Some Combined Results from Eneström–Kakeya and Rouché Theorems on the Generalized Schur Stability of Polynomials and the Stability of Quasi-Polynomials-Application to Time-Delay Systems

Abstract

This paper derives some generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström–Kakeya theorem combined with the Rouché theorem. It is first investigated, under sufficiency-type conditions, the derivation of the eventually generalized Schur stability sufficient conditions which are not necessarily related to the zeros of the polynomial lying in the unit open circle. In a second step, further sufficient conditions were introduced to guarantee that the above generalized Schur stability property persists within either the same above complex nominal stability region or in some larger one. The classical weak and, respectively, strong Schur stability in the closed and, respectively, open complex unit circle centred at zero are particular cases of their corresponding generalized versions. Some of the obtained and proved results are further generalized “ad hoc” for the case of quasi-polynomials whose zeros might be interpreted, in some typical cases, as characteristic zeros of linear continuous-time delayed time-invariant dynamic systems with commensurate constant point delays.

1. Introduction

It has to firstly be pointed out that the stability of polynomials is an identical property to the stability of linear time-invariant systems in the sense that a continuous-time linear time-invariant system is asymptotically stable if the poles of its transfer function, that is the zeros of its denominator polynomial are in the open complex half-plane $\mathit{Re}s<0$. In this case, the denominator polynomial of the transfer function is said to be Hurwitz-stable. In the same way, a discrete-time linear time-invariant system is asymptotically stable if the poles of its discretized transfer function, that is, the zeros of its denominator polynomial are in the open complex unit circle centred at zero, $\left|z\right|<1$. In such a case, the denominator polynomial is said to be Schur-stable. The Rouché theorem on zeros is a very useful tool to investigate if the stability of a polynomial persists under a given eventually unstructured perturbation in the event that such a perturbation does not imply increase in the degree of the polynomial. See, for instance, [1] and the references therein.

The idea to test the stability is simple since it suffices to elucidate if all the zeros of the perturbed polynomial in the open stability region are the same as those of the unperturbed one, or equivalently, if there is no extra zero on the open instability region based on the knowledge of the absence of zeros of the unperturbed polynomial in such a region. For that purpose, it suffices to check just a simple inequality on the boundary of the region while it is not necessary to know the exact allocation of the zeros of both polynomials, the unperturbed one (say, the nominal one) and the perturbed one. The same basic idea is extendable to any open region under minimum constraints on the boundary. On the other hand, the Eneström–Kakeya theorem is very popular to elucidate if all the zeros of a polynomial lie within the closed complex unit circle centred at zero, $\left|z\right|\le 1$, provided that all the polynomial coefficients are real and non-negative and they have a non-strict decreasing ordering condition from their highest to their lowest power [2]. Dually, it is also seen as a direct consequence that its open complementary region $\left|z\right|>1$ in the complex plane is zero-free. The above property is in fact a weak Schur stability condition since the eventual existence of zeros at the boundary of the complex unit circle means that the polynomial is not Schur-stable (or strictly Schur-stable) but, in fact, critically stable. Some extensions of the above theorem are given in [2] by considering zeros within an annulus of prescribed radii which can be characterized. In [3,4], some further extensions are given for the case of complex coefficients whose moduli are ordered in decreasing order. Some generalizations of the Eneström–Kakeya theorem are given in [5,6] for the case of polynomials with complex coefficients. In particular, in [5], the conditions of positive realness of the coefficients are dropped and, in addition, the hypothesis related to the coefficients monotonically increasing is weakened. In [7], several generalizations of the Eneström–Kakeya are provided which remove the non-negativity conditions of the coefficients of the polynomial. One of them applies to the case of real non-necessarily non-negative coefficients under a condition involving two combined mixed constraints of both non-strictly decreasing ordering and non-strictly increasing ordering types. The two above ordering conditions involve two groups of consecutive modified coefficients which weight the polynomial coefficients through the powers of a positive prescribed real number. Another relevant generalization given in that paper was concerned with mixed non-strict decreasing/increasing ordering conditions leading the zeros of the polynomial to belong to a certain closed annulus. On the other hand, the Cauchy’s classical bound of the moduli of the zeros of a polynomial with complex coefficients is weakened in [8]. See also [9,10,11]. A necessary and sufficient condition which is easy to test to determine whether all the zeros of a polynomial with complex coefficients to lie on the unit circle is provided in [9] as an alternative to previous similar classic theorems stated by Cohn and Schur. The condition is based on the existence of another polynomial with zeros on the unit circle such that it and its reciprocal together satisfy an additive condition which equalizes the tested polynomial. In [11], it is assumed that the ordering of coefficients is strictly decreasing from the highest to the lowest power and that there is a threshold real number larger than the quotient between any two consecutive coefficients in a decreasing order of powers. In that way, it is ensured that there is a zero-free complex region characterized by such a threshold by inspecting those coefficients. In [12], the stability of multidimensional discrete polynomials is discussed by proposing a modified Jury tabulation and a sign criterion. The proposed criterion extends Jury´s necessary and sufficient criterion on the Schur stability of single-valued polynomials to multivalued discrete polynomials.

Complex functions defined by polynomials whose coefficients are analytic transcendent complex functions, which come from a ring, in the same complex argument as the polynomials, for instance, exponential functions are commonly referred to as quasi-polynomials. Therefore, the coefficients of the quasi-polynomials are periodic functions with an integral period. A typical example of quasi-polynomials are those defining the characteristic equation of linear continuous-time time-invariant systems with commensurate constant point delays; that is, all the delays are integer multiple of the basic minimum one. See, for instance, Refs. [13,14,15,16,17] and some of the references therein. These quasi-polynomials might be described by two-variable polynomials, one variable being the argument of the Laplace transform “$s$”, formally equivalent to the time-derivative operator ”$D$” ($D=d/dt$), while the other one is “$z$” with $z=z\left(s\right)=e^{-hs}$. However, the description in two variables gives only eventually sufficiency-type stability conditions since the variable $z$ is in fact a function of $s$, although it greatly simplifies the stability tests, [13,14,15,16,17], related to the involvement in the stability tests of just the independent Laplace transform argument [16]. It is well-known that the Möbius transformation, which is a bilinear-type transformation in the complex plane, relates the stability of characteristic polynomials associated with linear discrete-time systems (Schur stability) with the stability of polynomials which describe the characteristic equation of linear continuous-time systems (Hurwitz stability). See, for instance, Refs. [17,18,19,20,21] and some references therein. This technique is also worked on in the paper for deriving stability results for characteristic polynomials associated with continuous-time systems associated with discrete-time ones which satisfy the hypotheses for guaranteeing Schur stability based on Eneström–Kakeya-type theorems or their generalizations.

Some recent related results are now briefly described. In [22], some extensions of the Eneström–Kakeya theorem are given under non-strict decreasing conditions from the highest power coefficient to the lowest power one which involve either the even or the odd real polynomial coefficients depending on whether the highest power coefficient is odd. In addition, a supplementary respective positivity condition exists on the first or second coefficient corresponding, respectively, to the zero or first powers of the polynomial. Extensions of the above result are given in [23] by considering complex coefficients which do not follow increasing or decreasing ordering rules.

In [24], some generalizations of the Eneström–Kakeya theorem are given related to the moduli and the real and imaginary parts of the complex coefficients of the polynomial. In particular, ordering conditions are assumed separately for the real and imaginary parts of the polynomial coefficients such that the intended zeros lie in a certain closed annulus. In [25], regions are obtained where no zeros of the polynomial are present and a disc which contains the zeros of the polynomial is also determined when the ordering condition of the coefficients is modified through scalar multiplicative weights for the first and last polynomial coefficients associated, respectively, to the highest and lowest powers of the polynomial variable.

Some further research on the stability of dynamics systems has been developed in [26,27,28] which is not based Eneström–Kakeya-type theorems. In particular, and based on the concept of stable functions, some sufficiency-type conditions for asymptotic stability, exponential stability, and uniform exponential stability of continuous linear time-varying systems are investigated in [26] through differential Lyapunov inequalities. In [27], it is discussed how higher-order averaging is useful to overcome certain drawbacks arising from the direct application of the averaging theorem. Two higher-order averaging methodologies are brought together, namely the perturbation theory through a near-identity transformation and the chronological calculus by means of Lie algebraic tools. Two applications are also discussed in that paper in light of the given theory, namely the Kapitza pendulum problem and the application of flapping flight dynamics associated with micro-air-vehicles or insect flight. On the other hand, some special types of time-varying and/or nonlinear systems are discussed in [28]. The performed research is mainly concerned with their stability properties. The systems dealt with are borrowed from certain industrial models such as complex nonlinear saturated systems, uncertain fuzzy differential systems, Lurie-type nonlinear systems subject to time-varying delay feedback, neutral-type delay differential equations, differential inclusions with nonlinear integral delays, and some chaotic communication systems. On the other hand, more recently, a control scheme for an unknown multivariable nonlinear system is proposed in [29] that is based on the implementation of an adaptive fuzzy controller together with a robust controller which possesses a Nussbaum-type nonlinear gain function. The scheme´s closed-loop stability is guaranteed. Also, in [30], an adaptive decentralized neural network control strategy based on an event-triggered mechanism is investigated for a class of nonlinear multi-input multi-output large-scale nonlinear systems subject to input saturation, external disturbances, and unmeasurable state. See also [31] for related control analysis and designs.

The main motivation of this research is to provide easy stability results of sufficiency type for discrete polynomials based on simply testing the positivity and monotonicity of the ordered polynomial coefficients based on basic and extended Eneström–Kakeya-type theorems with the eventual concourse of the Rouché theorem on zeros to evaluate robustness under deviations of the coefficients of the standard monotonicity conditions. It can be noticed that, although they are just of sufficiency type, the stability conditions provided by Eneström–Kakeya-type theorems are, in general, easier and faster and more practical to test than Jury´s stability tests or alternative tests based on the Möbius transformation which transforms the unit circle into the left-hand-side complex plane allowing the use of the Routh–Hurwitz stability test. This facility is obvious since Eneström–Kakeya-type theorems only need, in general, to test monotonicity conditions on the polynomial coefficients of generalized monotonicity conditions under the use of multiplicative constants for such coefficients. The main objective and contribution of this paper is to give and to prove some simple to test results which are attained by combining the celebrated Eneström–Kakeya and Rouché theorems of zeros for the generalized Schur stability of discrete polynomials. For such a purpose, it is supposed that the allocation of the polynomial zeros in a prescribed circle, not necessarily the unit circle of the complex plane centred at zero. In this context, the analysis first characterizes, under sufficiency-type conditions, an eventually generalized Schur stability property. Such a generalization is made under prescribed conditions on the coefficients of the polynomial by using different versions or extensions of the Eneström–Kakeya theorem. Those conditions are monotonicity constraints on the ordered polynomial coefficients. In a second step, it is investigated, under further sufficient conditions, how the generalized Schur stability property persists in either the same above complex nominal stability region or in some larger one which contains the nominal one, provided that the nominal polynomial is perturbed. The perturbation of the coefficients of the polynomial can consist, in particular, of a lot of the “a priori” conditions assumed about the coefficient of the nominal polynomial by either the Eneström–Kakeya theorem or some of its invoked generalizations. This two-step-based procedure gives a flexible “modus operandi” methodology to derive the sufficient conditions of generalized Schur stability since the first step can be addressed with convenience by using a reference (or nominal) polynomial which fulfils the decreasing ordering condition of the coefficients from the highest to the lowest power of the complex argument z. The second step rearranges the current polynomial as being a perturbed one of the above reference polynomial. The final stability conditions are easier to achieve as the perturbation between both polynomials becomes smaller. The degree of the perturbed polynomial is assumed not to be higher than that of the nominal one. A set of related results in this context are established and proved and discussed in Section 2. Some examples are discussed, some of them linked to classical control theory problems of continuous-time and discrete-time stability. Section 3 is devoted to the generalization of some of the former results to the case of quasi-polynomials whose zeros might be interpreted, in some cases, as being the characteristic zeros of linear continuous-time delayed dynamic systems with commensurate constant point delays. Some further examples are provided and discussed in Section 4. Finally, conclusions and potential guidelines for future related work end the paper.

Notation

$$R_{0+}=\left\{r\in R:r\ge 0\right\}=R_{+}\cup \left\{0\right\}$$

$$R_{-}=\left\{r\in R:r<0\right\}; R_{-0}=\left\{r\in R:r\le 0\right\}=R_{-}\cup \left\{0\right\}$$

$$Z_{+}=\left\{r\in Z:r>0\right\}; Z_{0+}=\left\{r\in Z:r\ge 0\right\}=Z_{+}\cup \left\{0\right\}$$

$$Z_{-}=\left\{r\in Z:r<0\right\}; Z_{-0}=\left\{r\in Z:r\le 0\right\}=Z_{-}\cup \left\{0\right\}$$

$$C_{+}=\left\{r\in C:\mathit{Re}r>0\right\}; C_{0+}=\left\{r\in C:\mathit{Re}r\ge 0\right\}=C_{+}\cup \left\{0\right\}$$

$$C_{-}=\left\{r\in C:\mathit{Re}r<0\right\}; C_{-0}=\left\{r\in C:\mathit{Re}r\le 0\right\}=C_{-}\cup \left\{0\right\}$$

$$\overline{n}=\left\{1,2,\cdots ,n\right\}$$

Let $M$ be a square real or complex matrix. Then, its set of eigenvalues is its spectrum set denoted by $spM$. If $M$ is non-Hurwitzian, equivalently referred to as not being a stability matrix, (i.e., it has unstable and or critically stable eigenvalues in $C_{0+}$), then its non-stable spectrum $s{p}_{-}M=\left\{s\in spM:\mathrm{Re}\mathrm{s}\le 0\right\}\subset spM$ is nonempty. If $M$ is non-convergent (i.e., non-stable in the discrete context), then its non-convergent spectrum, denoted similarly as above as $s{p}_{-}M=\left\{z\in spM:\left|z\right|\ge 1\right\}\subset spM$, is nonempty.

The complex imaginary unit is denoted by $i=\sqrt{-1}$.

Let $P\left(z\right)$ be a polynomial of degree $n\ge 1$ and real coefficients defined by $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j$. It is said that it fulfils the weak positivity ordering condition (WPOc) if $p_j\ge p_{j-1}\ge 0$; $j\in \overline{n}\cup \left\{0\right\}$, and it is said that it fulfils the strong positivity ordering condition (SPOc) if $p_j>p_{j-1}>0$; $j\in \overline{n}\cup \left\{0\right\}$. If $P\left(z\right)$ satisfies the SPOc, then it also trivially satisfies the WPOc. If $p_j=0$ for $j\in \overline{m}\cup \left\{0\right\}$ and $m<n$, then $P\left(z\right)=z^{m+1}\left({\displaystyle {\sum }_{j=0}^m{p}_j}{z}^j\right)$ so that the zeros of $P\left(z\right)$ are those of $P_1\left(z\right)={\displaystyle {\sum }_{j=0}^m{p}_j}{z}^j$ plus a zero $z=0$ of multiplicity $m+1$. And the eventual properties derived from the WPOc and SPOc for the nonzero zeros of $P\left(z\right)$ are those of $P_1\left(z\right)$.

$P\left(z\right)$ is weakly Schur $\rho$-stable ($\mathrm{WS}\left(\rho \right)\mathrm{S}$) if all its zeros of lie in $\left|z\right|\le \rho$,respectively, it is strongly Schur $\rho$-stable ($\mathrm{SS}\left(\rho \right)\mathrm{S}$) if all its zeros of lie in $\left|z\right|<\rho$. The concept of $P\left(z\right)$ being $\mathrm{SS}\left(1\right)\mathrm{S}$ coincides with the classical concept of Schur stability.

2. Main Results

2.1. Basic Results

In [10,11], the following results are summarized for a polynomial $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j$, of real coefficients and degree $n\ge 1$, concerning the classical Eneström–Kakeya theorem:

Theorem A 

(Eneström–Kakeya theorem [10,11]). If the WPOc holds, then all the zeros of $P\left(z\right)$ lie in $\left|z\right|\le 1$.

Theorem B 

([11]). If $p_j\in R_{+}$; $j\in \overline{n}\cup \left\{0\right\}$ and if the SPOc holds, then all the zeros $P\left(z\right)$ lie in $\left|z\right|\le t=\underset{j\in \overline{n}}{\mathit{max}}\left(p_{j-1}/p_j\right)<1$.

The subsequent definitions and lemma are useful for further use concerning its stability.

Definition 1. 

$P\left(z\right)$ is weakly Schur $\rho$-stable ($\mathrm{WS}\left(\rho \right)\mathrm{S}$) if all the zeros of $P\left(z\right)$ lie in $\left|z\right|\le \rho$.

Definition 2. 

$P\left(z\right)$ is strongly Schur $\rho$-stable ($\mathrm{SS}\left(\rho \right)\mathrm{S}$) if all the zeros of $P\left(z\right)$ lie in $\left|z\right|<\rho$.

From the above definitions, we directly obtain the following:

Lemma 1. 

If $P\left(z\right)$ is $\mathrm{SS}\left(\rho \right)\mathrm{S}$, then it is $\mathrm{WS}\left(\rho \right)\mathrm{S}$.

If $P\left(z\right)$ is $\mathrm{WS}\rho \mathrm{S}$, then it is $\mathrm{SS}\left(\rho +\epsilon \right)\mathrm{S}$ for any real $\epsilon >0$.

Remark 1. 

If $\rho =1$ and $P\left(z\right)$ is $\mathrm{SS}\left(1\right)\mathrm{S}$, then all its zeros of $P\left(z\right)$ lie in $\left|z\right|<1$ which leads to the usual definition of the Schur stability of polynomials. This property is “strong” in the sense that the zeros lie in an open complex unit circle centred at zero. Its real version is that $P\left(z\right)$ is $\mathrm{WS}\left(1\right)\mathrm{S}$ if its zeros lie in $\left|z\right|\le 1$. It can be pointed out that, on the other hand, it is well-known that a discrete linear time-invariant system whose characteristic polynomial is $\mathrm{SS}\left(\rho \right)\mathrm{S}$, for $\rho \in \left[0,1\right]$, is globally asymptotically stable in the sense that, for any given finite initial conditions, its solution sequence is bounded and it converges asymptotically to zero in the absence of forcing terms. If its characteristic polynomial has the weaker property of being $\mathrm{WS}\left(1\right)\mathrm{S}$ then the system is globally stable in the sense that, for any given finite initial conditions, its solution sequence is bounded in the absence of forcing terms [18].

Remark 2. 

Note from Theorem B, Definitions 1 and 2, and Lemma 1 that if the SPOc holds, then $P\left(z\right)$ is WS $\underset{j\in \overline{n}}{\mathit{max}}\left(a_{j-1}/a_j\right)$S, $\mathrm{SS}\left(1\right)\mathrm{S}$ and also SS$\left(\underset{j\in \overline{n}}{\mathit{max}}\left(a_{j-1}/a_j\right)-\epsilon \right)$S, for any given real constant $\epsilon \in \left(0,\underset{j\in \overline{n}}{\mathit{max}}\left(a_{j-1}/a_j\right)\right)$, since $\underset{j\in \overline{n}}{\mathit{max}}\left(a_{j-1}/a_j\right)<1$.

The subsequent result follows directly from the inspection of a polynomial $P\left(z\right)$, giving Theorems A and B:

Lemma 2. 

Consider the polynomial $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j$ of real coefficients and degree $n$ (i.e., $p_n\ne 0$). Thus, the following properties hold:

• If $p_0\ne 0$, then $z=0$ is not a zero of $P\left(z\right)$.
• If ${\displaystyle {\sum }_{j=0}^n{p}_j}>0$, then $z=1$ is not a zero of $P\left(z\right)$.
• Assume that the WPOc holds. If $n\ge 2$ is even and if there is at least one positive even coefficient $p_{2\mathcal{l}}$ for $\mathcal{l}\in \overline{m}\cup \left\{0\right\}$, then $z=-1$ is not a zero of $P\left(z\right)$.
• Assume that the WPOc holds. If $n\ge 1$ is odd and, if there is at least one positive even coefficient $p_{2\mathcal{l}+1}$ for $\mathcal{l}\in \overline{m}\cup \left\{0\right\}$, then $z=-1$ is not a zero of $P\left(z\right)$.
• If $p_n>{\displaystyle {\sum }_{j=0}^{n-1}{p}_j}>0$, then there is no zero in $\left|z\right|\ge 1$ so that all the zeros of $P\left(z\right)$ are in $\left|z\right|<1$; thus, $P\left(z\right)$ is $\mathrm{SS}\left(1\right)\mathrm{S}$.
• If $1>\rho \ge p_0/\left({\displaystyle {\sum }_{j=0}^{n-1}{p}_j}\right)$ under the necessary condition that $p_0\le \rho \left({\displaystyle {\sum }_{j=0}^{n-1}{p}_j}\right)<{\displaystyle {\sum }_{j=0}^{n-1}{p}_j}$, then all the zeros of $P\left(z\right)$ are in $\left|z\right|\le \rho <1$; thus, $P\left(z\right)$ is $\mathrm{WS}\left(\rho \right)\mathrm{S}$ and $\mathrm{SS}\left(1\right)\mathrm{S}$.
• Properties 6 and 7, respectively, are generalized if the polynomial coefficients are complex and $\left|p_n\right|>{\displaystyle {\sum }_{j=0}^{n-1}\left|p_j\right|}>0$, respectively, $1>\rho \ge \left|p_0\right|/\left({\displaystyle {\sum }_{j=0}^{n-1}\left|p_j\right|}\right)$.

Proof. 

Property 1 follows from $P\left(0\right)=a_0$. Property 2 follows from $P\left(1\right)={\displaystyle {\sum }_{j=0}^n{p}_j}$.

To prove Property 3, first note that ${\displaystyle {\sum }_{j=0}^n{p}_j=}{\displaystyle {\sum }_{j=0}^m{p}_{2k}+}{\displaystyle {\sum }_{j=0}^{m-1}{p}_{2k+1}>0}$ if at least one of the polynomial coefficients is positive. If $n$ is even and positive then $n=2m\ge 2$ for some $m\in Z_{+}$. Now, if there is at least one positive even coefficient $p_{2\mathcal{l}}$ for $\mathcal{l}\in \overline{m}\cup \left\{0\right\}$, then $z=-1$ is not a zero of $P\left(z\right)$, since then

$$\begin{array}{ll}P\left(-1\right)& ={\displaystyle {\sum }_{j=0}^n{p}_j}{\left(-1\right)}^j={\displaystyle {\sum }_{j=0}^m{p}_{2j}-}{\displaystyle {\sum }_{j=0}^{m-1}{p}_{2j+1}}\\ & =p_{2\mathcal{l}}+\left({\displaystyle {\sum }_{j\left(\ne \mathcal{l}\right)=0}^m{p}_{2j}-}{\displaystyle {\sum }_{j=1}^m{p}_{2j-1}}\right)\ge p_{2\mathcal{l}}+0>0\end{array}$$

(1)

then $P\left(-1\right)\ne 0$ and $z=-1$ is not a zero of $P\left(z\right)$.

To prove Property 4, note that if $n\ge 1$ is odd then $n=2m+1$ for some $m\in Z_{0+}$ so that:

$$\begin{array}{ll}P\left(-1\right)& ={\displaystyle {\sum }_{j=0}^n{p}_j}{\left(-1\right)}^j={\displaystyle {\sum }_{j=0}^m{p}_{2j}-}{\displaystyle {\sum }_{j=0}^m{p}_{2j+1}}\\ & =p_{2\mathcal{l}+1}+\left({\displaystyle {\sum }_{j=0}^m{p}_{2j}-}{\displaystyle {\sum }_{j\left(\ne \mathcal{l}\right)=0}^m{p}_{2j+1}}\right)\ge p_{2\mathcal{l}}+0>0\\ & =p_{2\mathcal{l}+1}+\left({\displaystyle {\sum }_{j=0}^m{p}_{2j}-}{\displaystyle {\sum }_{j\left(\ne \mathcal{l}\right)=1}^{m+1}{p}_{2j-1}}\right)\ge p_{2\mathcal{l}+1}+0>0\end{array}$$

(2)

since the property WPOc holds. Then $P\left(-1\right)\ne 0$ and $z=-1$ is not a zero of $P\left(z\right)$.

Property 5 is proved as follows by rewriting $P\left(z\right)=P\left({\rho }_z{e}^{i{\theta }_z}\right)$ for each complex $z={\rho }_z{e}^{i{\theta }_z}$ for some positive real numbers ${\rho }_z$, ${\theta }_z\in \left[0,2\pi \right)$; thus, if $z_0$ is a zero of $P\left(z\right)$, then $P\left(z_0\right)=0$ so that

$$-p_n{z}_0^n=-p_n{\rho }_{z_0}^n{e}^{in{\theta }_{z_0}}={\displaystyle {\sum }_{j=0}^{n-1}{p}_j}{z}_0^j={\displaystyle {\sum }_{j=0}^{n-1}{p}_j}{\rho }_{z_0}^j{e}^{ij{\theta }_{z_0}}$$

(3)

so that

$$\begin{gathered} \left|-p_n{\rho }_{z_0}^n{e}^{in{\theta }_{z_0}}\right|=\left|p_n{\rho }_{z_0}^n{e}^{in{\theta }_{z_0}}\right|=p_n{\rho }_{z_0}^n=\left|{\displaystyle {\sum }_{j=0}^{n-1}{p}_j}{\rho }_{z_0}^j{e}^{ij{\theta }_{z_0}}\right| \\ \le {\displaystyle {\sum }_{j=0}^{n-1}\left|e^{ij{\theta }_{z_0}}\right|}{p}_j{\rho }_{z_0}^j={\displaystyle {\sum }_{j=0}^{n-1}{p}_j{\rho }_{z_0}^j} \end{gathered}$$

(4)

Assume that ${\rho }_{z_0}=1$ ($z_0$ is on the boundary of $\left|z\right|=1$). Then $p_n\le {\displaystyle {\sum }_{j=0}^{n-1}{p}_j}$ so that if $p_n>{\displaystyle {\sum }_{j=0}^{n-1}{p}_j}$ then there is no zero in $\left|z\right|=1$. Now, assume that ${\rho }_{z_0}>1$ ($z_0$ is in $\left|z\right|>{\rho }_{z_0}>1$). Then, $p_n{\rho }_{z_0}^n\le {\displaystyle {\sum }_{j=0}^{n-1}{p}_j{\rho }_{z_0}^j}\le {\rho }_{z_0}^n\left({\displaystyle {\sum }_{j=0}^{n-1}{p}_j}\right)$, and again $p_n\le {\displaystyle {\sum }_{j=0}^{n-1}{p}_j}$, so that if $p_n>{\displaystyle {\sum }_{j=0}^{n-1}{p}_j}$ then there is no zero in $\left|z\right|>1$ and then Property 5 has been proved.

Now, assume that ${\rho }_{z_0}<1$ ($z_0$ is in $\left|z\right|<1$). Then,

$$\begin{gathered} \left|-p_0{\rho }_{z_0}^0{e}^{in{\theta }_{z_0}}\right|=\left|p_0{\rho }_{z_0}^0{e}^{i0{\theta }_{z_0}}\right|=p_0=\left|{\displaystyle {\sum }_{j=1}^n{p}_j}{\rho }_{z_0}^j{e}^{ij{\theta }_{z_0}}\right| \\ \le {\displaystyle {\sum }_{j=1}^n\left|e^{ij{\theta }_{z_0}}\right|}{p}_j{\rho }_{z_0}={\rho }_{z_0}\left({\displaystyle {\sum }_{j=0}^{n-1}{p}_j}\right) \end{gathered}$$

(5)

which fails if ${\rho }_{z_0}<p_0/\left({\displaystyle {\sum }_{j=0}^{n-1}{p}_j}\right)$ and which holds if $1>{\rho }_{z_0}\ge p_0/\left({\displaystyle {\sum }_{j=0}^{n-1}{p}_j}\right)$ under the necessary condition that $p_0<\left({\displaystyle {\sum }_{j=0}^{n-1}{p}_j}\right)$. Then, Property 6 has been proved. □

Note that Properties 5 and 6 of Lemma 2 are not a direct consequence of Theorem B since it does not invoke the constraint that the sequence of coefficients from the highest to the lowest power of $z$ are strictly decreasing. Thus, those properties are derived under weaker constraints than those of Theorem B. On the other hand, the extended version of Property 5, invoked in Property 7, for complex polynomial coefficients was firstly obtained for a monic polynomial, i.e., for the case $p_n=1$, that is, for the leading coefficient being unity, under the form $1>{\displaystyle {\sum }_{j=0}^{n-1}{p}_j}$ in [20], but the given proof was much longer and more involved than the above one. See also [21] and some references therein.

2.2. Extended Combined Results with Examples from Theorem A and Rouché’s Theorem

The subsequent result extends Theorem A for the case when the coefficients are not necessarily subject to a WPOc, but the polynomial is a sum of a polynomials subject to a WOPC plus another one which fulfils the Rouché theorem of zeros taking as a reference the one under the WOPc.

Theorem 1. 

Consider two polynomials $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j$ and $\tilde{P}\left(z\right)={\displaystyle {\sum }_{j=m_1}^m{\tilde{p}}_j}{z}^j$ of real coefficients and respective degrees $n\ge 1$ and $n\ge m\ge m_1\ge 1$ such that the WPOc holds. Then, the following properties hold:

(i) 

For any given real $\epsilon >0$, all the zeros of the polynomial $Q\left(z\right)=P\left(z\right)+\tilde{P}(z)$ are in $\left|z\right|<1+\epsilon$ provided that

$$\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}{\displaystyle \frac{\left|{\displaystyle {\sum }_{j=m_1}^m{\left(1+\epsilon \right)}^j{\tilde{p}}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}{\left|{\displaystyle {\sum }_{j=0}^n{\left(1+\epsilon \right)}^j{p}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}}<1$$

(6)

and a sufficient condition for (1) to hold is $\left({\displaystyle {\sum }_{j=m_1}^m\left|{\tilde{p}}_j\right|}\right)/\left|\left(\left|p_i\right|-{\displaystyle {\sum }_{j\left(\ne i\right)=0}^n\left|p_j\right|}\right)\right|<{\left(1+\epsilon \right)}^{-m}$ for any $i\in \overline{n}$.

(ii) 

Assume, furthermore, that $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j\ne 0$ for any complex $z$ on the boundary of the unit complex circle centred at zero $\left|z\right|=1$. Then, all the zeros of the polynomial $Q\left(z\right)=P\left(z\right)+\tilde{P}(z)$ are in $\left|z\right|<1$ provided that

$$\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}{\displaystyle \frac{\left|{\displaystyle {\sum }_{j=m_1}^m{\tilde{p}}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}{\left|{\displaystyle {\sum }_{j=0}^n{p}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}}<1$$

(7)

and a sufficient condition for (2) to hold is $\left({\displaystyle {\sum }_{j=m_1}^m\left|{\tilde{p}}_j\right|}\right)/\left|\left(\left|p_i\right|-{\displaystyle {\sum }_{j\left(\ne i\right)=0}^n\left|p_j\right|}\right)\right|<1$ for any $i\in \overline{n}$.

Proof. 

From the Eneström–Kakeya theorem, $P\left(z\right)$ has all its zeros in the closed complex circle centred at zero so that all of them also lie in $\left|z\right|\le 1+\epsilon$ for any given real constant $\epsilon >0$ See, for instance, [1,2]. Now, from the Rouché theorem on zeros, $Q\left(z\right)$ has the same number of zeros in $\left|z\right|\le 1+\epsilon$ as $P\left(z\right)$; that is, all of them, since both have the same degree, provided that $\left|\tilde{P}\left(z\right)/P\left(z\right)\right|<1$ for any z satisfying $\left|z\right|=1+\epsilon$ since $P\left(z\right)$ has no zero in $\left|z\right|=1+\epsilon$, that is, if (6) holds. Property (i) has been proved. Property (ii) is proved in a similar way to Property (ii) by noting that if $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j\ne 0$ for $\left|z\right|=1$ then from the Eneström Eneström–Kakeya theorem $P\left(z\right)$ has all its zeros in $\left|z\right|<1$ so that

$${\left|\tilde{P}\left(z\right)/P\left(z\right)\right|}_{\left|z\right|=1}\le \underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}{\displaystyle \frac{\left|{\displaystyle {\sum }_{j=m_1}^m{\tilde{p}}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}{\left|{\displaystyle {\sum }_{j=0}^n{p}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}}<1$$

(8)

for which a sufficient condition is $\left({\displaystyle {\sum }_{j=m_1}^m\left|{\tilde{p}}_j\right|}\right)/\left|\left(\left|p_i\right|-{\displaystyle {\sum }_{j\left(\ne i\right)}^n\left|p_j\right|}\right)\right|<1$ for any $i\in \overline{n}$. □

Note that if $p_0=0$, Theorem 1 holds trivially since $P\left(z\right)=z{P}_1\left(z\right)$, where $P_1\left(z\right)={\displaystyle {\sum }_{j=1}^n{p}_j}{z}^j$, has a zero $z=0$ in $\left|z\right|\le 1$ and the result applies directly to $P_1\left(z\right)={\displaystyle {\sum }_{j=1}^n{p}_j}{z}^j$. By generalizing the above idea, Theorem 1 is applicable with direct “ad hoc” modifications to any polynomial $P\left(z\right)=z^p{P}_p\left(z\right)$ with $p_i=0$; $i\in \overline{p-1}\cup \left\{0\right\}$, $P_p\left(z\right)={\displaystyle {\sum }_{j=p}^n{p}_j}{z}^j$, and $0\le p<n$.

An extended result from Theorem 1 under weaker constraints and also for guaranteeing the zeros to be in an annulus follows below:

Theorem 2. 

Consider the polynomials $P\left(z\right)$, $\tilde{P}\left(z\right)$, and $Q\left(z\right)$ of Theorem 1 under the weaker constraints $p_j\ge p_{j-1}$; $j\in \overline{n}$ than the WPOc. Let $\rho$ be defined as $\rho =\left(p_n-p_0+\left|p_0\right|\right)/\left|p_n\right|$. Then, the following properties hold:

(i) 

For any given real $\epsilon >0$, all the zeros of the polynomial $Q\left(z\right)=P\left(z\right)+\tilde{P}(z)$ are in $\left|z\right|<\rho +\epsilon$ provided that

$$\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}{\displaystyle \frac{\left|{\displaystyle {\sum }_{j=m_1}^m{\left(\rho +\epsilon \right)}^j{\tilde{p}}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}{\left|{\displaystyle {\sum }_{j=0}^n{\left(\rho +\epsilon \right)}^j{p}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}}<1$$

(9)

and a sufficient condition for (4) to hold is $\left({\displaystyle {\sum }_{j=m_1}^m\left|{\tilde{p}}_j\right|}\right)/\left|\left(\left|p_i\right|-{\displaystyle {\sum }_{j\left(\ne i\right)=0}^n\left|p_j\right|}\right)\right|<{\left(\rho +\epsilon \right)}^{-m}$ for any $i\in \overline{n}$.

If, in addition, $P\left(z\right)\ne 0$ for $\left|z\right|=\rho$, then the two above conditions also hold with $\epsilon =0$.

(ii) 

Assume, in addition, that $P\left(z\right)$ has no zeros in $\partial R_P=\left\{z\in C:\left(\left|z\right|=R_1\right)\vee \left(\left|z\right|=R_2\right)\right\}$, where

$$R_2={\displaystyle \frac{c}{2}}\left({\displaystyle \frac{1}{\left|p_n\right|}}-{\displaystyle \frac{1}{M_2}}\right)+{\left\{{\displaystyle \frac{c^2}{4}}{\left({\displaystyle \frac{1}{\left|p_n\right|}}-{\displaystyle \frac{1}{M_2}}\right)}^2+{\displaystyle \frac{M_2}{\left|p_n\right|}}\right\}}^{1/2}$$

(10)

$$R_1={\displaystyle \frac{1}{2M_1^2}}\left[-R_2^{2}b\left(M_1-\left|p_0\right|\right)+{\left\{R_2^4{b}^2{\left(M_1-\left|p_0\right|\right)}^2+4\left|p_0\right|R_2^2{M}_1^3\right\}}^{1/2}\right]$$

(11)

$$M_2=\rho p_n=p_n-p_0+\left|p_0\right|; M_1=R_2^n\left(\left|p_n\right|R_2+p_n-p_0\right)$$

(12)

$$c=p_n-p_{n-1}; b=p_1-p_0$$

(13)

where $0\le R_2\le 1$; $1\le R_1\le \left(p_n-p_0+\left|p_0\right|\right)/\left|p_n\right|$. Then, $P\left(z\right)$ has its zeros in the open annulus $R_1<\left|z\right|<R_2$. If (4) holds for $\epsilon =0$, or if its associate next sufficient condition for that holds with $\epsilon =0$, then $Q\left(z\right)$ also has its zeros in $R_1<\left|z\right|<R_2$.

Proof. 

It follows that, if $p_j\ge p_{j-1}$; $j\in \overline{n}$, then $P\left(z\right)$ has its zeros in $\left|z\right|\le \rho$ from an extended Eneström–Kakeya theorem in [2], so that they also lie in $\left|z\right|<\rho +\epsilon$, [2], and if (1) or the next sufficiency-type condition holds then $Q\left(z\right)$ also has all its zeros in $\left|z\right|<\rho +\epsilon$ from Rouché’s theorem. If, furthermore, $P\left(z\right)\ne 0$ for $\left|z\right|=\rho$, then $P\left(z\right)$ has its zeros in $\left|z\right|<\rho$ so that $Q\left(z\right)$ has also its zeros in $\left|z\right|<\rho$ if (9) holds with $\epsilon =0$. Property (i) has been proved. From an extended Eneström–Kakeya theorem provided in [2], $P\left(z\right)$ has its zeros in $R_1\le \left|z\right|\le R_2$ in the general case so that, if in addition, $P\left(z\right)\ne 0$ for $z\in \partial R_P$, then $R_1<\left|z\right|<R_2$. If, furthermore, (10)–(13) hold for $\epsilon =0$, or if its associate next sufficient condition for that holds with $\epsilon =0$, then $Q\left(z\right)$ also has its zeros in $R_1<\left|z\right|<R_2$. □

The subsequent result gives the relations between the first and the last polynomial coefficients to establish the sufficient conditions of the complex region where the zeros of the polynomial belong, under the decreasing order constraint of their moduli:

Corollary 1. 

In Theorem 2, $\rho =\left(p_n-p_0+\left|p_0\right|\right)/\left|p_n\right|\le 1$, 1 under the constraints $p_j\ge p_{j-1}$; $j\in \overline{n}$ of Theorem 2 if and only if $\left|p_n/p_0\right|\ge \left(1-\mathit{sgn}{p}_0\right)/\left(1-\mathit{sgn}{p}_n\right)$. Then, $P\left(z\right)$ is $\mathrm{WS}\left(\rho \right)\mathrm{S}$ and $\mathrm{WS}\left(1\right)\mathrm{S}$. Also, $\rho <1$ if and only if $\left|p_n/p_0\right|>\left(1-\mathit{sgn}{p}_0\right)/\left(1-\mathit{sgn}{p}_n\right)$. Then, $P\left(z\right)$ is $\mathrm{SS}\left(1\right)\mathrm{S}$.

Proof. 

The proof is direct by taking into account that

$$\left[\left|p_n/p_0\right|\ge \left(1-\mathit{sgn}{p}_0\right)/\left(1-\mathit{sgn}{p}_n\right)\right]\iff \left[\rho \le 1\right]$$

(14)

$$\left[\left|p_n/p_0\right|>\left(1-\mathit{sgn}{p}_0\right)/\left(1-\mathit{sgn}{p}_n\right)\right]\iff \left[\rho <1\right]$$

(15)

□

Example 1. 

Under the conditions of Corollary 1, one has the following:

(a)

If $\mathit{sgn}{p}_n=\mathit{sgn}{p}_0=-1$, then $\rho \le 1$ if and only if $-p_n=\left|p_n\right|\ge \left|p_0\right|=-p_0$ from the coefficient constraints and $\left|p_n/p_0\right|\ge \left(1-\mathit{sgn}{p}_0\right)/\left(1-\mathit{sgn}{p}_n\right)=2$ which is equivalent to $p_n\le 2p_0<0$. For instance, if $n=1$, then $P\left(z\right)=-p_{1}z-p_0=\left|p_1\right|z+\left|p_0\right|$ has a zero $-1\le z=-\left|p_0\right|/\left|2p_0\right|=-1/2<0$ with $\left|z\right|=1/2\le 1$. If the coefficients inequality is strict, that is, $p_1<2p_0<0$, then $-1<z=-\left|p_0\right|/\left|p_1\right|=-1/2<0$ and $\left|z\right|=1/2<1$. $P\left(z\right)$ is $\mathrm{WS}\left(1/2\right)\mathrm{S}$ and $\mathrm{SS}\left(1\right)\mathrm{S}$.

(b)

If $\mathit{sgn}{p}_n=-1$; $\mathit{sgn}{p}_0=1$, then $-p_n=\left|p_n\right|\ge \left|p_0\right|=2p_0>0$. For instance, if $P\left(z\right)=-\left|p_1\right|z+p_0$ has a zero, $1\ge z=p_0/\left|p_1\right|=\left|p_0\right|/\left|p_1\right|\ge \left(1-\mathit{sgn}{p}_0\right)/\left(1-\mathit{sgn}{p}_n\right)=0/2=0$. If the coefficients inequality is strict, that is, $-p_1=\left|p_1\right|>p_0>\left(1-\mathit{sgn}{p}_0\right)/\left(1-\mathit{sgn}{p}_n\right)=0/2=0$, then $1>z=p_0/\left|p_1\right|=\left|p_0\right|/\left|p_1\right|>0$.

(c)

If $\mathit{sgn}{p}_n=-\mathit{sgn}{p}_0=1$, then $p_n=\left|p_n\right|\ge \left|p_0\right|=-p_0>0$ from the coefficients constraint. However, the corollary constraint $\left[\left|p_n/p_0\right|\ge \left(1-\mathit{sgn}{p}_0\right)/\left(1-\mathit{sgn}{p}_n\right)\right]=2/0=\infty$ fails and this case is not covered by the corollary.

(d)

If $\mathit{sgn}{p}_n=\mathit{sgn}{p}_0=1$, then $p_n=\left|p_n\right|\ge \left|p_0\right|=p_0>0$ from the constraint on the coefficients. However, the corollary constraint $p_n\left(1-\mathit{sgn}{p}_n\right)\ge p_0\left(1-\mathit{sgn}{p}_0\right)$ gives $\left|z\right|\le \rho =1$ and this case guarantees only weak 1-Schur stability from the corollary but not strong 1-Schur stability since $\rho =\left(p_n-p_0+\left|p_0\right|\right)/\left|p_n\right|=1$.

Theorem 1 can be extended as follows by relaxing the monotonicity condition on the coefficients:

Theorem 3. 

Consider two polynomials $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{t}^j{p}_j}{z}^j$ and $\tilde{P}\left(z\right)={\displaystyle {\sum }_{j=m_1}^m{\tilde{p}}_j}{z}^j$ of real coefficients and respective degrees $n\ge 1$ and $m$ with $n\ge m\ge m_1\ge 1$. Then, the following properties hold:

(i) 

Assume that for some nonzero real constant $t$, $p_j\ge t^{-1}{p}_{j-1}\ge 0$; $j\in \overline{n}$. Then, all the zeros of $P\left(z\right)$ lie in $\left|z\right|\le 1$. For any given real $\epsilon >0$, all the zeros of the polynomial $Q\left(z\right)=P\left(z\right)+\tilde{P}(z)$ are in $\left|z\right|<1+\epsilon$ provided that

$$\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}{\displaystyle \frac{\left|{\displaystyle {\sum }_{j=m_1}^m{\left(1+\epsilon \right)}^j{\tilde{p}}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}{\left|{\displaystyle {\sum }_{j=0}^n{\left(1+\epsilon \right)}^j{t}^j{p}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}}<1$$

(16)

and a sufficient condition for (9) to hold is $\left({\displaystyle {\sum }_{j=m_1}^m\left|{\tilde{p}}_j\right|}\right)/\left|\left(\left|t^i{p}_i\right|-{\displaystyle {\sum }_{j\left(\ne i\right)=0}^n\left|t^j{p}_j\right|}\right)\right|<{\left(1+\epsilon \right)}^{-m}$ for any $i\in \overline{n}$.

(ii) 

Define the polynomial $P^{\prime }\left(tz\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{\left(tz\right)}^j$, which gives the same value as $P\left(z\right)$ for each $z\in C$, subject to $p_j\ge p_{j-1}\ge 0$; $j\in \overline{n}$ with $p_0>0$. Then all the zeros of $P\left(z\right)$ lie in $\left|z\right|\le t^{-1}$ so that $P^{\prime }\left(tz\right)$ is $\mathrm{SS}\left(1\right)\mathrm{S}$ if $\left|t\right|>1$. Define the polynomial ${\tilde{P}}^{\prime }\left(tz\right)={\displaystyle {\sum }_{j=m_1}^m{\tilde{p}}_j}{\left(tz\right)}^j$. Then, $Q^{\prime }\left(tz\right)=P\left(tz\right)+{\tilde{P}}^{\prime }(tz)$ has all its zeros in $\left|z\right|\le t^{-1}$ if

$$\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}{\displaystyle \frac{\left|{\displaystyle {\sum }_{j=m_1}^m{t}^j{\tilde{p}}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}{\left|{\displaystyle {\sum }_{j=0}^n{t}^j{p}_j}\left(\mathit{cos}j\theta +i\mathit{sin}j\theta \right)\right|}}<1$$

(17)

and a sufficient condition for (17) to hold is

$$\left({\displaystyle {\sum }_{j=m_1}^m\left|{\tilde{p}}_j\right|}\right)/\left|\left(\left|p_i\right|-{\displaystyle {\sum }_{j\left(\ne i\right)=0}^n\left|p_j\right|}\right)\right|<{\displaystyle \frac{\mathit{min}\left(1,t^n\right)}{\mathit{max}\left(t^{m_1},t^m\right)}}.$$

(18)

for any $i\in \overline{n}$.

Proof. 

The first part of Property (i) follows from Eneström–Kakeya theorem as in Theorem 1. The second part follows from the Rouché theorem on zeros. □

In a very similar way as for addressing the above result, we can now obtain an extended version of the Eneström–Kakeya theorem of the original polynomial to guarantee when the zeros lie within closed circles of the complex plane centred at zero.

If a weighted sequence with a powered arbitrary nonzero weight sequence of the polynomial coefficients of the highest to the lowest power is non-increasing, then the zeros are in an open circle of radius defined by the weight. This idea is addressed in the next result:

Theorem 4. 

Consider the polynomials $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j$ of degree $n\ge 1$ with real coefficients. If $p_j\ge t^{-1}{p}_{j-1}\ge 0$; $j\in \overline{n}$ for any given real constant $t>0$, then all the zeros of $P\left(z\right)$ lie in $\left|z\right|\le t^{-1}$.

Consider the polynomial $Q\left(z\right)=P\left(z\right)+\tilde{P}\left(z\right)$, where $\tilde{P}\left(z\right)={\displaystyle {\sum }_{j=m_1}^m{\tilde{p}}_j}{z}^j$ has a degree $m$ such that $n\ge m\ge m_1\ge 0$. If $p_j\ge t{p}_{j-1}\ge 0$; $j\in \overline{n}$ for any given real constant $t>0$ and, furthermore, ${\left|{\displaystyle \frac{\tilde{P}\left(z\right)}{P\left(z\right)}}\right|}_{\left|z\right|=t^{-1}+\tilde{t}}<1$ for any arbitrary real constant $\tilde{t}>0$, then all the zeros of $Q\left(z\right)$ lie in $\left|z\right|\le t^{-1}+\tilde{t}$. If $t>1$ and $\tilde{t}<1-t^{-1}$, then $P\left(z\right)$ is $\mathrm{SS}1\mathrm{S}$ with zeros in $\left|z\right|\le t^{-1}<1$ and $Q\left(z\right)$ is $\mathrm{SS}\left(1\right)\mathrm{S}$ with zeros in $\left|z\right|\le t^{-1}+\tilde{t}<1$.

Proof. 

Note that for any positive real constant $t$

$$P\left(z\right)={\displaystyle {\sum }_{j=0}^n\frac{p_j{t}^{n-j}}{t^n}}{\left(tz\right)}^j$$

(19)

Then, $P\left(z\right)=P^{\prime }\left(tz\right)={\displaystyle {\sum }_{j=0}^n{p}_j^{\prime }}{\left(tz\right)}^j$ where $p_j^{\prime }={\displaystyle \frac{p_j{t}^{n-j}}{t^n}}$. From the Eneström–Kakeya-theorem, and since $p_j^{\prime }\ge p_{j-1}^{\prime }\ge 0$; $j\in \overline{n}$ is equivalent to $p_j\ge t{p}_{j-1}\ge 0$; $j\in \overline{n}$, this last condition implies that the zeros of $P^{\prime }\left(tz\right)$, and also those of $P\left(z\right)$, lie in $\left|z\right|\le t^{-1}$. This proves the first part of the theorem. On the other hand, note that the zeros of $P\left(z\right)$ also lie in the open circle $\left|z\right|<t^{-1}+\tilde{t}$ centred at zero for any $\tilde{t}>0$. Then, if ${\left|{\displaystyle \frac{\tilde{P}\left(z\right)}{P\left(z\right)}}\right|}_{\left|z\right|=t^{-1}+\tilde{t}}<1$, then the zeros of $Q\left(z\right)$ lie in $\left|z\right|\le t^{-1}+\tilde{t}$ from the Rouché theorem on zeros. As a result, if $t>1$ and $\tilde{t}<1-t^{-1}$ with zeros in $\left|z\right|\le t^{-1}<1$, $Q\left(z\right)$ is $\mathrm{SS}\left(1\right)\mathrm{S}$ with zeros in $\left|z\right|\le t^{-1}+\tilde{t}<1$. □

Remark 3. 

Theorem 4 can be used to check the stability of polynomials which can be stable but do not fulfil the decreasing ordering of the polynomial coefficients in order to apply the Eneström–Kakeya theorem. Assume that the polynomial $Q\left(z\right)$ does not meet the decreasing ordering coefficient constraints to apply such a theorem but a polynomial $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j$ which is $\mathrm{WS}\left({\mathrm{t}}^{-1}\right)\mathrm{S}$, then $\mathrm{WS}\left({\mathrm{t}}^{-1}+\epsilon \right)\mathrm{S}$ for any real $\epsilon \ge 0$ fulfils them while $\tilde{P}\left(z\right)=Q\left(z\right)-P\left(z\right)$ fulfils ${\left|{\displaystyle \frac{\tilde{P}\left(z\right)}{P\left(z\right)}}\right|}_{\left|z\right|=t^{-1}+\epsilon }<1$. Thus, $Q\left(z\right)$ is $WS\left(t^{-1}+\epsilon \right)S$ too (and also $\mathrm{SS}\left({\mathrm{t}}^{-1}+\epsilon \right)\mathrm{S}$ is stable as a result) according to the Rouché theorem of zeros.

Example 2. 

The polynomial $Q\left(z\right)=z^2+1.1z+c=0$ is $\mathrm{SS}\left(1\right)\mathrm{S}$, and then $\mathrm{WS}\left(1\right)\mathrm{S}$ as well, for $c\in \left(0,1\right)$. Since the coefficients do not fulfil the constraint $1\ge 1.1\ge c>0$, the Eneström–Kakeya theorem cannot be applied. Now, we put $Q\left(z\right)=P\left(z\right)+\tilde{P}\left(z\right)$, where $P\left(z\right)=p_2{z}^2+p_{1}z+a_0$ with $p_i=t^i$; $i=1,2$ and $p_0=c$ for some given real $t>1$. Thus, $p_2>p_1>p_0>0$ which is $\mathrm{SS}\left(t\right)\mathrm{S}$ according to Eneström–Kakeya theorem, and

$$\tilde{P}\left(z\right)=Q\left(z\right)-P\left(z\right)=\left(1-t^2\right)z^2+\left(1-t\right)z=\left(1-t\right)z\left[\left(1+t\right)z+1\right]$$

(20)

so that

$${\left|{\displaystyle \frac{\tilde{P}\left(z\right)}{P\left(z\right)}}\right|}_{\left|z\right|=t^{-1}}={\left|{\displaystyle \frac{\left(1-t\right)z\left[\left(1+t\right)z+1\right]}{t^2{z}^2+tz+1}}\right|}_{\left|z\right|=t^{-1}}<1$$

(21)

By taking $t=1.1$, one obtains ${\left|{\displaystyle \frac{\tilde{P}\left(z\right)}{P\left(z\right)}}\right|}_{\left|z\right|=t^{-1}}={\left|{\displaystyle \frac{\left(1-t\right)z\left[\left(1+t\right)z+1\right]}{t^2{z}^2+tz+1}}\right|}_{\left|z\right|=t^{-1}}\le 0.27<1$ and one concludes that $Q\left(z\right)$ is $\mathrm{SS}\left(t^{-1}\right)\mathrm{S}$ for $t>1$ so that it is $\mathrm{SS}\left(1\right)\mathrm{S}$ as well, as it follows from Lemma 1.

Theorem 5. 

Consider the polynomials $P\left(z\right)=z^n+{\displaystyle {\sum }_{j=0}^{n-1}{p}_j}{z}^j$ of degree $n\ge 1$ with complex coefficients and ${\displaystyle {\sum }_{j=0}^{n-1}{p}_j}{z}^j$ not being identically null for all $z\in C$, $\tilde{P}\left(z\right)=-{\displaystyle {\sum }_{j=0}^{n-1}\left(p_j+\left|p_j\right|\right)}{z}^j$, and $Q\left(z\right)=P\left(z\right)+\tilde{P}\left(z\right)$ of Theorem 1 under the weaker constraints $p_j\ge p_{j-1}$; $j\in \overline{n-1}\cup \left\{0\right\}$ of Theorem 2. Then, $P\left(z\right)$ has just one zero in $C_{+}$ provided that ${\left|\tilde{P}\left(z\right)/P(z)\right|}_{z=i\omega ,\omega \in R_{0+}}<1$.

Proof. 

Note that

$$Q\left(z\right)=P\left(z\right)+\tilde{P}\left(z\right)=z^n-{\displaystyle {\sum }_{j=0}^{n-1}\left|p_j\right|}{z}^j$$

(22)

has a unique positive real root $p$ from a classical Cauchy’s theorem, [8], and, furthermore, all the zeros of $P\left(z\right)$ have the following property:

$$Z\left(\left[P\left(z\right)\right]\right)\subset \overline{B}\left(p\right)\subset B\left(1+\overline{p}\right)$$

(23)

with $\overline{p}=\underset{11\le n-1}{\mathit{max}}\left|p_i\right|$ where $B\left(r\right)=\left\{z\left|:z<r\right|\right\}$ and $\overline{B}\left(r\right)=\left\{z\left|:z\le r\right|\right\}$ are open and closed balls in $R$. Take the open right-hand-side complex plane $C_{+}=\left\{z\in C:\mathit{Re}z>0\right\}$. From the above reasoning, $Q\left(z\right)$ only has a root in $C_{+}$ which is $p$. Thus, from Rouché theorem $P\left(z\right)=Q\left(z\right)-\tilde{P}\left(z\right)$, only has a root in $C_{+}$ if ${\left|\tilde{P}\left(z\right)/P(z)\right|}_{z\in \partial C_{+}}<1$ with the boundary of $C_{+}$ being $\partial C_{+}=\left\{i\omega :\omega \in R_{0+}\right\}\cup \left\{z\in C_{+}:\left|z\right|\to \infty \right\}$. Since $\mathit{deg}\left(\tilde{P}\left(z\right)\right)\le n-1<\mathit{deg}\left(P\left(z\right)\right)=n$, then ${\left|\tilde{P}\left(z\right)/P(z)\right|}_{\left|z\right|\to \infty }=0<1$ so that it suffices to check the conditions for ${\left|\tilde{P}\left(z\right)/P(z)\right|}_{z=i\omega :\omega \in R_{0+}}<1$ to guarantee that $P\left(z\right)$ has only one zero, as $Q\left(z\right)$ has, in $C_{+}$. □

2.3. Extended Results from Theorem B and Rouché’s Theorem

It has been seen that the standard (strong) Schur stability, that is, strong 1-Schur stability of a polynomial consists of all its zeros lying in $\left|z\right|<1$ (see Definition 2 and Remark 1). However, the Eneström–Kakeya theorem (Theorem A) only gives sufficient conditions for the polynomial to be $\mathrm{WS}\left(1\right)\mathrm{S}$. A combination of Theorem B with the Rouché theorem now gives sufficient conditions for the polynomial to be $\mathrm{SS}\left(1\right)\mathrm{S}$ as follows:

Theorem 6. 

Let polynomials $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j=P^{*}\left(z\right)+\left(P\left(z\right)-P^{*}\left(z\right)\right)$ and $P^{*}\left(z\right)={\displaystyle {\sum }_{j=0}^m{p}_j^{*}}{z}^j$ be of real coefficients and respective degrees $n\ge m$ with $n\ge 1$ and $m\ge 0$, where $p_j^{*}>p_{j-1}^{*}>0$; $j\in \overline{n}$. Then, the following properties hold:

(i) 

If ${\left|1-P\left(z\right)/P^{*}\left(z\right)\right|}_{\left|z\right|=1}<1$, then $P\left(z\right)$ is $\mathrm{SS}\left(1\right)\mathrm{S}$ and WS $\underset{j\in \overline{n}}{\mathit{max}}\left|p_{j-1}/p_j\right|\left(<1\right)$. The sufficiency-type constraint ${\left|1-P\left(z\right)/P^{*}\left(z\right)\right|}_{\left|z\right|=1}<1$ is equivalent to $2>{\left|\left({\displaystyle {\sum }_{j=0}^n{p}_j}{e}^{ij\theta }\right)/\left({\displaystyle {\sum }_{j=0}^m{p}_j^{*}}{e}^{ij\theta }\right)\right|}_{\theta \in \left[0,2\pi \right)}>0$.

(ii) 

Assume that $n=m$ and $p_j={\rho }_j{p}_j^{*}$, with ${\rho }_j\in \left[\underset{¯}{\rho },\overline{\rho }\right]$ for $j\in \overline{n}\cup \left\{0\right\}$. Then, the property (i) holds if $\underset{¯}{\rho }>0$ and $\overline{\rho }<2$.

Proof. 

Note $m=n-q$ for some $0\le q\le n$ if and only if $p_j=p_j^{*}>p_{j-1}^{*}>0$ for $j=n-q+1,n-q+2,\cdots ,n-1,n$ and $p_{n-q}\ne p_{n-q}^{*}$. It turns out that $P^{*}\left(z\right)$ is $\mathrm{SS}\left(1\right)\mathrm{S}$ (and also WS $\underset{j\in \overline{n}}{\mathit{max}}\left|p_{j-1}/p_j\right|\left(<1\right)$(1)S) from Theorem B. As a result, $P\left(z\right)$ is guaranteed to be SS(1)S provided that ${\left|1-P\left(z\right)/P^{*}\left(z\right)\right|}_{\left|z\right|=1}<1$, equivalently $0<{\left|P\left(e^{i\theta }\right)/P^{*}\left(e^{i\theta }\right)\right|}_{\theta \in \left[0,2\pi \right)}<2$, since then all the zeros of $P\left(z\right)$ lie in $\left|z\right|\le \underset{j\in \overline{n}}{\mathit{max}}\left|p_{j-1}/p_j\right|<1$ from the Rouché theorem. Property (i) has been proved. If $m=m$ and $p_j={\rho }_j{p}_j^{*}$ for $j\in \overline{n}\cup \left\{0\right\}$, then

$$0<{\left|{\displaystyle \frac{P\left(e^{i\theta }\right)}{P^{*}\left(e^{i\theta }\right)}}\right|}_{\theta \in \left[0,2\pi \right)}={\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^n{\rho }_j}{p}_j^{*}\left(e^{ij\theta }\right)}{{\displaystyle {\sum }_{j=0}^n{p}_j^{*}}\left(e^{ij\theta }\right)}}\right|}_{\theta \in \left[0,2\pi \right)}<2$$

(24)

holds if

$$0<\underset{¯}{\rho }={\left|{\displaystyle \frac{\overline{\rho }\left({\displaystyle {\sum }_{j=0}^n{p}_j^{*}}\left(e^{ij\theta }\right)\right)}{{\displaystyle {\sum }_{j=0}^n{p}_j^{*}}\left(e^{ij\theta }\right)}}\right|}_{\theta \in \left[0,2\pi \right)}\le {\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^n{\rho }_j}{p}_j^{*}\left(e^{ij\theta }\right)}{{\displaystyle {\sum }_{j=0}^n{p}_j^{*}}\left(e^{ij\theta }\right)}}\right|}_{\theta \in \left[0,2\pi \right)}\le {\left|{\displaystyle \frac{\overline{\rho }\left({\displaystyle {\sum }_{j=0}^n{p}_j^{*}}\left(e^{ij\theta }\right)\right)}{{\displaystyle {\sum }_{j=0}^n{p}_j^{*}}\left(e^{ij\theta }\right)}}\right|}_{\theta \in \left[0,2\pi \right)}=\overline{\rho }<2$$

(25)

and Property (ii) follows. □

Remark 4. 

Define the polynomial $q_{n-\mathcal{l}}\left(tz\right)={\displaystyle {\sum }_{k=0}^{n-\mathcal{l}}{a}_k}{\left(tz\right)}^k$ of degree $n-\mathcal{l}>0$ defined for any given positive real constant $t$ with real coefficients satisfying the WPOc with $a_0>0$ of the reciprocal polynomial $q_{n-\mathcal{l}}^{*}\left(tz\right)={\left(tz\right)}^{m-\mathcal{l}}{q}_{m-\mathcal{l}}{\left(tz\right)}^{-1}$. Then, all the zeros of the subsequent polynomial are as follows:

$$p_n\left(tz\right)={\left(tz\right)}^{\mathcal{l}}{q}_{n-\mathcal{l}}\left(tz\right)+e^{i\theta }{q}_{n-\mathcal{l}}^{*}\left(tz\right)$$

(26)

for any given integer $\mathcal{l}$ and any real $\theta$ lie on $\left|z\right|=t^{-1}$.

Proof. 

Since $a_{n-\epsilon }\ge a_{n-\mathcal{l}-1}\ge \dots \ge a_0>0$, all the zeros of $q_{n-\mathcal{l}}\left(tz\right)$ lie in $\left|z\right|\le t^{-1}$. □

2.4. Results for the Polynomial Coefficients in Known Intervals

The generalization of the Eneström–Kakeya theorem and its extensions to the case when the coefficients (or their moduli) belong to known intervals while the known coefficients are unknown is of interest. Some of the above theorems might be extended, supported by the Rouché theorem, provided that the coefficient ordered sequences hold for coefficients (or their moduli) within the ranges of variation which are not necessarily the current coefficients (or their modulus).

Theorem 7. 

Consider a polynomial $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j$ of real coefficients and degree $n\ge 1$ such that $p_j\in \left[{\underset{¯}{p}}_j,{\overline{p}}_j\right]$, such that ${\underset{¯}{p}}_j\ge 0$ and ${\overline{p}}_j<+\infty$; $j\in \overline{n}$. Assume that $p_j^{*}\ge p_{j-1}^{*}$; $j\in \overline{n}$, where $p_j^{*}\in \left[{\underset{¯}{p}}_j,{\overline{p}}_j\right]$; $j\in \overline{n}\cup \left\{0\right\}$. Then, $P\left(z\right)$ has all its zeros in $\left|z\right|\le 1$ if any of the subsequent conditions given from the weaker one to the stronger one hold:

$${\left|1-{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j}{{\displaystyle {\sum }_{j=0}^n{p}_j^{*}}{z}^j}}\right|}_{\left|z\right|=1}<1$$

(27)

$$\underset{0\le j\le n}{\mathit{max}}\left({\overline{p}}_j-{\underset{¯}{p}}_j\right)<{\displaystyle \frac{1}{n+1}}{\left|{\displaystyle {\sum }_{j=0}^{n-1}{p}_j^{*}{z}^j}\right|}_{\left|z=1\right|}$$

(28)

$$\underset{0\le j\le n}{\mathit{max}}\left({\overline{p}}_j-{\underset{¯}{p}}_j\right)<{\displaystyle \frac{1}{n+1}}\left|p_n^{*}-{\displaystyle {\sum }_{j=0}^{n-1}{p}_j^{*}}\right|$$

(29)

Proof. 

Note that

$$P\left(z\right)=P^{*}\left(z\right)+\left(P\left(z\right)-P^{*}\left(z\right)\right)$$

(30)

where $P^{*}\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j^{*}}{z}^j$ and $\left|p_j-p_j^{*}\right|\le \left|{\overline{p}}_j-{\underset{¯}{p}}_j\right|$; $j\in \overline{n}\cup \left\{0\right\}$. Since $p_j^{*}\ge p_{j-1}^{*}\ge 0$ for $j\in \overline{n}$ and since ${\underset{¯}{p}}_j\ge 0$ for $j\in \overline{n}\cup \left\{0\right\}$, one has from Eneström–Kakeya theorem that all the zeros of $P^{*}\left(z\right)$ lie in $\left|z\right|\le 1$. Now, one concludes from the Rouché theorem that all the zeros of $P\left(z\right)$ lie in $\left|z\right|\le 1$ as well if (27) holds

$${\left|{\displaystyle \frac{P\left(z\right)-P^{*}\left(z\right)}{P^{*}\left(z\right)}}\right|}_{\left|z\right|=1}={\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^n\left(p_j-p_j^{*}\right)}{z}^j}{{\displaystyle {\sum }_{j=0}^n{p}_j^{*}}{z}^j}}\right|}_{\left|z\right|=1}={\left|1-{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j}{{\displaystyle {\sum }_{j=0}^n{p}_j^{*}}{z}^j}}\right|}_{\left|z\right|=1}<1$$

For the above condition to hold, and since $\left|p_j-p_j^{*}\right|\le \left|{\overline{p}}_j-{\underset{¯}{p}}_j\right|$; $j\in \overline{n}\cup \left\{0\right\}$, it suffices that (28) holds. And, for (28) to hold, it suffices that (29) holds since $p_n^{*}\ge p_j^{*}$ for $j\in \overline{n}\cup \left\{0\right\}$, and

$${\left|{\displaystyle {\sum }_{j=0}^n{p}_j^{*}}{z}^j\right|}_{\left|z\right|=1}\ge \left|\left|p_n^{*}\right|{\left|z^n\right|}_{\left|z\right|=1}-{{\displaystyle {\sum }_{j=0}^{n-1}\left|p_j^{*}{z}^j\right|}}_{\left|z\right|=1}\right|\ge \left|\left|p_n^{*}\right|-{\displaystyle {\sum }_{j=0}^{n-1}\left|p_j^{*}{z}^j\right|}\right|$$

(31)

$${\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^n\left({\overline{p}}_j-{\underset{¯}{p}}_j\right)}{z}^j}{{\displaystyle {\sum }_{j=0}^n{p}_j^{*}}{z}^j}}\right|}_{\left|z\right|=1}<1$$

(32)

and, furthermore, for the last condition to hold, it suffices that

$$\underset{0\le j\le n}{\mathit{max}}\left({\overline{p}}_j-{\underset{¯}{p}}_j\right)<{\displaystyle \frac{1}{n+1}}{\left|{\displaystyle {\sum }_{j=0}^{n-1}{p}_j^{*}{z}^j}\right|}_{\left|z=1\right|}$$

(33)

or that

$$\underset{0\le j\le n}{\mathit{max}}\left({\overline{p}}_j-{\underset{¯}{p}}_j\right)<{\displaystyle \frac{1}{n+1}}\left|p_n^{*}-{\displaystyle {\sum }_{j=0}^{n-1}{p}_j^{*}}\right|$$

(34)

□

Corollary 2. 

Theorem 7 holds if $P^{*}\left(z\right)=\overline{P}\left(z\right)={\displaystyle {\sum }_{j=0}^n{\overline{p}}_j}{z}^j$ or if $P^{*}\left(z\right)=\underset{¯}{P}\left(z\right)={\displaystyle {\sum }_{j=0}^n{\underset{¯}{p}}_j}{z}^j$.

2.5. Some Simple Examples Linked to Control Theory

Three examples related to control theory problems follow below:

Example 3 

(linear discrete time-invariant dynamic system). Consider a single-input single-output (SISO) linear discrete dynamic system of dimension $n$ described by the state-space controllability form

$$x_{k+1}=A{x}_k+b{u}_k, y_k=x_{1k}; x\left(0\right)=x_0$$

(35)

where ${\left\{x_k\right\}}_{k=0}^{\infty }\subset R^n$ is the state vector, ${\left\{y_k\right\}}_{k=0}^{\infty }\subset R$ and ${\left\{u_k\right\}}_{k=0}^{\infty }\subset R$ are the scalar output and input sequences; and $A\in R^{n\times n}$ and $b\in R^n$ are as follows:

$$A_0=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1\\ -a_0& -a_1& -a_2& \cdots & -a_{n-1}\end{array}\right]; b=\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ 1\end{array}\right]$$

(36)

The above description is a state-space description of the open-loop (that is, control-free) difference equation $y_{k+n}=-{\displaystyle {\sum }_{i=1}^n{a}_{n-i}}{y}_{k+n-i}$ for $k\in Z_{0+}$, under an identically zero control sequence, subject to any set of initial conditions ${\left(y_0,y_1,\cdots ,y_{n-1}\right)}^T=x_0$. The characteristic polynomial of $A$ is $p_0\left(z\right)=z^n+{\displaystyle {\sum }_{i=1}^n{a}_{n-i}}{z}^{n-i}$. The above discrete system is controllable, (or, in short, the pair $\left(A_0,b\right)$ is controllable), that is, $rank\lfloor b,A_{0}b,\cdots ,A_0^{n-1}b\rfloor =n$ [18,19]. Now, assume that the control sequence is generated through the linear difference equation $u_{k+n}={\displaystyle {\sum }_{i=1}^n{k}_{n-i}}{u}_{k+n-i}$, where $k_i\in R$ for $i\in \overline{n-1}\cup \left\{0\right\}$ are the controller gains, which combined with the above open-loop control difference equation yields the closed loop (that is, subject to linear feedback control)

$$y_{k+n}={\displaystyle {\sum }_{i=1}^n\left(k_{n-i}-a_{n-i}\right)}{y}_{k+n-i}$$

(37)

whose characteristic polynomial is $p_c\left(z\right)=z^n+{\displaystyle {\sum }_{i=1}^n\left(a_{n-i}-k_{n-i}\right)}{z}^{n-i}$ and the associated matrix of dynamics of the closed-loop difference system becomes

$$A_c=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1\\ k_0-a_0& k_1-a_1& k_2-a_2& \cdots & k_{n-1}-a_{n-1}\end{array}\right]$$

(38)

Since $\left(A_0,b\right)$ is controllable, then the spectrum is freely assignable to any prescribed positions in the complex plane by the controller gains. Note that $sp\left(A_c\right)=\left\{z\in C:p_c\left(z\right)=0\right\}$. Without calculating the zeros of $p_c\left(z\right)$, we can apply Theorem B to conclude that, if $k_{n-i}<a_{n-i}-a_{n-i-1}+k_{n-i-1}$ for $i\in \overline{n}$ and $k_n\left(=0\right)<a_n\left(=1\right)-a_{n-1}+k_{n-1}$, equivalently $k_{n-1}>a_{n-1}-1$, subject to the initialization $k_{-1}=a_{-1}=0$; then, ${\left\{a_{n-i}-k_{n-i}\right\}}_{i=1}^n$ is positive and strictly decreasing, that is, the $SPOc$ property is satisfied. Then, all the zeros of $p_c\left(z\right)$ lie in $\left|z\right|\le {\rho }_c=\underset{i\in \overline{n-1}\cup \left\{0\right\}}{\mathit{max}}{\displaystyle \frac{a_{i-1}-k_{i-1}}{a_i-k_i}}<1$ and the discrete linear dynamic closed-loop system is $\mathrm{WS}\left({\rho }_c\right)\mathrm{S}$ and then $\mathrm{SS}\left(1\right)\mathrm{S}$ as a result. This closed-loop stabilization is achievable by the choice of the control gains even if, in the control-free case, ${\rho }_0=\underset{i\in \overline{n-1}\cup \left\{0\right\}}{\mathit{max}}{\displaystyle \frac{a_{i-1}}{a_i}}\ge 1$ so that the open-loop system is not $\mathrm{SS}\left(1\right)\mathrm{S}$.

If the decreasing condition for the coefficients is not strict although they are non-negative, that is, $k_{n-i}\le a_{n-i}-a_{n-i-1}+k_{n-i-1}$ for $i\in \overline{n}$ together with $k_{n-1}\ge a_{n-1}-1$, then all the zeros of $p_c\left(z\right)$ lie in $\left|z\right|\le 1$ from Theorem A and then the discrete linear dynamic system is only guaranteed to be $WS(1)S$.

Example 4 

(discretized transfer function under a zero-order hold). The ordering conditions WPOc and SPOc can also be invoked for investigating the stability properties of continuous-time linear systems derived from their discretized counterparts. Consider a single-input single-output (SISO) linear continuous dynamic system of dimension $n$ whose transfer function, that is, the Laplace quotient transforms $Y\left(s\right)=Ly\left(t\right)$ of the output “y(t)” to that input “u(t)”, with $U\left(s\right)=Lu\left(t\right)$ under zero initial conditions, is $G\left(s\right)=Ly\left(t\right)/Lu\left(t\right)=B\left(s\right)/A\left(s\right)$, where $A\left(s\right)={\displaystyle {\sum }_{i=0}^n{a}_i}{s}^i$ and $B\left(s\right)={\displaystyle {\sum }_{i=0}^m{b}_i}{s}^i$ are polynomials of real coefficients in the Laplace argument “s” of respective degrees $n\ge 1$ and $m\le n$. Since the Laplace transform argument and the time derivative operator $D=d/dt$ are formally equivalent, the transfer function represents the ordinary differential equation $A\left(D\right)y\left(t\right)=B\left(D\right)u\left(t\right)$ of order $n$ whose solution $y:R_{0+}\to R$ is unique for any given set of finite initial conditions $y^{\left(i\right)}\left(0\right)={D^{i}y\left(t\right)\rfloor }_{t=0}$ for $i\in \overline{n-1}\cup \left\{0\right\}$, with $D^i=D^{i-1}D$ and $D^0=1$, and for any given piecewise constant input $u:R_{0+}\to R$. If $u\left(t\right)=u\left(kT\right)=u_k$, $t\in \left[kT,\left(k+1\right)T\right)$ for any given sampling period $T>0$; then, the discrete-transfer function $G\left(z\right)$ in the argument “z”, with $z=e^{Ts}$ is the quotient of the Laplace transforms of the output $y\left(t\right)$ to the above piecewise constant input with finite discontinuities at the sampling instants $kT$, $k\in Z_{0+}$. The discretization device transforms a sequence ${\left\{u_k\right\}}_{k=0}^{\infty }$ into a piecewise constant function. $u:R_{0+}\to R$ defined by

$$u\left(t\right)=u\left(kT\right)=u_k={\displaystyle {\sum }_{j=1}^{n\left(t\right)}u\left(jT\right)}\left[1\left(jT\right)-1\left(\left(j-1\right)T\right)\right]; t\in \left[kT,\left(k+1\right)T\right), n\left(t\right)=\left\{\mathit{max}z\in Z_{0+}:z\le t\right\}$$

(39)

where $1\left(t\right)$ is the Heaviside step function, referred to as a sampling and zero-order hold device (ZOH) whose transfer function is $G_{0T}\left(s\right)=\left(1-e^{-Ts}\right)/s$. Thus, we have

$$G\left(z\right)=Zy\left(t\right)/Zu\left(t\right)=Z\left(\left(1-e^{-Ts}\right)G\left(s\right)/s\right)=\left(1-z^{-1}\right)Z\left(G\left(s\right)/s\right)$$

(40)

Assume that $A\left(0\right)\ne 0$, that is, $G\left(s\right)$ has no integrators, and note that there is a unique polynomial $B^{\prime }\left(s\right)={\displaystyle {\sum }_{i=0}^{m^{\prime }}{b}_i^{\prime }}{s}^i$ of degree $m\prime =n-1$ such that

$$\begin{array}{ll}{\displaystyle \frac{G\left(s\right)}{s}}& ={\displaystyle \frac{B\left(s\right)}{sA\left(s\right)}}={\displaystyle \frac{a}{s}}+{\displaystyle \frac{B^{\prime }\left(s\right)}{A\left(s\right)}}={\displaystyle \frac{aA\left(s\right)+s{B}^{\prime }\left(s\right)}{sA\left(s\right)}}\\ & ={\displaystyle \frac{{\displaystyle {\sum }_{i=0}^{n}a{a}_i{s}^i+{\displaystyle {\sum }_{i=0}^{n-1}{b}_i^{\prime }}{s}^{i+1}}}{{\displaystyle {\sum }_{i=0}^{n+1}{a}_i{s}^{i+1}}}}={\displaystyle \frac{{\displaystyle {\sum }_{i=0}^m{b}_i}{s}^i}{{\displaystyle {\sum }_{i=0}^{n+1}{a}_i{s}^{i+1}}}}\end{array}$$

(41)

with

$$\begin{array}{l}a=b_0/a_0\\ b_0^{\prime }=b_1-a{a}_1=b_1-b_0{a}_1/a_0\\ b_1^{\prime }=b_2-a{a}_2=b_2-b_0{a}_2/a_0\end{array}$$

(42)

$$\begin{array}{l}{b}_{m-1}^{\prime }=b_m-b_0{a}_m/a_0\\ b_m^{\prime }=b_{m+1}-b_0{a}_{m+1}/a_0=-b_0{a}_{m+1}/a_0\end{array}$$

$$b_{n-1}^{\prime }=b_n-b_0{a}_n/a_0=-b_0{a}_n/a_0$$

Define the auxiliary transfer function $G^{\prime }\left(s\right)=B\prime \left(s\right)/A\left(s\right)$. Then, by the linearity of the $z$-transform,

$$G\left(z\right)={\displaystyle \frac{Q\left(z\right)}{P\left(z\right)}}={\displaystyle \frac{z-1}{z}}\left(aZ\left({\displaystyle \frac{1}{s}}\right)+Z\left(G^{\prime }\left(s\right)\right)\right)=a+{\displaystyle \frac{z-1}{z}}{G}^{\prime }\left(z\right)$$

(43)

Thus, the poles of $G\prime \left(z\right)$, which are the zeros of the polynomial $P\left(z\right)$, are $z_i=e^{s_{i}T}$; $i\in \overline{n}$ (since $G^{\prime }\left(s\right)$ and $G\left(s\right)$ have the same denominator polynomial $A\left(s\right)$, which is repeated in the case that the respective multiplicities are greater than one. The poles of $G\left(z\right)$ are the above plus an additional one at $z=0$. This null pole can also be interpreted as a numerator factor $z^{-1}$ of the discrete-transfer function which supplies a one-sample delay of the output sequence with respect to the input sequence. Therefore, all the above results based on the ordering conditions WPOc and/or SPOc of the polynomial coefficients being applicable to $P\left(z\right)$ conclude that its weak and/or strong Schur stability properties lead to the critical or Hurwitz stability of the polynomial $A\left(s\right)$ and, thus, to the corresponding stability properties of the continuous-transfer function which has been discretized through the ZOH.

If the input is the sequence ${\left\{u_k\right\}}_{k=0}^{\infty }$, instead the above piecewise constant function, that is, a ZOH is not involved then the resulting discrete-transfer function is $G_d\left(z\right)=Z\left(G\left(s\right)\right)$ and its poles are the zeros of the polynomial $P_d\left(z\right)$ which are those of $P\left(z\right)$ except $z=0$ derived from the discretization process itself through the ZOH. Therefore, the above recalled or established results based on the ordering conditions WPOc and/or SPOc of the polynomial coefficients still apply to conclude that some degree of Schur stability of the denominator polynomials of the discrete-transfer function translate in a corresponding stability property of the denominator of the continuous-transfer function.

Finally, it has to be pointed out that (a) the used notation of the z-transforms of the continuous-transfer function is commonly used in the literature but it is an abuse of notation since z-transforms are Laplace transforms of time sequences, rather than transforms of Laplace transforms; (b) if some kind of sampling and hold device, such as the mentioned zero-order one or another one of higher order, is removed from the discretization process, as proposed in the last part of this example, then the resulting discrete-transfer function is not useful in many applications since the sampled input sequence does not supply the necessary power [19].

Example 5 

(root locus). The root locus is a popular analysis of stability of the closed-loop system in both linear time-invariant continuous-time and discrete-time systems. To fix the ideas, consider a discrete-transfer function $G\left(z\right)=B\left(z\right)/A(z)$, where $B\left(z\right)$ and $A\left(z\right)$ are polynomials with $\mathit{deg}B\left(z\right)\le \mathit{deg}A\left(z\right)$. Assume that such a transfer function is controlled under a constant feed-forward controller of scalar gain $K>0$ and unity negative feedback. Thus, the closed-loop transfer function becomes $F_{ff}\left(z\right)=KB\left(z\right)/\left(A\left(z\right)+KB\left(z\right)\right)$. If the same constant controller is a feedback controller under negative feedback then the closed-loop transfer function is $F_{fb}\left(z\right)=B\left(z\right)/\left(A\left(z\right)+KB\left(z\right)\right)$. In both cases, the closed-loop poles are the zeros of the denominator polynomial $P_K\left(z\right)=A\left(z\right)+KB\left(z\right)$. The so-called root locus, [18], is the curve in the complex plane which allocates the position of those zeros as $K$ ranges from $K=0$ to $K=+\infty$ so that the zeros of $P_K\left(z\right)$ are arbitrarily close to those of $A\left(z\right)$ as $K\to 0$ and arbitrarily close to those of $B\left(z\right)$ as $K\to +\infty$. As a result of the continuity of the locus as a function of $K$, if both $B\left(z\right)$ and $A\left(z\right)$ are SS(1)S then there exist $K_1,K_2\in \left[K_1,+\infty \right)\subset \left(0,+\infty \right)$ such that $P_K\left(z\right)$ is SS(1)S for $K\in \left[0,K_1\right)\cup \left[K_2,+\infty \right)$ since all its zeros are in $\left|z\right|<1$ and the closed-loop transfer functions $F_{ff}\left(z\right)$ and $F_{fb}\left(z\right)$ are then stable. Such a conclusion does not guarantee that $P_K\left(z\right)$ is SS(1)S, and then the closed-loop transfer functions $F_{ff}\left(z\right)$ and $F_{fb}\left(z\right)$ are then stable, for $K\in \left[0,+\infty \right)$.

Now, note that

$$P_K\left(z\right)={\displaystyle {\sum }_{i=0}^n{a}_i}{z}^i+{\displaystyle {\sum }_{i=0}^{n}K{b}_i}{z}^i={\displaystyle {\sum }_{i=0}^n\left(a_i+K{b}_i\right)}{z}^i$$

(44)

Assume that $a_i+K{b}_i>0$; $i\in \overline{n}\cup \left\{0\right\}$ and that there exists a real constant $t$ satisfying

$$1>t\ge \underset{K\in \left[0,+\infty \right]}{\mathit{sup}}\underset{i\in \overline{n}}{\mathit{max}}\left(a_{i-1}+K{b}_{i-1}\right)/\left(a_i+K{b}_i\right)$$

(45)

Then, from Theorem 3(i), $P_k\left(z\right)$ is SS(1)S and SS(t)S with zeros in $\left|z\right|\le t<1$, and the closed-loop transfer functions $F_{ff}\left(z\right)$ and $F_{fb}\left(z\right)$ are then stable, for $K\in \left[0,+\infty \right)$. As expected, the necessary conditions from (45) to hold are the fulfilment for $K=0$ and $K=+\infty$ so that (45) implies that

$$1>t\ge \mathit{max}\left(\underset{i\in \overline{n}}{\mathit{max}}{a}_{i-1}/a_i,\mathit{max}{b}_{i-1}/b_i\right)$$

(46)

with the closed-loop transfer functions $F_{ff}\left(z\right)$ and $F_{fb}\left(z\right)$ then being stable, for $K\in \left[0,+\infty \right)$. Condition (45) is relaxed to a weaker one, which is simpler to test, if $a_i>0$, $b_i>0$; $i\in \overline{n}\cup \left\{0\right\}$ and there exists $t$, subject to $1>t\ge \mathit{max}\left(\underset{i\in \overline{n}}{\mathit{max}}{a}_{i-1}/a_i,\mathit{max}{b}_{i-1}/b_i\right)$, since then for $K\in \left[0,+\infty \right)$, one has from (46) that

$$t\left(a_i+K{b}_i\right)=t{a}_i+tK{b}_i\ge a_{i-1}+K{b}_{i-1}$$

(47)

Note that $\left(45\right)$ implies $\left(46\right)$ as a necessary condition and, conversely, (46) implies (45), since (47) holds for the whole scalar controller range of positive real gains $K\in \left[0,+\infty \right)$. Thus, the closed-loop transfer functions $F_{ff}\left(z\right)$ and $F_{fb}\left(z\right)$ are stable with poles in $\left|z\right|\le t<1$, for $K\in \left[0,+\infty \right)$ if (46) holds with $a_i>0$, $b_i>0$; $i\in \overline{n}\cup \left\{0\right\}$. If (46) is modified as follows,

$$1\ge t>\mathit{max}\left(\underset{i\in \overline{n}}{\mathit{max}}{a}_{i-1}/a_i,\mathit{max}{b}_{i-1}/b_i\right)$$

(48)

then the closed-loop transfer functions $F_{ff}\left(z\right)$ and $F_{fb}\left(z\right)$ are stable with poles in $\left|z\right|<t\le 1$, for $K\in \left[0,+\infty \right)$.

Now, assume that $A\left(z\right)$ and $B\left(z\right)$ are perturbed as follows $A\left(z\right)\to A\left(z\right)+\tilde{A}\left(z\right)=\left(1+{\lambda }_A\left(z\right)\right)A\left(z\right)$ and $B\left(z\right)\to B\left(z\right)+\tilde{B}\left(z\right)=\left(1+{\lambda }_B\left(z\right)\right)B\left(z\right)$ and define $\lambda =\mathit{max}\left(\underset{0\le \tilde{t}\le 1-t}{\mathit{sup}}\underset{\left|z\right|=t+\tilde{t}}{\mathit{sup}}\left|{\lambda }_A\left(z\right)\right|,\underset{0\le \tilde{t}\le 1-t}{\mathit{sup}}\underset{\left|z\right|=t+\tilde{t}}{\mathit{sup}}\left|{\lambda }_B\left(z\right)\right|\right)$. If $\lambda <1$, then

$$\left|{\displaystyle \frac{{\tilde{P}}_K\left(z\right)}{P_K\left(z\right)}}\right|=\left|{\displaystyle \frac{\tilde{A}\left(z\right)+K\tilde{B}\left(z\right)}{A\left(z\right)+KB\left(z\right)}}\right|\le \lambda <1;\left|z\right|\le 1$$

so that all the zeros of $P_K\left(z\right)+{\tilde{P}}_k\left(z\right)=\left(1+{\lambda }_A\left(z\right)\right)A\left(z\right)+\left(1+{\lambda }_B\left(z\right)\right)B\left(z\right)$ are in $\left|z\right|<1$ for $K\in \left[0,+\infty \right)$ from the Rouché theorem (Theorem 3(ii)) since those of $P\left(z\right)$ are in $\left|z\right|\le t<1$ for $K\in \left[0,+\infty \right)$ (Theorem 3(i)).

2.6. A Result by Using a Previous Cauchy’s Theorem on Bounds of Zeros

The subsequent elementary result is supported by a celebrated theorem of Cauchy on the upper-bounds of the zeros of a polynomial of complex coefficients, combined with the Rouché theorem of zeros. It neither involves any ordering condition on the real and imaginary parts of the coefficients of the polynomial nor on their moduli.

Theorem 8. 

Consider the polynomials $P\left(z\right)=z^n+{\displaystyle {\sum }_{j=0}^{n-1}{p}_j}{z}^j$ of degree $n\ge 1$ with complex coefficients, such that ${\displaystyle {\sum }_{j=0}^{n-1}{p}_j}{z}^j$ is not identically null for all $z\in C$, $\tilde{P}\left(z\right)=-{\displaystyle {\sum }_{j=0}^{n-1}\left(p_j+\left|p_j\right|\right)}{z}^j$ and $Q\left(z\right)=P\left(z\right)+\tilde{P}\left(z\right)$ of Theorem 1 under the constraint $\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}{n}_{\eta }\left(\theta \right)/d_{\eta }\left(\theta \right)<1$, where

$$n_{\eta }\left(\theta \right)=2\sqrt{{\left({\displaystyle {\sum }_{j\in poscoP}{p}_j}{\eta }^j\mathit{cos}j\theta \right)}^2+{\left({\displaystyle {\sum }_{j\in poscoP}{p}_j}{\eta }^j\mathit{sin}j\theta \right)}^2}$$

(49)

$$d_{\eta }\left(\theta \right)=\sqrt{{\left({\eta }^n\mathit{cos}n\theta +{\displaystyle {\sum }_{j\in poscoP}{p}_j}{\eta }^j\mathit{cos}j\theta -{\displaystyle {\sum }_{j\in negcoP}\left|p_j\right|}{\eta }^j\mathit{cos}j\theta \right)}^2+{\left({\eta }^n\mathit{sin}n\theta +{\displaystyle {\sum }_{j\in poscoP}{p}_j}{\eta }^j\mathit{sin}j\theta -{\displaystyle {\sum }_{j\in negcoP}\left|p_j\right|}{\eta }^j\mathit{sin}j\theta \right)}^2}$$

(50)

where

$$poscoP=\left\{j\in \overline{n-1}\cup \left\{0\right\}:p_j\in R_{+}\right\}; negcoP=\left\{j\in \overline{n-1}\cup \left\{0\right\}:p_j\in R_{-}\right\}$$

(51)

Then, all the zeros of $P\left(z\right)$ are in $\left|z\right|\le \eta$.

Also, all the zeros of $P\left(z\right)$ are in $\left|z\right|<\overline{\eta }=1+\underset{i\in \overline{n-1}\cup \left\{0\right\}}{\mathit{max}}\left|p_i\right|$, under the weaker condition $\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}{n}_{\eta }\left(\theta \right)/d_{\eta }\left(\theta \right)<1$.

Proof. 

From the classical Cauchy’s theorem [8], it is well-known that if $\eta$ is the unique positive zero of $Q\left(z\right)=z^n-{\displaystyle {\sum }_{j=0}^{n-1}\left|p_j\right|}{z}^j$ then all the zeros of $P\left(z\right)$ lie in the closed ball $\overline{B}\left(\eta \right)$ centred at $z=0$ which in turn is contained in the open ball $B\left(1+a\right)$ centred at $z=0$, where $a=\underset{i\in \overline{n-1}\cup \left\{0\right\}}{\mathit{max}}\left|p_i\right|$. Thus, all the zeros of $Q\left(z\right)$ are in $\left|z\right|<\eta \le \overline{\eta }=1+\underset{i\in \overline{n-1}\cup \left\{0\right\}}{max}\left|p_i\right|$. Note that $\left[p_j>0\right]\Rightarrow$$\left[p_j+\left|p_j\right|=2p_j\right]$ and $\left[p_j<0\right]\Rightarrow$$\left[p_j+\left|p_j\right|=0\right]$. Then, from the Rouché theorem, $P\left(z\right)$ has all its zeros in $\left|z\right|<\eta$ if

$${\left|{\displaystyle \frac{\tilde{P}\left(z\right)}{P\left(z\right)}}\right|}_{\left|z\right|=\eta }\le \underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}{\displaystyle \frac{n\left(\theta \right)}{d\left(\theta \right)}}<1$$

(52)

which proves the result. □

2.7. A Result for Alternating Signs of Coefficients and Further Related Results

It is of interest to consider the case when the positivity and negativity of the real coefficients alternate on different adjacent sequences of powers of $z$, in general, of distinct sizes and at the same time keep the alternate increasing/decreasing ordering sequences of consecutive coefficients. In that context, we state the subsequent result:

Theorem 9. 

Consider the polynomial $P\left(z\right)={\displaystyle {\sum }_{j=0}^n{p}_j}{z}^j$ which has real coefficients and degree $n$, i.e., $p_n\ne 0$ so that $\mathit{deg}P\left(z\right)=n$. Assume that there exist integers $\left\{n_1=0,n_2,\cdots ,n_{m+1}=n\right\}$, $\left\{m_1,m_2,\cdots ,m_m\right\}$ satisfying the following constraints:

$$\begin{array}{l}{n}_{m+1}=n={\displaystyle {\sum }_{i=1}^m{m}_i}={\displaystyle {\sum }_{i=1}^m\left(n_{i+1}-n_i\right)}\\ n_1=0\end{array}$$

(53)

$$m_i=n_{i+1}-n_i\ge 2$$

and define, in general, non-connected disjoint sets of integers which contain the powers of the argument $z$ according to the decreasing rules of the coefficients of $P\left(z\right)$ as follows:

$$S_1=\left\{i\in \overline{n}\cup \left\{0\right\}:p_{i,j+1}>p_{ij}>0,j\in \left[n_i,n_{i+1}-1\right]\right\}$$

(54)

$$S_2=\left\{i\in \overline{n}\cup \left\{0\right\}:p_{i,j+1}<p_{ij}<0,j\in \left[n_i,n_{i+1}-1\right]\right\}$$

(55)

such that $S_1$ and $S_2$ can be expressed as disjoint unions of their respective connected components as follows:

$$S_1={\displaystyle {\cup }_{i\in {\overline{q}}_1}{S}_{1i}}, S_2={\displaystyle {\cup }_{i\in {\overline{q}}_2}{S}_{2i}}; m=q_1+q_2 \mathrm{and} n=card{S}_1+card{S}_2$$

(56)

so that the coefficients of each connected component of $S_1$ (respectively, of those of $S_2$) are followed in decreasing order of powers of $z$ by those of $S_2$ (respectively, of those of $S_1$) until the set of coefficients ends with $p_0$. Assume that each absolute value of the leading coefficient of one connected component of $S_1$ or, respectively, $S_2$ is larger than the last coefficient of the preceding connected component of $S_2$ or, respectively, $S_1$, that is, $\left|p_{i+1,n_{i+2}}\right|>\left|p_{i{n}_i}\right|$.

Note that the sets of coefficients verify the notation equality

$$\left(p_0,p_1,\cdots ,p_n\right)=\left(p_{00},p_{10}\cdots ,p_{1n_2},p_{2n_2+1},\cdots ,p_{2n_3},\cdots ,p_{m{n}_{m+1}}\right)$$

(57)

In view of the above relations, we have that $P\left(z\right)$ can be rewritten equivalently as follows:

$$\begin{array}{l}P\left(z\right)={\displaystyle {\sum }_{i=1}^m{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}{p}_{ij}}}{z}^j={\displaystyle {\sum }_{i\in S_1}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{ij}\right|}}{z}^j-{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{ij}\right|}}{z}^j\\ ={\displaystyle {\sum }_{i\in S_1}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{ij}\right|}}{z}^j+{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{ij}\right|}}{z}^j-2{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{ij}\right|}}{z}^j\\ =P^{*}\left(z\right)-2\tilde{P}\left(z\right)\\ ={\displaystyle {\sum }_{j=0}^n\left|p_j\right|}{z}^j-2{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{ij}\right|}}{z}^j\end{array}$$

(58)

Note that $P^{*}\left(z\right)={\displaystyle {\sum }_{j=0}^n\left|p_j\right|}{z}^j$, defined with the absolute values of the coefficients of $P\left(z\right)$, has all its zeros in $\left|z<1\right|$ from Theorem B. From the Rouché theorem, the zeros of $P\left(z\right)$ lie in $\left|z<1\right|$ if, furthermore,

$${\left|{\displaystyle \frac{\tilde{P}\left(z\right)}{P^{*}\left(z\right)}}\right|}_{\left|z\right|=1}={\left|{\displaystyle \frac{{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{ij}\right|}}{z}^j}{{\displaystyle {\sum }_{j=1}^n\left|p_j\right|}{z}^j}}\right|}_{\left|z\right|=1}<{\displaystyle \frac{1}{2}}$$

(59)

If the coefficients of $\tilde{P}\left(z\right)$ are expressed as $\left|p_{\left(.\right)}\right|=\left|\epsilon p_{{\left(.\right)}_r}\right|$ with respect to a reference polynomial ${\tilde{P}}_r\left(z\right)$ of the same degree, so that $\tilde{P}\left(z\right)=\epsilon {\tilde{P}}_r\left(z\right)$ for some real constant $\epsilon \in \left(0,\overline{\epsilon }\right)$, then the above condition holds if

$$\overline{\epsilon }<{\displaystyle \frac{1}{2}}\underset{\left|z\right|=1}{\mathit{inf}}\left|{\displaystyle \frac{P^{*}\left(z\right)}{\tilde{P}\left(z\right)}}\right|=\underset{\left|z\right|=1}{\mathit{inf}}\left|{\displaystyle \frac{P^{*}\left(z\right)}{P^{*}\left(z\right)-P\left(z\right)}}\right|={\displaystyle \frac{1}{2}}\underset{\left|z\right|=1}{\mathit{inf}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^n\left|p_j\right|}{z}^j}{{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{i{j}_r}\right|}}{z}^j}}\right|$$

(60)

and the proof is complete since $P\left(z\right)$ is SS(1)S.

The above result is now directly extended to a polynomial $Q\left(z\right)$ which satisfies the conditions of $P\left(z\right)$ in the above theorem but is affected by an unstructured (in the sense that it does not fulfil ordering conditions on its coefficients) additive polynomial.

Corollary 3. 

Assume that $Q\left(z\right)=P\left(z\right)-\Delta \left(z\right)$ where $P\left(z\right)$ is the polynomial of Theorem 9 and $\Delta \left(z\right)$ is a polynomial with $\mathit{deg}\Delta \left(z\right)\le \mathit{deg}P\left(z\right)=n$. Then, the zeros of $Q\left(z\right)$ lie in $\left|z\right|<1$, and then $Q\left(z\right)$ is SS(1)S, if

$$\underset{\left|z\right|=1}{\mathit{inf}}\left|{\displaystyle \frac{P^{*}\left(z\right)}{2\tilde{P}\left(z\right)+\Delta \left(z\right)}}\right|<1$$

(61)

Corollary 4. 

Assume that the ordering conditions of coefficients in Theorem 9 are modified accordingly to the following redefinitions of the sets $S_1$ and $S_2$ for some given real constant $t\in R_{+}$:

$$S_1\left(t\right)=\left\{i\in \overline{n}\cup \left\{0\right\}:p_{i,j+1}>t{p}_{ij}>0,j\in \left[n_i,n_{i+1}-1\right]\right\}$$

(62)

$$S_2\left(t\right)=\left\{i\in \overline{n}\cup \left\{0\right\}:p_{i,j+1}<t{p}_{ij}<0,j\in \left[n_i,n_{i+1}-1\right]\right\}$$

(63)

and that $\left|p_{i+1,n_{i+2}}\right|>\left|t{p}_{i{n}_i}\right|$ when linking adjacent components of powers of the complex argument z. Then, the zeros of $P\left(z\right)$ lie in $\left|z\right|<t^{-1}$, and $P\left(z\right)$ is SS $\left(t^{-1}\right)$ S, if $\epsilon \in \left(0,{\overline{\epsilon }}_t\right)$ such that

$${\overline{\epsilon }}_t<{\displaystyle \frac{1}{2}}\underset{\left|z\right|=t^{-1}}{\mathit{inf}}\left|{\displaystyle \frac{P^{*}\left(z\right)}{\tilde{P}\left(z\right)}}\right|={\displaystyle \frac{1}{2}}\underset{\left|z\right|=t^{-1}}{\mathit{inf}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^n\left|p_j\right|}{z}^j}{{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{i{j}_r}\right|}}{z}^j}}\right|$$

(64)

If $t\ge 1$ then $P\left(z\right)$ is SS $\left(t^{-1}\right)$ S as well.

Corollary 5. 

Assume that $Q\left(z\right)=P\left(z\right)-\Delta \left(z\right)$ where $P\left(z\right)$ is the polynomial of Theorem 9 and $\Delta \left(z\right)$ is a polynomial with $\mathit{deg}\Delta \left(z\right)\le \mathit{deg}P\left(z\right)=n$. Then, the zeros of $Q\left(z\right)$ lie in $\left|z\right|<t^{-1}$, and then $Q\left(z\right)$ is SS( $t^{-1}$ )S, and also SS(1)S if $t\ge 1$, for some given $t\in R_{+}$, if

$$\underset{\left|z\right|=t^{-1}}{\mathit{inf}}\left|{\displaystyle \frac{P^{*}\left(z\right)}{2\tilde{P}\left(z\right)+\Delta \left(z\right)}}\right|<1$$

(65)

provided that (a) the powers of the complex argument z of the coefficients of $P\left(z\right)$ are redefined by replacing $S_i\to S_i\left(t\right)$ for $i=1,2$; and (b) that $\left|p_{i+1,n_{i+2}}\right|>\left|t{p}_{i{n}_i}\right|$ when linking adjacent components of powers of coefficients.

Fast tests of the above relevant inequalities to guarantee the Rouché theorem for a tolerance to the perturbed polynomial which keeps the Schur stability property can be performed under sufficiency-type conditions by checking on real values of the complex argument $z$. For instance, for the Theorem 9, we have the following:

Corollary 6. 

The inequality (60) is guaranteed if

$$\overline{\epsilon }<{\displaystyle \frac{1}{2}}{\displaystyle \frac{\left|{\displaystyle {\sum }_{j=0}^m\left|p_{2j}\right|}-{\displaystyle {\sum }_{j=1}^m\left|p_{2j-1}\right|}\right|}{\left|{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{i{j}_r}\right|}}\right|}}$$

(66)

Proof. 

Consider two cases as follows:

(a)

If $n$ is even then $n=2m$ for $m\in Z_{0+}$ and (60) takes the form

$$\overline{\epsilon }<{\displaystyle \frac{1}{2}}\underset{\left|z\right|=1}{\mathit{inf}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^{2m}\left|p_j\right|}{z}^j}{{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{i{j}_r}\right|}}{z}^j}}\right|={\displaystyle \frac{1}{2}}\underset{\left|z\right|=1}{\mathit{inf}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^m\left|p_{2j}\right|}{z}^{2j}+{\displaystyle {\sum }_{j=1}^m\left|p_{2j-1}\right|}{z}^{2j-1}}{{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{i{j}_r}\right|}}{z}^j}}\right|$$

(67)

which holds, by taking a lower-bound of the above last term, if (66) holds since a lower bound of the numerator of (67) for $\left|z\right|=1$ is reached when the two additive terms are opposed and an upper bound of the denominator for $\left|z\right|=1$ is reached at $z=1$.

(b)

If $n$ is odd then $n=2m+1$ for $m\in Z_{0+}$ and * takes the form

$$\overline{\epsilon }<{\displaystyle \frac{1}{2}}\underset{\left|z\right|=1}{\mathit{inf}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^{2m+1}\left|p_j\right|}{z}^j}{{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{i{j}_r}\right|}}{z}^j}}\right|={\displaystyle \frac{1}{2}}\underset{\left|z\right|=1}{\mathit{inf}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{j=0}^m\left|p_{2j}\right|}{z}^{2j}+{\displaystyle {\sum }_{j=1}^m\left|p_{2j+1}\right|}{z}^{2j+1}}{{\displaystyle {\sum }_{i\in S_2}{\displaystyle {\sum }_{j=n_i}^{n_{i+1}}\left|p_{i{j}_r}\right|}}{z}^j}}\right|$$

(68)

which is also guaranteed if (66) holds. □

3. Some Extensions to the Allocation of the Zeros of a Class of Quasi-Polynomials Linked to Dynamic Systems with Constant Point Delays

3.1. Introductory Notes on Quasi-Polynomials and Delay Systems

Quasi-polynomials are referred to as transcendent complex equations which are very related to polynomials of several variables. A typical class of quasi-polynomials includes those of the form

$$Q_h\left(s\right)={\displaystyle {\sum }_{k=0}^n{\displaystyle {\sum }_{j=0}^m{a}_{kj}}}{s}^j{e}^{-khs}={\displaystyle {\sum }_{j=0}^m{b}_j}\left(e^{-hs}\right)s^j={\displaystyle {\sum }_{k=0}^n{a}_k}\left(s\right)e^{-khs}$$

(69)

where $h\in R$ and $a_{kj}$ are real or complex coefficients for $k\in \overline{n}\cup \left\{0\right\}$ and $k\in \overline{m}\cup \left\{0\right\}$.

$$a_k\left(s\right)={\displaystyle {\sum }_{j=0}^m{a}_{kj}}{s}^j; b_j\left(e^{-hs}\right)={\displaystyle {\sum }_{k=0}^n{a}_{kj}}{e}^{-khs}$$

(70)

Note that $Q_h\left(s\right)$ is a transcendent function in a complex variable “$s$”, but it depends jointly on $s$ and $e^{-hs}$ (and on a finite number of their powers) as mutually dependent arguments and, for that reason, $Q_h\left(s\right)$ is not a polynomial of two complex variables but a quasi-polynomial.

However, $Q_h\left(s\right)$ can be represented by a two-variable polynomial $P\left(s,z^{-1}\right)$ with the identification of $z^{-1}=z^{-1}\left(s\right)$ with $e^{-hs}$. Since $z^{-1}$ depends on $s$ then the zeros of $P\left(s,z^{-1}\right)$ are zeros of $Q_h\left(s\right)$ if and only if $z=e^{hs}$, that is $s=h\mathit{ln}z$, if $h\ne 0$, and $s=0$ if $z=1$, that is, if $P\left(0,1\right)=0$ then $Q_h\left(0\right)=0$. In other words, if $\left(s_0,z_0\right)=\left(0,1\right)$ is a zero of $P\left(s,z^{-1}\right)$ then $s_0=0$ is a zero of $Q\left(s\right)$ but if $\left(0,z_0\right)$ is a zero of $P\left(s,z^{-1}\right)$ for some $z_0\ne 1$ then $Q_h\left(0\right)\ne 0$, namely, $s_0=0$ is not a zero of $Q_h\left(s\right)$. And also, if $h\ne 0$ and $\left(s_0,z_0^{-1}\right)$ is a zero of $P\left(s,z^{-1}\right)$ then $s_0$ is a zero of $Q_h\left(s\right)$ if and only if $s_0=h^{-1}\mathit{ln}{z}_0$. It can be pointed out that derived potential stability results for the two-variable polynomials “s” and “z” obtained with the above change are just sufficiency-type stability conditions for the original quasi-polynomials which just depend on the argument “s”. Those conditions are not, in general, necessary but they are easier to test than the necessary and sufficient ones related to the quasi-polynomials of a single argument “s” [16,17].

If $h\in R_{0+}$, and $a_{kj}$ are real for $k\in \overline{n}\cup \left\{0\right\}$ and $k\in \overline{m}\cup \left\{0\right\}$, then $Q_h\left(s\right)$ is the characteristic equation of a time-delay invariant dynamic system of dimension $m$ with $n$ commensurate point delays $h_k=kh$ for $k\in \overrightarrow{n}\cup \left\{0\right\}$ (i.e., all delays are integer multiples of a basic delay $h$) which define a differential system of $m$ differential functional equations of first-order:

$$\dot{x}\left(t\right)={\displaystyle {\sum }_{k=0}^n{A}_k}x\left(t-kh\right)$$

(71)

where $A_k\in R^{m\times m}$ so that $Q_h\left(s\right)=\det \left(s{I}_m-{\displaystyle {\sum }_{k=0}^n{A}_k{e}^{-khs}}\right)$ is the characteristic quasi-polynomial of (71), $Q_h\left(s\right)=0$ is its characteristic equation, and $s$ is the Laplace transform argument, formally equivalent to the time-derivative operator $D=d/dt$. If (23) is subject to any given piecewise continuous function $\phi :\left[-nh,0\right]\to R^m$ then the solution $x:\left[-nh,0\right]\cup R_{+}\to R^m$ of (71) is unique with $x\left(t\right)=\phi \left(t\right)$; $t\in \left[-nh,0\right]$. See, for instance, [13,14]. It is well-known that a quasi-polynomial $Q_h\left(s\right)$ has infinitely many zeros but its number is finite within any finite band $\eta \ge \mathit{Re}s\ge {\eta }_0$ of the complex plane and that infinitely many of them are stable fulfilling $\mathit{Re}s\to -\infty$, see [14,15,16,17]. Therefore, if $h=0$ then all the point delays are null, the initial condition is a point initial condition at $t=0$, and (71) becomes its particular delay-free version which follows:

$$\dot{x}\left(t\right)=\left({\displaystyle {\sum }_{k=0}^n{A}_k}\right)x\left(t\right)$$

(72)

In the following, a direct extension of one of the former basic Eneström–Kakeya results is given for quasi-polynomials.

3.2. Main Result of This Section

The main result of this section is now stated:

Theorem 10. 

Let $Q\left(s\right)={\displaystyle {\sum }_{k=0}^n{a}_k}\left(s\right)e^{hks}$ be a quasi-polynomial of degree $m\ge 1$ in the argument s, that is, the highest power of the polynomials $a_k\left(s\right)$ is $m$, with an associated two-variable polynomial $P\left(s,z\right)={\displaystyle {\sum }_{k=0}^n{a}_k}\left(s\right)z^k$, via the replacement $e^{hs}\to z$, of degrees $n\ge 1$ in the argument $z$ and $m\ge 1$ in the argument $s$, where $a_k\left(s\right)={\displaystyle {\sum }_{j=0}^m{a}_{kj}}{s}^j$ is a polynomial of degree $m$ such that the coefficients $a_{kj}$ are real for $k\in \overline{n}\cup \left\{0\right\}$, $j\in \overline{m}\cup \left\{0\right\}$. Then, the following properties hold:

(i) 

Assume that for any $s={\gamma }_s+i{\omega }_s\in C_{-}$, $0<a_0\left({\gamma }_s\right)<a_1\left({\gamma }_s\right)<\dots <a_n\left({\gamma }_s\right)$, with $a_k\left({\gamma }_s\right)={\displaystyle {\sum }_{j=0}^m{a}_{kj}}{\gamma }_s^j$ for $k\in \overline{m}\cup \left\{0\right\}$.

Then, all the zeros of $P\left({\gamma }_s,z\right)={\displaystyle {\sum }_{k=0}^n{a}_k}\left({\gamma }_s\right)z^k={\displaystyle {\sum }_{k=0}^n{\displaystyle {\sum }_{j=0}^m{a}_{kj}}{\gamma }_s^j}{z}^k$ lie in $t_s=t_s\left({\gamma }_s\right)=\underset{k\in \overline{m}}{\mathit{max}}\left(a_{k-1}\left({\gamma }_s\right)/a_k\left({\gamma }_s\right)\right)=\underset{k\in \overline{m}}{\mathit{max}}\left(\left({\displaystyle {\sum }_{j=0}^m{a}_{k-1,j}{\gamma }_s^j}\right)/\left({\displaystyle {\sum }_{j=0}^m{a}_{kj}{\gamma }_s^j}\right)\right)<1$. As a result, one has the following:

(1)

$P\left({\gamma }_s,z\right)$ is $SS\left(\alpha \right)S$ for any real constant $\alpha \in \left(t,1\right]$, where $t=\underset{{\gamma }_s\in R_{-}}{\mathit{sup}}{t}_s\left({\gamma }_s\right)$;

(2)

$P\left(s,z\right)$ is $SS\left(\beta \right)S$ for any real constant $\beta \in \left[\alpha ,1\right]$ if, furthermore,

$$\underset{{\gamma }_s\in R_{-},{\omega }_s\in R_{0+},\left|z\right|=\beta }{\mathit{sup}}\left|{\displaystyle \frac{P\left({\gamma }_s+i{\omega }_s,z\right)-P\left({\gamma }_s,z\right)}{P\left({\gamma }_s,z\right)}}\right|<1$$

(73)

equivalently,

$$\underset{{\gamma }_s\in R_{-},{\omega }_s\in R_{0+},\theta \in \left[0,2\pi \right)}{\mathit{sup}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{k=0}^n{\displaystyle {\sum }_{j=0}^m{a}_{kj}}\left({\gamma }_s^2+{\omega }_s^2\right)k/2{\beta }^k}{e}^{ik\left(arctg\left({\omega }_s/{\gamma }_s\right)+\theta \right)}}{{\displaystyle {\sum }_{k=0}^n{\displaystyle {\sum }_{j=0}^m{a}_{kj}}{\gamma }_s^j}{\beta }^k{e}^{ik\theta }}}\right|<1$$

(74)

so that it is $SS\left(1\right)S$;

(3)

If all the zeros of $Q\left(s\right)$ lie in $C_{-}$, then $Q\left(s\right)$ is Hurwitz-stable;

(ii) 

$Q\left(s\right)$ is still Hurwitz-stable if $0<a_0\left({\gamma }_s\right)<a_1\left({\gamma }_s\right)<\dots <a_n\left({\gamma }_s\right)$ does not hold in some nonempty set $C_v\subset C_{-}$, but $Q\left(s\right)\ne 0$ for any $s\in C_v$ with $\mathit{Re}s={\gamma }_s$, provided that in (73) the supremum is taken on ${\gamma }_s\in C_{-}\backslash C_{\mathrm{v}},{\omega }_s\in R_{0+},\left|z\right|=\beta$, or in (74), on ${\gamma }_s\in C_{-}\backslash C_{\mathrm{v}},{\omega }_s\in R_{0+},\theta \in \left[0,2\pi \right)$;

(iii) 

Assume real constants $a_k^{*}$ for $k\in \overline{n}\cup \left\{0\right\}$ satisfying the constraints $0<a_0^{*}<a_1^{*}<\dots <a_n^{*}$. Then, all the zeros of $P\left(s,z\right)$, for $s\in C_{-}$, lie in $\left|z\right|<\rho$ for some real $\rho \in \left(t^{*},1\right]$, where $t^{*}=\underset{k\in \overline{m}}{\mathit{max}}\left(a_{k-1}^{*}/a_k^{*}\right)<1$ if

$$\underset{\mathit{Re}s<0,\theta \in \left[0,2\pi \right)}{\mathit{sup}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{k=0}^n\left({\displaystyle {\sum }_{j=0}^m{a}_{kj}}{s}^j-a_k^{*}\right)}{\rho }^k{e}^{ik\theta }}{{\displaystyle {\sum }_{k=0}^n{a}_k^{*}}{\rho }^k{e}^{ik\theta }}}\right|<1$$

(75)

and $P\left(s,z\right)$ is $\mathrm{SS}\left(\alpha \right)\mathrm{S}$ for any real constant $\alpha \in \left[\rho ,1\right]$ so that it is $\mathrm{SS}\left(1\right)\mathrm{S}$. Also, if all the zeros of $Q\left(s\right)$ lie in $C_{-}$, then $Q\left(s\right)$ is Hurwitz-stable. Sufficient conditions for (75) to hold are

$$\underset{\gamma \in R_{-}}{\mathit{sup}}\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{k=0}^n\left|{\displaystyle {\sum }_{j=0}^m{a}_{kj}}{\gamma }^j-a_k^{*}\right|}{e}^{ik\theta }}{{\displaystyle {\sum }_{k=0}^n{a}_k^{*}}{e}^{ik\theta }}}\right|<{\rho }^n$$

(76)

$$\underset{\gamma \in R_{-}}{\mathit{sup}}\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}\left|{\displaystyle \frac{\underset{k\in \overline{n}\cup \left\{0\right\}}{\max }\left|{\displaystyle {\sum }_{j=0}^m{a}_{kj}}{\gamma }^j-a_k^{*}\right|e^{ik\theta }}{{\displaystyle {\sum }_{k=0}^n{a}_k^{*}}{e}^{ik\theta }}}\right|<{\displaystyle \frac{{\rho }^n}{n+1}}$$

(77)

$$\underset{\gamma \phi R_{\phi }}{\mathit{sup}}\underset{j\phi \overline{n}\phi \left\{0\right\}}{\mathit{max}}{\displaystyle \frac{\underset{k\phi \overline{n}\phi \left\{o\right\}}{\mathit{max}}\left|{\displaystyle {\sum }_{j=0}^m{a}_{kj}{\gamma }^i-a_k^{*}}\right|}{\left|a_j^{*}-\left|{\displaystyle {\sum }_{k\left(\phi j\right)=0}^n{a}_k^{*}}\right|\right|}}={\displaystyle \frac{\underset{k\phi \overline{n}\phi \left\{o\right\}}{\mathit{max}}\left|{\displaystyle {\sum }_{j=0}^m{a}_{kj}{\gamma }^i-a_k^{*}}\right|}{\underset{j\phi \overline{n}\phi \left\{o\right\}}{\mathit{min}}\left|a_j^{*}-\left|{\displaystyle {\sum }_{k\left(\phi j\right)=0}^n{a}_k^{*}}\right|\right|}}<{\displaystyle \frac{{\rho }^n}{n+1}}$$

(78)

Proof. 

Property (i) follows from Theorem B which guarantees that, for any given ${\gamma }_s=\mathit{Re}s<0$, $P\left({\gamma }_s,z\right)$ has all its zeros in $\left|z\right|\le t_s<1$ since $0<a_0\left({\gamma }_s\right)<a_1\left({\gamma }_s\right)<\dots <a_n\left({\gamma }_s\right)$, $t_s=t_s\left({\gamma }_s\right)$. Note also that $\left|z\right|\le t_s<1$ implies by taking the principal part of the complex Neperian logarithm $\mathit{ln}z$ that $\left|z\right|<1$ is only possible with ${\gamma }_s<0$ since

$$\mathit{ln}z-i\theta =\mathit{ln}\left|z\right|=\mathit{ln}\left|e^{hs}\right|=h\left(\mathit{Re}s+i\mathrm{Im}s\right)=h\left({\gamma }_s+i{\omega }_s\right)\le \mathit{ln}{t}_s<0+i0=0, \mathrm{some} \theta \in \left[0,2\pi \right)$$

(79)

Then, by using the Rouché theorem combined with the above result, the constraint (73), equivalently (74), guarantees that the zeros of $P\left(s,z\right)$ for any $s={\gamma }_s+i{\omega }_s\in C_{-0}$ (since $\mathit{Re}s={\gamma }_s<0$) remain in $\left|z\right|<1$. Property (ii) follows from Property (i) since all points of $C_v$ which do not fulfil the constraint are not zeros of $Q\left(s\right)$. Property (iii) is proved closely to the proof of Property (i) by taking as a first reference for Schur stability through the application of Theorem B a set of real constants fulfilling the decreasing ordering constraints instead of establishing the constraint for all values of the open left-half complex plane. □

Remark 5. 

Theorem 10 is easily applicable to the time-delay dynamic system (71) whose characteristic quasi-polynomial is $Q_h\left(s\right)=\det \left(s{I}_m-{\displaystyle {\sum }_{k=0}^n{A}_k{e}^{-khs}}\right)$. Note, by simple inspection, that the quasi-polynomial ${\overline{Q}}_h\left(s\right)=e^{nhs}{Q}_h\left(s\right)$ has the same number of finite zeros in $R_{0+}$ as $Q_h\left(s\right)$. Therefore, $Q_h\left(s\right)$ is Hurwitz-stable if and only if ${\overline{Q}}_h\left(s\right)$ is Hurwitz-stable.

3.3. Some Further Results

A corollary of Theorem 10 follows below by taking into account Remark 5. Its proof is direct, and then omitted, from that of Theorem 8 by replacing $Q\left(s\right)\to Q_h\left(s\right),{\overline{Q}}_h\left(s\right)$ and $P\left(s,z\right)\to \overline{P}\left(s,z\right)$. The use of those replacements by using the factor $e^{hns}$ implies that the decreasing strictly ordering condition of the coefficients of Theorem 10 is now re-organized to be fixed in reverse order.

Corollary 7. 

Let $Q_h\left(s\right)$ be the characteristic quasi-polynomial of (72) and let ${\overline{Q}}_h\left(s\right)$ be defined as

$$\begin{gathered} {\overline{Q}}_h\left(s\right)=e^{nhs}{Q}_h\left(s\right)={\displaystyle {\sum }_{k=0}^n{\displaystyle {\sum }_{j=0}^m{a}_{kj}}}{s}^j{e}^{\left(n-k\right)hs}={\displaystyle {\sum }_{k=0}^n{a}_k}\left(s\right)e^{\left(n-k\right)hs} \\ ={\displaystyle {\sum }_{k=0}^n{a}_{n-k}}\left(s\right)e^{khs}={\displaystyle {\sum }_{k=0}^n{\displaystyle {\sum }_{j=0}^m{a}_{\left(n-k\right)j}}}{s}^j{e}^{khs} \end{gathered}$$

(80)

of degree $m\ge 1$ in the argument s with an associated two-variable polynomial $\overline{P}\left(s,z\right)={\displaystyle {\sum }_{k=0}^n{a}_{n-k}}\left(s\right)z^k$, via the replacement $e^{hs}\to z$, of degrees $n\ge 1$ in the argument $z$ and $m\ge 1$ in the argument $s$, where $a_{n-k}\left(s\right)={\displaystyle {\sum }_{j=0}^m{a}_{\left(n-k\right)j}}{s}^j$ is a polynomial of degree $m$ such that the coefficients $a_{kj}$ are real for $k\in \overline{n}\cup \left\{0\right\}$, $j\in \overline{m}\cup \left\{0\right\}$. Then, the following properties hold:

(i) 

Assume that for any $s={\gamma }_s+i{\omega }_s\in C_{-}$, $0<a_n\left({\gamma }_s\right)<a_{n-1}\left({\gamma }_s\right)<\dots <a_0\left({\gamma }_s\right)$, with $a_{n-k}\left({\gamma }_s\right)={\displaystyle {\sum }_{j=0}^m{a}_{\left(n-k\right)j}}{\gamma }_s^j$ for $k\in \overline{m}\cup \left\{0\right\}$.

Then, all the zeros of $\overline{P}\left({\gamma }_s,z\right)={\displaystyle {\sum }_{k=0}^n{a}_{n-k}}\left({\gamma }_s\right)z^k={\displaystyle {\sum }_{k=0}^n{\displaystyle {\sum }_{j=0}^m{a}_{\left(n-\right)kj}}{\gamma }_s^j}{z}^k$ lie in $t_s=t_s\left({\gamma }_s\right)=\underset{k\in \overline{m}}{\mathit{max}}\left(a_k\left({\gamma }_s\right)/a_{k-1}\left({\gamma }_s\right)\right)=\underset{k\in \overline{m}}{\mathit{max}}\left(\left({\displaystyle {\sum }_{j=0}^m{a}_{kj}{\gamma }_s^j}\right)/\left({\displaystyle {\sum }_{j=0}^m{a}_{\left(k-1\right)j}{\gamma }_s^j}\right)\right)<1$. As a result, one has

(1)

$\overline{P}\left({\gamma }_s,z\right)$ is $\mathrm{SS}\left(\alpha \right)\mathrm{S}$ for any real constant $\alpha \in \left(t,1\right]$, where $t=\underset{{\gamma }_s\in R_{-}}{\mathit{sup}}{t}_s\left({\gamma }_s\right)$;

(2)

$\overline{P}\left(s,z\right)$ is $\mathrm{SS}\left(t^{-1}\right)\mathrm{S}$ for any real constant $\beta \in \left[\alpha ,1\right]$ if, furthermore,

$$\underset{{\gamma }_s<0,{\omega }_s\in R_{0+},\left|z\right|=\beta }{\mathit{sup}}\left|{\displaystyle \frac{ \overline{P}\left({\gamma }_{\mathrm{s}}+i{\omega }_s,z\right)- \overline{P}\left({\gamma }_{\mathrm{s}},z\right)}{ \overline{P}\left({\gamma }_{\mathrm{s}},z\right)}}\right|<1$$

(81)

equivalently,

$$\underset{{\gamma }_s<0,{\omega }_s\in R_{0+},\theta \in \left[0,2\pi \right)}{\mathit{sup}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{k=0}^n{\displaystyle {\sum }_{j=0}^m{a}_{kj}}\left({\gamma }_s^2+{\omega }_s^2\right)k/2{\beta }^k}{e}^{ik\left(arctg\left({\omega }_s/{\gamma }_s\right)+\theta \right)}}{{\displaystyle {\sum }_{k=0}^n{\displaystyle {\sum }_{j=0}^m{a}_{kj}}{\gamma }_s^j}{\beta }^k{e}^{ik\theta }}}\right|<1$$

(82)

so that it is SS(1)S;

(3)

If all the zeros of ${\overline{Q}}_h\left(s\right)$ and $Q_h\left(s\right)$ lie in $C_{-}$, then $Q_h\left(s\right)$ is Hurwitz-stable;

(ii) 

$Q_h\left(s\right)$ is still Hurwitz-stable if $a_0\left({\gamma }_s\right)>a_1\left({\gamma }_s\right)>\dots >a_n\left({\gamma }_s\right)>0$ does not hold in some nonempty set $C_v\subset C_{-}$, but $Q\left(s\right)\ne 0$ for any $s\in C_v$ with $\mathit{Re}s={\gamma }_s$, provided that the supremum is taken in (76) on ${\gamma }_s\in C_{-}\backslash C_{\mathrm{v}},{\omega }_s\in R_{0+},\left|z\right|=\beta$, or in (77), on ${\gamma }_s\in C_{-}\backslash C_{\mathrm{v}},{\omega }_s\in R_{0+},\theta \in \left[0,2\pi \right)$.

(iii) 

Assume real constants $a_k^{*}$ for $k\in \overline{n}\cup \left\{0\right\}$ satisfying the constraints $0<a_n^{*}<a_{n-1}^{*}<\dots <a_0^{*}$. Then, all the zeros of $\overline{P}\left(s,z\right)$, for $s\in C_{-}$, lie in $\left|z\right|<\rho$ for some real $\rho \in \left(t^{*},1\right]$, where $t^{*}=\underset{k\in \overline{m}}{\mathit{max}}\left(a_k^{*}/a_{k-1}^{*}\right)<1$ if

$$\underset{\mathit{Re}s<0,\theta \in \left[0,2\pi \right)}{\mathit{sup}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{k=0}^n\left({\displaystyle {\sum }_{j=0}^m{a}_{\left(n-k\right)j}}{s}^j-a_{n-k}^{*}\right)}{\rho }^k{e}^{ik\theta }}{{\displaystyle {\sum }_{k=0}^n{a}_{n-k}^{*}}{\rho }^k{e}^{ik\theta }}}\right|<1$$

(83)

and $\overline{P}\left(s,z\right)$ is $\mathrm{SS}\left(\alpha \right)\mathrm{S}$ for any real constant $\alpha \in \left[\rho ,1\right]$ so that it is $\mathrm{SS}\left(1\right)\mathrm{S}$. Also, if all the zeros of ${\overline{Q}}_h\left(s\right)$ and those of $Q_h\left(s\right)$ lie in $C_{-}$, then $Q_h\left(s\right)$ is Hurwitz-stable. Sufficient conditions for (76) to hold are

$$\underset{\gamma \in R_{-}}{\mathit{sup}}\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}\left|{\displaystyle \frac{{\displaystyle {\sum }_{k=0}^n\left|{\displaystyle {\sum }_{j=0}^m{a}_{\left(n-k\right)j}}{\gamma }^j-a_k^{*}\right|}{e}^{ik\theta }}{{\displaystyle {\sum }_{k=0}^n{a}_{n-k}^{*}}{e}^{ik\theta }}}\right|<{\rho }^n$$

(84)

$$\underset{\gamma \in R_{-}}{\mathit{sup}}\underset{\theta \in \left[0,2\pi \right)}{\mathit{sup}}\left|{\displaystyle \frac{\underset{k\in \overline{n}\cup \left\{0\right\}}{\mathit{max}}\left|{\displaystyle {\sum }_{j=0}^m{a}_{kj}}{\gamma }^j-a_k^{*}\right|e^{ik\theta }}{{\displaystyle {\sum }_{k=0}^n{a}_k^{*}}{e}^{ik\theta }}}\right|<{\displaystyle \frac{{\rho }^n}{n+1}}$$

(85)

$$\stackrel{\underset{\gamma \in R_{-}}{\mathit{sup}}\underset{j\in \overline{n}\cup \left\{0\right\}}{\mathit{max}}}{{\displaystyle \frac{\underset{k\in \overline{n}\cup \left\{0\right\}}{\mathit{max}}\left|{\displaystyle {\sum }_{j=0}^{m}a\left(n-kj\right)}{\gamma }^j-a_{n-k}^{*}\right|}{\left|a_{n-j}^{*}-\left|{\displaystyle {\sum }_{k\left(\ne j\right)=0}^n{a}_{n-k}^{*}}\right|\right|}}={\displaystyle \frac{\underset{k\in \overline{n}\cup \left\{0\right\}}{\mathit{max}}\left|{\displaystyle {\sum }_{j=0}^m{a}_{\left(n-k\right)j}}{\gamma }^j-a_{n-k}^{*}\right|}{\underset{j\in \overline{n}\cup \left\{0\right\}}{\mathit{min}}\left|a_{n-j}^{*}-\left|{\displaystyle {\sum }_{k\left(\ne j\right)=0}^n{a}_{n-k}^{*}}\right|\right|}}<{\displaystyle \frac{{\rho }^n}{n+1}}}$$

(86)

The subsequent result formally states the following expected result. The decreasing ordering condition for the coefficients from its highest to its lowest power cannot hold in the closed complex right-hand-side for the particular case of the delay-free system, provided that it is asymptotically stable.

Proposition 1. 

Assume that the delay-free system (72) is asymptotically stable. Then, the constraint $0<a_0\left(\gamma \right)<a_1\left(\gamma \right)<\dots <a_n\left(\gamma \right)$ does not hold for $\gamma \ge 0$.

Proof. 

The delay-free system (72) is defined for $h=0$. Thus, take $z=e^{-hs}=1$ for any $s\in C$ since $h=0$. Thus, the characteristic quasi-polynomial $Q_h\left(s\right)=P\left(s,e^{-hs}\right)$ of (21) and (22) becomes $Q_0\left(s\right)=P\left(s,1\right)$ for the delay-free system (72). If (72) is asymptotically stable then $Q_0\left(s\right)=P\left(s,1\right)=0$ only for at most $m$ complex values $s$ (the characteristic zeros of (72)) with $\mathit{Re}s\le -\sigma <0$ for some real constant $\sigma >0$. Then, $Q_0\left(s\right)=P\left(s,1\right)\ne 0$ for any $s\in C$ with $\mathit{Re}s\ge 0$. Now, proceed by contradiction arguments. If $0<a_0\left(\gamma \right)<a_1\left(\gamma \right)<\dots <a_n\left(\gamma \right)$ holds for some real constant $\gamma \ge 0$ then, from Theorem 10 (ii), $P_{\gamma }\left(z\right)=P\left(\gamma ,z\right)$ has all its zeros in $\left|z\right|=\left|z\left(s\right)\right|=\left|e^{-0.s}\right|=1<1$, since $h=0$, leading to a contradiction. □

4. Further Examples

Now, the basic results are applied to the stability of two fourth and fifth perturbed polynomials under the Schur stability of their nominal versions.

Example 6. 

Consider the fourth-degree nominal polynomial:

$$P_4\left(z\right)=1.1z^4+z^3+0.99z^2+0.8712z+0.766656$$

The coefficients ordered from the highest to the lowest power of $z$ form an ordered strictly decreasing sequence of positive real numbers. By simple inspection, it follows from Theorem A that all its zeros are in $\left|z\right|\le 1$ since the coefficients satisfy the SPOc and then, trivially, the WPOc. Furthermore, it can be seen that $t=\underset{j\in \overline{4}}{\mathit{min}}\left(p_j/p_{j-1}\right)=1.01010$, where $p_{\left(.\right)}$ are the polynomial coefficients, is an upper bound of all the ratios between consecutive coefficients from the highest to the lowest power of $z$. Therefore, all its zeros lie also in $\left|z\right|\le t^{-1}=0.99<1$ from Theorem B. The moduli upper bound is calculated without a need to factorize the polynomial in zeros as this theorem states. Concerning the weak and strong Schur stability properties, one concludes from Definition 1 that $P_4\left(z\right)$ is $\mathrm{WS}\left({\mathrm{t}}^{-1}\right)$S, since all its zeros satisfy $\left|z\right|\le t^{-1}$ and it is also SS(1)S, since all its zeros satisfy $\left|z\right|<1$. The calculations of the zeros yield that they are

$$z_{1,2}=-0.79124\pm 0.54392i; z_{3,4}=0.24124\pm 0.87944i$$

of the respective moduli  $\left|z_{1,2}\right|=0.9601604<t^{-1}$ and $\left|z_{3,4}\right|=0.9192733<t^{-1}$. From the Rouché theorem of zeros, it follows that any perturbed polynomials  $Q_4\left(z\right)=P_4\left(z\right)+{\tilde{P}}_4\left(z\right)$ satisfying ${\left|{\tilde{P}}_4\left(z\right)/P_4\left(z\right)\right|}_{\left|z\right|=1}<1$ are also SS(1)S, that is, strongly 1-Schur-stable from Theorem 1 or from Theorem 3, which is the classical concept of stability in the discrete framework; that is, all the zeros of $P_4\left(z\right)$ are in the open complex unit circle centred at zero.

Consider the particular polynomial

$$Q_4\left(z\right)=P_4\left(z\right)+{\tilde{P}}_4\left(z\right)=z^4+0.9z^3+0.99z^2+0.8712z+0.9134$$

which does not fulfil the constraints SPOc and WPOc, and

$${\tilde{P}}_4\left(z\right)=Q_4\left(z\right)-P_4\left(z\right)=-0.1\left(z^4+z^3\right)+0.147744$$

so that ${\left|{\tilde{P}}_4\left(z\right)/P_4\left(z\right)\right|}_{\left|z\right|=1}\le 0.15433$< 1. Then $Q_4\left(z\right)$ is also SS (1)S. The calculation of the zeros of $Q_4\left(z\right)$ yields 

$$z_{1,2}=-0.755087\pm 0.616743i; z_{3,4}=0.305097\pm 0.9315591i$$

 of the respective moduli $\left|z_{1,2}\right|=0.975$ and $\left|z_{3,4}\right|=0.980$.

On the other hand, all the polynomials belonging to the family

$$Q_4\left(z,\lambda \right)=P_4\left(z\right)+\lambda {\tilde{P}}_4\left(z\right); \lambda \in \left(-{\lambda }_0,{\lambda }_0\right)$$

 are SS (1)S for some  ${\lambda }_0>1.$

Also, if any perturbation polynomial  ${\tilde{\widehat{P}}}_4\left(z\right)$ of a degree of at most four is prefixed, then all members of the family of polynomials

$${\widehat{Q}}_4\left(z,\lambda \right)=P_4\left(z\right)+\lambda {\tilde{\widehat{P}}}_4\left(z\right)$$

are SS(1)S for all $\lambda \in \left(-{\lambda }_a,{\lambda }_a\right)$ with ${\lambda }_a=\underset{\left|z=1\right|}{\mathit{sup}}\left|{\displaystyle \frac{P_4\left(z\right)}{{\tilde{\widehat{P}}}_4\left(\mathrm{z}\right)}}\right|$, since $\lambda \underset{\left|z=1\right|}{\mathit{sup}}\left|{\displaystyle \frac{{\tilde{\widehat{P}}}_4\left(\mathrm{z}\right)}{P_4\left(z\right)}}\right|<1$.

Example 7. 

Consider the fifth-degree nominal polynomial

$$P_5\left(z\right)=2.1z^5+2.0405z^4+z^3+1.46124997z^3+0.46701z^2+0.066752z+0.00338$$

The coefficients ordered from the highest to the lowest power of $z$ form an ordered strictly decreasing sequence of positive real numbers. By simple inspection, it follows from Theorem A that all its zeros are in $\left|z\right|\le 1$ since the coefficients satisfy the SPOc and then, trivially, the WPOc.

Furthermore, it can be seen that $t=\underset{j\in \overline{5}}{\mathit{min}}\left(p_j/p_{j-1}\right)>1.026894$, so that $t^{-1}\le 0.971666$, is an upper-bound of all the ratios between consecutive coefficients from the highest to the lowest power of $z$. Therefore, all its zeros lie also in $\left|z\right|\le 0.971666<1$ from Theorem B. Concerning the weak and strong Schur´s stability properties, one concludes from Definition 1 that $P_5\left(z\right)$ is $\mathrm{WS}\left({\mathrm{t}}^{-1}\right)$S, since all its zeros satisfy $\left|z\right|\le t^{-1}\le 0.971666<1$ and it is also SS(1)S, since all its zeros satisfy $\left|z\right|<1$.

Again, the upper-bound of the zeroes moduli is established without performing a polynomial factorization. The calculations of the zeros conclude that they are as follows:

$$z_{1,2}=0.1785874\pm 0.0773172i; z_{3,4}=-0.2516989\pm 0.5649814i; z_5=-0.1110939.$$

If the polynomial is perturbed with a polynomial perturbation $\tilde{P}\left(z\right)$, then $Q_5\left(z\right)=P_5\left(z\right)+{\tilde{P}}_5\left(z\right)$ has its zeros in $\left|z\right|<1$ if ${\left|{\tilde{P}}_5\left(z\right)/P_5\left(z\right)\right|}_{\left|z\right|=1}<1$ from Theorem 3. For instance, consider the perturbation polynomial

$${\tilde{P}}_5\left(z\right)=-\left(1.1z^5+0.1408z^4+0.23995762z^3+0.11518z^2+0.02057z\right)+0.0018$$

so that $\left|\tilde{P}\left(z\right)/P\left(z\right)\right|\le 0.99037<1$ for $\left|z\right|=1$; then

$$Q_5\left(z\right)=P_5\left(z\right)+{\tilde{P}}_5\left(z\right)=z^5+2.1813z^4+1.70120759z^3+0.58219z^2+0.09732z+0.00456$$

is SS(1)S. Its zeros are $-0.1151$, $-0.2147$, $-0.3236$, $-0.8743$, and $-0.6536$.

On the other hand, all the polynomials belonging to the family

$$Q_5\left(z,\lambda \right)=P_5\left(z\right)+\lambda {\tilde{P}}_5\left(z\right); \lambda \in \left(-{\lambda }_0,{\lambda }_0\right)$$

are SS (1)S for some  ${\lambda }_0>1.$

Also, if any perturbation polynomial ${\tilde{\widehat{P}}}_5\left(z\right)$ of a degree of at most five is prefixed, then all members of the family of polynomials

$${\widehat{Q}}_5\left(z,\lambda \right)=P_5\left(z\right)+\lambda {\tilde{\widehat{P}}}_5\left(z\right)$$

 are SS(1)S for all $\lambda \in \left(-{\lambda }_a,{\lambda }_a\right)$ with ${\lambda }_a=\underset{\left|z=1\right|}{\mathit{sup}}\left|{\displaystyle \frac{P_5\left(z\right)}{{\tilde{\widehat{P}}}_5\left(\mathrm{z}\right)}}\right|$.

Example 8. 

It is well-known that the particular parameterization of the Möbius transformation defined by $z=\left(s+1\right)/\left(s-1\right)$ transforms the complex region $\left|z\right|\le 1$ into the closed left-half plane $\mathit{Re}s\le 0$ with corresponding boundaries almost everywhere, that is, $\left|z\right|=1$ for all $z\ne 1$ into $\mathit{Re}s=0$, [32]. One concludes that the continuous polynomial $P_c\left(s\right)={\left.{\left(s-1\right)}^{n}P\left(z\right)\right]}_{z=\left(s+1\right)/\left(s-1\right)}$ is Hurwitz-stable, that is with roots in $\mathit{Re}s\le 0$ (respectively, strictly Hurwitz with roots in $\mathit{Re}s<0$), if and only if the associated discrete one $P\left(z\right)$ is WS(1)S (respectively, SS(1)S). Therefore, concerning the two above examples, we have that the following polynomials are Hurwitz:

$$P_{4c}\left(s\right)={\left(s-1\right)}^4{P}_4\left(\left(s+1\right)/\left(s-1\right)\right); Q_{4c}\left(s\right)={\left(s-1\right)}^4{Q}_4\left(\left(s+1\right)/\left(s-1\right)\right)$$

$$P_{5c}\left(s\right)={\left(s-1\right)}^5{P}_5\left(\left(s+1\right)/\left(s-1\right)\right); Q_{5c}\left(s\right)={\left(s-1\right)}^5{Q}_5\left(\left(s+1\right)/\left(s-1\right)\right)$$

Also, if one assumes that a perturbed discrete polynomial has the same degree $n$ as that of the nominal one, as it is the case in both above examples, and that the nominal one satisfies the Schur stability condition according to some of the given results then, by linearity, one has the following:

$$Q_{nc}\left(s\right)=P_{nc}\left(s\right)+{\tilde{P}}_{nc}\left(s\right)={\left(s-1\right)}^n\left[P_n\left(\left(s+1\right)/\left(s-1\right)\right)+{\tilde{P}}_n\left(\left(s+1\right)/\left(s-1\right)\right)\right]$$

so that

(1)

$P_{nc}\left(s\right)={\left(s-1\right)}^n{P}_n\left(\left(s+1\right)/\left(s-1\right)\right)$ is Hurwitz-stable if $P_n\left(\left(s+1\right)/\left(s-1\right)\right)$ is SS(1)S;

(2)

The Rouché test guaranteeing the maintenance of the Hurwitz stability of $Q_{nc}\left(s\right)$ from that of $P_{nc}\left(s\right)$ becomes

$$\underset{\omega \in R_{0+}}{\mathit{sup}}\left|{\tilde{P}}_{nc}\left(i\omega \right)/P_{nc}\left(i\omega \right)\right|<1$$

5. Conclusions and Future Work

This article has presented and proved the generalized Schur-type stability results of polynomials based on several forms and generalizations of the Eneström Eneström–Kakeya theorem combined with the Rouché theorem of zeros. The extensions of the Eneström Eneström–Kakeya theorem are applied for a given nominal polynomial which satisfies some decreasing conditions on the polynomial coefficients from the highest to the lowest power of the polynomial expression. Some variants are also derived for the case when the coefficients are not decreasing but consist of a worst-case known deviation from a decreasing rule. Then, Rouché s theorem on zeros is applied to guarantee that the zeros of a perturbed polynomial from the nominal one will be contained in the same complex region as those of the nominal polynomial. The perturbed polynomial is not necessarily structured in the sense that it can be arbitrary but with sufficiently small absolute values of its coefficients. Sufficiency-type conditions are also derived to guarantee the general Schur-type stability of a perturbed polynomial on arbitrary regions which do not necessarily coincide with the open unit circle centred at zero. In a second step, further sufficient conditions are introduced to guarantee that the above generalized Schur stability property persists for the perturbed polynomial within either the same above complex nominal stability region or in some larger one. It is seen that the known weak and, respectively, strong Schur stability in the closed and, respectively, open complex unit circle centred at zero are particular cases of the generalized versions. Some generalizations of the previous results for polynomials are given for the case of the zeros to be included in some annulus instead of within some disc centred at zero. Some further generalizations are provided for the case of quasi-polynomials whose zeros may be interpreted as the characteristic ones of linear continuous-time delayed time-invariant dynamic systems with commensurate constant point delays.

Also, some potential applications related to some control theory stability problems in the discrete framework are also discussed as well as some numerical examples of application for concrete polynomials also being performed. The stability of associated polynomials in the complex open left-half plane, which are related to the Schur-type stability of the original polynomials, via the Möbius transformation of the complex region $\left|z\right|\le 1$ into the complex region $\mathit{Re}s\le 0$.

It is foreseen that we will extend these ideas in future works to formalize the stability of the associated continuous-time systems based on that of the original discrete ones obtained from the various versions of the Eneström Eneström–Kakeya type theorems and their given worked extensions. It is also foreseen, based on the proposals and the results in [33,34] that we will develop further study extensions for the case of constraints on the polynomial coefficients if they are eventually complex under separate sufficiency-type explicit conditions on their real and imaginary parts and under the removal of the monotonicity conditions on the real coefficients of such polynomials by satisfying alternative conditions by the involvement of multiplicative constants on some of the coefficients. Finally, Eneström Eneström–Kakeya theorem-type extensions could be performed combined with the Rouché theorem of zeros to derive robust stability results for quasi-polynomials.