Robust Hyperexponential Stabilization via Nested Exponential Conformable Derivatives

Abstract

This paper presents a novel class of conformable integro-differential operators designed to model systems with rapid and ultra-rapid dynamics. This class of local operators enables the design of controllers and observers that induce hyperexponential convergence and provide robustness against bounded disturbances and dynamic uncertainties. The proposed method leverages Nested Exponential Functions (NEFs) and Nested Exponential Factorial Functions (NEFFs) to capture fast dynamics effectively. Additionally, the proposed study examines the Fundamental Theorem of Calculus in the context of Nested Exponential Conformable (NEC) operators, unveiling structural properties, such as stability and robustness, that produce dynamical systems with enhanced hyperexponential convergence and faster dynamics compared to existing approaches. Stability results for NEC systems are established, and some illustrative examples based on numerical simulations are presented to demonstrate the reliability of the proposed approach.

1. Introduction

Conformable calculus centers on defining a class of local operators capable of approximating fractional derivatives over a sufficiently small bounded interval [1]. These conformable operators are commonly used to describe fractional dynamics in complex systems through local derivatives, thereby simplifying system analysis. Unlike traditional fractional derivatives, conformable derivatives focus solely on information near a specific point, where the derivative is evaluated, rather than over an entire interval. Conformable derivatives were defined by altering the traditional integer-order derivative, introducing a time-dependent argument in the limit evaluation. This study revisits this concept from a different perspective, emphasizing the capture of fast dynamics characterized by hyperexponential convergence.

In recent years, the study of dynamic systems that exhibit hyperexponential stability [2] has attracted significant attention. In the innovative study [3], the authors introduced a time-varying linear differentiator for signals with essentially bounded second-order time derivatives. The proposed differentiator achieves convergence at a rate exceeding any other exponential function. The key idea involves crafting an increasing time-varying feedback gain to ensure the system dynamics achieve a faster convergence rate. This concept was further elaborated in [4], while broader aspects of hyperexponential stability were examined in [5]. Hyperexponential stabilizing methods that rely on time-varying feedback were explored in [6,7,8].

A different perspective is considered in the inspiring work [9], also considering a class of nested exponential functions. However, this work relies on the implicit functions method, relying on the online recursive computation of the Lyapunov function, and thus of the control signal. Additional interesting results have been presented in [10,11,12].

To the authors’ knowledge, there is no calculus tool that explicitly addresses hyperexponential stabilization. In this context, the conformal derivative technique stands out as a central tool that allows the problem to be studied clearly and rigorously.

One of the most useful definitions of conformable operators was introduced by Khalil et al. in [13], which stands for an operator that extends the classical definition, such that, for the case of a differentiable function, results in the product of a conventional derivative with a corresponding time-varying function. In [14], Abdeljawad illustrated several properties of the operator proposed by Khalil, including the chain rule, Gronwall’s inequality, and integration by parts, among others, which are attractive features of classical operator definitions. Additional formulations of conformable derivatives are available in [15,16,17,18]. In addition, several authors studied additional properties in [19,20], and recent applications of conformable operators have demonstrated the practical feasibility of these techniques [21,22,23,24].

Conformable derivatives are appealing for conducting Lyapunov-like analysis, as they allow the use of quadratic functions, extending some classical stability results to novel classes of dynamical systems [25]. This in particular motivates the present study.

Statement of Contribution. Inspired by the principles of hyperexponential stability and the concept of conformable derivatives, the main focus of this paper consists of investigating the benefits of utilizing a novel set of conformable operators to assess rapid and ultra-rapid dynamics with an expanded range of hyperexponential convergence capabilities.

The motivation of the proposed study is the extension of systems of the form $\dot{x}\left(t\right)=(a+bt)x\left(t\right)$, where a and b are constant parameters, and their solution is $x\left(t\right)=x\left(0\right)exp(at+\frac{b}{2}{t}^2)$, to a class of systems with enhanced convergence capabilities. The problem becomes even more compelling and engaging in the context of higher-order systems, where the derivatives of the time-varying feedback gains introduce non-trivial intricacies into both stability analysis and control design.

It is also worth noting that the limit case $\dot{x}\left(t\right)=-a\left(\frac{1}{0!}+\frac{t}{1!}+\frac{t^2}{2!}+\dots \right)x\left(t\right)$, with $a>0$, leads to a solution whose convergence rate is bounded by a nested exponential function [26], which fits into a particular case of the approach proposed in this paper. Moreover, standard techniques introduce additional complexities that would make stability analysis and control design highly intricate, particularly for higher-order dynamics and strongly coupled systems.

The following section introduces nested exponential and nested exponential factor functions. Section 3 details the proposed nested exponential conformable derivative. In Section 4, the stability of nested exponential conformable systems is examined. Section 5 presents the numerical simulation. Finally, Section 6 presents the main conclusions.

2. Nested Exponential Function and Nested Exponential Factorial Function

Two functions, Nested Exponential Function and Nested Exponential Factorial Function, which are central to the analysis and design of dynamic systems exhibiting hyperexponential stability, are introduced and studied below.

Definition 1

(Nested Exponential Function).  Consider a finite sequence of real numbers ${\left\{a_i\right\}}_{i=0}^n$. The n-order Nested Exponential Function (NEF) is defined as

$${\mathcal{E}}_n\left(t\right)=\mathcal{E}\left(t,{\left\{a_i\right\}}_{i=0}^n\right)=a_{n}exp\left(a_{n-1}exp\left(a_{n-2}exp\left(\dots a_{1}exp\left(a_{0}t\right)\right)\right)\right).$$

(1)

For $n=0$, the definition is taken as ${\mathcal{E}}_0\left(t\right)=a_0$.

The behavior of the NEF in Definition 1 is analyzed as follows:

Proposition 1.

Consider a sequence ${\left\{a_i\right\}}_{i=0}^n$ of real numbers. Then, the function ${\mathcal{E}}_n\left(t\right)$ is

• Uniformly bounded for $t\ge 0$ if $a_i\le 0$ for some $i\in \{0,1,\dots ,n-1\}$, or $a_n=0$.
• Unbounded for $t\ge 0$ if $a_n\ne 0$ and $a_i>0$ for all $i\in \{0,1,\dots ,n-1\}$.

Proof. 

The proof is split in two cases of interest.

Case I: Uniformly bounded

The case $a_n=0$ is straightforward because ${\mathcal{E}}_n\left(t\right)=0\forall t\ge 0$. Additionally, if $a_i=0$ for some $i\in \{0,1,\dots ,n-1\}$, the expression of ${\mathcal{E}}_n\left(t\right)$ simplifies to

$${\mathcal{E}}_n\left(t\right)=a_{n}exp\left(a_{n-1}exp\left(\dots a_{i+1}exp\left(0\right)\right)\right),$$

(2)

which results in the constant evaluation ${\mathcal{E}}_n\left(t\right)=a_{n}exp\left(a_{n-1}exp\left(\dots a_{i+1}\right)\right)\forall t\ge 0$.

Next, consider the case of $a_i\ne 0$ for all $i\in \{0,1,\dots ,n\}$ and $a_i<0$ for some $i\in \{0,1,\dots ,n-1\}$. Let i be the smallest index such that $a_i<0$. In this scenario,

$$exp(a_{i-1}exp(\dots a_{1}exp\left(a_{0}t\right)))\to \infty \mathrm{as}t\to \infty .$$

Let $L>0$ be an arbitrary real number. The above limit implies that

$$\exists T_L>0\mathrm{such}\mathrm{that}t>T_L⟹exp(a_{i-1}exp(\dots a_{1}exp\left(a_{0}t\right)))>L.$$

Now, let $\delta >0$ be arbitrarily small. Then, there is some $L>0$, such that $exp\left(a_i\xi \right)<\delta$, whenever $\xi >L$ (remembering that $a_i<0$), and consequently, $t>T_L$ implies that $exp(a_{i}exp(\dots a_{1}exp\left(a_{0}t\right)))<\delta$. Therefore,

$$exp(a_{i}exp(\dots a_{1}exp\left(a_{0}t\right)))\to 0\mathrm{as}t\to \infty .$$

Using similar arguments, one finds that the limit of ${\mathcal{E}}_n\left(t\right)$ becomes

$${\mathcal{E}}_n\left(t\right)\to a_{n}exp\left(a_{n-1}exp\left(\dots a_{i+1}exp\left(0\right)\right)\right)\mathrm{as}t\to \infty .$$

Since ${\mathcal{E}}_n\left(t\right)$ is continuous, it is real-valued and bounded on any compact support. In addition, the limit above implies that for any $\epsilon >0$, there is some $T_{\epsilon }>0$, such that

$$|{\mathcal{E}}_n\left(t\right)|\le \epsilon +|a_n|exp\left(a_{n-1}exp\left(\dots a_{i+1}exp\left(0\right)\right)\right),$$

for $t>T_{\epsilon }$, which also indicates that ${\mathcal{E}}_n\left(t\right)$ is uniformly bounded for any $t\ge 0$, with an upper bound given by

$$\left|{\mathcal{E}}_n\left(t\right)\right|\le \epsilon +\left|a_n\right|exp\left(a_{n-1}exp\left(\dots a_{i+1}exp\left(0\right)\right)\right)+\underset{\tau \in [0,T_{\epsilon }]}{max}\left|{\mathcal{E}}_n\left(\tau \right)\right|.$$

Case II: Unbounded

Finally, if $a_n\ne 0$ and $a_i>0$ for all $i\in \{0,1,\dots ,n-1\}$, it follows trivially that $|{\mathcal{E}}_n\left(t\right)|\to \infty$ as $t\to \infty$ because each nested exponential term diverges to ∞. □

Definition 2

(Nested Exponential Factorial Function). The Nested Exponential Factorial Function (NEFF) is defined as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{F}}_n\left(t\right)& =\prod _{i=0}^n{\mathcal{E}}_i\left(t\right)\hfill \\ \hfill & ={\mathcal{E}}_n\left(t\right){\mathcal{E}}_{n-1}\left(t\right)\dots {\mathcal{E}}_1\left(t\right){\mathcal{E}}_0\left(t\right).\hfill \end{array}\end{array}$$

(3)

For the case $n=0$, the definition is taken as ${\mathcal{F}}_0\left(t\right)={\mathcal{E}}_0\left(t\right)=a_0$.

The above definitions lead to the following result:

Proposition 2.

The derivative of the NEF satisfies

$${\mathcal{E}}_n^{\prime }\left(t\right)={\mathcal{F}}_n\left(t\right),$$

for each $n\in \mathbb{N}=\{1,2,\dots \}$.

Proof. 

The proof can be derived using mathematical induction.

For the base case: $n=1$, one has that ${\mathcal{E}}_1\left(t\right)=a_{1}exp\left(a_{0}t\right)$, whose derivative is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_1^{\prime }\left(t\right)& =a_0{a}_{1}exp\left(a_{0}t\right)\hfill \\ \hfill & =a_0{\mathcal{E}}_1\left(t\right)\hfill \\ \hfill & ={\mathcal{E}}_1\left(t\right){\mathcal{E}}_0\left(t\right)\hfill \\ \hfill & ={\mathcal{F}}_1\left(t\right).\hfill \end{array}\end{array}$$

(4)

Now, suppose that ${\mathcal{E}}_n^{\prime }\left(t\right)={\mathcal{F}}_n\left(t\right)$ is sustained for some $n\ge 1$. Then, the expression in (1) for $n+1$ can be rewritten as

$${\mathcal{E}}_{n+1}\left(t\right)=a_{n+1}exp\left({\mathcal{E}}_n\left(t\right)\right).$$

(5)

Furthermore, since $n\ge 1$, taking the derivative of (5) produces

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{n+1}^{\prime }\left(t\right)& =a_{n+1}exp\left({\mathcal{E}}_n\left(t\right)\right){\mathcal{E}}_n^{\prime }\left(t\right)\hfill \\ \hfill & ={\mathcal{E}}_{n+1}\left(t\right){\mathcal{E}}_n^{\prime }\left(t\right).\hfill \end{array}\end{array}$$

(6)

This establishes that the derivatives of the NEF can be found recursively. Moreover, using the supposition ${\mathcal{E}}_n^{\prime }\left(t\right)={\mathcal{F}}_n\left(t\right)$ leads to

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{E}}_{n+1}^{\prime }\left(t\right)& ={\mathcal{E}}_{n+1}\left(t\right){\mathcal{F}}_n\left(t\right)\hfill \\ \hfill & ={\mathcal{E}}_{n+1}\left(t\right){\Pi }_{i=0}^n{\mathcal{E}}_i\left(t\right)\hfill \\ \hfill & ={\Pi }_{i=0}^{n+1}{\mathcal{E}}_i\left(t\right)\hfill \\ \hfill & ={\mathcal{F}}_{n+1}\left(t\right),\hfill \end{array}\end{array}$$

(7)

which concludes the proof. □

Remark 1.

An alternative analysis proceeds as follows: Consider the non-autonomous differential equation

$$\dot{y}\left(t\right)={\mathcal{F}}_n\left(t\right)y\left(t\right),$$

(8)

with initial condition $y\left(0\right)=y_0\in \mathbb{R}$ and $n\in \mathbb{N}$. The unique solution to this equation is given by

$$y\left(t\right)=y_{0}exp\left({\int }_0^t{\mathcal{F}}_n\left(\tau \right)d\tau \right).$$

(9)

Now, assuming ${\mathcal{F}}_n\left(t\right)={\mathcal{E}}_n^{\prime }\left(t\right)$, the expression of the solution can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y\left(t\right)& =y_{0}exp\left({\left.{\mathcal{E}}_n\left(\tau \right)\right|}_{\tau =0}^{\tau =t}\right)\hfill \\ \hfill & =y_{0}exp\left({\mathcal{E}}_n\left(t\right)-{\mathcal{E}}_n\left(0\right)\right)\hfill \\ \hfill & =y_{0}exp\left(-{\mathcal{E}}_n\left(0\right)\right)exp\left({\mathcal{E}}_n\left(t\right)\right).\hfill \end{array}\end{array}$$

(10)

By setting $a_{n+1}=y_{0}exp\left(-{\mathcal{E}}_n\left(0\right)\right)$, the solution simplifies to

$$y\left(t\right)={\mathcal{E}}_{n+1}\left(t\right),$$

(11)

with initial condition $y\left(0\right)={\mathcal{E}}_{n+1}\left(0\right)$. Furthermore, directly from the differential equation, one has $\dot{y}\left(t\right)={\mathcal{F}}_n\left(t\right){\mathcal{E}}_{n+1}\left(t\right)={\mathcal{F}}_{n+1}\left(t\right)$. Therefore, ${\mathcal{E}}_{n+1}^{\prime }\left(t\right)={\mathcal{F}}_{n+1}\left(t\right)$.

3. Nested Exponential Conformable Derivative

Definition 3

(Nested Exponential Conformable Derivative). The Nested Exponential Conformable Derivative (NECD) with respect to the sequence ${\left\{a_i\right\}}_{i=0}^n$ is defined as

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)=\underset{\epsilon \to 0}{lim}{\displaystyle \frac{f\left(t+\frac{\epsilon }{{\mathcal{F}}_n\left(t\right)}\right)-f\left(t\right)}{\epsilon }},$$

(12)

whenever the above limit exists.

A more practical form of this definition is presented as follows.

Proposition 3.

Let $f\left(t\right)$ be a differentiable function, and ${\left\{a_i\right\}}_{i=0}^n$ be a finite sequence of real numbers, such that ${\mathcal{F}}_n\left(t\right)\ne 0\forall t\ge 0$. Then,

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)={\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}{f}^{\prime }\left(t\right).$$

(13)

Proof. 

Consider the substitution $\epsilon \left(t\right)=h{\mathcal{F}}_n\left(t\right)$, for some corresponding value h. Then, for a fixed t, expression (12) becomes

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)& =\underset{h\to 0}{lim}{\displaystyle \frac{f\left(t+h\right)-f\left(t\right)}{h{\mathcal{F}}_n\left(t\right)}}\hfill \\ \hfill & ={\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}\underset{h\to 0}{lim}{\displaystyle \frac{f\left(t+h\right)-f\left(t\right)}{h}},\hfill \end{array}\end{array}$$

(14)

and the result follows when considering the limit. □

Additional properties of the NECD are crucial for the methods discussed in subsequent sections.

Theorem 1.

Let f and g be differentiable functions for $t\ge 0$ and let a and $b\in \mathbb{R}$ be arbitrary constants. Then, for $n\ge 1$ and ${\left\{a_i\right\}}_{i=0}^n$ a sequence of positive real numbers, the following holds:

1. 

Derivative of a constant:

${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}a=0$.

2. 

Derivative of a time-power function:

${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{t}^a={\displaystyle \frac{a}{{\mathcal{F}}_n\left(t\right)}}{t}^{a-1}$ provided $a\ne 0$.

3. 

Linearity:

${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}[af\left(t\right)+bg\left(t\right)]=a{\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)+b{\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}g\left(t\right)$.

4. 

Product rule:

${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\left[f\left(t\right)g\left(t\right)\right]=f\left(t\right){\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}g\left(t\right)+g\left(t\right){\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)$.

5. 

Quotient rule:

${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{\displaystyle \frac{f\left(t\right)}{g\left(t\right)}}={\displaystyle \frac{g\left(t\right){\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)-f\left(t\right){\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}g\left(t\right)}{g^2\left(t\right)}}$.

6. 

Chain rule:

${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(g\left(t\right)\right)=f^{\prime }\left(g\left(t\right)\right){\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}g\left(t\right)$.

Proof. 

The proof is straightforward. □

An interesting property can be observed in the following results.

Proposition 4.

Let $f:[0,\infty )\to \mathbb{R}$ be a differentiable function, and ${\left\{a_i\right\}}_{i=0}^n$, with $n\ge 1$, be a sequence of positive real numbers. If ${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)\ge 0$ (resp. $\le 0$), then $f\left(t\right)$ is nondecreasing (resp. nonincreasing).

Proof. 

The condition ${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)\ge 0$ is equivalent to $f^{\prime }\left(t\right)\ge 0$, since

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)={\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}{f}^{\prime }\left(t\right),$$

where ${\mathcal{F}}_n\left(t\right)>0$ as those $a_i$ are assumed to be positive. Similarly, ${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)\le 0$ implies $f^{\prime }\left(t\right)\le 0$; therefore, $f\left(t\right)$ is nondecreasing when ${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)\ge 0$ and nonincreasing when ${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)\le 0$. □

Definition 4

(Nested Exponential Conformable Integral). Let $f\left(t\right)$ be a locally integrable function. The Nested Exponential Conformable Integral (NECI) with respect to the sequence ${\left\{a_i\right\}}_{i=0}^n$ is defined as

$${\mathcal{I}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)={\int }_0^t{\mathcal{F}}_n\left(\tau \right)f\left(\tau \right)d\tau .$$

(15)

NEC operators satisfy a version of the Fundamental Theorem of Calculus, as stated below:

Theorem 2.

Let $f\left(t\right)$ be a differentiable function, and ${\left\{a_i\right\}}_{i=0}^n$ a finite sequence of real numbers with ${\mathcal{F}}_n\left(t\right)\ne 0\forall t\ge 0$. Then,

$$\begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{\mathcal{I}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)& =f\left(t\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\mathcal{I}}^{{\left\{a_i\right\}}_{i=0}^n}{\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)& =f\left(t\right)-f\left(0\right).\hfill \end{array}$$

(17)

Proof. 

The proof of the first part is as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{\mathcal{I}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)& ={\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{\int }_0^t{\mathcal{F}}_n\left(\tau \right)f\left(\tau \right)d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}{\displaystyle \frac{d}{dt}}{\int }_0^t{\mathcal{F}}_n\left(\tau \right)f\left(\tau \right)d\tau \hfill \\ \hfill & ={\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}{\mathcal{F}}_n\left(t\right)f\left(t\right)\hfill \\ \hfill & =f\left(t\right).\hfill \end{array}\end{array}$$

(18)

The second part is as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{I}}^{{\left\{a_i\right\}}_{i=0}^n}{\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}f\left(t\right)& ={\mathcal{I}}^{{\left\{a_i\right\}}_{i=0}^n}\left[{\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}{f}^{\prime }\left(t\right)\right]\hfill \\ \hfill & ={\int }_0^t{\mathcal{F}}_n\left(\tau \right)\left[{\displaystyle \frac{1}{{\mathcal{F}}_n\left(\tau \right)}}{f}^{\prime }\left(\tau \right)\right]d\tau \hfill \\ \hfill & ={\int }_0^t{f}^{\prime }\left(\tau \right)d\tau \hfill \\ \hfill & =f\left(t\right)-f\left(0\right).\hfill \end{array}\end{array}$$

(19)

These two results complete the proof. □

An important example is provided below: Consider the linear NEC differential equation

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}y\left(t\right)=\alpha y\left(t\right),$$

(20)

with the initial condition $y\left(0\right)=y_0\in \mathbb{R}$, where $\alpha \in \mathbb{R}$ is a constant. This equation can be rewritten as

$${\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}\dot{y}\left(t\right)=\alpha y\left(t\right).$$

(21)

Alternatively, noting that ${\mathcal{F}}_n\left(t\right)={\mathcal{E}}_n^{\prime }\left(t\right)$, the above simplifies to

$${\displaystyle \frac{\dot{y}\left(t\right)}{y\left(t\right)}}=\alpha {\mathcal{E}}_n^{\prime }\left(t\right),$$

(22)

whose solution can be expressed as

$$y\left(t\right)=y_{0}exp\left(-\alpha {\mathcal{E}}_n\left(0\right)\right)exp\left(\alpha {\mathcal{E}}_n\left(t\right)\right),$$

(23)

or equivalently, in its expanded form:

$$y\left(t\right)=y_{0}exp\left(-\alpha {\mathcal{E}}_n\left(0\right)\right)exp\left(\alpha a_{n}exp\left(a_{n-1}exp\left(\dots a_{1}exp\left(a_{0}t\right)\right)\right)\right).$$

(24)

It follows that $y\left(t\right)$ is also a NEF. When some terms in the sequence ${\left\{a_i\right\}}_{i=0}^n$ are zero, the solution $y\left(t\right)$ remains a constant. On the other hand, if none of the terms in the sequence are zero, but some of them are negative, $y\left(t\right)$ remains uniformly bounded and converges to a constant value. An example of this is the Gompertz function [27,28], which is a special case with a finite sequence $\{a_0,a_1\}$. It is interesting to consider that the Gompertz function can result as a solution to an autonomous nonlinear differential equation or a nonautonomous linear differential equation. These cases are simplified by considering the approach of this paper, by means of NECD operators.

The most relevant case for modeling and control occurs when ${\left\{a_i\right\}}_{i=0}^n$ is a sequence of positive real numbers. In this scenario, if $\alpha \le 0$, then the solution $y\left(t\right)$ is uniformly bounded for any initial condition $y_0\ne 0$. Moreover, the solution $y\left(t\right)$ converges to zero with a hyperexponential rate if and only if $\alpha <0$.

4. Stability of NEC Dynamical Systems

Consider the NEC system

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\mathit{x}\left(t\right)=\mathit{f}(t,\mathit{x}\left(t\right)),$$

(25)

where ${\left\{a_i\right\}}_{i=0}^n$ is a sequence of positive real numbers, $\mathit{x}\in {\mathbb{R}}^m$ is the state vector, $\mathit{f}\in {\mathbb{R}}^m$ is a smooth real vector field, and $\mathit{x}\left(0\right)={\mathit{x}}_0\in \Omega \subseteq {\mathbb{R}}^m$ is the initial condition, for $\Omega$ a convex domain containing the origin.

Definition 5.

System (25) is Locally Nested Exponentially Stable (LNES) if there are constants ${\gamma }_1>0$ and ${\gamma }_2>0$, and a sequence of positive numbers ${\left\{a_i\right\}}_{i=0}^n$ such that

$$\left|\right|\mathit{x}\left(t\right)\left|\right|\le {\gamma }_{1}exp\left(-{\gamma }_2{\mathcal{E}}_n\left(t;{\left\{a_i\right\}}_{i=0}^n\right)\right).$$

(26)

The system (25) is Globally Nested Exponentially Stable (GNES) when the above conditions remain in the entire domain $\Omega ={\mathbb{R}}^m$.

Theorem 3.

Let $V\left(\mathit{x}\right)$ be a continuous and differentiable almost everywhere in Ω, such that

1. 

$V\left(\mathit{x}\right)>0$ for $\mathit{x}\ne \mathbf{0}$, and $V\left(\mathbf{0}\right)=0$,

2. 

$V\left(\mathit{x}\right)\ge {\left|\right|\mathit{x}\left|\right|}^b$, for $b\ge 1$, and

3. 

${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V\left(\mathit{x}\right)\le -\gamma V^q\left(\mathit{x}\left(t\right)\right),$

in Ω, for some $q\in (0,1]$ and $\gamma >0$. Thus, the origin of (25) is LNES. Furthermore, if $\Omega ={\mathbb{R}}^m$, then the origin of (25) is the GNES.

Proof. 

Because ${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V\left(\mathit{x}\left(t\right)\right)\le 0$, one gets that $V\left(\mathit{x}\right(t\left)\right)$ is nonincreasing and, consequently, $V\left(\mathit{x}\left(t\right)\right)\le V\left({\mathit{x}}_0\right)$. This means that $\mathit{x}=\mathbf{0}$ is an equilibrium, and ${\mathit{x}}_0=\mathbf{0}$ implies $\mathit{x}\left(0\right)=\mathbf{0}$ for all $t\ge 0$.

Therefore, without loss of generality, we consider ${\mathit{x}}_0\ne \mathbf{0}$. Then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V\left(\mathit{x}\left(t\right)\right)& \le -\gamma V^q\left(\mathit{x}\left(t\right)\right)\hfill \\ \hfill & =-\gamma V^q\left({\mathit{x}}_0\right){\displaystyle \frac{V^q\left(\mathit{x}\left(t\right)\right)}{V^q\left({\mathit{x}}_0\right)}}.\hfill \end{array}\end{array}$$

(27)

Thus, as $q\in (0,1]$ and ${\displaystyle \frac{V\left(\mathit{x}\right(t\left)\right)}{V\left({\mathit{x}}_0\right)}}\le 1$, one gets

$${\displaystyle \frac{V\left(\mathit{x}\right(t\left)\right)}{V\left({\mathit{x}}_0\right)}}\le {\displaystyle \frac{V^q\left(\mathit{x}\left(t\right)\right)}{V^q\left({\mathit{x}}_0\right)}},$$

(28)

whereby

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V\left(\mathit{x}\left(t\right)\right)& \le -\gamma V^q\left({\mathit{x}}_0\right){\displaystyle \frac{V\left(\mathit{x}\right(t\left)\right)}{V\left({\mathit{x}}_0\right)}}\hfill \\ \hfill & =-\gamma V^{q-1}\left({\mathit{x}}_0\right)V\left(\mathit{x}\left(t\right)\right),\hfill \end{array}\end{array}$$

(29)

where $V^{q-1}\left({\mathit{x}}_0\right)$ is well-defined, as $V^{q-1}\left({\mathit{x}}_0\right)>0$ since ${\mathit{x}}_0\ne \mathbf{0}$.

The above NEC differential inequality is equivalent to the conventional differential inequality,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{\dot{V}\left(\mathit{x}\left(t\right)\right)}{V\left(\mathit{x}\right(t\left)\right)}}& \le -\gamma V^{q-1}\left({\mathit{x}}_0\right){\mathcal{F}}_n\left(t\right)\hfill \\ \hfill & =-\gamma V^{q-1}\left({\mathit{x}}_0\right){\mathcal{E}}_n^{\prime }\left(t\right).\hfill \end{array}\end{array}$$

(30)

Integrating with respect to t produces

$$ln\left[{\displaystyle \frac{V\left(\mathit{x}\right(t\left)\right)}{V\left({\mathit{x}}_0\right)}}\right]\le -\gamma V^{q-1}\left({\mathit{x}}_0\right)\left[{\mathcal{E}}_n\left(t\right)-{\mathcal{E}}_n\left(0\right)\right],$$

(31)

and consequently, using the monotonicity of the exponential function,

$$V\left(\mathit{x}\left(t\right)\right)\le V\left({\mathit{x}}_0\right)exp\left(\gamma V^{q-1}\left({\mathit{x}}_0\right){\mathcal{E}}_n\left(0\right)\right)exp\left(-\gamma V^{q-1}\left({\mathit{x}}_0\right){\mathcal{E}}_n\left(t\right)\right).$$

(32)

The remaining of this proof follows from $V\left(\mathit{x}\right)\ge {\left|\right|\mathit{x}\left|\right|}^b$, meaning that

$$\left|\right|\mathit{x}\left(t\right)\left|\right|\le V^{1/b}\left({\mathit{x}}_0\right)exp\left({\displaystyle \frac{\gamma }{b}}{V}^{q-1}\left({\mathit{x}}_0\right){\mathcal{E}}_n\left(0\right)\right)exp\left(-{\displaystyle \frac{\gamma }{b}}{V}^{q-1}\left({\mathit{x}}_0\right){\mathcal{E}}_n\left(t\right)\right),$$

(33)

thereby completing the proof. □

A less restrictive version of the previous result can be obtained, but at the expense of sacrificing the LNES (resp. GNES) condition.

Theorem 4.

Let $V(t,\mathit{x}(t\left)\right)$ be a differentiable function satisfying the following conditions:

1. 

$k_1(\parallel \mathit{x}\parallel )\le V(t,\mathit{x})\le k_2(\parallel \mathit{x}\parallel )$,

2. 

${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V(t,\mathit{x})\le -k_3(\parallel \mathit{x}\parallel )$,

for all $t\ge 0$ and $\mathit{x}\in {\mathbb{R}}^m$, where $k_i\in \mathcal{K}$ for $i=1,2,3$. Then, the origin of system (25) is locally asymptotically stable. Furthermore, if $k_i\in {\mathcal{K}}^{\infty }$, for $i=1,2,3$, the origin of (25) is globally asymptotically stable.

Proof. 

It is possible to determine that $k_2^{-1}\left(V(t,\mathit{x}\left(t\right))\right)\le \left|\right|\mathit{x}\left(t\right)\left|\right|$ since $k_2$ is a strictly monotonically increasing function with respect to its argument. Then,

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V(t,\mathit{x}\left(t\right))\le -k_3\left(k_2^{-1}\left(V(t,\mathit{x}\left(t\right))\right)\right).$$

(34)

Moreover, because ${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V(t,\mathit{x}\left(t\right))\le 0$, one has that $V(t,\mathit{x}(t\left)\right)$ is nonincreasing over time. Thus, the range of $V(t,\mathit{x}(t\left)\right)$ is confined to the compact set $[0,V(0,{\mathit{x}}_0)]$.

Consider the closed disk ${\mathcal{B}}_{\epsilon }=\{\mathit{x}\in \Omega :|\left|\mathit{x}\right||\le \epsilon \}$, and suppose ${\mathit{x}}_0\in \Omega \setminus {\mathcal{B}}_{\epsilon }$. It is clear that ${inf}_{\mathit{x}\notin {\mathcal{B}}_{\epsilon }}V(t,\mathit{x})\ge k_1\left(\epsilon \right)$. Then,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V(t,\mathit{x}\left(t\right))& \le -k_3\left(k_2^{-1}\left(V(t,\mathit{x}\left(t\right))\right)\right)\hfill \\ \hfill & \le -k_3\left(k_2^{-1}\left(k_1\left(\epsilon \right)\right)\right)\hfill \\ \hfill & \le -k_3\left(k_2^{-1}\left(k_1\left(\epsilon \right)\right)\right){\displaystyle \frac{V(t,\mathit{x}(t\left)\right)}{V(0,{\mathit{x}}_0)}}.\hfill \end{array}\end{array}$$

(35)

Similarly to the proof of the previous theorem, one finds that $V(t,\mathit{x}(t\left)\right)$ converges to $\mathsf{\partial }{\mathcal{B}}_{\epsilon }$, that is, the boundary of ${\mathcal{B}}_{\epsilon }$, with a hyperexponential rate.

Moreover, since $\epsilon >0$ was arbitrarily small, one obtains $V(t,\mathit{x}(t\left)\right)\to 0$ as $t\to \infty$, which in turns implies $\mathit{x}\left(t\right)\to \mathbf{0}$ as $t\to \infty$.

The case of $k_i\in {\mathcal{K}}^{\infty }$ with $i=1,2,3$, ensures that the origin is globally asymptotically stable. □

The application of the above results is as follows.

Theorem 5.

Consider the NEC system

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\mathit{x}\left(t\right)=A\mathit{x}\left(t\right),$$

(36)

where ${\left\{a_i\right\}}_{i=0}^n$ is a sequence of positive real numbers, $\mathit{x}\in {\mathbb{R}}^m$, and $A\in {\mathbb{R}}^{m\times m}$ is a constant matrix. If A is a Hurwitz matrix, then the origin ${\mathit{x}}^{*}=\mathbf{0}$ is GNES, and $\mathit{x}\left(t\right)\to 0$ as $t\to \infty$.

Proof. 

Consider the quadratic function $V\left(\mathit{x}\left(t\right)\right)={\mathit{x}}^T\left(t\right)P\mathit{x}\left(t\right)$, for some positive definite constant matrix P, whose NECD with respect to ${\left\{a_i\right\}}_{i=0}^n$ and along (36) is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V\left(\mathit{x}\left(t\right)\right)& =2{\mathit{x}}^T\left(t\right)P{\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\mathit{x}\left(t\right)\hfill \\ \hfill & =2{\mathit{x}}^T\left(t\right)PA\mathit{x}\left(t\right)\hfill \\ \hfill & ={\mathit{x}}^T\left(t\right)(PA+A^{T}P)\mathit{x}\left(t\right).\hfill \end{array}\end{array}$$

(37)

Because A is a Hurwitz matrix, there are positive definite matrices P and Q that resolve the Lyapunov equation $PA+A^{T}P+Q=0$. Then,

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V\left(\mathit{x}\left(t\right)\right)=-{\mathit{x}}^T\left(t\right)Q\mathit{x}\left(t\right).$$

(38)

The proof follows by noticing that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\lambda }_m{\left(P\right)\left|\right|\mathit{x}\left(t\right)\left|\right|}^2\le V\left(\mathit{x}\left(t\right)\right)\le {\lambda }_M{\left(P\right)\left|\right|\mathit{x}\left(t\right)\left|\right|}^2\hfill \\ \hfill & {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V\left(\mathit{x}\left(t\right)\right)\le -{\lambda }_m{\left(Q\right)\left|\right|\mathit{x}\left(t\right)\left|\right|}^2\hfill \end{array}\end{array}$$

(39)

where ${\lambda }_m\left(\right)$ and ${\lambda }_M\left(\right)$ are the minimum and maximum proper value functions, respectively. From this, asymptotic stability follows directly.

In this particular case, one has

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V(t,\mathit{x}\left(t\right))\le -\gamma V(t,\mathit{x}\left(t\right)),$$

(40)

for $\gamma ={\displaystyle \frac{{\lambda }_m\left(Q\right)}{{\lambda }_M\left(P\right)}}$, meaning that ${\mathit{x}}^{*}=\mathbf{0}$ is a GNES, and $\mathit{x}\left(t\right)\to 0$ as $t\to \infty$, possibly faster than exponentially. □

It is usually the case that dynamic systems are subject to bounded but unknown disturbances. In this case, the stability analysis is conducted in a slightly different manner.

Theorem 6.

Consider the uncertain NEC system

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\mathit{x}\left(t\right)=A\mathit{x}\left(t\right)+\mathit{\xi }\left(t\right),$$

(41)

where ${\left\{a_i\right\}}_{i=0}^n$ is a sequence of positive real numbers, $\mathit{x}\in {\mathbb{R}}^m$, $A\in {\mathbb{R}}^{m\times m}$ is a constant matrix, and $\mathit{\xi }\left(t\right)$ is a bounded uncertainty. If A is a Hurwitz matrix, it satisfies

$$PA+A^{T}P+ϵP^2+\kappa P\prec 0,$$

(42)

for some $ϵ,\kappa >0$ and $P\succ 0$. Then, $\mathit{x}\left(t\right)$ remains uniformly ultimately bounded.

Proof. 

Consider $ϵ=\kappa =0$, then, because A is Hurwitz, $P\succ 0$ exists such that $PA+A^{T}P=-Q\prec -\delta I\prec 0$ for some $\delta >0$. Furthermore, since $F(ϵ,\kappa )={\lambda }_M\left(PA+A^{T}P+ϵP^2+\kappa P\right)$ is continuous in $ϵ$ and $\kappa$, there are some $ϵ,\kappa >0$, such that $F(ϵ,\kappa )\le -{\displaystyle \frac{\delta }{2}}$.

Consider again the quadratic function $V(t,\mathit{x}\left(t\right))={\mathit{x}}^T\left(t\right)P\mathit{x}\left(t\right)$. Thus,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V(t,\mathit{x}\left(t\right))& =2{\mathit{x}}^T\left(t\right)P{\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\mathit{x}\left(t\right)\hfill \\ \hfill & =2{\mathit{x}}^T\left(t\right)PA\mathit{x}\left(t\right)+2{\mathit{x}}^T\left(t\right)P\mathit{\xi }\left(t\right)\hfill \\ \hfill & ={\mathit{x}}^T\left(t\right)(PA+A^{T}P)\mathit{x}\left(t\right)+2{\mathit{x}}^T\left(t\right)P\mathit{\xi }\left(t\right).\hfill \end{array}\end{array}$$

(43)

Taking into account that $2{\mathit{x}}^T\left(t\right)P\mathit{\xi }\left(t\right)\le ϵ{\mathit{x}}^T\left(t\right)P^2\mathit{x}\left(t\right)+{ϵ}^{-1}{\mathit{\xi }}^T\left(t\right)\mathit{\xi }\left(t\right)$, for arbitrary $ϵ>0$, and considering the quadratic matrix inequality (QMI) (42), one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}V(t,\mathit{x}\left(t\right))& <-\kappa {\mathit{x}}^T\left(t\right)P\mathit{x}\left(t\right)+{ϵ}^{-1}{\mathit{\xi }}^T\left(t\right)\mathit{\xi }\left(t\right)\hfill \\ \hfill & =-\kappa V(t,\mathit{x}\left(t\right))+{ϵ}^{-1}{\left|\right|\mathit{\xi }\left(t\right)\left|\right|}^2.\hfill \end{array}\end{array}$$

(44)

An upper bound for the solution $V\left(t\right)$ of the differential inequality of NEC can be found from the method of variation of parameters, resulting in

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & V(t,\mathit{x}\left(t\right))\le V_{0}exp\left(\kappa {\mathcal{E}}_n\left(0\right)\right)exp\left(-\kappa {\mathcal{E}}_n\left(t\right)\right)\hfill \\ \hfill & +{ϵ}^{-1}{\int }_0^{t}exp\left(-\kappa [{\mathcal{E}}_n\left(t\right)-{\mathcal{E}}_n\left(\tau \right)]\right){\left|\right|\mathit{\xi }\left(\tau \right)\left|\right|}^2{\mathcal{F}}_n\left(\tau \right)d\tau ,\hfill \end{array}\end{array}$$

(45)

for $V_0=V(0,\mathit{x}\left(0\right))$. The first term on the right-hand side of the previous inequality converges to zero when $t\to \infty$. The limit for the second term can be found by using the L’Hôpital rule, as follows,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \underset{t\to \infty }{lim}{\int }_0^{t}exp\left(-\kappa [{\mathcal{E}}_n\left(t\right)-{\mathcal{E}}_n\left(\tau \right)]\right){\left|\right|\mathit{\xi }\left(\tau \right)\left|\right|}^2{\mathcal{F}}_n\left(\tau \right)d\tau \hfill \\ \hfill & =\underset{t\to \infty }{lim}{\displaystyle \frac{{\int }_0^{t}exp\left(\kappa {\mathcal{E}}_n\left(\tau \right)\right){\left|\right|\mathit{\xi }\left(\tau \right)\left|\right|}^2{\mathcal{F}}_n\left(\tau \right)d\tau }{exp\left(\kappa {\mathcal{E}}_n\left(t\right)\right)}}\hfill \\ \hfill & ={\overline{lim}}_{t\to \infty }{\displaystyle \frac{exp\left(\kappa {\mathcal{E}}_n\left(t\right)\right){\left|\right|\mathit{\xi }\left(\tau \right)\left|\right|}^2{\mathcal{F}}_n\left(t\right)}{\kappa {\mathcal{E}}_n^{\prime }\left(t\right)exp\left(\kappa {\mathcal{E}}_n\left(t\right)\right)}}\hfill \\ \hfill & ={\displaystyle \frac{{\ell }_{\xi }}{\kappa }},\hfill \end{array}\end{array}$$

(46)

for ${\ell }_{\xi }=\overline{lim}{\left|\right|\mathit{\xi }\left(t\right)\left|\right|}^2$. Therefore,

$$\underset{t\to \infty }{lim}V(t,\mathit{x}\left(t\right))\le {\displaystyle \frac{{\ell }_{\xi }}{ϵ\kappa }},$$

(47)

completing the proof. □

Remark 2.

The QMI (42) is difficult to solve analytically. However, there are computational tools to numerically solve the equivalent Linear Matrix Inequality (LMI) problem for matrix P, given A, ϵ and κ,

$$\left[\begin{array}{ccccc}PA+A^{T}P+\kappa P,& & P& & 0\\ P& & -{ϵ}^{-1}I& & 0\\ 0& & 0& & -P\end{array}\right]\prec 0.$$

(48)

If the solution P to the above problem exists, the dynamic system (41) shows the stability properties discussed for the given ϵ and κ.

Remark 3.

In the case of the NEC control system

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\mathit{x}\left(t\right)=A\mathit{x}\left(t\right)+B\mathit{u}\left(t\right)+\mathit{\xi }\left(t\right),$$

(49)

where the pair $(A,B)$ is controllable. The controller $\mathit{u}\in {\mathbb{R}}^p$ is proposed as the state feedback $\mathit{u}=-K\mathit{x}$, which drives $\mathit{x}$ to a vicinity of the origin as accurately as possible, despite the presence of $\mathit{\xi }$. In these conditions, the QMI becomes

$$PA+A^{T}P-PBK-K^T{B}^{T}P+ϵP^2+\kappa P,⪯0.$$

(50)

Now, defining $X=P^{-1}$ and $Y=KX$, pre- and post-multiplying by X yields

$$AX+X{A}^T-BY-Y^T{B}^T+ϵI+\kappa X\prec 0.$$

(51)

Then, X and Y, and consequently, $P=X^{-1}$ and $K=Y{X}^{-1}$, can be resolved from

$$\left[\begin{array}{ccc}AX+X{A}^T-BY-Y^T{B}^T+ϵI+\kappa X& & 0\\ 0& & -X\end{array}\right]\prec 0.$$

(52)

The solution to the above LMI proves the stability conditions of the closed-loop system and provides adequate feedback gains for control implementation.

5. Simulations

5.1. Comparisons

With the purpose of highlighting the advantages and limitations of the proposed scheme in a clearer way, the following system is considered:

$$\dot{y}\left(t\right)=-\alpha \left(t\right)y\left(t\right),$$

(53)

where the feedback gain function $\alpha \left(t\right)$ is defined according to different schemes:

• Exponential: $\alpha \left(t\right)=1$.
• NEC1: $\alpha \left(t\right)={\mathcal{F}}_1\left(t\right)=exp\left(t\right)$.
• NEC2: $\alpha \left(t\right)={\mathcal{F}}_2\left(t\right)=exp(exp\left(t\right))exp\left(t\right)$.
• Exp(Polynomial): $\alpha \left(t\right)=1+t$.

The Super-Twisting (ST) dynamical system is also considered, as it is a popular and effective chatter-less scheme, with finite-time convergence:

$$\begin{array}{cc}\hfill \dot{y}\left(t\right)& =-1.5L^{0.5}{\left|y\left(t\right)\right|}^{1/2}+z\left(t\right),\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill \dot{z}\left(t\right)& =-1.1L\mathrm{sign}\left(y\right(t\left)\right),\hfill \end{array}$$

(55)

with gain $L=100$ to ensure a comparable transient behavior. It should be noted that increasing the value of parameter L produces a faster transient, at the expense of larger self-sustained oscillations during the steady state, something that negatively affects the accuracy in the invariant $y\left(t\right)=0$.

Figure 1 illustrates the comparison results for five different convergence regimes. It can be observed that the NEC1 and NEC2 produce the highest accuracy. Besides, it is worth mentioning that, from a theoretical point of view, the solution $y\left(t\right)$ in the ST algorithm converges to zero after a finite time, resulting in an almost vertical line. However, from a practical point of view, the ST algorithm relies on an infinite “linear” gain value at $y\left(t\right)=0$, leading to the so-called chattering phenomenon.

From a numerical point of view, linear schemes, although they do not produce a theoretical finite-time convergence, render more accurate results during the steady state period. In addition, the high-gain phenomenon can be avoided by saturating the argument $t_{\arg }$ in $\alpha \left(t_{\arg }\right)$.

5.2. Robust Differentiator Based on NEC Estimator

The problem consists of a quick estimate of the derivative of the signal $y\left(t\right)$, that is, $\dot{y}\left(t\right)$, which may be subject to measurement noise.

Consider, for this purpose, the NEC estimator.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{x}_1\left(t\right)& =2\omega z_1\left(t\right)+x_2\left(t\right),\hfill \\ \hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{x}_2\left(t\right)& ={\omega }^2{z}_1\left(t\right),\hfill \end{array}\end{array}$$

(56)

where $z_1\left(t\right)=y\left(t\right)-x_1\left(t\right)$ is the deviation from the estimator $x_1\left(t\right)$ to the time signal $y\left(t\right)$, and $\omega >0$ is the feedback gain. Furthermore, one can consider the auxiliary NEC error $z_2\left(t\right)={\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{y}_1\left(t\right)-x_2\left(t\right)$.

The NEC dynamics of the estimation errors $z_1\left(t\right)$ and $z_2\left(t\right)$ is

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{z}_1\left(t\right)& =-2\omega z_1\left(t\right)+z_2\left(t\right),\hfill \\ \hfill {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}{z}_2\left(t\right)& =-{\omega }^2{z}_1\left(t\right)+{\left({\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\right)}^{2}y\left(t\right).\hfill \end{array}\end{array}$$

(57)

Alternatively, considering $\mathit{z}\left(t\right)={[z_1\left(t\right),z_2\left(t\right)]}^T$, one has

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\mathit{z}\left(t\right)=A\mathit{z}\left(t\right)+B{\left({\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\right)}^{2}y\left(t\right).$$

(58)

for $A=\left[\begin{array}{ccc}-2\omega & & 1\\ -{\omega }^2& & 0\end{array}\right]$ and $B=\left[\begin{array}{c}0\\ 1\end{array}\right]$.

Therefore, whenever ${\left({\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\right)}^{2}y\left(t\right)$ is a uniformly bounded function, the estimation error $\mathit{z}\left(t\right)$ also remains ultimately uniformly bounded.

As long as ${\left({\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\right)}^{2}y\left(t\right)$ vanishes at $t\to \infty$, so does $\mathit{z}\left(t\right)$. This case occurs when $y\left(t\right)$ has bounded first- and second-order time-derivatives.

Notice that for $z_2\approx 0$, one has $x_2\left(t\right)\approx {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}y\left(t\right)$, which drives $\dot{y}\left(t\right)\approx {\mathcal{F}}_n\left(t\right)x_2\left(t\right)$, allowing us to estimate the derivative of $y\left(t\right)$.

The following simulations consider the signal,

$$y=sin\left(2t\right)+cos\left(2t\right).$$

(59)

The NECDs are based on the NEF with a sequence of the terms $a_0=2$, $a_1=1$, and $a_2=1$, and the observer design considers the feedback parameter $\omega =5$. Furthermore, to ensure numerical stability, the observer’s time was limited to $t_s=1/a_0$. The simulator runs at 100 kHz based on the Euler method.

Figure 2 presents the results for the noise-free case, where both the estimated signal and its derivative converge to their ideal values, and the estimation error converges to zero. The proposed design produces an Integral Squared Error (ISE) ${\int }_0^{t_{sim}}{z}_1^2\left(t\right)dt=8.875\times {10}^{-3}$, for $t_{sim}=5$ s the simulation time.

The super-twist (ST) sliding-mode differentiator below is considered for comparison.

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{x}}_1\left(t\right)& =1.5L^{1/2}{\left|z_1\left(t\right)\right|}^{1/2}\mathrm{sign}\left(z_1\left(t\right)\right)+x_2\left(t\right),\hfill \\ \hfill {\dot{x}}_2\left(t\right)& =1.1L\mathrm{sign}\left(z_1\left(t\right)\right).\hfill \end{array}\end{array}$$

(60)

However, this is a nonlinear scheme that provides theoretically infinite gain around equilibrium, expecting better steady-state regimes. Therefore, the ST algorithm gains are adjusted to match the ISE performance index, producing $L=730.3$. Figure 3 shows the performance of the ST differentiator, which renders the same ISE performance of the proposed design, at the expense of a high gain L.

The noisy case is also considered for uniformly distributed random noise of amplitude $0.001$. The same gains are used for both schemes. Figure 4 shows the results obtained using the NEC differentiator, and Figure 5 shows the performance of the ST scheme. Both methods perform similarly during the steady state, as both rely on high-gain approximations, that is, estimating both the signal $y\left(t\right)$ and its derivative $\dot{y}\left(t\right)$ with a high-degree of accuracy, even under noise conditions. The ISE index slightly increases, less than 1%, in both cases. However, the high peaks and aggressive oscillations during the transient period are detrimental of the ST scheme, as it also happens in the noise-free case.

5.3. Nested Exponential Stabilization of a Pendulum System

The control case offers some motivation for the proposed tools, as it enforces robust stability under the presence of bounded disturbances. Consider for instance the simplified dynamics of an uncertain pendulum system,

$$\ddot{y}\left(t\right)+sin\left(y\left(t\right)\right)=u\left(t\right)+\phi \left(t\right),$$

(61)

for $\phi \left(t\right)=sin\left(2t\right)+3sin\left(4t\right)$ an unknown but bounded disturbance.

The control objective is to enforce the GNES dynamics.

$${\left({\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\right)}^{2}y\left(t\right)+2\omega {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}y\left(t\right)+{\omega }^{2}y\left(t\right)=0,$$

(62)

for $\omega >0$ a control parameter.

The controller $u\left(t\right)$ is then designed to enforce (62). Thus, considering,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\left({\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\right)}^{2}y\left(t\right)& ={\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\left[{\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}\dot{y}\left(t\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}{\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}\dot{y}\left(t\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{\ddot{y}\left(t\right)}{{\mathcal{F}}_n^2\left(t\right)}}+{\displaystyle \frac{\dot{y}\left(t\right)}{{\mathcal{F}}_n\left(t\right)}}{\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}\right],\hfill \end{array}\end{array}$$

(63)

the controller $u\left(t\right)$ can be resolved from

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\displaystyle \frac{1}{{\mathcal{F}}_n^2\left(t\right)}}[u\left(t\right)-sin\left(y\left(t\right)\right)]+{\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}{\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{1}{{\mathcal{F}}_n\left(t\right)}}\right]\dot{y}\left(t\right)\hfill \\ \hfill & =-2\omega {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}y\left(t\right)-{\omega }^{2}y\left(t\right),\hfill \end{array}\end{array}$$

(64)

where the term $\phi \left(t\right)$ is not included because it is assumed to be unknown.

The controller results as a Proportional-Integral controller with time-varying gains and feedforward compensation, that is,

$$u\left(t\right)=sin\left(y\left(t\right)\right)-\left[2\omega {\mathcal{F}}_n\left(t\right)-{\displaystyle \frac{{\mathcal{F}}_n^{\prime }\left(t\right)}{{\mathcal{F}}_n\left(t\right)}}\right]\dot{y}\left(t\right)-{\omega }^2{\mathcal{F}}_n^2\left(t\right)y\left(t\right).$$

(65)

The substitution of the proposed controller in the system dynamics leads to the closed-loop equation

$${\left({\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\right)}^{2}y\left(t\right)+2\omega {\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}y\left(t\right)+{\omega }^{2}y\left(t\right)={\displaystyle \frac{\phi \left(t\right)}{{\mathcal{F}}_n^2\left(t\right)}},$$

(66)

where it can be seen that the term ${\displaystyle \frac{\phi \left(t\right)}{{\mathcal{F}}_n^2\left(t\right)}}$ vanishes hyperexponentially fast as $t\to \infty$, producing $y\left(t\right)\to 0$ as $t\to \infty$. This can be realized by performing the change of variables

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x_1& =y,\hfill \\ \hfill x_2& ={\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}y,\hfill \end{array}\end{array}$$

(67)

leading to

$${\mathcal{D}}^{{\left\{a_i\right\}}_{i=0}^n}\mathit{x}\left(t\right)=A\mathit{x}\left(t\right)+\mathit{\xi }\left(t\right)$$

(68)

with $\mathit{x}={[x_1,x_2]}^T$, $A=\left[\begin{array}{cc}0& 1\\ -{\omega }^2& -2\omega \end{array}\right]$, and $\mathit{\xi }\left(t\right)={\left[0,{\displaystyle \frac{\phi \left(t\right)}{{\mathcal{F}}_n^2\left(t\right)}}\right]}^T$.

Using Theorem 6, one can find $P\succ 0$ and $ϵ,\kappa >0$, such that $PA+A^{T}P+ϵP^2+\kappa P⪯0$, meaning that $\mathit{x}\left(t\right)$ converges inside a ball centered at the origin, whose radius is proportional to the upper limit of $\left|\right|\mathit{\xi }\left(t\right)\left|\right|$, which in this case is zero. Therefore, $\mathit{x}\left(t\right)\to \mathbf{0}$ as $t\to \mathbf{0}$.

The simulation conditions were set as $a_0=a_1=a_2=\omega =1$, and the initial conditions $y\left(0\right)=\pi$. The noticeable problem with controller (65) is that the gains become very large after some time. However, the time value is saturated as $t_s=1/a_0$ in the computation of the NECD to alleviate this effect.

Figure 6 presents simulation results. The output evolves close to zero despite disturbance, and the control signal remains within acceptable limits.

For the sake of comparisons, a conventional proportional-integral (PI) controller with feedforward compensation is also considered; that is,

$$u\left(t\right)=sin\left(y\left(t\right)\right)-2\omega \dot{y}\left(t\right)-{\omega }^{2}y\left(t\right).$$

(69)

The feedback gains for the PI controller were chosen in accordance with $\omega =4.36$, which assure that both control signals, the PI and the NEC, have the same energy through the Integral Squared Control $ISC={\int }_0^{t_{\mathrm{sim}}}{u}^2\left(t\right)dt=249.2$.

Figure 7 shows the results of the numerical implementation of a conventional Proportional-Integral controller with feedforward compensation. The output signal exhibits a behavior similar to that of the proposed NEC controller; however, the control signal exhibits high-amplitude transient dynamics, with also a large overshoot, which could be detrimental to the process.

It can be argued that NEC controllers perform better since the gains are designed and programmed in the sense that these produce an output signal that converges with guaranteed hyperexponential stability. The quantitative results below show that in fact the NEC controller outperforms the conventional PI controller:

• NEC Controller: $ITAE=0.3728$, $ISE=2.994$, $ITAC=106.9$, $ISC=249.2$.
• PI Controller: $ITAE=3.349$, $ISE=2.99$, $ITAC=114.5$, $ISC=249.7$.

6. Conclusions

This paper introduced a new class of conformable operators, leading to dynamic systems with advanced characteristics, such as fast hyperexponential stabilization. The operators under consideration can be used to study and delineate a broader category of robust control systems, enabling robust and fast stabilization, even under the presence of matched bounded disturbances. Furthermore, it has been shown that these methodologies can be extended to develop robust observation techniques, which have potential applications in fault detection and diagnostics. Future studies will consider the use of similar techniques in the control and observation of complex engineering processes. Some problems remain open, the most notable being large gains during steady state, which prevents exact numerical calculations. Another problem is the need for a synchronous clock signal, if identical derivatives are to be computed on different systems, which can be interesting for fast synchronization protocols in secure communication.