The Routh–Hurwitz stability criterion is a necessary and sufficient method to establish the stability of a single-input, single-output (SISO), linear time invariant (LTI) control system. More generally, given a polynomial, some calculations using only the coefficients of that polynomial can lead to the conclusion that it is not stable. For the discrete case, see the Jury test equivalent.
The criterion establishes a systematic way to show that the linearized equations of motion of a system have only stable solutions exp(pt), that is where all p have negative real parts. It can be performed using either polynomial divisions or determinant calculus.
The criterion is derived through the use of the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices.
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The criterion is related to Routh–Hurwitz theorem. Indeed, from the statement of that theorem, we have where:
By the fundamental theorem of algebra, each polynomial of degree n must have n roots in the complex plane (i.e., for an ƒ with no roots on the imaginary line, p + q = n). Thus, we have the condition that ƒ is a (Hurwitz) stable polynomial if and only if p − q = n (the proof is given below). Using the Routh–Hurwitz theorem, we can replace the condition on p and q by a condition on the generalized Sturm chain, which will give in turn a condition on the coefficients of ƒ.
Let f(z) be a complex polynomial. The process is as follows:
Notice that we had to suppose b different from zero in the first division. The generalized Sturm chain is in this case . Putting , the sign of is the opposite sign of a and the sign of by is the sign of b. When we put , the sign of the first element of the chain is again the opposite sign of a and the sign of by is the opposite sign of b. Finally, -c has always the opposite sign of c.
Suppose now that f is Hurwitz-stable. This means that (the degree of f). By the properties of the function w, this is the same as and . Thus, a, b and c must have the same sign. We have thus found the necessary condition of stability for polynomials of degree 2.
A tabular method can be used to determine the stability when the roots of a higher order characteristic polynomial are difficult to obtain. For an nth-degree polynomial
the table has n + 1 rows and the following structure:
where the elements and can be computed as follows:
When completed, the number of sign changes in the first column will be the number of non-negative poles.
Consider a system with a characteristic polynomial
We have the following table:
1 | 2 | 3 | 0 |
4 | 5 | 6 | 0 |
0.75 | 1.5 | 0 | 0 |
−3 | 6 | 0 | |
3 | 0 | ||
6 | 0 |
In the first column, there are two sign changes (0.75 → −3, and −3 → 3), thus there are two non-negative roots where the system is unstable. " Sometimes the presence of poles on the imaginary axis creates a situation of marginal stability. In that case the coefficients of the "Routh Array" become zero and thus further solution of the polynomial for finding changes in sign is not possible. Then another approach comes into play. The row of polynomial which is just above the row containing the zeroes is called "Auxiliary Polynomial".
We have the following table:
1 | 8 | 20 | 16 |
2 | 12 | 16 | 0 |
2 | 12 | 16 | 0 |
0 | 0 | 0 | 0 |
In such a case the Auxiliary polynomial is which is again equal to zero. The next step is to differentiate the above equation which yields the following polynomial. . The coefficients of the row containing zero now become "8" and "24". The process of Routh array is proceeded using these values which yield two points on the imaginary axis. These two points on the imaginary axis are the prime cause of marginal stability.[1]