- Routh-Hurwitz stability criterion
The Routh-Hurwitz stability criterion is a necessary (and frequently sufficient) method to establish the
stabilityof 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 motionof 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 determinantcalculus.
Using Euclid's algorithm
The criterion is related to
Routh-Hurwitz theorem. Indeed, from the statement of that theorem, we have where:
* "p" is the number of roots of the polynomial "f"("z") located in the left
* "q" is the number of roots of the polynomial "f"("z") located in the right half-plane (let us remind ourselves that "f" is supposed to have no roots lying on the imaginary line);
* "w"("x") is the number of variations of the generalized Sturm chain obtained from and (by successive Euclidean divisions) where for a real "y".By the
fundamental theorem of algebra, each polynomial of degree "n" must have "n" roots in the complex plane (i.e., for an "f" with no roots on the imaginary line, "p"+"q"="n"). Thus, we have the condition that "f" is a (Hurwitz) stable polynomialif 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 "f".
Let "f"("z") be a complex polynomial. The process is as follows:
# Compute the polynomials and such that where "y" is a real number.
# Compute the
Sylvester matrixassociated to and .
# Rearrange each row in such a way that an odd row and the following one have the same number of leading zeros.
# Compute each principal minor of that matrix.
# If at least one of the minors is negative (or zero), then the polynomial "f" is not stable.
* Let (for the sake of simplicity we take real coefficients) where (to avoid a root in zero so that we can use the Routh-Hurwitz theorem). First, we have to calculate the real polynomials and :Next, we find divide those polynomials to obtain the generalizes Sturm chain:
** yields and the Euclidean division stops.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 order polynomial
* the table has 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:In the first column, there are two sign changes (0.75 -> -3, and -3 -> 3), thus there are two non-negative poles and the system is unstable.
Suppose "f" is stable. Then, we must have "q"=0. Since "p"+"q"="n", we find "p"-"q"="n". Suppose now that "p"-"q"="n". Since "p"+"q"="n", subtracting the two equations, we find 2"q"=0, that is "f" is stable.
Derivation of the Routh array
Nyquist stability criterion
Jury stability criterion
* cite journal
author = Hurwitz, A.
year = 1964
title = ‘On the conditions under which an equation has only roots with negative real parts
journal = Selected Papers on Mathematical Trends in Control Theory
* cite book
author = Routh, E.J.
year = 1877
title = A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion
publisher = Macmillan and co.
* cite journal
author = Gantmacher, F.R.
year = 1959
title = Applications of the Theory of Matrices
journal = Interscience, New York
volume = 641
issue = 9
pages = 1–8
* cite journal
author = Pippard, A.B.
coauthors = Dicke, R.H.
year = 1986
title = Response and Stability, An Introduction to the Physical Theory
journal = American Journal of Physics
volume = 54
pages = 1052
url = http://link.aip.org/link/?AJPIAS/54/1052/1
accessdate = 2008-05-07
doi = 10.1119/1.14826
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