The Routh stability criterion provides a convenient method of determining a control systems stability.   It determines the number of characteristic roots within the unstable right half of the s-plane, and the number of characteristic roots in the stable left half, and the number of roots on the imaginary axis.   It does not locate the roots.  The method may also be used to establish limiting values for a variable factor beyond which the the system would become unstable...

The characteristic equation being tested for stability is generally of the form

a _ns ⁿ + a _n-1s ^n-1 +....a ₁s + a ₀ = 0...

The Routh Array is constructed as follows>

The elements in rows 3 downwards are calulated as shown below..

The numerator in each case is formed from elements in the two rows above the element being calculated. Using the value in column 1(pivot column) and the value in the column to the right of the element being calculated. The denominator is the element in the pivot column in the row above the element being calculated   The calculated element is made 0 if the row is too short to complete the calculation.

b ₁= ( a_n-1.a_n-2 - a_n. a _n-3 )/ a_n-1
b ₂= ( a_n-1 .a_n-4 - a_n. a _n-5)/ a _n-1 ..etc..

c ₁= ( b₁ .a_n-3  - a_n-1. b ₂ ) / b₁
c ₂= ( b₁ .a_n-5  - a_n-1. b ₃ ) / b₁ .....etc.

The last row will have just one element.

For system stability the primary requirement is that all of the roots of the characteristic equation have negative real parts....
All of the roots of the characteristic equation have negative real parts only if the elements in column 1 of the routh array are the same sign,
The number of sign changes in column 1 is equal to the number of roots of the characteristic equation with positive real parts..

Example ..
To test the stability of a system having a characteristic equation

F(s) = s ³ + 6 s ² + 12 s + 8 = 0

The Routh Array is constructed as follows..

Column 1 (pivot column) includes no changes of sign and therefore the roots of the characteristic equation have only real parts and the system is stable.

1)....
If a zero appears in the first column 1 of any row marginal stability or instability is indicated.  The normal method of constructing the array cannot be continued because the divisor would be zero. A convenient method or resolving this method is to simply replace the zero by a small number δ and continue as normal.  The limit as δ -> 0 is then determined and the first column is checked for sign changes..

Example.
To test the stability of a system having a characteristic equation

F(s) = s ⁵ + 2 s ⁴ + 2 s ³ + 4 s ² + s + 1

The Routh Array is constructed as follows

There are two sign changes in column 1 and there are therefore 2 positive roots and the system is unstable..

2)....If a all of the elements in a row is zero (two rows are proportional)
This indicates that the characteristic polynomial is divided exactly by the polynomial one row above the all zero row (always even-ordered polynomial).    Call this polynomial N(s).

This also indicates the presence of a divisor polynomial N(s) whose roots are all symmetrically arranged about the origin i.e. they are of the form

s = ±α ..or s = ± j ω ..or s = - α ± j ω and s = + α ± j ω

An all zero row will always be associated with and odd power of s

In order to complete the array the previous row is differentiated with respect to s and the array is completed in the normal way..

When assessing this modified array the number of sign changes in the first column (before the all zero row) indicates the number of roots of the remainder polynomial with positive real parts..... From the all zero row down, each change of sign in column 1 indicates the number of roots in the divisor polynomial with positive real roots and as the roots are symmetrical this would indicate the number of roots in the right half s plane and the number of roots in the left- hand s plane.    Root not accounted for in this way i.e no sign change, must lie of the imaginary axis


Example .. Consider a closed loop control system with negative feedback which has an open loop transfer function.

KGH(s) = K / (s (s+1). ( s ² + s + 1) ..

The closed loop characteristic equation =

F(s) = s ⁴ + 2s ³ + 2s ² + s + K = 0

The Routh Array is constructed as follows=

In this array row 3 becomes an all zero row if K = 3/4 and the divisor polynomial of row 2 = (3/2) s ² + 3/4 = 0 ...= 2 s ² + 1
By dividing F(s) by (2 s² + 1) the equation is obtained....... F(s) = ( 1/2 s ² + s + 3/4)
N(s)= (2 s ² + 1)

The array is completed when K = 3/4 by differentiating N(s) with respect to s.
The coefficients of N'(s) are used to replace the zero coefficients in row 3
An all zero row

There are no sign changes up to/including row 2 indicating the roots of the remainder polynomial are in the right hand s plane.
As there are no changes of sign from row 2 down the roots of the divisor polynomial must lie on the imaginary axis.
To locate these roots set s in N(s) to jω

i.e 2 ( j ω ) ² + 1 = 0 therefore .... ω = 1/ √2 rads/unit time

It is sometimes required to find a range of values of a parameter for which the system is stable. This can be achieved by use of the Routh Criteria using the method illustrated by the following example...

The system characteristic equation =

F(s) = s ³ + 3 s ² + 3s + K = 0

In order for the system to be stable there should be no sign change in column 1. To achieve this K must be greater than -1 and K must be less than 8. Therefore for system stability .... -1 < K < 8..