Fixed-Time Stability of Random Nonlinear Systems

Dear Editor,

This letter embarks on an examination of fixed-time stability (FxTS) for random nonlinear systems (RNSs) governed by random differential equations. This endeavor encompasses a multifaceted analysis of FxTS, commencing with its rigorous definition and its integration with Lyapunov theory, along which a consequential corollary emerges. Particularly, the positive definiteness of the expectation of settling time is established, and a less conservative upper bound is derived. The effectiveness of the proposed fixed-time theorem is verified by an example.

In numerous instances, physical systems necessitate operation within stochastic environments to attain the desired level of performance. Depending on the manner in which stochastic disturbances are characterized, they can be broadly categorized into two distinct classes: stochastic nonlinear systems (SNSs) featuring white noise and RNSs characterized by colored noise. However, SNSs face limitations, primarily stemming from the ambiguity surrounding the physical interpretation of the Hessian term in Ito integrals and the non-differentiability of white noise at all points, rendering them less suitable for modeling a diverse spectrum of practical engineering systems. Conversely, RNSs are defined as random differential equations entailing colored noise, characterized by an uneven spectral density. This framework finds extensive application in both theoretical and practical domains [1]–[4], owing to its versatility and capacity to accurately model a wide range of real-world engineering systems.

Within the framework of RNSs, stability theory stands out as a pivotal and ever-evolving area of focus. A seminal milestone in this pursuit was marked by the seminal work in [5], which introduced the definitions of asymptotic stability and noise-to-state stability in probability, along with the corresponding Lyapunov criterion. This foundational work laid the groundwork for subsequent research endeavors, resulting in a wealth of outstanding contributions [6]–[8]. It is worth noting that the predominant focus of current research within this domain revolves around asymptotic stability. However, recognizing the need for expedited convergence rates, recent efforts have turned toward investigating finite-time stability in the context of RNSs. Notably, in [9], the Lyapunov criteria for noise-to-state practically finite-time stability in RNSs was introduced. Furthermore, the study presented in [10] delves into event-triggered finite-time control strategies for a specific class of high-order RNSs.

Nevertheless, finite-time stability, despite its advantages, exhibits a limitation tied to the dependence of its settling time on the initial state, a constraint that hinders its practical utility, particularly in scenarios where critical information like the initial value remains unknown. To address this issue, the concept of FxTS was introduced in [11], offering the invaluable attribute of a finite settling time without being contingent on knowledge of the initial conditions. Subsequently, the field witnessed remarkable advancements in the theory of FxTS for deterministic systems [12]–[14]. It’s essential to note that when dealing with the non-deterministic counterpart, a fundamental distinction arises, namely, the settling time of these systems becomes a random variable. The fixed-time stable theorem for SNSs is studied in [15], and the fixed-time control for SNSs is implemented in [16]–[18]. However, to the best of our knowledge, no systematic proposal has been made for the FxTS of RNSs. This compels us to put forth a fixed-time theorem for RNSs in order to enable the implementation of subsequent fixed-time observation and control. On the other hand, as inspired by [19], the characteristic of random settling time piques our curiosity, driving us to explore and analyze the nuanced properties of the settling time within the context of RNSs.

In this letter, the FxTS for RNSs is proposed. In particular, the following problems are mainly solved.

$1)$ How to provide the fixed-time theorem of RNSs for stability analysis?

$2)$ What are the sufficient conditions for ensuring that the expectation of settling time is bounded and positive definite?

$3)$ How to estimate the upper bound of settling time with reduced conservatism?

Drawing inspiration from the aforementioned studies, this letter introduces, for the first time, both the definition and theorem pertaining to FxTS in the context of RNSs. Notably, it is rigorously established that the expectation of settling time is unequivocally positive definite, while concurrently presenting an explicit upper bound that exhibits reduced conservatism.

Notation: Let $\mathrm{K}_\infty$ denote a class of continuous and strictly increasing function satisfying $\lim_{t \to \infty}y(t) = \infty$ and $y(0) = 0$. Let $\mathrm{C}^2$ be the set of twice continuously differentiable functions. $a \wedge b = \min\{a,b\}$. $I\{\cdot \}$ denotes the indicator function.

Problem Formulation: Consider a RNS described by

where $x \in \mathrm{R}^n$ is the state, $\xi \in R^l$ is a $F_t$-adapted and piecewise continuous stationary process defined on a complete probability space $(\Omega, F, F_t, P)$, and $f:\mathrm{R}^n \times \mathrm{R}_{+} \to \mathrm{R}^n$ and $\phi:\mathrm{R}^n \times \mathrm{R}_{+} \to \mathrm{R}^{n \times l}$ are piecewise continuous and locally Lipschitz in x for each $t \in [t_0, T]$.

Assumption 1: There exists a positive constant ϖ such that $\sup_{t \le t_0} E\{|\xi|^2\} < \varpi$.

Lemma 1 [5]: If Assumption 1 holds and $| f(0,t) | = \|\phi(0,t) \| = 0$, $\forall t \in [t_0, \infty)$, then (1) has a pathwise strong unique solution.

Assumption 2: Suppose $|\xi(t)|^2$ satisfies the week law of large numbers, i.e., for sufficiently small positive constants $\epsilon, \sigma$, there exists $T>t_0$ such that for any $t \ge T$,

Definition 1: If system (1) admits a solution $x(t; x_0)$ for any initial value $x_0 \in \mathrm{R}^n$, and the following conditions hold:

${\rm{i}})$ for $\epsilon \in (0, 1)$ and $r > 0$, there exists a constant $\delta(\epsilon, r) > 0$ such that $P\{|x(t, x_0)| < r\} \ge 1-\epsilon$ for any $t>0$ whenever $|x_0| < \delta$,

${\rm{ii}})$ and the first hitting time $\tau = \inf\{t \ge 0 : x(t, x_0) = 0\}$, which called random settling time, is bounded almost surely. That is $P(\tau< \infty) = 1$ and satisfies $E\tau\le T$ with a constant $T>0$,

then the trivial zero solution of (1) is said to be fixed-time stable in probability.

Main results: A novel criteria of FxTS for (1) in this section.

Theorem 1: If there exists $\mathrm{K}_{\infty}$ class functions $\nu_1$, $\nu_2$, real numbers $h_1,h_2 > 0$, and a $\mathrm{C}^2$ function $V : \mathrm{R}^n \to \mathrm{R}_{+}$ such that for any $x \in \mathrm{R}^n$,

where $r:\mathrm{R}_{+} \to \mathrm{R}_{+}$ is continuous differentiable with the derivative $r^{\prime}(s) \ge 0$, $r(s) >0$ for any $s>0$ and $\int_{0}^{\epsilon}\frac{ds}{r(s)} < M, \quad \forall \epsilon >0,$ and if the parameters satisfy then

${\rm{i}})$ the trivial solution of (1) is fixed-time stable in probability,

${\rm{ii}})$ and random settling time is bounded by $E\tau \le M/(h_1-2h_2\sqrt{\varpi})$, and its expectation is positive definite.

Proof. Let $0<\epsilon<1, s>0$ be arbitrary constants. Define stopping time $\gamma_s = \inf\{t;|x(t;x_0)|>s\}$, which gives

By (2), we have

Taking $\vartheta = \nu_2^{-1}(\mu_1(s)\epsilon)$, it follows from (1) that $P(\gamma_s \le t) \le \epsilon$ whenever $|x_0| \le \vartheta$. When $t \to \infty$, it follows that $P(\gamma_s < \infty) \le \epsilon$, leading to $P(\sup_{t \ge 0}|x(t;x_0)|\le s)\ge 1-\epsilon.$ Hence, stability in probability is guaranteed. Next, we concentrate on proving FxTS in probability.

From (3), $\dot{V}(x)$ is given by

We only need to be proved the case $x_0 \in \mathrm{R}^n \setminus \{0\}$. Because the assertion holds evidently for $x_0 = 0$, since $x(t; x_0) \equiv 0$. Let the function ${\cal{R}}(x) = \int_{0}^{V(x)} \frac{ds}{r(s)},$ which is twice continuously differentiable in $\mathrm{R}^n \setminus\{0\}$. Define the first hitting time $\tau = \inf\{t \ge 0; |x(t;x_0)| = 0\}.$ Then,

According to Assumption 2, there exists a $T>0$ such that for each sufficiently small $\epsilon >0$ and $\sigma \in (0,3\varpi )$,

Combining with Assumption 1 and Cauthy-Schitz inequation, it follows that

Hence, (7) can be rewritten to

Combining (10) and ${\cal{R}}(s) \ge 0$ leads to $E(t\wedge \tau) \le {{\cal{R}}(x_0)}/(h_1-$ $2h_2\sqrt{\varpi}).$ Letting $t \to \infty$ and using Fatou lemma, one has $E\tau \le$ ${{\cal{R}}(x_0)}/({h_1-2h_2\sqrt{\varpi }})<{M}/({h_1-2h_2\sqrt{\varpi}}),$ which implies that $P(\tau_r< \infty) = 1$. Hence, FxTS is guaranteed.

Then, we prove that the settling time $E\tau$ is positive definite. Because (1) is stable in probability, there exists a $0<\delta = \delta(s)\le 1$ and an open neighborhood $\Omega_\delta = \{x\in \mathrm{R}^n: |x|<\delta \}$ such that

Due to the continuity of functions $f(x,t),$ and $\phi(x,t)$, there exists an $s = s(z)>0$ for $z>0$ such that

It is clear that $\{\gamma_s = \infty\} = \{\sup\limits_{t\ge 0} |x(t)|\le s\}$. Because $x(\tau) = 0$, one has $|x_0|I_{\gamma_s = \infty}\le I_{\gamma_s = \infty} \int_0^{\tau} (|f(x,t)|+\rho_1|\phi(x,t)|^2 + \rho_2 |\xi(t)|^2)dt,$ where $\rho_1, \rho_2$ are positive constants. It follows from (11) and (12) that

which implies that

where $c > 0$ is an arbitrary constant. The proof is complete.

A corollary is provided next.

Corollary 1: If there exists functions $\nu_1, \nu_2 \in \mathrm{K}_{\infty}$, positive real numbers $h_1, h_2$, and a $\mathrm{C}^2$ positive definite function $V : \mathrm{R}^n \to \mathrm{R}_+$ such that

where $0<{\varrho_1}<1<{\varrho_2}$, and if $h_1 >2h_2\sqrt{\varpi}$, then the trivial solution of (1) is fixed-time stable in probability, and the random settling time is upper bounded by $E\tau \le {M}/{(h_1-2h_2\sqrt{\varpi})}.$

Proof. Let $r(s) = \alpha s^{\varrho_1} +\beta s^{\varrho_2}$, (6) is rewritten as

Similar to the proof of Theorem 1, Corollary 1 can be proved.

Then, we give an estimate of settling time with reduced conservatism. The function ${\cal{R}}(x)$ satisfies

Though mathematical analysis, when the parameter is taken as $\theta^* = {(\alpha/\beta)}^{\varrho_1-\varrho_2}$, the minimum value of function ${\Theta}(\theta)$ is given by

Furthermore, the random settling time is given by $E\tau \le \frac{M}{h_1-2h_2\sqrt{\varpi}}.$ This implies that FxTS is achieved.

Remark 1: Similar to Lemma 1 in the deterministic systems [12], by selecting parameter $\theta = 1$, the settling time $E \tau_n = {1}/\alpha(1- \varrho_1)(h_1- 2h_2\sqrt{\varpi})+{1}/{\beta(\varrho_2-1)(h_1-2h_2\sqrt{\varpi})}$ can be obtained. In contrast, $E \tau$ has less conservatism in estimating the settling time because $E \tau < E \tau_n$. It is also worth noting that the parameter selections for α and β. One is that the function ${\cal{R}}(s) \le M$ must be guaranteed, which removes the reliance on initial values for the trivial solution of (1). The other is that the settling time should not be overestimated, i.e., if the parameters are $\alpha = \rho$, $\beta = 1/\rho$, $E \tau \to \infty$ as $\rho \to \infty$.

Numerical example: This section presents an example illustrating the validity of the FxTS results. Consider the RNS

According to [5], we consider $\xi(t)$ as a stationary process generated by $\kappa \dot{\xi}(t) = -\xi(t)+\pi(t), \xi(0) = 0,$ where $\kappa>0$. Let $\pi \in R$ represent a zero-mean bandlimited white noise with power spectrum satisfying $F_{\omega}(j\mathfrak{b}) = \Xi, |\mathfrak{b}| \le \mathfrak{b}_w,$ otherwise, $F_{\omega}(j\mathfrak{b}) = 0,$ where $\mathfrak{b}_w > 0$ is bandwidth and Ξ is noise power. Fig. 1 depicts the random noise variation curve of $\xi(t)$.

Figure  1.  Random noise variation curve.

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Select Lyapunov function $V(x) = \frac{1}{2}x^2$. It follows that

where $h_1 = 3/2$ and $h_2 = 1/2$. Then initial values are given by $x_0 = 3, x_0 = 1$, and $x_0 = -2$. It is shown in Fig. 2 that the state responses can converge to zero after $2.75$s regardless of how the initial value is determined. Due to the possible errors in the simulation, we think that the order of magnitude is in the range of $10^{-6}$, which can be regarded as states converging to zero. Therefore, the validity of the proposed fixed-time stability theorem is verified.

Figure  2.  The state responses of system (19).

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Conclusion: In this letter, we have introduced both the definition and Lyapunov criterion pertaining to FxTS for RNSs. Additionally, we have contributed a valuable corollary, which facilitates the practical implementation of Lyapunov functions in various applications. Furthermore, we have rigorously established the positive definiteness of the expectation associated with settling time while presenting a more refined upper-bound estimate that mitigates conservatism.