Routh's stability criterion

Routh's stability criterion

1. This is usefull to tells us wether or not there are unstable roots in a polynomial equation.

   • This criterion only applies to plolynomials with a finite number of terms.

2. When is applied to a control system, information about absolute stability can be obtained directly from the coefficients of the characteristic equation

3. Steps to compute Routh's stability criterion:

   • 1.- Write the polynomial as follows:
     • 2.- If any of the coefficients are zero or negative in the presence of at least one positive coefficient, a root ore roots exist that are iimaginary or that have positive real parts
       • 3.- If all coefficients are positive arrenge the terms in the same pattern as shown
         • 4.- The process of forming rows continues until we run out of elements, to calculate them we use is this
           • • To calculate cn, dn,en and so on we use the same pattern

4. Special Cases

   • If a first-column term in any row is zero but not all of them we proce to replace that zero with a very small quantity that aproches to zero
     • then we procede to solve it normaly, and if both roots sorround that small quantity are positive then there are two imaginary roots
   • If all the elements on each column in any row are zero then we proceed to replace that row values with the ones obtained by the derivative of the polynomial equation that precceds that row

5. It is possible to determine the effects of changing one or two parameters of a system by examining the values that cause instability.

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