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Process Control: Modeling, Design, and Simulation by B. Wayne Bequette

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5.5. Routh Stability Criterion

The stability of the system is determined from the values of the roots (poles) of its characteristic equation. Finding roots is easy for first- and second-order equations (and not too hard for third) since there is an analytical solution for the roots of polynomials through third order. If the polynomial is fourth order or higher, the roots must be determined numerically. There is a method for determining whether any of the roots are positive (unstable) without actually calculating the roots (Routh, 1905). This method involves an analysis of the coefficients of the characteristic polynomial by setting up the Routh array. The test of the coefficients in the Routh array is called the Routh stability criterion.

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