Gujarat TechnicalUniversity
BIRLA VISHWAKARMA MAHAVIDYALAYA
ET Department
ROUTH-STABILITY CRITERSIONS U BMI T ED BY : -
A PA R T RI V ED I : 1 3 00 8 011 20 5 7VATS A L B O D I WA LA : 1 4 00 8 3 111 0 0 2
Under the Guidance :Prof. Amit ChokasiET department, BVM
CONTENT
The concept of stability
The Routh-Hurwitz stability criterion
The relative stability of feedback systems
Design examples and MATLAB simulation
Summary
The concept of stability
A stable system is a dynamic system with a
bounded output to a bounded input (BIBO).
absolute stability
relative stability
Stability for LTI system
A necessary and sufficient condition for a feedback system to be stable is that all the poles of the system transfer function have negative real parts.
Impulse response approach to zero when time lead to infinite.
Routh-Hurwitz criterion
Routh criterion:
This criterion states that the number
of roots of characteristic equation with positive
real parts is equal to the number of changes in
sign of the first column of the Routh array.
Routh-Hurwitz Stability Criterion – Generate Routh Table
Given Routh Table
Routh-Hurwitz Stability Criterion – Generate Routh Table
Routh Table
The value in a row can be divided for easy calculation
Routh-Hurwitz Stability Criterion – Generate Routh Table Example
Given
+10+31S+1030
R-H Criterion – Special Cases
1. Zero in the first column
2. Zero in the entire row
Special Case 1: There are cases when the first element of the
Routh array is zero and the rest of the row is non-zero. In such a case we cannot proceed to the next stage without making some modifications.
Example:
++2+2+3S+5=0
2
3
4
5
S
S
S
S
011
222
053
Special Case 2:
There are cases when all the elements of a row in the Routh array are zero. Obtaining a row of zeros implies one of the four conditions.
i. Real roots but symmetrically located about the j-axis.
ii. Conjugate roots on imaginary axis.
iii. Symmetrically located complex roots about the j-axis.
iv. Repeated conjugate roots on the j-axis.
+2+8+12+20+16+16=0
=1, =8, =12, =20, =16, =16
The Routh array is,
row is zero,
Take auxiliary equation using row,
A(s)=2+12+16
=8+24S
3
4
5
6
S
S
S
S
0221
012128
0161620
0016
Application of R-H criterion One of the important application of the Routh
array to determine the value of the gain K for stability. In many practical examples, an amplifier of gain K is introduced to control the overall system.
1 + k G(s) H(s) = 0H(s)
G(s)+ - KR(s) C(s)
ADVANTAGES
Stability of the system can be determined without actually solving the characteristic equation.
No evaluation of determinants is required which saves calculation time.
It is not tedious or time consuming. It progresses systematically. For unstable system, it gives the number of roots
of characteristic equation having positive real part.
ADVANTAGES
Relative stability of the system can be ascertained. It helps in finding out the range of values of K for
system stability. It helps in finding out intersection point of root
locus with imaginary axis. By using this criterion, critical value of gain can
be determined and hence the frequency of sustained oscillations.
DISADVANTAGES
It is valid only for real coefficient of the characteristic equation.
It does not provide exact locations of the closed-loop poles in left or right half of s-plane.
It does not suggest methods of stabilizing an unstable system.
Applicable only to linear systems.
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