Order-Independent Algorithm for the Asymptotic Stability of Complex Polynomials

The Extended Routh Array (ERA) settles the asymptotic stability of complex polynomials. The ERA is a natural extension of the Routh Array which applies only to real polynomials. Although the ERA is a nice theoretical algorithm for stability testing, it has its limitations. Unfortunately, as the order of the polynomial increases, the size of calculations increases dramatically as will be shown below. In the current work, we offer an alternative algorithm which is basically equivalent to the ERA, but has the extra advantage of being simpler, more efficient, and easy to apply even to large order polynomials. In all the steps required in the construction of the new algorithm, only one single and simple algebraic operation is needed, which makes it a polynomial order-independent algorithm. 2010 AMS Subject Classification: Primary 37C75, 93C15. Secondary 93D, 30E


Introduction
The problem of determining conditions under which all the roots of a given polynomial lie in the left-half plane is one of the fundamental problems in the study of stability of a dynamic system. Such polynomials are called Hurwitz polynomials, or asymptotically stable polynomials, and they arise in a variety of applications such as control systems, circuit analysis, numerical computations, systems theory, and digital signal processing to name a few. For some references in this respect see (Gutman, 1979), (Krein & Neimark, 1981) and (Zahreddine, 1992).
Most of the existing methods for testing the Hurwitz stability are restricted to the real case. For a variety of references in this context, see (Howland, 1971), (Lipatov & Sokolov, 1979), and (Zahreddine, 1999). Since modern communication and information theory uses both complex signals and complex envelopes in signal analysis and detection, it is becoming more and more imperative to address the theory of stability essentially when dealing with systems having complex coefficients. With this in mind, the Extended Routh Array (ERA) which is the complex counterpart of the Routh Array for real systems was developed in (Zahreddine, 1993). The ERA settles the Hurwitz stability of complex polynomials.
In (Zahreddine, 1994), we introduced the concept of wide sense stability and we showed how the ERA can handle the appearance of vanishing leading array elements (singularities).
The problem of root distribution of a polynomial in some sub-regions of the complex plane like sectors, ellipses, parabolas has also been investigated (Gutman, 1979), and (Zahreddine, 1996).
Most of the approaches used in the derivation of stability criteria involve elaborate techniques such as: index theory, Sturm chains, Rouche's theorem, Lyaponov equations, or generalized bezoutians. See for example (Heinig & Jungnickel, 1984) (Horn & Johnson, 1991) and (Mawhin, 1997). In the same vein, the ERA also suffers a major defect. As the order of the polynomial increases, the size of calculations increases dramatically as will be made clear in Section 2 below.
it. We end up in Section 5 by some concluding remarks.

The Extended Routh Array (ERA)
The definitions and results stated in this section are based on (Zahreddine, 1993) and (Zahreddine, 2017 Obviously, h can be written as (2) This function is sometimes referred to as the test fraction (Hovstad, 1989).
Call h 1 and h 2 the numerator and denominator of h respectively. If Re a 1  0, denote by h 3 the remainder of the division of h 1 by h 2 , and in general denote by h k the remainder obtained upon dividing h k2 by h k1 for k = 3, …, n+1. It is worthwhile to mention that Lemma 4.1 of [Zahreddine, 1993) gives direct expressions of the coefficients of h k in terms of those of and so on.
In the above ERA, the k-th row represents the coefficients of h k for k = 1, 2, 3, …, n+1 and all rows have the same size by completing by zeros.
Theorem 4.1 of (Zahreddine, 1993) shows that the ERA is an algorithm to test the asymptotic stability of complex polynomials.
Theorem 2.1 (Theorem 4.1, (Zahreddine, 1993)) The complex polynomial f in (1) is Hurwitz if and only if each term of the first column of the extended Routh array is positive.

An Alternative Form of the ERA
Let's consider the following arrangement, in which the 1 st and 2 nd rows match those of the ERA. The notations used in this arrangement will be justified in due context.
The next theorem shows that the above array can well serve as an algorithm to test the Hurwitness of a complex polynomial.

Theorem 3.1
The complex polynomial f in (1)  Lemma 4.1 (Zahreddine, 1993) shows how the b k 's in the above expression can be written in terms of the elements of the 1 st column of the ERA. In fact,  (Zahreddine, 1993), the polynomials f 1 , f 2 , …, f n+1 participating in the construction of the ERA satisfy the following relations:   (Zahreddine, 1993), and it can easily be seen that the terms arising from these relations are the same as those forming our new algorithm (3). That leads to the following expression of h (s), The uniqueness of the continued fraction expansion of h(s) implied by Section 3 of (Zahreddine, 1993) leads to the conclusion that b j =  j for 0  j  n  1. We want to prove the relations ,1 1,1 1,1 2,1 1 for 0 1, where we define 1 and Re .
Obviously these relations hold for j = 0, and j = 1, since 0,1 1,1 1 1 and Re by (4). p p a  Now the relation b 1 =  1 combined with (5) and (6)  and that completes the proof of the theorem.

Order-Independent Algorithm
We introduce the following arrangement based on the ERA, and arrangement (3) defined in Section 3.  i  a  a  ia  i  a  a  i  a  a  ia  a  i  a  a  i  a  a  a  i  a  a  i  a  ia  a  i  a  a i a a i a Theorem 4.1 The complex polynomial f in (1) From the 2k th row, subtract 1/Rea 1 multiplied by the (2k  1) st row for 1 ≤ k ≤ j  1.
Using Arrangement 3 of Section (3) Obviously, the newly defined determinant D j is of order 2j  1 for 2 ≤ j ≤ n. Again, from the (2k  1) st row, subtract q 11 /p 11 multiplied by the 2k th row for 1 ≤ k ≤ j  1.