Routh's theorem - Hurwitz

The Routh – Hurwitz Theorem provides an opportunity to determine whether a given polynomial is Hurwitz stable . It was proved in 1895 by A. Hurwitz and named after E.J. Routh (who proposed in 1876 another - but equivalent to the Hurwitz criterion - criterion for the stability of a polynomial) and A. Hurwitz ^[1] .

Content

Conventions

Let be $f(z)$ ${\ displaystyle f (z)}$ - polynomial (with complex coefficients) of degree $n$ ${\ displaystyle n}$ . Moreover, among its roots there are no two roots on the same imaginary line (i.e., on the line $z=ic$ ${\ displaystyle z = ic}$ Where $i$ ${\ displaystyle i}$ - imaginary unit and $c$ ${\ displaystyle c}$ Is a real number ). Let's denote $P_{0}(y)$ ${\ displaystyle P_ {0} (y)}$ (polynomial degree $n$ ${\ displaystyle n}$ ) and $P_{1}(y)$ ${\ displaystyle P_ {1} (y)}$ (a nonzero polynomial of degree strictly less than $n$ ${\ displaystyle n}$ ) through $f(iy)=P_{0}(y)+iP_{1}(y)$ ${\ displaystyle f (iy) = P_ {0} (y) + iP_ {1} (y)}$ , relative to the real and imaginary parts $f$ ${\ displaystyle f}$ imaginary line.

We introduce the following notation:

$p$ ${\ displaystyle p}$ - number of roots $f$ ${\ displaystyle f}$ in the left half-plane (taken taking into account multiplicities);
$q$ ${\ displaystyle q}$ - number of roots $f$ ${\ displaystyle f}$ in the right half-plane (taken taking into account multiplicities);
$\Delta \arg f(iy)$ ${\ displaystyle \ Delta \ arg f (iy)}$ - change argument $f(iy)$ ${\ displaystyle f (iy)}$ when $y$ ${\ displaystyle y}$ runs from $-\infty$ ${\ displaystyle - \ infty}$ before $+\infty$ ${\ displaystyle + \ infty}$ ;
$w(x)$ ${\ displaystyle w (x)}$ Is the number of changes in the generalized Sturm chain obtained from $P_{0}(y)$ ${\ displaystyle P_ {0} (y)}$ and $P_{1}(y)$ ${\ displaystyle P_ {1} (y)}$ using the Euclidean algorithm ;

$I_{-\infty }^{+\infty }r(x)$ ${\ displaystyle I _ {- \ infty} ^ {+ \ infty} r (x)}$ - Cauchy index of rational function $r(x)$ ${\ displaystyle r (x)}$ on the real line.

Let be $f(z)$ ${\ displaystyle f (z)}$ Is the Hurwitz polynomial over the field of complex numbers (i.e. $f$ ${\ displaystyle f}$ it has no complex coefficients and all its roots lie in the left half-plane). Decompose $f$ ${\ displaystyle f}$ in the amount of:

f(z)=g(z^{2})+zh(z)

{\ displaystyle f (z) = g (z ^ {2}) + zh (z)}

.

Denote the coefficients $g$ ${\ displaystyle g}$ as $a_{j}^{0}$ ${\ displaystyle a_ {j} ^ {0}}$ , but $h$ ${\ displaystyle h}$ - as $a_{j}^{1}$ ${\ displaystyle a_ {j} ^ {1}}$ . Attention! They are numbered “from the end,” that is, the free coefficient of the polynomial $g$ ${\ displaystyle g}$ is an $a_{0}^{0}$ ${\ displaystyle a_ {0} ^ {0}}$ .

Wording

In the notation introduced above, the Routh - Hurwitz theorem is formulated as follows:

p-q={\frac {1}{\pi }}\Delta \arg f(iy)=-I_{-\infty }^{+\infty }{\frac {P_{1}(y)}{P_{0}(y)}}=w(+\infty )-w(-\infty ).

{\ displaystyle pq = {\ frac {1} {\ pi}} \ Delta \ arg f (iy) = - I _ {- \ infty} ^ {+ \ infty} {\ frac {P_ {1} (y)} {P_ {0} (y)}} = w (+ \ infty) -w (- \ infty).}

From the first equality, for example, we can conclude that when the argument change $f(iy)$ ${\ displaystyle f (iy)}$ positive then $f(z)$ ${\ displaystyle f (z)}$ has more roots to the left of the imaginary axis than to the right. Equality $p-q=w(+\infty )-w(-\infty )$ ${\ displaystyle pq = w (+ \ infty) -w (- \ infty)}$ can be considered as a complex analogue of Sturm's theorem . However, there is a difference: in the Sturm theorem, the left side $p+q$ ${\ displaystyle p + q}$ , but $w$ ${\ displaystyle w}$ from the right side there is the number of changes in the Sturm chain (while in this case $w$ ${\ displaystyle w}$ refers to the generalized Sturm chain).

Hurwitz sustainability criterion

We define the Hurwitz matrix as odd and even coefficients arranged by a “ladder”:

H_{f}={\begin{pmatrix}a_{1}&a_{3}&\dots &a_{1+2\cdot [{\frac {n-1}{2}}]}&&\\a_{0}&a_{2}&\dots &a_{2\cdot [{\frac {n}{2}}]}&&\\&a_{1}&a_{3}&\dots &a_{1+2\cdot [{\frac {n-1}{2}}]}&\\&a_{0}&a_{2}&\dots &a_{2\cdot [{\frac {n}{2}}]}&\\&\vdots &&&\vdots &\\&&&&\dots &a_{n}\end{pmatrix}},

{\ displaystyle H_ {f} = {\ begin {pmatrix} a_ {1} & a_ {3} & \ dots & a_ {1 + 2 \ cdot [{\ frac {n-1} {2}}]} && \\ a_ {0} & a_ {2} & \ dots & a_ {2 \ cdot [{\ frac {n} {2}}]} && \\ & a_ {1} & a_ {3} & \ dots & a_ {1 + 2 \ cdot [{\ frac {n-1} {2}}]} & \\ & a_ {0} & a_ {2} & \ dots & a_ {2 \ cdot [{\ frac {n} {2}}]} & \\ & \ vdots &&& \ vdots & \\ &&&& \ dots & a_ {n} \ end {pmatrix}},}

depending on the degree of the polynomial, the last line will have even or odd coefficients. All major minors of this matrix are positive if $f$ ${\ displaystyle f}$ - Hurwitz polynomial, and vice versa.

Routh stability criterion

Sturm chain starting with polynomials $g$ ${\ displaystyle g}$ and $h$ ${\ displaystyle h}$ determines the sequence $a_{0}^{1},a_{0}^{2},\dots ,a_{0}^{n}$ ${\ displaystyle a_ {0} ^ {1}, a_ {0} ^ {2}, \ dots, a_ {0} ^ {n}}$ leading coefficients of polynomials in a chain. All elements of this sequence have exactly the same sign if $f$ ${\ displaystyle f}$ - Hurwitz polynomial, and vice versa.

There is a more general version of the Routh criterion: the number of roots in the right half - plane is equal to the number of sign changes in the chain.
Please also note that in the entry $a_{0}^{i}$ ${\ displaystyle a_ {0} ^ {i}}$ number $i$ ${\ displaystyle i}$ Is the index of the variable, not an exponent.

Equivalence

Hurwitz and Routh criteria are equivalent. They both characterize Hurwitz-stable polynomials.

Proof

Applying the Gauss Method to the Matrix $H_{f}$ ${\ displaystyle H_ {f}}$ we get the diagonal matrix $H_{f}^{*}$ ${\ displaystyle H_ {f} ^ {*}}$ . However, now the Hurwitz criterion meets the requirement “all elements $h_{j,j}^{*}$ ${\ displaystyle h_ {j, j} ^ {*}}$ transformed matrices have the same sign. ” If we consider in detail how the Gauss method transforms the matrix $H_{f}$ ${\ displaystyle H_ {f}}$ , we obtain the conditions for the generation of the Sturm chain. Making sure that the coefficients $h_{j,j}^{*}$ ${\ displaystyle h_ {j, j} ^ {*}}$ correspond to the coefficients $a_{0}^{j}$ ${\ displaystyle a_ {0} ^ {j}}$ , we get the Routh criterion.

Routh Criterion - Hurwitz

The stability criterion easily follows from this theorem, since $f(z)$ ${\ displaystyle f (z)}$ - Hurwitz stable if and only if $p-q=n$ ${\ displaystyle pq = n}$ . Thus, we obtain conditions on the coefficients $f(z)$ ${\ displaystyle f (z)}$ imposing additional conditions $w(+\infty )=n$ ${\ displaystyle w (+ \ infty) = n}$ and $w(-\infty )=0$ ${\ displaystyle w (- \ infty) = 0}$ .

Along with the Stieltjes theorem, the Routh – Hurwitz theorem gives methods for characterizing stable polynomials. Stability is a property that is important not only in the theory of functions of complex variables. For example, in control theory, a rational filter is stable if and only if its z-transformation is stable. It is such if the Laurent polynomial in the denominator has no roots outside the unit circle . The solution to this problem can, however, be reduced to the stability problem of the “ordinary” polynomial in the formulation stated in this article.

In addition, the correspondence of the Routh and Hurwitz criteria gives more information about the structure of the simple Routh criterion, which is visible when studying the more complex Hurwitz criterion.

Notes

↑ Postnikov, 1981 , p. 15-16.

Literature

Gantmakher F.R. Matrix Theory. - M .: Nauka , 1967 .-- 576 p.
Postnikov M. M. Stable polynomials. - M .: Nauka, 1981. - 176 p.

Links

Weisstein, Eric W. Routh-Hurwitz Theorem on the Wolfram MathWorld website.

[_01abb9fb1f84fdb2-1] Postnikov, 1981 , p. 15-16.