The Routh-Hurwitz Stability Criterion, Revisited
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چکیده
In the mid-nineteenth century James C. Maxwell, and others, became interested in the stability of motion of dynamic systems. Maxwell’s interest in stability stemmed in part from his work with an automatic control system a speed governor he and his colleagues were using in laboratory measurements to establish the definition of the ohm. Maxwell was the first to publish a dynamic analysis of this feedback system using differential equations [ 1] , [2 ] . In this analysis he defined the types of responses one could expect from the solutions of linearized equations of motion having constant coefficients. He identified the conditions which must prevail on the roots of the characteristic polynomial corresponding to the linear differential equation in order that the solution of the homogeneous equation be stable. (They must lie inside the left-half plane.) In this paper he also urged mathematicians to address the question of how the coefficients of the polynomial are related to its roots which, for polynomials of degree higher than three or four, was a difficult question in those days, but a vital one in the study of stability, as it is today. In the mid-1870s Maxwell was on the judging committee for the Adams Prize, a biennial competition for the best essay on a scientific subject selected by the committee. The topic for the 1877 Adams prize was The Stability of Motion E.J. Routh won the competition that year for his essay which showed how the number of roots of the characteristic polynomial lying in the right half plane could be determined from the coefficients of the polynomial [3]. Some twenty years following Routh, the Swiss mathematician A. Hurwitz, unaware of Routh’s work, but also inspired by a stability problem in a control system advanced by his
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Comments on "Routh Stability Criterion"
— In this note, we have shown special case on Routh stability criterion, which is not discussed, in previous literature. This idea can be useful in computer science applications.
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