New Versions of the Hermite Bieler Theorem in Stability Contexts

The Hermite-Bieler theorem played key roles in several control theory problems including the proof of Kharitonov’s theorem and derivations of elementary proofs of the Routh’s algorithm for determining the Hurwitz stability of a real polynomial. In the present work, we explore the stability of complex continuous-time systems of differential equations. Using the theory of positive paraodd functions, we obtain Hermite-Bieler like conditions for the Routh-Hurwitz stability of such systems. We also look at the problem of stability of discrete-time systems of difference equations. By using suitable conformal mappings, we also establish Hermite-Bieler like conditions for the Schur-Cohn stability of these systems. In both cases, the conditions are necessary as well as sufficient.


Introduction
The Routh-Hurwitz criterion for the stability of continuous-time systems of differential equations, and the Schur-Cohn criterion for the stability of discrete-time systems of difference equations have long been treated. The Routh-Hurwitz criterion settles the stability of continuoustime systems with real coefficients. The Extended Routh Array (ERA) which proved to be a natural extension of the Routh Array was introduced [13]. The ERA settles the asymptotic stability of systems with complex coefficients which arise in modern communication, information theory, signal analysis and detection. The main argument in the construction of the ERA dwells on the use of positive paraodd functions which continue to play a highly effective role in the stability of complex continuous-time systems. The Schur-Cohn criterion settles the stability of discrete-time systems, see for example [2]. Using conformal mappings, explicit relationships between the Routh-Hurwitz and the Schur-Cohn criteria were established, [14]. Recently, further connections between the two stability types were explored, [10]. Postnikov gave rich contributions in these directions, including Cauchy indices, Sturm chains, amplitude-phase interpretation of stability, principle of the argument, continued fractions, and the Hermite-Biehler theorem [6].
Kharitonov's famous theorem relied heavily on the Hermite-Bieler theorem [4]. That led to tremendous interest in the latter which also provided elementary derivations of Routh's algorithm for determining the Hurwitz stability of a real polynomial. A version of the Hermite-Bieler theorem was used to determine the set of all stabilizing proportionalintegral PI controllers [1]. Using similar approaches, results on the stability of fractional order polynomials are obtained [7]. By means of the square transform, a class of symmetric indefinite Hermite-Bieler functions was obtained from positive definite ones [5]. The Hermite-Bieler theorem was also used to study the real-rootedness of Eulerian polynomials of different types [8]. Some of the author's contributions were done in the spirit of Kharitonov's theorem [11,12].
In this paper, we provide further characterizations of the Routh-Hurwitz and the Schur-Cohn stability criteria in terms of the Hermite-Bieler theorem.

Routh-Hurwitz Case
We recall the definitions of [9].

Definition 2.1
A non-constant polynomial is a Hurwitz polynomial if all its roots have negative real parts.

Definition 2.2
The paraconjugate of a rational function ( ) Therefore, if h is positive and satisfies h h * = − , we say that h is positive para-odd.
Let ( ) f z be as defined in (1) and let it represent the characteristic polynomial of a continuous-time system of differential equations.
Define the function if odd Obviously, h can be written as This function is sometimes referred to as the test fraction, [3].
The following theorem was established in [13]. Therefore, for such z, Re (z) and Re (1/z) have the same sign, from which it follows that 1/g is positive if and only if g is positive.
The following is a new version of the Hermite-Bieler theorem in the Routh-Hurwitz stability context. Proof Assume that f is Hurwitz, then by Theorem 2.1 above, h(z) can be written as We may assume that 0 0. a = Otherwise, we define the function 0 , k h a = − and we continue with k. Let r(t) = −ih(it), then r(t) can be written as It follows that the equation r(t) = 0 has only one root on each such interval and that completes the first part of the proof.
Conversely, assume 1 2 , , , n it it it … are the roots of N(z), and 1 2 , , , n iw iw iw … the roots of D(z) such that. Re a is now assumed to be positive, it follows that 0 0. d > Again, we may assume that 0 0. a = Reconsider the function Therefore, the polynomial

Schur-Cohn Case
We make the following definitions Definition 3.1 A non-constant polynomial is a Schur polynomial if all its roots lie inside the unit disc. Furthermore, the zeros of g g τ − interlock with those of g g τ + .
considered, the Routh-Hurwitz stability problem for continuous-time systems of differential equations and the Schur-Cohn stability problem for discrete-time systems of difference equations. In both cases, we establish Hermite-Bieler like conditions which guarantee the stability of the system. The conditions are necessary as well as sufficient. In the continuous-time case, we invoked the theory of positive para-odd functions which proved highly effective in stability analysis. In the discrete-time case, the use of suitable conformal mappings permits a smooth transfer of results from the continuous to the discrete case.