Routh-Hurwitz stability criterion

An inherently stable process can be destabilised by the addition of a feedback control loop—particularly where integral action is included. This is illustrated in the following example using the characteristic equation and the Routh-Hurwitz stability criterion. 

Bilinear transformation to map the interior of the unit circle in the z-plane onto the left half of the complex variable -plane. (Application of the Routh-Hurwitz stability criterion). ... 

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. 

Stability analysis methods Routh-Hurwitz criterion Apply the Routh test on the closed-loop characteristic polynomial to find if there are closed-loop poles on the right-hand-plane. 

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. 

We first introduce the time honored (/.< ., ancient ) Routh-Hurwitz criterion for stability testing. 

The complete Routh array analysis allows us to find, for example, the number of poles on the imaginary axis. Since BIBO stability requires that all poles lie in the left-hand plane, we will not bother with these details (which are still in many control texts). Consider the fact that we can calculate easily the exact roots of a polynomial with MATLAB, we use the Routh criterion to the extent that it serves its purpose.1 That would be to derive inequality criteria for proper selection of controller gains of relatively simple systems. The technique loses its attractiveness when the algebra becomes too messy. Now the simplified Routh-Hurwitz recipe without proof follows. 

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. 

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... 

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... 

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] ... 

The criterion of stability for closed-loop systems does not require calculation of the actual values of the roots of the characteristic polynomial. It only requires that we know if any root is to the right of the imaginary axis. The Routh-Hurwitz procedure allows us to test if any root is to the right of the imaginary axis and thus reach quickly a conclusion as to the stability of the closed-loop system without computing the actual values of the roots. 

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. 

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

---