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Stable Polynomial
In the context of the characteristic polynomial of a differential equation or difference equation, a polynomial is said to be stable if either: * all its roots lie in the open left half-plane, or * all its roots lie in the open unit disk. The first condition provides stability for continuous-time linear systems, and the second case relates to stability of discrete-time linear systems. A polynomial with the first property is called at times a Hurwitz polynomial and with the second property a Schur polynomial. Stable polynomials arise in control theory and in mathematical theory of differential and difference equations. A linear, time-invariant system (see LTI system theory) is said to be BIBO stable if every bounded input produces bounded output. A linear system is BIBO stable if its characteristic polynomial is stable. The denominator is required to be Hurwitz stable if the system is in continuous-time and Schur stable if it is in discrete-time. In practice, stability is det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Equation (calculus)
In mathematics, the characteristic equation (or auxiliary equation) is an algebraic equation of degree upon which depends the solution of a given th- order differential equation or difference equation. The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. Such a differential equation, with as the dependent variable, superscript denoting ''n''th-derivative, and as constants, :a_y^ + a_y^ + \cdots + a_y' + a_y = 0, will have a characteristic equation of the form :a_r^ + a_r^ + \cdots + a_r + a_ = 0 whose solutions are the roots from which the general solution can be formed. Analogously, a linear difference equation of the form :y_=b_1y_ + \cdots + b_ny_ has characteristic equation :r^n - b_1r^ - \cdots - b_n =0, discussed in more detail at Linear recurrence with constant coefficients#Solution to homogeneous case. The characteristic roots (roots of the characteristic equation) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Routh–Hurwitz Theorem
In mathematics, 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. Notations Let ''f''(''z'') be a polynomial (with complex coefficients) of degree ''n'' with no roots on the imaginary axis (i.e. the line ''Z'' = ''ic'' where ''i'' is the imaginary unit and ''c'' is a real number). Let us define P_0(y) (a polynomial of degree ''n'') and P_1(y) (a nonzero polynomial o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |