In the control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system.
A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as time goes on.
The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in 1876 to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts.
[1] German mathematician Adolf Hurwitz independently proposed in 1895 to arrange the coefficients of the polynomial into a square matrix, called the Hurwitz matrix, and showed that the polynomial is stable if and only if the sequence of determinants of its principal submatrices are all positive.
[2] The two procedures are equivalent, with the Routh test providing a more efficient way to compute the Hurwitz determinants (
The importance of the criterion is that the roots p of the characteristic equation of a linear system with negative real parts represent solutions ept of the system that are stable (bounded).
Thus the criterion provides a way to determine if the equations of motion of a linear system have only stable solutions, without solving the system directly.
With the advent of computers, the criterion has become less widely used, as an alternative is to solve the polynomial numerically, obtaining approximations to the roots directly.
The Routh test can be derived through the use of the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices.
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).
(for the sake of simplicity we take real coefficients) where
Next, we divide those polynomials to obtain the generalized Sturm chain:
Notice that we had to suppose b different from zero in the first division.
Finally, −c has always the opposite sign of c. Suppose now that f is Hurwitz-stable.
We have thus found the necessary condition of stability for polynomials of degree 2.
(When this is derived you do not know all coefficients should be positive, and you add
In general the Routh stability criterion states a polynomial has all roots in the open left half-plane if and only if all first-column elements of the Routh array have the same sign.
All coefficients being positive (or all negative) is necessary for all roots to be located in the open left half-plane.
from fourth-order polynomial, and conditions for fifth- and sixth-order can be simplified.
A tabular method can be used to determine the stability when the roots of a higher order characteristic polynomial are difficult to obtain.
In the first column, there are two sign changes (0.75 → −3, and −3 → 3), thus there are two roots whose real part are non-negative and the system is unstable.
for stability, all the elements in the first column of the Routh array must be positive when
The system cannot have jω poles since a row of zeros did not appear in the Routh table.
We have the following table with zero appeared in the first column which prevents further calculation steps:
Sometimes the presence of poles on the imaginary axis creates a situation of marginal stability.
In that case the coefficients of the "Routh array" in a whole row become zero and thus further solution of the polynomial for finding changes in sign is not possible.
The next step is to differentiate the above equation which yields the polynomial
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.