Search results
Results from the WOW.Com Content Network
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 ...
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.
Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or from the coefficients without solving the equation by the Routh–Hurwitz stability criterion .
Thus the theorem provides a mathematical test, the Routh–Hurwitz stability criterion, 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.
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
If there exists an eigenvalue λ of A with Re(λ) > 0 then the solution is unstable for t → ∞. Application of this result in practice, in order to decide the stability of the origin for a linear system, is facilitated by the Routh–Hurwitz stability criterion. The eigenvalues of a matrix are the roots of its characteristic polynomial.
In control theory, and especially stability theory, a stability criterion establishes when a system is stable. A number of stability criteria are in common use: Circle criterion; Jury stability criterion; Liénard–Chipart criterion; Nyquist stability criterion; Routh–Hurwitz stability criterion; Vakhitov–Kolokolov stability criterion
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).