Updated 11 Nov Routh-Hurwitz stability criterion identifies the conditions when the poles of a polynomial cross into the right hand half plane and hence would be considered as unstable in control engineering. Farzad Sagharchi Retrieved June 5,

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In 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 control system. The Routh test is an efficient recursive algorithm that English mathematician Edward John Routh proposed in to determine whether all the roots of the characteristic polynomial of a linear system have negative real parts. A polynomial satisfying the Routh—Hurwitz criterion is called a Hurwitz polynomial.

The importance of the criterion is that the roots p of the characteristic equation of a linear system with negative real parts represent solutions e pt 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. For discrete systems, the corresponding stability test can be handled by the Schur—Cohn criterion, the Jury test and the Bistritz test.

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. Hurwitz derived his conditions differently.

The criterion is related to Routh—Hurwitz theorem. By the fundamental theorem of algebra , each polynomial of degree n must have n roots in the complex plane i. 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.

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 n th-degree polynomial. When completed, the number of sign changes in the first column will be the number of non-negative roots.

In the first column, there are two sign changes 0. The system is unstable, since it has two right-half-plane poles and two left-half-plane poles. 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.

Then another approach comes into play. The row of polynomial which is just above the row containing the zeroes is called the "auxiliary polynomial".

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. From Wikipedia, the free encyclopedia. This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.

Bellman and R. Kalaba Eds. New York: Dover, pp. Control Systems: Principles and Design, 2nd Ed. Tata McGraw-Hill Education.

PHI Learning. Control Systems Engineering. Automatic Control Systems. Delhi: Katson Publishers. Categories : Stability theory Electronic feedback Electronic amplifiers Signal processing Polynomials.

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## Routh–Hurwitz stability criterion

In mathematics , the Routh—Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh-Hurwitz theorem is important in dynamical systems and control theory , because the characteristic polynomial of the differential equations of a stable linear system has roots limited to the left half plane. Thus the theorem provides a test for whether a linear dynamical system is stable. Let f z be a polynomial with complex coefficients of degree n with no roots on the imaginary line i.

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