Routh-Hurwitz Stability Test

STABILITY ANALYSIS

So, A(s) or for the simplified feedback system (1+GH)

Properties of the Ruth-Hurwitz table:

1. Polynomial A(s) is stable (i.e. all roots of A(s) have negative real parts) if there is no sign change in the first column.

2. The number of sign changes in the first column is equal to the number of roots of A(s) with positive real parts.

s 3 + 4s2 + 8s + 12 = 0

Example1

2 roots with positive real part

Example2

Example3

Example4

Special cases:1.The properties of the table do not change when all the coefficients of a row are multiplied by the same positive number.

2. If the first-column term becomes zero, replace 0 by and ��continue;

• If the signs above and below are the same, thenthere is a pair of (complex) imaginary roots.

Example 4

Example 5

• If there is a sign change, then there are roots withpositive real parts.

3. If all coefficients in a line become 0, then A(s) has roots of equal magnitude radially opposed on the real or imaginary axis. Such roots can be obtained from the roots of the auxiliary polynomial.

Example 5

Cont. Example 5

Application of Routh’s stability criterion to control system analysis

•Example 6: find the range of K for the system stability

The steady state error will be studied for the following input signals:Unit step inputUnit ramp inputUnit parabolic input

First of all we must know the type of the system to know the response

Static Error Constants: Kp, Kv, Ka

Summary