Full Text
MECHANICS
VLADISLAV JAROMÍNEK
ON THE EQUIVALENCE OF THE ROUTH–HURWITZ AND MARKOV STABILITY CRITERIA
(Presented by Academician V. S. Kulebakin, 5 VIII 1959)
In the present work, proceeding from the works of P. L. Chebyshev and A. A. Markov on continued fractions \((^{1,2})\) and their connection with the Routh–Hurwitz problem, proved by F. R. Gantmacher \((^3)\), the task was posed of finding a direct connection between the Hurwitz determinants and the Markov determinants. As a result this led to the formulation of a stability criterion equivalent to the Routh–Hurwitz criterion, called the Markov criterion, and also to the solution of the inverse stability problem for linear systems. Here the latter was interpreted as the selection of values of the coefficients of the characteristic equation satisfying the conditions of the required stability margin in terms of determinants.
§ 1. In order to reveal the connection between the Hurwitz determinants and the Markov determinants, let us consider the matrices \(S^{(1)}=\{s_{iq}\}_1^n\), \(A^{(1)}=\{a_{iq}\}_1^n\), and \(H_n=\{h_{iq}\}_1^n\), where, in particular, \(H_n\) is the Hurwitz matrix composed of the coefficients \(a_s=h_{iq}\) of the characteristic polynomial* \(f(z)\).
Starting from \(S^{(1)}\), we write:
\[ \det\{s_{iq}\}_1^n = S^{(1)} \begin{pmatrix} 12\ldots n\\ 12\ldots n \end{pmatrix} = \left| \begin{array}{cccccccc} 1 & s_{-1} & 0 & 0 & 0 & 0 & \cdots \\ 0 & s_0 & 1 & s_{-1} & 0 & 0 & \cdots \\ 0 & -s_1 & 0 & s_0 & 1 & s_{-1} & \cdots \\ 0 & s_2 & 0 & -s_1 & 0 & s_0 & \cdots \\ 0 & -s_3 & 0 & s_2 & 0 & -s_1 & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \end{array} \right|_{1}^{n}. \tag{1} \]
It can be proved for any \(n\) that the principal minors \(S_s^*=S^{(1)}\binom{12\ldots s}{12\ldots s}\) \((s=1,2,\ldots,n)\), generated by the matrix \(S^{(1)}\), are both groups of Markov determinants (1):
\[ \begin{aligned} &S_1^*=1; \qquad S_2^*=s_0; \qquad S_4^*=\left|\begin{array}{cc} s_0 & s_1\\ s_1 & s_2 \end{array}\right|; \qquad S_6^*=\left|\begin{array}{ccc} s_0 & s_1 & s_2\\ s_1 & s_2 & s_3\\ s_2 & s_3 & s_4 \end{array}\right|; \ \ldots \\[1em] &S_3^*=s_1; \qquad S_5^*=\left|\begin{array}{cc} s_1 & s_2\\ s_2 & s_3 \end{array}\right|; \qquad S_7^*=\left|\begin{array}{ccc} s_1 & s_2 & s_3\\ s_2 & s_3 & s_4\\ s_3 & s_4 & s_5 \end{array}\right|; \ \ldots \end{aligned} \tag{2} \]
The lower triangular matrix \(A^{(1)}\) is composed only of the odd coefficients of the characteristic equation, and
\[ \det\{a_{iq}\}_1^n = A^{(1)} \begin{pmatrix} 12\ldots n\\ 12\ldots n \end{pmatrix} = \left| \begin{array}{cccc} a_1 & \cdot & \cdot & \cdots \\ a_3 & a_1 & \cdot & \cdots \\ a_5 & a_3 & a_1 & \cdots \\ \cdot & \cdot & \cdot & \cdots \end{array} \right|_{1}^{n}. \tag{3} \]
The last element \(a_{iq}\) in (3) different from zero will be \(a_{2k+1}\), if
* Real polynomials with constant coefficients are considered.
\(n=2k+1\), and \(a_{2k-1}\), if \(n=2k\). Taking into account that
\[ \Delta_n=\det\{h_{iq}\}_1^n =H\binom{12\ldots n}{12\ldots n} = \left| \begin{array}{ccccc} a_1 & a_0 & 0 & \ldots & {}^n\\ a_3 & a_2 & a_1 & \ldots & \\ a_5 & a_4 & a_3 & \ldots & \\ \cdot & \cdot & \cdot & \cdot & \cdot \end{array} \right|_1, \tag{4} \]
where \(h_{iq}=a_s\ne0\), if \(0\le s\le n\), and \(h_{iq}=a_s\equiv0\), if \(s<0\) or \(s>n\), we write
\[ \Delta_n=H\binom{12\ldots n}{12\ldots n} =A^{(1)}\binom{12\ldots n}{12\ldots n}\cdot S^{(1)}\binom{12\ldots n}{12\ldots n}. \tag{5} \]
By imposing the corresponding conditions on the values of the elements of the matrix \(S^{(1)}\), one may require that, for any \(n\ge1\), the following determinant identities hold identically with respect to the index \(s\):
\[ \Delta_s=H\binom{12\ldots s}{12\ldots s} \equiv A^{(1)}\binom{12\ldots s}{12\ldots s}\cdot S^{(1)}\binom{12\ldots s}{12\ldots s} \quad \text{for } s=1,2,\ldots,n. \tag{6} \]
The equalities (6), taking into account the properties of the matrices \(A^{(1)}\) and \(S^{(1)}\), lead to the following dependence between the Hurwitz determinants \(\Delta_s\) and the Markov determinants \(S_s^*\):
\[ \Delta_s=a_1^s S_s^* \qquad (s=1,2,\ldots,n). \tag{7} \]
At the same time, the order of the determinants \(S_s^*\) is considerably lower than the order of \(\Delta_s\). (For the computation of Markov determinants of higher order one may apply the convenient “Krakovian method” \((^7,^8)\).)
The conditions necessary and sufficient for uniquely determining all elements \(s_\beta\) of the matrix \(S^{(1)}\) through the coefficients of the characteristic polynomial are obtained as a result of multiplying the determinants on the right-hand side of equality (5) according to the formula
\[ h_{iq}=\sum_{\alpha=1}^{n} a_{i\alpha}s_{\alpha q}. \tag{8} \]
They constitute \(n\) linear equations, which we shall call the independent correspondence conditions (between the coefficients of the characteristic polynomial and the Markov parameters \(s_\beta\)). It is expedient to divide them into two groups. The first of them corresponds to the significant elements of the matrix \(H_n\), and the second to the zero elements of this matrix.
\[ \varkappa=k+1 \left\{ \begin{array}{l} a_0=a_1s_{-1},\\ a_2=a_3s_{-1}+a_1s_0,\\ a_4=a_5s_{-1}+a_3s_0-a_1s_1,\\ \cdots\\ a_{2k}=a_{2k+1}s_{-1}+a_{2k-1}s_0-a_{2k-3}s_1+\cdots+(-1)^{k-1}a_1s_{k-1} \end{array} \right., \tag{9} \]
where \(k=E(n/2)\) and \(a_{2k+1}\equiv0\), if \(n=2k\);
\[ 0=\sum_{i=0}^{\,n-k-1}(-1)^{(q-i)}a_{1+2i}s_{q-i} \qquad (q=k,k+1,\ldots,n-2). \tag{10} \]
Equations (9), whose number is \(\alpha=k+1\), represent the first group, and equations (10), whose number is \(n-\alpha\), the second group of independent correspondence conditions. We illustrate the joint notation of expressions (9) and (10) by the example \(n=2k+1=5\):
\[ \alpha=k+1 \left\{ \begin{array}{l} a_0=a_1s_{-1},\\ a_2=a_3s_{-1}+a_1s_0,\\ a_4=a_5s_{-1}+a_3s_0-a_1s_1; \end{array} \right. \]
\[ n-\alpha \left\{ \begin{array}{l} 0=\phantom{-\,}a_5s_0-a_3s_1+a_1s_2,\\ 0=-a_5s_1+a_3s_2-a_1s_3. \end{array} \right. \]
On the basis of the results obtained, in particular the dependences (7) and the relations (15)*, given below, we formulate stability criteria equivalent to the Routh—Hurwitz and Liénard—Chipart criteria (³,⁴,⁶). Their foundations were laid by A. A. Markov in his work “On functions obtained when converting series into continued fractions” (¹), published in 1894; we shall therefore call them the Markov and Liénard—Chipart—Markov criteria. In doing so we shall proceed from the necessary stability conditions for linear systems established by A. Stodola, consisting in the constancy of the signs of all coefficients of the polynomial \(f(z)\).
Markov stability criterion. In order that all roots of the characteristic real polynomial \(f(z)=a_0z^n+a_1z^{n-1}+\cdots+a_n\) have negative real parts, it is necessary and sufficient that all its coefficients be of one sign and that all the corresponding Markov determinants*** be positive.
Liénard—Chipart—Markov stability criterion. In order that all roots of the characteristic real polynomial \(f(z)=a_0z^n+a_1z^{n-1}+\cdots+a_n\) have negative real parts, it is necessary and sufficient that all its coefficients be of one sign and that only the corresponding even or only the odd Markov determinants be positive.
A consequence of the latter is the lemma:
Lemma. The determinantal inequalities \(S_s^*>0\) or \(\overline{S}_s^*>0\) \((s=1,2,\ldots,n)\), where \(S_s^*\) and \(\overline{S}_s^*\) correspond to a characteristic real polynomial satisfying the Stodola conditions, are not independent, since from the positivity of the even Markov determinants follows the positivity of the odd ones, and conversely.
§ 2. The determinants \(S_s^*\) are expressed linearly in terms of the senior Markov parameters****. Consequently, starting from the prescribed values of the Hurwitz determinants \(\Delta_s\) and the dependences (7), one can find the values of all parameters \(s_\beta\). This makes it possible to determine uniquely the values of all coefficients of the characteristic equation. In doing so it is necessary to consider separately the cases of odd and even degree \(n\).
Case \(n=2k+1\). The odd coefficients of the characteristic equation are determined by the formula
\[ a_{2i+1}=(-1)^{i+1}a_1 \frac{ C_q^{(0)} \begin{pmatrix} 12\ldots\ldots k\\ 12\ldots q\ldots k \end{pmatrix} }{ S_k^{(0)} \begin{pmatrix} 12\ldots k\\ 12\ldots k \end{pmatrix} }; \quad i=1,2,\ldots,k=\mathrm{E}\left(\frac{n}{2}\right);\ q=1+k-i, \tag{11} \]
* The dependences (7), as well as the dependences (15), in contrast to the dependence known in the literature (³–⁵)
\[
R_p=a_0^{2p}\left|s_{k+l}\right|_0^{p-1}=\nabla_{2p},
\]
determine a direct and the most general connection between the Hurwitz and Markov determinants, valid for both even and odd degrees of the characteristic equation.
** The formulated criterion is obtained from the theorems of A. A. Markov on determinants and roots ((¹), p. 96). It represents a generalization of Theorem 17 of F. R. Gantmakher from ((³), p. 468) to all both even and odd degrees of the characteristic equation under a single law for forming the Markov parameters.
*** In the case of the initial dependence (7) these will be the determinants \(S_s^*\), and in the case of dependence (15) the determinants \(\overline{S}_s^*\). The determinants \(S_s^*\) and \(\overline{S}_s^*\) differ essentially from one another in the manner of forming their elements (the Markov parameters).
It should be noted that in the case of the determinants \(\overline{S}_s^*\) one may in general disregard the Stodola conditions, and in the case of \(S_s^*\) replace these conditions only by \(\operatorname{sign} a_0a_1>0\). However, the Stodola conditions have been included in the formulation of the criterion in connection with the fact that their nonfulfillment is the first indication of instability.
**** A senior Markov parameter is the element \(s_\beta\) of the determinant \(S_s^*\) with the largest value of the index \(\beta\).
where
\[ S_k^{(0)}= \begin{vmatrix} s_0 & s_1 & \ldots & s_{k-1}\\ s_1 & s_2 & \ldots & s_k\\ \cdot & \cdot & \cdot & \cdot\\ s_{k-1} & s_k & \ldots & s_{2k-2} \end{vmatrix}; \qquad C^{(0)}= \begin{vmatrix} s_0 & s_1 & \ldots & s_{k-1} & s_k\\ s_1 & s_2 & \ldots & s_k & s_{k+1}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ s_{k-1} & s_k & \ldots & s_{2k-2} & s_{2k-1} \end{vmatrix} \tag{12} \]
and \(C_q^{(0)}\binom{12\ldots\ldots k}{12\ldots q\ldots k}\) is a determinant of order \(k\), which is obtained from the augmented matrix \(C^{(0)}\) by transposing its \(q\)-th column to the last, i.e., the \((k+1)\)-st. After the values of all odd coefficients \(a_{2i+1}\) have been found, it is easy, with the aid of the first group of \(\alpha=k+1\) independent correspondence conditions, to determine all even coefficients \(a_{2i}\).
Case \(n=2k\). The odd coefficients are found by the formula
\[ a_{2i+1}=(-1)^{i+1}a_1 \frac{ C_q^{(1)}\binom{12\ldots\ldots k-1}{12\ldots q\ldots k-1} }{ S_{k-1}^{(1)}\binom{12\ldots k-1}{12\ldots k-1} }, \qquad i=1,2,\ldots,k-1;\quad q=k-i, \tag{13} \]
where
\[ S_{k-1}^{(1)}= \begin{vmatrix} s_1 & s_2 & \ldots & s_{k-1}\\ s_2 & s_3 & \ldots & s_k\\ \cdot & \cdot & \cdot & \cdot\\ s_{k-1} & s_k & \ldots & s_{2k-3} \end{vmatrix}; \qquad C^{(1)}= \begin{vmatrix} s_1 & s_2 & \ldots & s_{k-1} & s_k\\ s_2 & s_3 & \ldots & s_k & s_{k+1}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ s_{k-1} & s_k & \ldots & s_{2k-3} & s_{2k-2} \end{vmatrix} \tag{14} \]
and \(C_q^{(1)}\binom{12\ldots\ldots,k-1}{12\ldots q\ldots,k-1}\) is a determinant of order \((k-1)\), which is obtained from the augmented matrix \(C^{(1)}\) by replacing its \(q\)-th column by the last, i.e., the \(k\)-th. The even coefficients are determined, analogously to the preceding case, with the aid of the first group of independent correspondence conditions.
It can also be proved that the relation
\[ \Delta_s=a_0^{s}\,\bar S_s^{*}\qquad (s=1,2,\ldots,n), \tag{15} \]
holds, which is obtained as a result of solving the determinant equality
\[ \Delta_n = H\binom{12\ldots n}{12\ldots n} \equiv A^{(0)}\binom{12\ldots n}{12\ldots n} \cdot S^{(0)}\binom{12\ldots n}{12\ldots n}, \tag{16} \]
where
\[ A^{(0)}= \begin{vmatrix} a_0 & \cdot & \cdot & \ldots\\ a_2 & a_0 & \cdot & \ldots\\ a_4 & a_2 & a_0 & \ldots\\ \cdot & \cdot & \cdot & \cdot \end{vmatrix}_{1}^{n}; \qquad S^{(0)}= \begin{vmatrix} \bar s_0 & 1 & 0 & 0 & \ldots\\ -\bar s_1 & 0 & \bar s_0 & 1 & \ldots\\ \bar s_2 & 0 & -\bar s_1 & 0 & \ldots\\ \cdot & \cdot & \cdot & \cdot & \cdot \end{vmatrix}_{1}^{n}; \]
\[ \bar S_1^{*}=\bar s_0;\quad \bar S_2^{*}=\bar s_1;\quad \bar S_3^{*}= \begin{vmatrix} \bar s_0 & \bar s_1\\ \bar s_1 & \bar s_2 \end{vmatrix}; \quad \bar S_4^{*}= \begin{vmatrix} \bar s_1 & \bar s_2\\ \bar s_2 & \bar s_3 \end{vmatrix}; \quad \bar S_5^{*}= \begin{vmatrix} \bar s_0 & \bar s_1 & \bar s_2\\ \bar s_1 & \bar s_2 & \bar s_3\\ \bar s_2 & \bar s_3 & \bar s_4 \end{vmatrix}; \ldots \]
In practical applications, relations (7) are more convenient than (15).
The author expresses his deep gratitude to Corresponding Member of the Academy of Sciences of the USSR Prof. B. N. Petrov for a number of valuable suggestions in writing this article.
Received
15 VII 1959
CITED LITERATURE
- A. A. Markov, Selected Works on the Theory of Continued Fractions and the Theory of Functions Least Deviating from Zero, 1948, p. 78.
- P. L. Chebyshev, Complete Collected Works, 3, Mathematical Analysis, Publishing House of the Academy of Sciences of the USSR, 1948, p. 307.
- F. R. Gantmacher, Theory of Matrices, 1954, p. 419.
- M. Krein, M. Naimark, The Method of Symmetric and Hermitian Forms in the Theory of Separation of the Roots of Algebraic Equations, 1936.
- A. Hurwitz, Math. Ann., 46, 273 (1895).
- Liènard, Chipart, J. de Math. pure et appl. (6), 10, 291 (1914).
- T. Banachiewicz, Biul. Polsk. Akad. Umiejętności, Ser. A, 109 (1937).
- W. Sierpiński, Zasady Algebry Wyższej, Monografie matematyczne, 11, Warszawa—Wrocław, 1951.