Routh criterion

For given coefficients of the characteristic equation the method of Routh, which is an alternative to the method of Hurwitz, can be applied, see section A.6. Here the coefficients will be arranged in the first two rows of the Routh schema, which contains rows:

The coefficients in the third row are the results from cross multiplication the first two rows according to

Building the cross products one starts with the elements of the first row. The calculation of these values will be continued until all remaining elements become zero. The calculation of the values are performed accordingly from the two rows above as follows:

From these new rows further rows will be built in the same way, where for the last two rows finally

and

follows. Now the Routh criterion is:

A polynomial is Hurwitzian, if and only if the following three conditions are valid:

a)

all coefficients are positive,

b)

all coefficients in the first column of the Routh schema are positive.

Example 5.3.3  

The Routh schema is:

As in the first row of the Routh schema a coefficient is negative the system is unstable.

For proving instability it is sufficient to build the Routh schema only until negative or zero value occurs in the first column. In the example given above the schema could have been stopped at the 5th row.

Another interesting property of the Routh scheme says, that the number of roots with positive real parts is equal to the number of changes of sign of the values in the first column.