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.