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In the control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system. A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as ...
In mathematics, the Routh–Hurwitz matrix, [1] or more commonly just Hurwitz matrix, corresponding to a polynomial is a particular matrix whose nonzero entries are coefficients of the polynomial. Hurwitz matrix and the Hurwitz stability criterion
The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial.Central to the field of control systems design, the Routh–Hurwitz theorem and Routh array emerge by using the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices.
Routh, E. J. (1877). A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion. Macmillan and co. Hurwitz, A. (1964). "On The Conditions Under Which An Equation Has Only Roots With Negative Real Parts". In Bellman, Richard; Kalaba, Robert E. (eds.). Selected Papers on Mathematical Trends in Control Theory. New York: Dover.
A polynomial in one variable with real coefficients is called a Hurwitz polynomial if the real parts of all roots are strictly negative. The Routh–Hurwitz theorem implies a characterization of Hurwitz polynomials by means of an algorithm that avoids computing the roots.
The Routh–Hurwitz theorem provides an algorithm for determining if a given polynomial is Hurwitz stable, which is implemented in the Routh–Hurwitz and Liénard–Chipart tests. To test if a given polynomial P (of degree d) is Schur stable, it suffices to apply this theorem to the transformed polynomial
Independently, Adolf Hurwitz analyzed system stability using differential equations in 1877, resulting in what is now known as the Routh–Hurwitz theorem. [7] [8] A notable application of dynamic control was in the area of crewed flight.
Kharitonov's theorem is a result used in control theory to assess the stability of a dynamical system when the physical parameters of the system are not known precisely. When the coefficients of the characteristic polynomial are known, the Routh–Hurwitz stability criterion can be used to check if the system is stable (i.e. if all roots have negative real parts).