RouthHurwitzTests

RouthHurwitzTests refers to a collection of methods used to assess the stability of linear time-invariant continuous systems by examining the characteristic polynomial of a system’s transfer function. The core components are the Routh array, developed by Edward Routh, and Hurwitz’s criterion, developed by Adolf Hurwitz. The tests determine whether all poles of the closed-loop transfer function lie in the left half of the complex plane, i.e., whether all roots have negative real parts.

Construction and interpretation: Given a characteristic polynomial a0 s^n + a1 s^{n-1} + ... + an with real coefficients, the Routh array is built using the coefficients in a structured table. The first two rows are formed from the even- and odd-index coefficients, and subsequent elements are computed from determinants of previous rows. The stability test hinges on the first column: the number of sign changes in that column equals the number of roots with positive real parts. For a stable system, there must be no sign changes (and typically all first-column elements are positive). If a zero appears in the first column, an auxiliary polynomial from the previous row is formed; if an entire row becomes zero, a derived polynomial is used, or a small positive epsilon may be substituted to continue the process.

Relation to Hurwitz criterion: The Routh-Hurwitz test is equivalent to Hurwitz’s criterion, which states that all leading principal minors of the Hurwitz matrix must be positive for stability. The two approaches provide complementary perspectives on the same stability question.

Applications and limitations: The methods are standard tools in control system design and analysis for verifying stability without explicit root finding. They are most practical for moderate polynomial orders; for high-order systems, the procedure can become algebraically intensive, and the tests do not yield pole locations.