The Routh-Hurwitz criterion

It can be readily checked that the above definition gives a good result for regular sets. For instance, for the point N(e) = 1 = e°, the section (0, 1) can be covered with sections of length e each, if the number of these sections is equal to N(e) = e 1, etc. Hence, it follows from (A43) that the dimension of such a point, section, will be 0, 1, etc., respectively. 

The Cantor set is an example of a self-similar set (fractal) of a fractional fractal dimension. 

The Routh-Hurwitz criterion decides when a given polynomial has roots with a negative real part. Such information proves useful in the analysis of stability of stationary solutions to systems of ordinary differential equations. The examined polynomial is a characteristic polynomial of the stability matrix atj 

From the coefficients of equation (A46), 1, al5.a , we construct the so-called Hurwitz matrices 

The Routh-Hurwitz criterion states that the characteristic polynomial (A45) has the roots with negative real parts if and only if (ifl) the following requirements  

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. 

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

The characteristic equation is already in the form of equation 7.122 and all the coefficients are positive. Thus, to examine the stability of the system further, the Routh array must be constructed, viz.  

All the elements in the first column of the array are positive and thus the system is bounded and stable. 

Destabilising a Stable Process with a Feedback Loop 

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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. 

For a more complex problem, the characteristic polynomial will not be as simple, and we need tools to help us. The two techniques that we will learn are the Routh-Hurwitz criterion and root locus. Root locus is, by far, the more important and useful method, especially when we can use a computer. Where circumstances allow (/.< ., the algebra is not too ferocious), we can also find the roots on the imaginary axis—the case of marginal stability. In the simple example above, this is where Kc = a/K. Of course, we have to be smart enough to pick Kc > a/K, and not Kc < a/K. 

Example 7.2 If we have only a proportional controller (i.e., one design parameter) and real negative open-loop poles, the Routh-Hurwitz criterion can be applied to a fairly high order system with ease. For example, for the following closed-loop system characteristic equation ... 

With the Routh-Hurwitz criterion, we need immediately xr > 0 and Kc > 0. (The, v term requires Kc > -1, which is overridden by the last constant coefficient.) The Routh array for this third order polynomial is... 

When the system has dead time, we must make an approximation, such as the Pade approximation, on the exponential dead time function before we can apply the Routh-Hurwitz criterion. The result is hence only an estimate. Direct substitution allows us to solve for the ultimate gain and ultimate frequency exactly. The next example illustrates this point. 

Let us first use the first order Pade approximation for the time delay function and apply the Routh-Hurwitz criterion. The approximate equation becomes... 

The points at which the loci cross the imaginary axis can be found by the Routh-Hurwitz criterion or by substituting s = jco in the characteristic equation. (Of course, we can also use MATLAB to do that.)... 

For as instructive as root locus plots appear to be, this technique does have its limitations. The most important one is that it cannot handle dead time easily. When we have a system with dead time, we must make an approximation with the Pade formulas. This is the same limitation that applies to the Routh-Hurwitz criterion. 

Show by means of the Routh-Hurwitz criterion that two conditions of controller parameters define upper and lower bounds on the stability of the feedback system incorporating this process. 

Hence with Kc= 1.8 and r, = 3.5, the polar plot of the open-loop transfer function passes through the point (-1, 0). This confirms the result obtained in Example 7.6 using the Routh-Hurwitz criterion, i.e. that with these controller parameters, the response of the controlled variable is conditionally stable. Figure 7.55 shows polar plots of the open-loop transfer function Gr(i) for different values of Kc and t>. 

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 ... 

In order to apply the Routh-Hurwitz criterion, the transformation given by equation 7.230 must be applied. Hence equation 7.236 becomes ... 

The proof of Theorem 5.3 consists in showing that the underlying semi-group is analytic (because of the degeneracy of the equation for a and t in (22)), and then in localizing the spectrum by the Routh-Hurwitz criterion. 

Since f p) < 0, the constant term is positive, so the Routh-Hurwitz criterion (Appendix A) says that Ec will be asymptotically stable if and only if... 

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... 

Consider a situation wherein the stationary state (a, b/a) of the equation without diffusion is stable, that is both the roots of equation (6.160) have negative real parts. It follows from the Routh-Hurwitz criterion (Appendix A5.8) that in this case the control parameters must satisfy the relationships... 

It follows from the Routh-Hurwitz criterion applied to (6.159 ) and from inequality (6.161) that the diffusional instability takes place when C < 0 (since always B > 0), that is when the inequality... 

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... 

What is the major advantage of the Routh-Hurwitz criterion for examining the stability of a system ... 

According to the first test of the Routh-Hurwitz criterion for stability (see Section 15.3), eq. (24.14) has at least one root with positive real part if any of its coefficients are negative. Thus the closed-loop behavior of the process is unstable if Kc i and Kc2 take on such values that make the last term of eq. (24.14) negative [all other terms in eq. (24.14) are always positive] ... 

Note. The allowable range of values for Kc and Kc2, which render stable responses when both loops are closed, can be found by applying the second test of the Routh-Hurwitz criterion. The reader is encouraged to complete this example and find the range of values of Kc and Kc2, which render stable responses. 

Example 15.4 Stability Analysis with the Routh-Hurwitz Criterion... 

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... 

Clarke (1974a, b, 1980) gives a detailed analysis of the stability of the steady state using the structure of the Vol pert graph and several derived graphs to check the Routh-Hurwitz criterion. 

Stable solutions require the amplification factor to be on or within the unit circle in the complex plane. The Mobius transformation maps the unit circle on the left complex half-plane, and thus in general stability can be analyzed in terms of the variable z by the Routh-Hurwitz criterion. 

One can see that since the constant term is negative, it follows immediately from the Routh-Hurwitz criterion that the origin is an unstable equilibrium state. Furthermore, it may have no zero characteristic roots when a and b are positive. The codimension-2 point (a = b = 0) requires special considerations. We postpone its analysis to the last section, where we discuss the bifurcation of double zeros in systems with symmetry. 

Assuming all of the system parameters to be non-negative numbers, the stability conditions based on the Routh-Hurwitz criterion are found to be... 

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