Routh-Hurwitz determinant

Use the Routh-Hurwitz eriterion to determine the number of roots with positive real parts in the following eharaeteristie equations... 

In the same way that the Routh-Hurwitz criterion offers a simple method of determining the stability of continuous systems, the Jury (1958) stability test is employed in a similar manner to assess the stability of discrete systems. 

By using the Routh-Hurwitz criterion determine whether or not the system is stable. 

In order to determine the number of roots of the z-transformed characteristic equation that lie outside the unit circle, a procedure analogous to the Routh-Hurwitz approach for continuous systems (Section 7.10.2) can be used. The Routh-Hurwitz criterion cannot be applied directly to the characteristic equation f(z) = 0. However, by mapping the interior of the unit circle in the z-piane on to the left half of a new complex variable -plane, the Routh-Hurwitz criterion can be applied as for continuous systems to the corresponding characteristic equation in terms of the new variable<4,). This mapping can be achieved using the bilinear transformation07 ... 

For stability at a rest point one wishes to show that the eigenvalues of the linearization lie in the left half of the complex plane. There is a totally general result, the Routh-Hurwitz criterion, that can determine this. It is an algorithm for determining the signs of the real parts of the zeros of a polynomial. Since the eigenvalues of a matrix A are the roots of a polynomial... 

Examination of the characteristic equation indicates that it is not necessary to compute the actual values of the roots. All that is required is a knowledge of the location of the roots, i.e., if the roots lie on the right- or left-hand side of the imaginary axis. A simple test known as the Routh-Hurwitz test allows one to determine if any root is located on the right side of the imaginary axis, therefore rendering the system unstable. 

To determine acceptable values, employ the Routh-Hurwitz approach ... 

The stability of the steady state is determined using the Routh-Hurwitz criterion which states that the Eigenvalues of the Jacobian have all negative real parts when... 

The Routh-Hurwitz conditions are well known and can be used to determine, in principle, the stability properties of the steady state of any n-variable system. This advantage is, however, balanced by the fact that in practice their use is very cumbersome, even for n as small as 3 or 4. The evaluation, by hand, of all the coefficients Cl of the characteristic polynomial and the Hurwitz determinants A constitutes a rather arduous task. It is for this reason that in the past this tool of linear stability analysis could hardly be found in the literature of nonlinear dynamics. The situation changed with the advent of computer-algebra systems or symbolic computation software. Software such as Mathematica (Wolfram Research, Inc., Champaign, IL) or Maple (Waterloo Maple Inc., Waterloo, Ontario) makes it easy to obtain exact, analytical expressions for the coefficients C/ of the characteristic polynomial (1.12) and the Hurwitz determinants A . 

To determine the stability of the th mode, we conduct a Routh-Hurwitz analysis. All eigenvalues k have a negative real part, if... 

Generally, inhomogeneities in parameters of an array of reactors lead to nonuniform steady states. This is not the case for Lengyel-Epstein networks with inhomogeneities in the parameter a, as is clear from the structure of (13.139). The network still has a unique uniform steady state given by (13.52). We use the Routh-Hurwitz criterion to determine the stability boundaries of this USS. Note that the Routh-Hurwitz analysis is general and can deal with the case where inhomogeneities in parameters lead to nonuniform steady states. Let... 

It is often difficult to determine quickly the roots of the characteristic equation. Hurwitz(I,) and Routh( 0) developed an algebraic procedure for finding the number of roots with positive real parts and consequently whether the system is unstable or not. 

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