- Routh-Hurwitz stability criterion
The

**Routh-Hurwitz stability criterion**is a necessary (and frequently sufficient) method to establish thestability of a single-input, single-output (SISO), lineartime invariant (LTI)control system . More generally, given apolynomial , 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 negativereal part s. It can be performed using either polynomial divisions ordeterminant 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 $p-q=w(+infty)-w(-infty)$ where:

* "p" is the number of roots of the polynomial "f"("z") located in the lefthalf-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 $P\_0(y)$ and $P\_1(y)$ (by successive Euclidean divisions) where $f(iy)=P\_0(y)+iP\_1(y)$ for a real "y".By thefundamental 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 $P\_0(y)$ and $P\_1(y)$ such that $f(iy)=P\_0(y)+iP\_1(y)$ where "y" is a real number.

# Compute theSylvester matrix associated to $P\_0(y)$ and $P\_1(y)$.

# 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 $f(z)=az^2+bz+c$ (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 $P\_0(y)$ and $P\_1(y)$:$f(iy)=-ay^2+iby+c=P\_0(y)+iP\_1(y)=-ay^2+c+i(by).$Next, we find divide those polynomials to obtain the generalizes Sturm chain:

** $P\_0(y)=((-a/b)y)P\_1(y)+c,$ yields $P\_2(y)=-c,$

** $P\_1(y)=((-b/c)y)P\_2(y),$ yields $P\_3(y)=0$ 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 $(P\_0(y),P\_1(y),P\_2(y))=(c-ay^2,by,-c)$. 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 $w(+infty)-w(-infty)=2$ (the degree of "f"). By the properties of the function "w", this is the same as $w(+infty)=2$ and $w(-infty)=0$. 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

* $D(s)=a\_ns^n+a\_\{n-1\}s^\{n-1\}+cdots+a\_1s+a\_0$the table has $n\; +\; 1$ rows and the following structure:where the elements $b\_i$ and $c\_i$ can be computed as follows:

* $b\_i=frac\{a\_\{n-1\}\; imes\{a\_\{n-2i-a\_n\; imes\{a\_\{n-2i-1\}\{a\_\{n-1$

* $c\_i=frac\{b\_1\; imes\{a\_\{n-2i-1-b\_\{i+1\}\; imes\{a\_\{n-1\}\{b\_1\}$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

* $D(s)=s^5+4s^4+2s^3+5s^2+3s+6$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

url = http://link.aip.org/link/?AJPIAS/54/1052/1

accessdate = 2008-05-07

doi = 10.1119/1.14826

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