SKEDSOFT

Control Systems - 1

Routh's stability criterion

 3.   If all coefficients are positive, arrange the coefficients of the polynomial in rows and columns according to the following pattern:

                The evaluation of the b's is continued until the remaining ones are all zero. The same pattern of cross-multiplying the coefficients of the two previous rows is followed in evaluating the c's, d's, e's, and so on. That is,

 

 

This process is continued until the nth row has been completed. The complete array of coefficients is triangular. Note  in developing the array an entire row may be divided or multiplied by a positive number in order to simplify the subsequent numerical calculation without altering the stability conclusion.

Routh's stability criterion states that the number of roots of Equation in 1 with positive real parts is equal to the number of changes in sign of the coefficients of the first column of the array. It should be noted that the exact values of the terms in the first column need not be known; instead, only the signs are needed.

The necessary and sufficient condition that all roots of Equation in 1 lie in the left-half s plane is that all the coefficients of Equation in 1 be positive and all terms in the first column of the array have positive signs.