Caputo Barrier Functions and Their Applications to the Safety, Safety-and-Stability, and Input-to-State Safety of a Class of Fractional-Order Systems

Abstract

Safety control based on barrier functions has gradually become one of the emerging and more important directions in the field of safety. Scholars are attempting to apply barrier functions to integer-order dynamical systems, such as general nonlinear systems, hybrid systems, linear systems, etc. Moreover, the introduction of barrier functions has even expanded the research approaches on safe reinforcement learning. However, there is very little research on the safety control problem of fractional-order dynamical systems. Based on our previous work, this article further explores, in depth, the problem of the transfer and adaptability of barrier functions for integer-order systems in fractional-order systems, and it also proposes the Caputo reciprocal barrier function and Caputo zeroing barrier function. And we established two theorems, which proved that we can also achieve uniform asymptotic stability or exponential stability with guaranteed safety. In the end, we created a new description for the definition of input-to-state safety under Caputo’s fractional-order systems, and we used this description and the above two Caputo barrier functions to construct two criteria of the Caputo input-to-state safety. Thus, we, finally, established the embryonic form of the theoretical framework of safety control based on barrier functions for fractional-order systems.

1. Introduction

1.1. Motivation

Safety control is one of the fundamental problems for many kinds of dynamical systems, especially large-scale and complex systems. Recently, in the last two decades, many scholars have verified that (control) barrier functions are effective for realizing safety control.

The research on safety control based on barrier functions can be roughly divided into three categories. The first category is the research on pure barrier functions for safety criteria. A number of prevalent barrier functions have emerged, which were contributed to by Prajna [1], Kong [2], Zhu [3], Dai [4], Ames [5,6], and Xu [7]. In addition, there are many other interesting works about barrier functions, such as vector barrier functions [8], extreme-point barrier functions [9,10], and barrier functions with converse theorems [11,12]. The second category is the research on safety-derived theories based on barrier functions, which mainly consider issues such as the synchronization of safety and stability [13] and the safety robustness and input-to-state safety (ISSf) issues [7,14,15]. The third category is the research on the application of safety control based on barrier functions [16,17,18,19,20,21,22].

The abovementioned barrier functions have realized the construction and application of safety control theories for general nonlinear systems, hybrid systems, stochastic systems, and multi-agent systems. Their remarkable common feature is that all these studies are carried out around integer-order systems. Therefore, the theory of fractional-order safety control is still a virgin land and has great research potential.

Fractional-order systems (FOSs) represent a unique class of systems characterized by their distinct integration and differentiation laws. Unlike traditional integer-order systems (IOSs), FOSs are defined by fractional-order differential equations, which have been demonstrated to provide more accurate descriptions of complex systems compared to their integer-order counterparts [23,24]. A notable feature of FOSs is their inherent infinite memory and hereditary properties, which fundamentally differentiate them from classical IOSs. As a result, the well-established theories of integer-order systems may not be directly applicable to the analysis and design of fractional-order systems. Factually, the modeling and control of FOSs has emerged as a significant research direction in the field of control theory. The stability theories of FOSs have been well established, as demonstrated in several studies [25,26,27]. Additionally, there has been growing interest in fault estimation and accommodation for FOSs [23]. However, the fundamental theories and practical applications of safety control for FOSs remain largely unexplored. Due to the special differential rules of fractional-order systems, such as memory effects, barrier functions are applicable under integer-order differentiation, but they may not be fully applicable in the case of fractional orders. In this regard, by comparing them with our previous work, it can be found that there are differences in details between the two theorems in [3,28]. With the abovementioned challenges, this article aims to extend and adapt the state safety theories based on barrier functions, which have been widely adopted for integer-order systems (IOSs), to the realm of FOSs. Given the increasing demand for robust safety mechanisms in complex dynamic systems, this endeavor is both timely and necessary.

In the past, our prior work [28,29] involved an initial attempt to demonstrate that certain special barrier functions can be employed to ensure the safety of Caputo’s fractional-order systems (FOSs). In these two articles, we proposed Caputo less-zero barrier function and Caputo exponential barrier function, as well as a theorem of asymptotic stability with guaranteed safety. However, these two articles have not yet established a comprehensive fundamental theoretical framework for fractional-order barrier functions. This article devotes more efforts to exploring the issues of uniform asymptotic stability with guaranteed safety, exponential stability with guaranteed safety, and the ISSf, so as to effectively supplement the theoretical framework of safety control for fractional-order systems.

1.2. Contributions

Our main contributions in this article are as follows.

(1)

We demonstrate the possibility of transferring the reciprocal barrier function (RBF) and the zeroing barrier function (ZBF) to Caputo fractional-order systems (CFOS), and we also propose Caputo RBF and Caputo ZBF. Based on two innovative Caputo BFs, we systematically derive the state safety criteria for fractional-order nonlinear dynamic systems. Our established state-safety theorems provide rigorous guarantees that all system states will remain within an known available state set, given that the initial conditions adhere to the set constraint.

(2)

On the basis of the theorem of asymptotical stability with guaranteed safety for CFOSs [29], we further propose the theorems of uniformly asymptotical stability with guaranteed safety and exponential stability with guaranteed safety for CFOSs. These two theorems demonstrate the possibility and solvability of achieving the synchronization of safety and uniformly asymptotical stability (or exponential stability).

(3)

We constructed a new description for the definition of Caupto input-to-state safety. The inspiration for this new description comes from the final product of the Caputo reciprocal barrier function proof process. The core inequality of our proposed definition of the Caputo input-to-state safety not only provides a unified representation of safety and ISSf for CFOSs, but it also facilitates the design and derivation of ISSf controllers. Then, under the definition of Caputo ISSf, using Caputo reciprocal barrier function and Caputo zeroing barrier function, we establish two ISSf criteria that can be directly applied to design Caputo ISSf controllers.

1.3. Organizations

This article is structured as follows. It consists of five sections that systematically unfold the research on fractional-order system safety control. Section 1 serves as the introduction, where the motivation and contributions of this study are presented. In Section 2, we introduce a class of Caputo fractional-order systems (CFOSs) and construct two Caputo barrier functions. These functions lay a solid theoretical foundation for subsequent analyses. Section 3 is devoted to a thorough exploration of the synchronous Caputo safety and stability theorems. Through meticulous mathematical derivations and in-depth discussions, we elucidate the compatibility between safety and stability within the framework of CFOSs. In Section 4, we shift our focus to the input-to-state safety and the corresponding ISSf barrier functions of CFOSs. This section offers the relationship between external inputs and the safety performance of CFOSs. In the end, Section 5 presents the conclusions, summarizing the key findings, contributions, and potential directions for future research.

2. Caputo Barrier Functions

This section focuses on extending safety theories for CFOSs with order $\alpha \in \left(0,1\right)$ using barrier functions. The objective is to transform the RBF and the ZBF into the Caputo RBF and the Caputo ZBF so as to be suitable for the CFOS. Specifically, we analyze a CFOS governed by Caputo fractional derivatives [26], which is defined as

$${}^{C}D^{\alpha }x\left(t\right)=f\left(x\left(t\right)\right),$$

(1)

where $\alpha \in \left(0,1\right)$, $x\in {\mathbb{R}}^n$, $t\ge t_0\in \left[0,+\infty \right)$, and $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$ are locally Lipschitz. And there is a safe zone $\mathcal{C}$ for CFOS (1), which can be defined by the following (2)–(4):

$$\begin{array}{c}\hfill \mathcal{C}=\left\{x\in {\mathbb{R}}^n:h\left(x\right)\ge 0\right\},\end{array}$$

(2)

$$\begin{array}{c}\hfill \partial \mathcal{C}=\left\{x\in {\mathbb{R}}^n:h\left(x\right)=0\right\},\end{array}$$

(3)

$$\begin{array}{c}\hfill Int\left(\mathcal{C}\right)=\left\{x\in {\mathbb{R}}^n:h\left(x\right)>0\right\},\end{array}$$

(4)

with a smooth function $h:{\mathbb{R}}^n\to \mathbb{R}$. And other states have $\forall x\in {\mathbb{R}}^n\setminus \mathcal{C},h\left(x\right)<0$. In addition, we can call CFOS (1) safe, if all the states beginning from $x\left(t_0\right)=x_0$ are in the set $\mathcal{C}$.

2.1. Caputo RBF

For IOSs, RBFs have proven to be highly effective in designing state-safety controllers, offering less restrictive constraints compared to the first generation barrier functions [1] and exponential barrier functions [2,3]. This naturally raises the question: can a RBF be developed to ensure the safety of CFOSs?

Definition 1.

For CFOS (1) having a set $\mathcal{C}$ defined by (2)–(4) with some continuously differentiable function $h:{\mathbb{R}}^n\to \mathbb{R}$, we can consider a function $B:\mathcal{C}\to \mathbb{R}$ as a Caputo reciprocal barrier function for the set $\mathcal{C}$ if there exist locally Lipschitz class $\mathcal{K}$ functions. (A continuous function $\alpha :\left[0,a\right)\to \left[0,\infty \right)$ is said to belong to class $\mathcal{K}$ if it is strictly increasing and $\alpha \left(0\right)=0$ [30]) ${\beta }_1,{\beta }_2,{\beta }_3$, for all $x\in Int\left(\mathcal{C}\right)$, has

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{{\beta }_1\left(h\left(x\right)\right)}}\le B\left(x\right)\le {\displaystyle \frac{1}{{\beta }_2\left(h\left(x\right)\right)}}& ,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {}^{C}D^{\alpha }B\left(x\left(t\right)\right)\le {\beta }_3\left(h\left(x\right)\right)& .\hfill \end{array}$$

(6)

Theorem 1.

Given a set $\mathcal{C}\subseteq {\mathbb{R}}^n$ for CFOS (1) with Conditions (2)–(4), if there exists a Caputo reciprocal barrier function $B:\mathcal{C}\to \mathbb{R}$, it can be defined by Definition 1. In addition, we can guarantee the set $\mathcal{C}$ is forward-invariant and that CFOS (1) can be said to be safe for any $x_0\in Int\left(\mathcal{C}\right)$.

Proof of Theorem 1.

Let ${}^{C}D^{\alpha }y={\beta }_3\left({\beta }_2^{-1}\left({\displaystyle \frac{1}{y}}\right)\right)\stackrel{def}{=}\beta \left({\displaystyle \frac{1}{y}}\right)$; hence, $\beta$ belongs to Class $\mathcal{K}$. By the comparison theorem for fractional-order System [26], we have $infy\left(t\right)\ge B\left(x\left(t\right)\right)$. According to Caputo’s fractional derivative [26], there is

$${}^{C}D^{\alpha }y={\displaystyle \frac{1}{\Gamma \left(1-\alpha \right)}}{\int }_{t_0}^t{\displaystyle \frac{y^{\prime }\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha }}}d\tau .$$

(7)

Then, let $z={\displaystyle \frac{1}{y}}$, which means

$${}^{C}D^{\alpha }z={\displaystyle \frac{1}{\Gamma \left(1-\alpha \right)}}{\int }_{t_0}^t{\displaystyle \frac{-\frac{y^{\prime }\left(\tau \right)}{y^2}}{{\left(t-\tau \right)}^{\alpha }}}d\tau .$$

(8)

By the Generalized First Mean Value Theorem for Integrals, there exists some point $\xi \in \left[t_0,t\right]$ satisfying

$${\int }_{t_0}^t{\displaystyle \frac{-\frac{y^{\prime }\left(\tau \right)}{y^2}}{{\left(t-\tau \right)}^{\alpha }}}d\tau ={\displaystyle \frac{-1}{y^2\left(\xi \right)}}{\int }_{t_0}^t{\displaystyle \frac{y^{\prime }\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha }}}d\tau .$$

Hence, (8) has

$$\begin{array}{cc}\hfill {}^{C}D^{\alpha }z& ={\displaystyle \frac{1}{\Gamma \left(1-\alpha \right)}}{\displaystyle \frac{-1}{y^2\left(\xi \right)}}{\int }_{t_0}^t{\displaystyle \frac{y^{\prime }\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha }}}d\tau \hfill \\ \hfill & =-{\displaystyle \frac{1}{y^2\left(\xi \right)}}{}^{C}D^{\alpha }y=-{\displaystyle \frac{1}{y^2\left(\xi \right)}}\beta \left({\displaystyle \frac{1}{y}}\right)\hfill \\ \hfill & =-z^2\left(\xi \right)\beta \left(z\right)\stackrel{def}{=}-\overline{\beta }\left(z\right).\hfill \end{array}$$

(9)

It is not difficult to find that $\overline{\beta }$ is also a class $\mathcal{K}$ function. By the equivalent Volterra Integral [26] and Bihari’s Inequality [31], we can obtain

$$z\left(t\right)\le {\eta }^{-1}\left[\eta \left(z\left(t_0\right)\right)+{\displaystyle \frac{{\left(t-t_0\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}}\right].$$

Following the proof methodology detailed in Theorem 3.1 of reference [26], we have

$$z\left(t\right)\le \sigma \left(z\left(t_0\right),t-t_0\right),$$

(10)

where $\sigma$ belongs to class $\mathcal{KL}$ function (A continuous function $\beta :\left[0,a\right)\times \left[0,\infty \right)\to \left[0,\infty \right)$ is said to belong to class $\mathcal{KL}$ if, when s is fixed, $\beta \left(r,s\right)$ belongs to class $\mathcal{K}$ with respect to r, and—when r is fixed—$\beta \left(r,s\right)$ is decreasing with respect to s and $\beta \left(r,s\right)\to 0$ as $s\to \infty$ [30]). As $y={\displaystyle \frac{1}{z}}$ and $B\left(x\left(t\right)\right)\le infy\left(t\right)$, then, by comparison theorem for fractional-order System [26], we can obtain

$${\displaystyle \frac{1}{\sigma \left(\frac{1}{B\left(x\left(t_0\right)\right)},t-t_0\right)}}\ge B\left(x\left(t\right)\right).$$

With (5), then, we have

$$h\left(x\left(t\right)\right)\ge {\beta }_1^{-1}\left(\sigma \left({\displaystyle \frac{1}{B\left(x\left(t_0\right)\right)}},t-t_0\right)\right)$$

(11)

for all $t\in I\left(x\left(t_0\right)\right)$. In Equation (11), ${\beta }_1^{-1}$ is the inverse of ${\beta }_1$ and belongs to class $\mathcal{K}$. As $x\left(t_0\right)\in \mathcal{C}$, then we have $B\left(x\left(t_0\right)\right)>0$. This means that $h\left(x\left(t\right)\right)\ge 0$ for all $t\in I\left(x\left(t_0\right)\right)$. Thus, $\mathcal{C}$ is forward-invariant and CFOS (1) is then safe. □

Remark 1.

For integer-order systems with $n=1$ ([5], Theorem 1), the dynamics are governed by $\dot{z}=-{\displaystyle \frac{\dot{y}}{y^2}}=-\beta \left(z\right)z^2\stackrel{def}{=}\overline{\beta }\left(z\right)$, where β is a class $\mathcal{K}$ function. However, for fractional-order systems with $\alpha \in \left(0,1\right)$, the analysis becomes significantly more complex. In this case, the Generalized First Mean Value Theorem for Integrals must be employed to establish the relationship ${}^{C}D^{\alpha }z=-z^2\left(\xi \right)\beta \left(z\right)\stackrel{def}{=}\overline{\beta }\left(z\right)$. A key distinction between the two cases lies in the nature of the nonlinear term. For the integer-order case, $z^2$ is a class $\mathcal{K}$ function. In contrast, for the fractional-order case, $z^2\left(\xi \right)$ represents a real positive value, which introduces additional challenges in the analysis. Consequently, the verification of safety theorems under Caputo’s fractional-order framework is more intricate compared to their integer-order counterparts.

2.2. Caputo ZBF

The ZBF is widely utilized in the analysis and control of input-to-state safety, serving as a critical quantitative framework for robust safety analysis. Consequently, transitioning from integer-order to fractional-order systems, it is imperative to verify whether a ZBF can be adapted into a Caputo ZBF.

Definition 2.

For CFOS (1) having a set $\mathcal{C}$ defined by (2)–(4) with some smooth function $h:{\mathbb{R}}^n\to \mathbb{R}$, we can consider h as a Caputo zeroing barrier function for the set $\mathcal{C}$, if there exists a locally Lipschitz class $\mathcal{K}$ function β for all $x\in Int\left(\mathcal{C}\right)$, we have

$${}^{C}D^{\alpha }h\left(x\left(t\right)\right)\ge -\beta \left(h\left(x\right)\right).$$

(12)

Theorem 2.

Given a set $\mathcal{C}\subseteq {\mathbb{R}}^n$ for CFOS (1) with Conditions (2)–(4), there exists a Caputo zeroing barrier function $h:\mathcal{C}\to \mathbb{R}$ defined by Definition 2. If so, we can guarantee the set $\mathcal{C}$ is forward-invariant and that CFOS (1) is safe for any $x_0\in Int\left(\mathcal{C}\right)$.

Proof of Theorem 2.

Let $B\left(x\right)={\displaystyle \frac{1}{h\left(x\right)}}$. Then, substitute it into (9); hence, $\exists \xi \in \left[t_0,t\right]$, such that

$${}^{C}D^{\alpha }{\displaystyle \frac{1}{h}}={\displaystyle \frac{1}{\Gamma \left(1-\alpha \right)}}{\int }_{t_0}^t{\displaystyle \frac{-\frac{h^{\prime }\left(\tau \right)}{h^2}}{{\left(t-\tau \right)}^{\alpha }}}d\tau ={\displaystyle \frac{-1}{h^2\left(\xi \right)}}{\int }_{t_0}^t{\displaystyle \frac{h^{\prime }\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha }}}d\tau =-h^2\left(\xi \right){}^{C}D^{\alpha }h.$$

Thus, with (12) and the proof of Theorem 1, the set $\mathcal{C}$ is forward-invariant. And then CFOS (1) is safe. □

In fact, the ZBF is employed to address the robustness of safety. Based on Theorem 2, we need to redefine the safe zone $\mathcal{C}$ by [7]. For any $\forall \delta \ge 0$, there exists a nonempty closed set ${\mathcal{C}}_{\delta }$, which is defined as

$$\begin{array}{c}\hfill {\mathcal{C}}_{\delta }=\left\{x\in {\mathbb{R}}^n:h\left(x\right)\ge -\delta \right\},\end{array}$$

(13)

$$\begin{array}{c}\hfill \partial {\mathcal{C}}_{\delta }=\left\{x\in {\mathbb{R}}^n:h\left(x\right)=-\delta \right\},\end{array}$$

(14)

$$\begin{array}{c}\hfill Int\left({\mathcal{C}}_{\delta }\right)=\left\{x\in {\mathbb{R}}^n:h\left(x\right)>-\delta \right\},\end{array}$$

(15)

where $h:{\mathbb{R}}^n\to \mathbb{R}$ is continuously differentiable.

Definition 3.

For CFOS (1) having a set ${\mathcal{C}}_{\delta }$ defined by (13)–(15) with some smooth $h:{\mathbb{R}}^n\to \mathbb{R}$, we can consider h as an extended Caputo zeroing barrier function for the set ${\mathcal{C}}_{\delta }$ if there exists a locally Lipschitz extended class $\mathcal{K}$ function (A continuous function $\beta :\left(-b,a\right)\to \left(-\infty ,\infty \right)$ for some $a,b>0$ is said to belong to extended class $\mathcal{K}$ if it is strictly increasing and $\beta \left(0\right)=0$ [7]) γ. Then, for all $x\in Int\left({\mathcal{C}}_{\delta }\right)$, we have

$${}^{C}D^{\alpha }h\left(x\left(t\right)\right)\ge -\gamma \left(h\left(x\right)\right).$$

(16)

Theorem 3.

Given a set ${\mathcal{C}}_{\delta }\subseteq {\mathbb{R}}^n$ for CFOS (1) with Conditions (13)–(15), if there exists an extended Caputo zeroing barrier function $h:{\mathcal{C}}_{\delta }\to \mathbb{R}$, which is defined by Definition 3, we can guarantee the set ${\mathcal{C}}_{\delta }$ is forward-invariant and that CFOS (1) can be said to be safe for any $x_0\in Int\left({\mathcal{C}}_{\delta }\right)$.

Proof of Theorem 3.

Let $l\left(x\right)=h\left(x\right)+\delta$. Then, the set ${\mathcal{C}}_{\delta }$ can transform into the set $\mathcal{C}$. By (16) and leveraging the properties of the extended class $\mathcal{K}$ function, we can obtain

$${}^{C}D^{\alpha }l\left(x\left(t\right)\right)={}^{C}D^{\alpha }h\left(x\left(t\right)\right)\ge -\gamma \left(h\left(x\right)\right)=-\gamma \left(l\left(x\right)-\delta \right)\ge -\gamma \left(l\left(x\right)\right).$$

By Theorem 2, we can easily ensure the safe zone ${\mathcal{C}}_{\delta }$ with the function is forward-invariant. Hence, CFOS (1) is safe. □

2.3. Comparison

In this study, we introduce two novel concepts of Caputo barrier functions: the Caputo ZBF (CZBF) and the Caputo RBF (CRBF). While the CZBF offers advantages in system safety analysis and controller design due to its computational tractability in practical scenarios, CRBF encounters limitations in real-world applications. Specifically, the derivative calculations required for CRBF implementation demand advanced fractional-order differentiation techniques, which significantly reduce their practical precedence compared to CZBF and other barrier function methodologies [28,29]. This observation highlights that, under fractional-order dynamics, the inherent computational complexity of reciprocal barrier functions is amplified, thereby compromising their operational efficiency relative to alternative safety control approaches.

3. Caputo Stability with Guaranteed Caputo Safety

For dynamic systems across engineering disciplines, safety control plays a fundamental role in ensuring operational integrity. Notwithstanding its importance, safety alone cannot guarantee other desirable properties such as stability, which is particularly critical for industrial systems where stability directly influences product quality consistency. Consequently, this study aims to develop control methodologies that simultaneously maintain both safety and stability for fractional-order systems. To this end, we first established Assumption 1, which provides the foundational conditions for subsequent analysis.

Assumption 1.

Assume the safe-state set $\mathcal{C}$ is a compact and closed (The word “closed" means that the boundary of the set is a closed curve, surface, etc., in a geometric sense) set, and the point $x=0\in \mathcal{C}$ is farthest from $\partial \mathcal{C}$.

In stability theory, the equilibrium point at the origin often serves as a canonical form through the translation of nonzero equilibria or system solutions [30]. In this context, we consider $x_e=0$ as the point farthest from $\partial \mathcal{C}$, implying $h\left(0\right)={max}_{x\in \mathcal{C}}h\left(x\right)$. For some systems with $x_e\ne 0$, define $p=x-x_e$. The transformed function $h\left(x\right)=h\left(p+x_e\right)\triangleq q\left(p\right)$ then satisfies $q\left(0\right)={max}_{p\in {\mathcal{C}}_p}q\left(p\right)$, where $\mathcal{C}=\left\{p\right|p+x_e\in \mathcal{C}\}$. This transformation allows the origin $x=0$ to act simultaneously as both the equilibrium point and the farthest point relative to the set boundary.

3.1. Uniformly Asymptotic Stability with Guaranteed Caputo Safety

Theorem 4.

For CFOS (1) with a set $\mathcal{C}$ satisfying Assumption 1, and which is well defined by (2)–(4) for some some function $h:{\mathbb{R}}^n\to \mathbb{R}$, then, for all $x\in Int\left(\mathcal{C}\right)\setminus \left\{0\right\}$, there exists a Caputo barrier function $B:\mathcal{C}\to \mathbb{R}$ satisfying

$$\begin{array}{cc}\hfill & -{\beta }_1\left(h\left(x\right)\right)\le B\left(x\left(t\right)\right)\le -{\beta }_2\left(h\left(x\right)\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & {}^{C}D^{\alpha }B\left(x\left(t\right)\right)\le \lambda {\beta }_3\left(h\left(x\right)\right),\hfill \end{array}$$

(18)

with ${\beta }_1,{\beta }_2,{\beta }_3$ being locally Lipschitz class $\mathcal{K}$ functions and $\lambda <0$. Meanwhile, $x=0$ also satisfies (17) but does not satisfy (18), then the set $\mathcal{C}$ is forward-invariant. Hence, CFOS (1) is safe and the origin of (1) is uniformly asymptotically stable with $x\left(t_0\right)=x_0\in Int\left(\mathcal{C}\right)\setminus \left\{0\right\}$.

Proof of Theorem 4.

The proof can be divided by two parts: one to prove the safety and the other to prove stability.

(1) For safety. From (18), set

$${}^{C}D^{\alpha }y={\beta }_3\left({\beta }_2^{-1}\left(-y\right)\right)\stackrel{def}{=}\beta \left(-y\right).$$

Hence, by the comparison theorem for fractional-order System [26], we have

$$infy\left(t\right)\ge B\left(x\left(t\right)\right).$$

Let $z=-y$, according to Caputo’s fractional derivative [26], then we can easily obtain

$${}^{C}D^{\alpha }z={}^{C}D^{\alpha }-y=-\beta \left(z\right).$$

This is very similar to Equation (9) in the proof of Theorem 1. Hence, we can also obtain Inequality (10), which is

$$z\left(t\right)\le \sigma \left(z\left(t_0\right),t-t_0\right)$$

with $\sigma$ a class $\mathcal{KL}$ function. As $y=-z$, we have

$$infy=-\sigma \left(z\left(t_0\right),t-t_0\right).$$

Due to $B\left(x\left(t\right)\right)\le infy\left(t\right)$, by the comparison theorem for fractional-order System [26], we can obtain

$$B\left(x\left(t\right)\right)\le -\sigma \left(-B\left(x\left(t_0\right)\right),t-t_0\right).$$

By (17), then, we have

$$h\left(x\left(t\right)\right)\ge {\beta }_1^{-1}\left(\sigma \left(-B\left(x\left(t_0\right)\right),t-t_0\right)\right)$$

for all $t\in I\left(x\left(t_0\right)\right)$. Moreover, ${\beta }_1^{-1}$ is the inverse of ${\beta }_1$ and belongs to class $\mathcal{K}$. As $x\left(t_0\right)\in \mathcal{C}$, we have $B\left(x\left(t_0\right)\right)<0$. This means that $h\left(x\left(t\right)\right)\ge 0$ for all $t\in I\left(x\left(t_0\right)\right)$. Thus, $\mathcal{C}$ is forward-invariant and then CFOS (1) is safe.

(2) For stability. By Assumption 1 and (17), $B\left(x\right)\ge B\left(0\right)$, there hence exists a real constant $b<0$, such that $b=essinfB\left(x\right)=B\left(0\right)$. According to [26], the Lapunov function V needs to be nonnegative; hence, we can set $V\left(x\right)=B\left(x\right)-b$. Thus, we have

$${}^{C}D^{\alpha }V\left(x\left(t\right)\right)={}^{C}D^{\alpha }B\left(x\left(t\right)\right).$$

By (17) and (18), we have

$$\begin{array}{cc}\hfill & -{\beta }_1\left(h\left(x\right)\right)-b\le V\le -{\beta }_2\left(h\left(x\right)\right)-b\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & {}^{C}D^{\alpha }V\left(x\left(t\right)\right)\le -\mu {\beta }_3\left(h\left(x\right)\right)\hfill \end{array}$$

(20)

with $\mu =-\lambda >0$. Let $W_1\left(x\right)=-{\beta }_1\left(h\left(x\right)\right)-b$, $W_2\left(x\right)=-{\beta }_2\left(h\left(x\right)\right)-b$ and $W_3\left(x\right)=\mu {\beta }_3\left(h\left(x\right)\right)$. We can then easily confirm that $W_1\left(x\right)$, $W_2\left(x\right)$, and $W_3\left(x\right)$ are continuous positive definite functions. Now, we can rewrite Inequalities (19) and (20) as follows:

$$\begin{array}{cc}\hfill & W_1\left(x\right)\le V\left(x\left(t\right)\right)\le W_2\left(x\right)\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & {}^{C}D^{\alpha }V\le -W_3\left(x\right).\hfill \end{array}$$

(22)

By Theorem 3.1 ([26]), we can confirm that $x=0$ is uniformly asymptotical stable.

Together with (1) and (2), Theorem 4 is established. □

3.2. Exponential Stability with Guaranteed Caputo Safety

Theorem 5.

For CFOS (1) with a set $\mathcal{C}$ satisfying Assumption 1 and being defined by (2)–(4) for some smooth function $h:{\mathbb{R}}^n\to \mathbb{R}$, for all $x\in Int\left(\mathcal{C}\right)\setminus \left\{0\right\}$ and if there exists a lower-bounded Caputo barrier function $B:\mathcal{C}\to \mathbb{R}$ satisfying

$$\begin{array}{cc}\hfill & -{\beta }_1\left(h\left(x\right)\right)\le B\left(x\left(t\right)\right)\le -{\beta }_2\left(h\left(x\right)\right),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & {}^{C}D^{\alpha }B\left(x\left(t\right)\right)\le \lambda \left(c-B\left(x\left(t\right)\right)\right),\hfill \end{array}$$

(24)

where ${\beta }_1,{\beta }_2,{\beta }_3$ are locally Lipschitz class $\mathcal{K}$ functions, c is a constant with $c=essinfB\left(x\right)$, and $\lambda >0$, then, the set $\mathcal{C}$ is forward-invariant. Hence, CFOS (1) is safe and the origin of (1) is exponentially stable with $x\left(t_0\right)=x_0\in Int\left(\mathcal{C}\right)\setminus \left\{0\right\}$.

Proof of the Theorem 5.

The proof also has two parts: first for safety, and then for stability.

(1) For safety. As $c=essinfB\left(x\right)$, then we have $c-B<0$. According to our previous work ([29], Theorem 1), it can be easily proved that the set $\mathcal{C}$ is forward-invariant and hence CFOS (1) is safe.

(2) For stability. Let $V\left(x\right)=B\left(x\right)-c$., then $\left|V\right|\le -c$ is implied. Hence, there exists positive constants $c_1,c_2$ such that V satisfies $c_1{\parallel x\parallel }^2\le V\le c_2{\parallel x\parallel }^2<-c$. Then, we have

$${}^{C}D^{\alpha }V\left(x\left(t\right)\right)={}^{C}D^{\alpha }B\left(x\left(t\right)\right).$$

Thus, we can obtain

$$\begin{array}{cc}\hfill {}^{C}D^{\alpha }V\left(x\left(t\right)\right)& ={}^{C}D^{\alpha }B\left(x\left(t\right)\right)\le \lambda \left(c-B\right)\hfill \\ \hfill & \le -\lambda V.\hfill \end{array}$$

(25)

Furthermore, (25) is more like Inequality (39) in the proof of Lemma 3.1 [26]. Hence, we can use the Gronwall–Bellman Inequality [32] and obtain the following inequality, which is similar with Inequality (42) in the proof of Lemma 3.1 [26]:

$$\begin{array}{cc}\hfill V\left(x\left(t\right)\right)& \le V\left(x_0\right)+{\displaystyle \frac{-\lambda }{\Gamma \left(\alpha \right)}}{\int }_{t_0}^t{\left(t-\tau \right)}^{\alpha -1}V\left(x\left(t\right)\right)d\tau \hfill \\ \hfill & \le V\left(x\left(t_0\right)\right)exp\left({\displaystyle \frac{-\lambda {\left(t-t_0\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}}\right).\hfill \end{array}$$

Thus, we have

$$\begin{array}{cc}\hfill \parallel x\parallel & \le {\left[{\displaystyle \frac{V\left(x\left(t\right)\right)}{c_1}}\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le {\left[{\displaystyle \frac{V\left(x\left(t_0\right)\right)}{c_1}}exp\left({\displaystyle \frac{-\lambda {\left(t-t_0\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}}\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le {\left[{\displaystyle \frac{c_2{\parallel x\left(t_0\right)\parallel }^2}{c_1}}exp\left({\displaystyle \frac{-\lambda {\left(t-t_0\right)}^{\alpha }}{\Gamma \left(\alpha +1\right)}}\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & ={\left({\displaystyle \frac{c_2}{c_1}}\right)}^{{\displaystyle \frac{1}{2}}}\parallel x\left(t_0\right)\parallel exp\left({\displaystyle \frac{-\lambda {\left(t-t_0\right)}^{\alpha }}{2\Gamma \left(\alpha +1\right)}}\right).\hfill \end{array}$$

(26)

Therefore, the origin $x=0$ is exponentially stable.

By all of the above, Theorem 5 is proved. □

Remark 2.

The abovementioned two kinds of synchronous Caputo safety and stability are all infinite-time stable. Since the proof of finite time stability under the fractional order is more difficult, we did not conduct in-depth research on synchronous Caputo safety and finite-time stability.

4. Caputo Input-to-State Safety

For integer-order systems, like the stability, the safety also needs to consider the robustness. And the concept of safety robustness was first discussed in [7]. Following this, M. Z. Romdlony [14] first proposed ISSf to describe the robustness of safety. He mainly established a notion of ISSf under the description of the distance from a point in the safe set to the unsafe set by less-zero barrier function. Meanwhile, Shishir Kolathaya [15] used the description function h of the set $\mathcal{C}$ to propose a different notion of ISSf via zeroing barrier function. In this section, we will list our latest theoretical research results using Caputo barrier function with respect to Caputo input-to-state safety, and we will then put forward some new notions of ISSf with different descriptions and different barrier functions.

Here, we consider a dynamic system with an additional disturbance:

$${}^{C}D^{\alpha }x=f\left(x\right)+g\left(x\right)d\left(t\right),$$

(27)

with disturbance $d\in {\mathbb{L}}_{\infty }^m$. If we involve forward-invariant sets, further interesting conclusions can be made. Here, there is a set $\mathcal{C}$ defined by (2)–(4) for System (1), which is System (27) without disturbance. In addition, there is a slightly larger set ${\mathcal{C}}_d\supseteq \mathcal{C}$ following the definition in [15] satisfying

$$\begin{array}{cc}\hfill {\mathcal{C}}_d& =\left\{x\in {\mathbb{R}}^n:h\left(x\right)+\delta \left({\parallel d\parallel }_{\infty }\right)\ge 0\right\},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \partial {\mathcal{C}}_d& =\left\{x\in {\mathbb{R}}^n:h\left(x\right)+\delta \left({\parallel d\parallel }_{\infty }\right)=0\right\},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill Int\left({\mathcal{C}}_d\right)& =\left\{x\in {\mathbb{R}}^n:h\left(x\right)+\delta \left({\parallel d\parallel }_{\infty }\right)>0\right\},\hfill \end{array}$$

(30)

with a class $\mathcal{K}$ function $\delta$ in $\left[0,a\right)$, ${\parallel d\parallel }_{\infty }\le \overline{d}\in \left[0,a\right)$. By [15], the set $\mathcal{C}$ is called an ISSf set if the set ${\mathcal{C}}_d$, which depends on d, is forward-invariant. Hence, on the basis of this aforementioned cognition, learning from Definition 4.7 (in [32]), we can obtain the following.

Definition 4.

System (27) with the set $\mathcal{C}$ defined by (2)–(4) and the ${\mathcal{C}}_d$ defined by (28)–(30) is said to be Caputo input-to-state safe if there exist a class $\mathcal{KL}$ function σ and a class $\mathcal{K}$ function δ such that, for any initial state $x\left(t_0\right)$ and any bounded disturbance $d\left(t\right)$, the solution $x\left(t\right)$ for all $t\ge t_0$ satisfies

$$h\left(x\left(t\right)\right)\ge \sigma \left(\eta \left(x\left(t_0\right),d\right),t-t_0\right)-\delta \left({∥d∥}_{\infty }\right),$$

(31)

with $\eta \left(x\left(t_0\right),d\right)=h\left(x\left(t_0\right)\right)+\delta \left({∥d∥}_{\infty }\right)$.

Remark 3.

In addition to the difference between fractional-order and integer-order systems, Definition 4 uses a new way to describe the input-to-state safety. This description is useful to the proofs of the following two theorems.

When $t\to \infty$, for Inequality (31) and function $\sigma \to 0$, then we have $h\left(x\right)\ge -\delta \left({\parallel d\parallel }_{\infty }\right)$, which produces $h\left(x\right)+\delta \left({\parallel d\parallel }_{\infty }\right)\ge 0$, meaning the solution $x\left(t\right)$ is still in set ${\mathcal{C}}_d$. Hence, Inequality (31) guarantees that, for any bounded disturbance, the set ${\mathcal{C}}_d$ will be forward-invariant all of the time. If $d\left(t\right)\equiv 0$, then (31) has

$$h\left(x\left(t\right)\right)\ge \sigma \left(h\left(x\left(t_0\right)\right),t-t_0\right).$$

This confirms that input-to-state safety implies the set $\mathcal{C}$ for the unforced autonomous System (1) without any disturbance is forward-invariant. It is a necessary condition or prerequisite for the establishment of (Caputo) ISSf.

The new question is how to use Definition 4 and how to judge the Caputo ISSf conveniently. We tried to use the Caputo reciprocal barrier function to explore some sufficient conditions for the Caputo ISSf.

Theorem 6.

System (27) is System (1) with the input of bounded disturbance $d\left(t\right)$. There are sets $\mathcal{C}$ defined by (2)–(4) and ${\mathcal{C}}_d$ defined by (28)–(30) satisfying $\mathcal{C}\subseteq {\mathcal{C}}_d$ and their continuously differentiable function $h:{\mathbb{R}}^n\to \mathbb{R}$. If there exists a Caputo reciprocal barrier function $B:{\mathcal{C}}_d\to \mathbb{R}$, locally Lipschitz class $\mathcal{K}$ functions ${\beta }_1,{\beta }_2,{\beta }_3$, and a class $\mathcal{K}$ function δ, with ${lim}_{r\to a}\delta \left(r\right)=b$ and b a finite-large real positive constant, such that for all $x\left(t_0\right)\in Int\left({\mathcal{C}}_d\right)$, then the following applies:

$$\begin{array}{cc}\hfill \eta \left(x,d\right)=h\left(x\right)+\delta \left({\parallel d\parallel }_{\infty }\right)& ,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{{\beta }_1\left(\eta \left(x,d\right)\right)}}\le B\left(x,d\right)\le {\displaystyle \frac{1}{{\beta }_2\left(\eta \left(x,d\right)\right)}}& ,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {}^{C}D^{\alpha }B\left(x,d\right)\le {\beta }_3\left(\eta \left(x,d\right)\right)& ,\hfill \end{array}$$

(34)

where ${\parallel d\parallel }_{\infty }\in \left[0,\overline{d}\right]$. Moreover, with a real positive and finite-large constant $\overline{d}$, System (27) can be said to be (locally) Caputo input-to-state safe, and the function B can be said to be a Caputo input-to-state safe reciprocal barrier function (Caputo ISSf-RBF).

Proof of the Theorem 6.

First, we need to prove the set $\mathcal{C}$ for System (1) is forward-invariant. As such, set $d\left(t\right)\equiv 0$; hence, the set ${\mathcal{C}}_d$ is equal to $\mathcal{C}$, and we thus have $\eta \left(x,d\right)=h\left(x\right)$. By Theorem 1, it can proved that, in this case, the set $\mathcal{C}$ is forward-invariant.

Now, we can rewrite (34) as

$${}^{C}D^{\alpha }B\le {\beta }_3\left({\beta }_2^{-1}\left({\displaystyle \frac{1}{B}}\right)\right).$$

By the proof of Theorem 1, we can obtain

$$\eta \left(x\left(t\right),d\right)\ge {\beta }_1^{-1}\left(\varphi \left(\eta \left(x\left(t_0\right),d\right),t-t_0\right)\right)$$

(35)

for all $t\in I\left(x\left(t_0\right)\right)$, where $\varphi$ is a class $\mathcal{KL}$ and $I\left(x\left(t_0\right)\right)$ is a maximum time interval. As ${\beta }_1^{-1}$ also belongs to class $\mathcal{K}$, then ${\beta }_1^{-1}\circ \varphi$ belongs to class $\mathcal{KL}$. Let $\sigma ={\beta }_1^{-1}\circ \varphi$. Thus, finally, Inequality (35) implies

$$h\left(x\left(t\right)\right)\ge \sigma \left(\eta \left(x\left(t_0\right),d\right),t-t_0\right)-\delta \left({∥d∥}_{\infty }\right).$$

Therefore, this satisfies Inequality (31). By Definition 4, System (27) is input-to-state safe. □

If we use a Caputo zeroing barrier function, can we also obtain Inequality (31) in Definition 4?

Theorem 7.

System (27) is System (1) with the input of bounded disturbance $d\left(t\right)$. There are sets $\mathcal{C}$ defined by (2)–(4), and ${\mathcal{C}}_d$ is defined by (28)–(30), satisfying $\mathcal{C}\subseteq {\mathcal{C}}_d$. If a continuously differentiable function $h:{\mathbb{R}}^n\to \mathbb{R}$ for the set $\mathcal{C}$ is a zeroing barrier function and there exists a locally Lipschitz class $\mathcal{K}$ function β and a class $\mathcal{K}$ function δ, then, with ${lim}_{r\to a}\delta \left(r\right)=b$ and b a finite-large real positive constant, there is the following such that, for all $x\left(t_0\right)\in Int\left({\mathcal{C}}_d\right)$, we have

$$\begin{array}{cc}\hfill \eta \left(x,d\right)=h\left(x\right)+\delta \left({\parallel d\parallel }_{\infty }\right)& ,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {}^{C}D^{\alpha }\eta \left(x,d\right)\ge -\beta \left(\eta \left(x,d\right)\right)& ,\hfill \end{array}$$

(37)

where ${\parallel d\parallel }_{\infty }\in \left[0,\overline{d}\right]$ has a real positive and finite-large constant $\overline{d}$, and System (27) can be said to be (locally) Caputo ISSf and the function h can be said to be a Caputo ISSf-ZBF.

Proof of the Theorem 7.

Let $B\left(x,d\right)={\displaystyle \frac{1}{\eta \left(x,d\right)}}$. According to the proof of Theorem 2, we have ${}^{C}D^{\alpha }B\left(x,d\right)=-k{}^{C}D^{\alpha }\eta \left(x,d\right)$ with a constant $k>0$. Thus, (37) can be rewritten as ${}^{C}D^{\alpha }B\left(x,d\right)\le k\beta \left(\eta \left(x,d\right)\right)$. As such, this satisfies Theorem 6. Then, we can obtain that the following Inequality (38) by (35) is

$$\eta \left(x\left(t\right),d\right)\ge \sigma \left(\eta \left(x\left(t_0\right)\right),t-t_0\right),$$

(38)

where $\sigma$ belongs to class $\mathcal{KL}$. Substituting (36) into (38), we can obtain

$$\begin{array}{cc}\hfill & h\left(x\left(t\right)\right)+\delta \left({\parallel d\parallel }_{\infty }\right)\ge \sigma \left(h\left(x\left(t_0\right)\right)+\delta \left({\parallel d\parallel }_{\infty }\right),t-t_0\right)\hfill \\ \hfill \Rightarrow & h\left(x\left(t\right)\right)\ge \sigma \left(h\left(x\left(t_0\right)\right)+\delta \left({\parallel d\parallel }_{\infty }\right),t-t_0\right)-\delta \left({\parallel d\parallel }_{\infty }\right).\hfill \end{array}$$

Therefore, by Definition 4, System (27) is Caputo input-to-state safe. □

5. Conclusions

This article established two novel Caputo barrier functions and employed them to derive corresponding safety criteria for a class of Caputo fractional-order systems (CFOSs). Additionally, synchronous safety-stability theorems were proposed for CFOSs under these barrier function frameworks. Finally, the concept of Caputo input-to-state safe barrier functions was introduced to analyze input-to-state safety properties. The key contributions and considerations are summarized as follows.

(1)

While CRBFs ensure the forward invariance of the set $\mathcal{C}$ for CFOSs, thereby guaranteeing system safety, a significant limitation arises: the computational complexity inherent in fractional-order differentiation makes CRBFs less tractable for real-world safety control applications compared to their integer-order counterparts. This shortcoming stems from the nonlocal nature of fractional derivatives, which complicates barrier function implementation under Caputo dynamics.

(2)

We renewed the description of ISSf based on [15] and then established two theorems of ISSf via using Caputo RBF and Caputo ZBF, respectively, which can be used to more conveniently achieve the analysis and control of Caputo ISSf.

Since this article and our two previous works [28,29] jointly established a theoretical framework of Caputo safety for CFOSs, future work will focus on the applications of the proposed methods to practical cases.