Routh–Hurwitz theorem

Routh–Hurwitz theorem This article includes a list of references, leitura relacionada ou links externos, mas suas fontes permanecem obscuras porque faltam citações em linha. Ajude a melhorar este artigo introduzindo citações mais precisas. (Marchar 2012) (Saiba como e quando remover esta mensagem de modelo) Na matemática, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh-Hurwitz theorem is important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable linear system has roots limited to the left half plane (negative eigenvalues). Thus the theorem provides a test to determine whether a linear dynamical system is stable without solving the system. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz.

Conteúdo 1 Notations 2 Declaração 3 Routh–Hurwitz stability criterion 4 Referências 5 External links Notations Let f(z) be a polynomial (with complex coefficients) of degree n with no roots on the imaginary axis (ou seja. the line Z = ic where i is the imaginary unit and c is a real number). Let us define {estilo de exibição P_{0}(y)} (a polynomial of degree n) e {estilo de exibição P_{1}(y)} (a nonzero polynomial of degree strictly less than n) por {estilo de exibição f(eu)=P_{0}(y)+iP_{1}(y)} , respectively the real and imaginary parts of f on the imaginary line.

Além disso, let us denote by: p the number of roots of f in the left half-plane (taking into account multiplicities); q the number of roots of f in the right half-plane (taking into account multiplicities); {displaystyle Delta arg f(eu)} the variation of the argument of f(eu) when y runs from −∞ to +∞; W(x) is the number of variations of the generalized Sturm chain obtained from {estilo de exibição P_{0}(y)} e {estilo de exibição P_{1}(y)} by applying the Euclidean algorithm; {estilo de exibição I_{-infty }^{+infty }r} is the Cauchy index of the rational function r over the real line. Statement With the notations introduced above, the Routh–Hurwitz theorem states that: {displaystyle p-q={fratura {1}{pi }}Delta arg f(eu)=left.{começar{casos}+EU_{-infty }^{+infty }{fratura {P_{0}(y)}{P_{1}(y)}}&{texto{for odd degree}}\[10pt]-EU_{-infty }^{+infty }{fratura {P_{1}(y)}{P_{0}(y)}}&{texto{for even degree}}fim{casos}}certo}=w(+infty )-W(-infty ).} From the first equality we can for instance conclude that when the variation of the argument of f(eu) é positivo, then f(z) will have more roots to the left of the imaginary axis than to its right. The equality p − q = w(+∞) − w(−∞) can be viewed as the complex counterpart of Sturm's theorem. Note the differences: in Sturm's theorem, the left member is p + q and the w from the right member is the number of variations of a Sturm chain (while w refers to a generalized Sturm chain in the present theorem).

Routh–Hurwitz stability criterion Main article: Routh–Hurwitz stability criterion We can easily determine a stability criterion using this theorem as it is trivial that f(z) is Hurwitz-stable iff p − q = n. We thus obtain conditions on the coefficients of f(z) by imposing w(+∞) = n and w(−∞) = 0.

References Routh, E.J. (1877). A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion. Macmillan and co. Hurwitz, UMA. (1964). "On The Conditions Under Which An Equation Has Only Roots With Negative Real Parts". In Bellman, Ricardo; Kalaba, Robert E. (ed.). Selected Papers on Mathematical Trends in Control Theory. Nova york: Dover. Gantmacher, F. R. (2005) [1959]. Applications of the Theory of Matrices. Nova york: Dover. pp. 226-233. ISBN 0-486-44554-2. Rahman, Q. EU.; Schmeisser, G. (2002). Analytic theory of polynomials. Monografias da Sociedade Matemática de Londres. Nova série. Volume. 26. Oxford: imprensa da Universidade de Oxford. ISBN 0-19-853493-0. Zbl 1072.30006. External links Mathworld entry Categories: Theorems about polynomialsTheorems in complex analysisTheorems in real analysis