Routh-Hurwitz stability criterion

Routh-Hurwitz stability criterion

The Routh-Hurwitz stability criterion is a necessary (and frequently 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 Euclidiean algorithm and Sturm's theorem in evaluating Cauchy indices.

Using Euclid's algorithm

The criterion is related to Routh-Hurwitz theorem. Indeed, from the statement of that theorem, we have $p-q=w\left(+infty\right)-w\left(-infty\right)$ where:
* "p" is the number of roots of the polynomial "f"("z") located in the left half-plane;
* "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 $P_0\left(y\right)$ and $P_1\left(y\right)$ (by successive Euclidean divisions) where $f\left(iy\right)=P_0\left(y\right)+iP_1\left(y\right)$ 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 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 "f".

Using matrices

Let "f"("z") be a complex polynomial. The process is as follows:
# Compute the polynomials $P_0\left(y\right)$ and $P_1\left(y\right)$ such that $f\left(iy\right)=P_0\left(y\right)+iP_1\left(y\right)$ where "y" is a real number.
# Compute the Sylvester matrix associated to $P_0\left(y\right)$ and $P_1\left(y\right)$.
# 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.

Example

* Let $f\left(z\right)=az^2+bz+c$ (for the sake of simplicity we take real coefficients) where $c eq 0$ (to avoid a root in zero so that we can use the Routh-Hurwitz theorem). First, we have to calculate the real polynomials $P_0\left(y\right)$ and $P_1\left(y\right)$:$f\left(iy\right)=-ay^2+iby+c=P_0\left(y\right)+iP_1\left(y\right)=-ay^2+c+i\left(by\right).$Next, we find divide those polynomials to obtain the generalizes Sturm chain:
** $P_0\left(y\right)=\left(\left(-a/b\right)y\right)P_1\left(y\right)+c,$ yields $P_2\left(y\right)=-c,$
** $P_1\left(y\right)=\left(\left(-b/c\right)y\right)P_2\left(y\right),$ yields $P_3\left(y\right)=0$ 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 $\left(P_0\left(y\right),P_1\left(y\right),P_2\left(y\right)\right)=\left(c-ay^2,by,-c\right)$. Putting $y=+infty$, the sign of $c-ay^2$ is the opposite sign of "a" and the sign of "by" is the sign of "b". When we put $y=-infty$, 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 $w\left(+infty\right)-w\left(-infty\right)=2$ (the degree of "f"). By the properties of the function "w", this is the same as $w\left(+infty\right)=2$ and $w\left(-infty\right)=0$. Thus, "a", "b" and "c" must have the same sign. We have thus found the necessary condition of stability for polynomials of degree 2.

Higher-order example

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 $n-th$ order polynomial
* $D\left(s\right)=a_ns^n+a_\left\{n-1\right\}s^\left\{n-1\right\}+cdots+a_1s+a_0$the table has $n + 1$ rows and the following structure:where the elements $b_i$ and $c_i$ can be computed as follows:
* $b_i=frac\left\{a_\left\{n-1\right\} imes\left\{a_\left\{n-2i-a_n imes\left\{a_\left\{n-2i-1\right\}\left\{a_\left\{n-1$
* $c_i=frac\left\{b_1 imes\left\{a_\left\{n-2i-1-b_\left\{i+1\right\} imes\left\{a_\left\{n-1\right\}\left\{b_1\right\}$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
* $D\left(s\right)=s^5+4s^4+2s^3+5s^2+3s+6$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.

Appendix A

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.

ee also

* Control engineering
* Derivation of the Routh array
* Nyquist stability criterion
* Routh–Hurwitz theorem
* Root locus
* Transfer function
* Jury stability criterion

References

* 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.
isbn =

* 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
accessdate = 2008-05-07
doi = 10.1119/1.14826

Wikimedia Foundation. 2010.

Поможем сделать НИР

Look at other dictionaries:

• Routh–Hurwitz theorem — In mathematics, Routh–Hurwitz theorem gives a test to determine whether a given polynomial is Hurwitz stable. It was proved in 1895 and named after Edward John Routh and Adolf Hurwitz.NotationsLet f(z) be a polynomial (with complex coefficients)… …   Wikipedia

• Jury stability criterion — The Jury stability criterion is a method of determining the stability of a linear discrete time system by analysis of the coefficients of its characteristic polynomial. It is the discrete time analogue of the Routh Hurwitz stability criterion.… …   Wikipedia

• Nyquist stability criterion — The Nyquist plot for . When designing a feedback control system, it is generally necessary to determine whether the closed loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that… …   Wikipedia

• Hurwitz — is a surname and may refer to:*Aaron Hurwitz, musician, see Live on Breeze Hill *Adolf Hurwitz (1859 1919), German mathematician **Hurwitz polynomial **Hurwitz matrix **Hurwitz quaternion **Hurwitz s automorphisms theorem **Hurwitz zeta function… …   Wikipedia

• Stability theory — In mathematics, stability theory deals with the stability of solutions (or sets of solutions) for differential equations and dynamical systems. Definition Let (R, X, Φ) be a real dynamical system with R the real numbers, X a locally compact… …   Wikipedia

• Hurwitz polynomial — In mathematics, a Hurwitz polynomial, named after Adolf Hurwitz, is a polynomial whose coefficients are positive real numbers and whose zeros are located in the left half plane of the complex plane, that is, the real part of every zero is… …   Wikipedia

• Polynôme de Hurwitz — Un polynôme de Hurwitz, ainsi nommé en l honneur du mathématicien allemand Adolf Hurwitz, est un polynôme d’une variable à coefficients réels dont les racines sont toutes à partie réelle strictement négative. En particulier, de tels polynômes… …   Wikipédia en Français

• BIBO stability — Bibo redirects here. For the Egyptian football player nicknamed Bibo, see Mahmoud El Khateeb. In electrical engineering, specifically signal processing and control theory, BIBO stability is a form of stability for signals and systems.BIBO stands… …   Wikipedia

• Derivation of the Routh array — The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial. Central to the field of control systems design, the Routh–Hurwitz theorem and Routh array… …   Wikipedia

• Edward Routh — Infobox Scientist name = Edward Routh caption = Edward John Routh (1831 1907) birth date = birth date|1831|1|20|df=y birth place = Quebec, Canada death date = death date and age|1907|6|7|1831|1|20|df=y death place = Cambridge, England residence …   Wikipedia