output signal with constant amplitude and constant 
frequency of oscillations for bounded input, then it is known as marginally stable 
system. The open loop control system is marginally stable if any two poles of the open 
loop transfer function is present on the imaginary axis. Similarly, the closed loop control 
system is  marginally stable if any two poles of the closed loop transfer function is present 
on the imaginary axis. 
n this chapter, let us discuss the stability analysis in the ‘s’ domain using the RouthHurwitz 
stability criterion. In this criterion, we require the characteristic equation to find the 
stability of the closed loop control systems. 
Routh-Hurwitz Stability Criterion 
Routh-Hurwitz stability criterion is having one necessary condition and one sufficient 
condition for stability. If any control system doesn’t satisfy the necessary condition, then 
we can say that the control system is unstable. But, if the control system satisfies the 
necessary condition, then it may or may not be stable. So, the sufficient condition is helpful 
for knowing whether the control system is stable or not. 
Necessary Condition for Routh-Hurwitz Stability 
The necessary condition is that the coefficients of the characteristic polynomial should be 
positive. This implies that all the roots of the characteristic equation should have negative 
real parts. 
Consider the characteristic equation of the order ‘n’ is - 
 
 
 
 
 
 
 
 
 
Note that, there should not be any term missing in the n
th 
order characteristic equation. 
This means that the n
th 
order characteristic equation should not have any coefficient that 
is of zero value. 
Sufficient Condition for Routh-Hurwitz Stability 
The sufficient condition is that all the elements of the first column of the Routh array 
should have the same sign. This means that all the elements of the first column of the 
Routh array should be either positive or negative. 
Routh Array Method 
If all the roots of the characteristic equation exist to the left half of the ‘s’ plane, then the 
control system is stable. If at least one root of the characteristic equation exists to the right 
half of the ‘s’ plane, then the control system is unstable. So, we have to find the roots of the 
characteristic equation to know whether the control system is stable or unstable. But, it is 
difficult to find the roots of the characteristic equation as order increases. 
So, to overcome this problem there we have the Routh array method. In this method, 
there is no need to calculate the roots of the characteristic equation. First formulate the 
Routh table and find the number of the sign changes in the first column of the Routh table. 
The number of sign changes in the first column of the Routh table gives the number of 
roots of characteristic equation that exist in the right half of the ‘s’ plane and the control 
system is unstable. 
Follow this procedure for forming the Routh table. 
 Fill the first two rows of the Routh array with the coefficients of the characteristic 
polynomial as mentioned in the table below. Start with the coefficient of sn and 
continue up to the coefficient of s0. 
 Fill the remaining rows of the Routh array with the elements as mentioned in the 
table below. Continue this process till you get the first column element of row s0s0 
is an. Here, an is the coefficient of s0 in the characteristic polynomial. 
Note - If any row elements of the Routh table have some common factor, then you can 
divide the row elements with that factor for the simplification will be easy. 
The following table shows the Routh array of the n
th 
order characteristic polynomial. 
 
 
 
 
 
 
 
 
Example 
Let us find the stability of the control system having characteristic equation, 
 
 
 
Step 1 - Verify the necessary condition for the Routh-Hurwitz 
stability. All the coefficients of the characteristic polynomial, 
are positive. So, the control system satisfies the necessary 
condition. 
Step 2 - Form the Routh array for the given characteristic polynomial. 
 
 
 
 
 
 
 
Step 3 - Verify the sufficient condition for the Routh-Hurwitz stability. 
All the elements of the first column of the Routh array are positive. There is no sign 
change in the first column of the Routh array. So, the control system is stable. 
Special Cases of Routh Array 
We may come across two types of situations, while forming the Routh table. It is difficult 
to complete the Routh table from these two situations. 
The two special cases are - 
 
 The first element of any row of the Routh’s array is zero. 
 All the elements of any row of the Routh’s array are zero. 
Let us now discuss how to overcome the difficulty in these two cases, one by one. 
First Element of any row of the Routh’s array is zero 
If any row of the Routh’s array contains only the first element as zero and at least one of 
the remaining elements have non-zero value, then replace the first element with a small 
positive integer, ?. And then continue the process of completing the Routh’s table. Now, 
find the number of sign changes in the first column of the Routh’s table by substituting ?? 
tends to zero.