The Hurwitz criterion

A polynomial

with k complex conjugate pairs of roots and real roots can always be represented as

If all roots of the polynomial of are in the left-half plane then for all constants and in Eq. (5.4) are positive. From this follows that all coefficients of the polynomial , which are products and sums of positive numbers, are also positive. This result is formulated in the so-called Stodola criterion:

For the polynomial to have all roots with negative real parts it is necessary that

These conditions are also sufficient for and as can be easily verified by calculating the roots. However, for this is no longer the case.

Example 5.3.1   The polynomial with positive coefficients

fulfills the Stodola criterion, but not all the roots , have negative real parts.

A polynomial for which all roots have negative real parts is called Hurwitzian. Therefore, according to the stability conditions introduced in section 5.2 a linear system is only asymptotically stable, if its characteristic polynomial is Hurwitzian. The Hurwitz criterion for the coefficients of a Hurwitz polynomial is as follows:

A polynomial is Hurwitzian, if and only if for all determinants

until

are positive.

The following schema of the coefficients can be used to build the Hurwitz determinants:

The Hurwitz determinants are characterised by the diagonal coefficients , ( ) and by the increasing indices from left to right. Coefficients with indices larger than are set to zero. For applying this criterion all determinants until have to be calculated. Calculation of the last determinant is trivial.

While for a 2nd-order system the conditions of the determinants are automatically fulfilled as soon as the coefficients are positive, for a 3rd-order system one obtains the Hurwitz conditions

It goes without saying that the determinant conditions will be only applied if the easily checkable conditions of Eq. (5.5) are fulfilled. The Hurwitz criterion is not only practical for the stability analysis of a system with given coefficients , but also of a system with free parameters. This is the task when the range of parameters must be determined for which the system is asymptotically stable. Therefore the following example is given.

Example 5.3.2   Figure 5.3 shows a control loop, for which the range of must be determined such that the closed loop is asymptotically stable.

Figure 5.3: Stability analysis of a simple control loop

The time constants and of both lag elements are known and positive. With the transfer function of the open loop

one obtains for the closed-loop transfer function

and by substituting

The characteristic equation of the closed loop is

According to the Stodola and Hurwitz criteria the following conditions must be met for asymptotic stability:

a)

Coefficients , , and must be positive. From this the lower limit follows.

b)

Furthermore

must be valid.

With the coefficients given above it follows that

and for the upper limit of

The closed loop is asymptotically stable for the range