Adaptive neural network control for a class of stochastic nonholonomic systems with asymmetric full-state constraints

Abstract

This paper proposes a novel adaptive neural network control scheme for a class of stochastic nonholonomic systems subject to asymmetric full-state constraints (AFSC), input saturation, and noise. Due to the limitations of physical structure and manufacturing technology, AFSC, input saturation, and noise widely exist in the actual system, which will lead to system performance degradation and even system instability. To facilitate controller design, we introduce a state-input scaling transformation to reconfigure the stochastic nonholonomic system into a more suitable format and adopt a novel barrier Lyapunov function (BLF) that simplifies the computation process by utilizing an adaptive neural network controller with a singular adaptive law. Furthermore, we implement a strategic switching control mechanism to mitigate the influence of uncontrollable phenomena. The proposed adaptive neural network controller, which requires only one adaptive law, can ensure that all states of the closed-loop system are uniformly ultimately bounded, while enforcing compliance with the specified asymmetric constraints on the system states and overcoming the influence of input saturation on the system. The efficacy of our proposed control approach is validated through a simulation example.

Keywords

stochastic nonholonomic systems asymmetric full-state constraints adaptive neural network control backstepping technique

Introduction

Nonholonomic systems have garnered sustained research attention owing to their broad applications in autonomous vehicles, robotic manipulators, and aerospace systems Reyhanoglu (1992). While foundational control strategies like the energy-momentum method Zenkov et al., (1998), robust exponential regulation Jiang (2000), and adaptive techniques Wang and Zhang (2012) have established critical theoretical frameworks, practical implementations face persistent challenges from physical constraints. Typical scenarios include wheeled mobile robots requiring simultaneous bounds on translational and angular velocities (Ding et al., 2017; Wu et al., 2022), where violating predefined operational limits may lead to system failure. This underscores the critical need for constraint-aware control designs that rigorously guarantee state/output boundedness without compromising stability.

There are many research results for the control of nonlinear systems featuring state or output constraints (Tee and Ge 2011; Zheng and Yan, 2025). These control methods share a common feature in their use of barrier Lyapunov function to prevent constraint violations. As the parameters of the barrier Lyapunov function approach certain limits, the value of the function tends towards infinity. By ensuring the boundedness of the barrier Lyapunov function within the closed-loop system, we are able to guarantee that these constraints are not violated. Mei, Ding and Chen introduced the barrier Lyapunov function to solve fixed-time control problems in a specific class of nonlinear systems with output constraints Mei et al., (2022). An innovative tangent-type barrier Lyapunov function was initially developed to specifically address the output constraint Chen and Sun (2020). Moreover, Gao et al. have expanded these study results to nonholonomic systems with output constraints Gao et al., (2021). However, there are two primary limitations: first, employing traditional barrier Lyapunov functions in nonholonomic systems can lead to increase the complexity of the computation process, and second, these methods only solve the control problem where the system is subject to symmetric state constraints. Notably, Yang et al., (2025a) developed an observer-based fuzzy adaptive controller for pneumatic polishing systems with full-state constraints, utilizing a logarithmic-type barrier Lyapunov function to enforce strict state bounds while handling input saturation. This work highlights the potential of intelligent control in asymmetric constraint scenarios, though its focus remains on symmetric constraints. It is evident that extending existing control schemes to stochastic nonholonomic systems with asymmetric constraints will meet the needs of practical engineering.

On the other hand, neural networks are utilized to approximate the nonlinearly uncertainties of nonlinear systems due to their robust approximation capabilities (Zhu et al., 2020; Sun et al., 2016; Zhu et al., 2025). Notably, in practical applications with complex constraints, Yang et al., (2025b) employed an adaptive neural network to approximate uncertain nonlinear dynamics in a pneumatic polishing system, achieving precise force tracking under actuator saturation. Their approach utilized a radial basis function (RBF) neural network to estimate unmeasurable states, reinforcing the capability of neural networks in handling real-world control challenges. A novel application of tangent-BLF has been introduced for the first time in stochastic nonlinear systems to ensure the preservation of full-state constraints Gao et al., (2019). Despite such advancements, the number of adaptive update laws in the aforementioned neural network-based adaptive control schemes either equals or exceeds the system order, which could result in increased consumption of computational resources and longer computational times. The design of the above controllers does not account for the impact of AFSC on the system. To alleviate the computational burden arising from neural network approximations, an adaptive controller contained only one adaptive law has been proposed Zhao et al., (2020). Additionally, Du and Wang (2024b) further extends this approach to adaptive event-triggered control for stochastic nonholonomic systems with unknown virtual control coefficients.

Inevitably, the presence of noise and state constraints in practical systems adds complexity to the design of controllers for stochastic systems. In Liu et al., (2018), both symmetric and asymmetric log-type barrier Lyapunov functions have been developed for the first time to address the control issues of nonlinear stochastic systems with full-state constraints. Subsequently, Hua et al. (2020) introduced an asymmetric tangent-type barrier Lyapunov function to ensure that all signals are semi-globally uniformly ultimately bounded (SGUUB) in probability. To guarantee the maintenance of system state constraints and the ultimate uniform boundedness (UUB) of all signals, a novel form of asymmetric barrier Lyapunov function, distinct from traditional log-type and tangent-type, was proposed Peng et al., (2024). However, constructing the controller for nonholonomic systems to makes maintaining state constraints is obviously more challenging and more realistic, because the dynamics of nonholonomic systems often exhibit Strong nonlinearity and high coupling, and the system’s sensitivity to disturbances and noise demands greater robustness from the controller, which has aroused the study interest of this article. To overcome these difficulties, we are inspired by (Du and Wang, 2024b; Peng et al., 2024), and propose an adaptive neural network control approach with only one adaptive law to address stochastic nonholonomic systems with AFSC. The primary contributions of this study are outlined as follows:

(i) Unlike traditional log-type or tangent-type barrier Lyapunov functions (Liu et al., 2018; Hua et al., 2020; Liu et al., 2025), we propose a novel asymmetric barrier Lyapunov function (BLF) for constrained nonholonomic systems. The proposed BLF not only improves system performance but also retains its validity when state constraints are relaxed, thus requiring no controller reconfiguration.

(ii) In contrast to previous works (Liu et al., 2018; Hua et al., 2020; Si et al., 2017; Ma and Liu, 2019) that focus on symmetric state constraints, real-world nonholonomic systems often involve AFSC, input saturation, and noise, which can degrade performance and cause instability. This paper simultaneously addresses these challenges within a unified framework, enhancing the controller’s versatility. Furthermore, unlike methods in (Liu and Zhu, 2021; Wang and Chen, 2013) that require multiple adaptive laws, the proposed strategy employs a single adaptive law, reducing computational complexity and ensuring that the number of adaptive laws remains independent of the system’s order and weight matrix dimension.

(iii) We develop a state-input scaling method that transforms constrained stochastic nonholonomic systems into a more tractable form, enabling simpler controller design. The proposed approach combines switching strategies and adaptive control to guarantee UUB stability while strictly enforcing all state constraints.

The structure of this paper is as follows: The Problem formulation introduces the basic concepts. The Controller design section details the adaptive neural network control scheme. The Stability analysis provides theoretical guarantees. The Simulation example validates the proposed method. Finally, the Conclusion summarizes the main findings.

Preliminaries

The following section introduces the stochastic theory and neural networks theory necessary for this paper.

Stochastic theory

Consider the following stochastic nonlinear systems

d x = f (x) d t + g (x) d ω,

(1)

where x ∈ Rⁿ is the measurable state of the system,

f (\cdot) : R^{n} \to R^{n}

and

g (\cdot) : R^{n} \to R^{n \times r}

are locally Lipschitz functions and satisfying

f (0) = 0

g (0) = 0

. ω represents an r-dimensional standard Wiener process.

Definition 1

Deng et al., (2001) For any specified C² function $V (x) \in R^{n}$ , the form of the defined differential operator V is as follows

L V (x) = \frac{\partial V (x)}{\partial x} f (x) + \frac{1}{2} Tr \{g^{T} (x) \frac{\partial^{2} V (x)}{\partial x^{2}} g (x)\},

(2)

where the term

\frac{1}{2} Tr \{g^{T} \frac{\partial^{2} V (x)}{\partial x^{2}} g\}

is called the Hessian term.

Lemma 1

Lin and Qian (2000) For any given x ∈ R and y ∈ R, there holds

α (x, y) x^{a} y^{b} \leq m {|x|}^{a + b} + \frac{b}{a + b} {(\frac{a + b}{a})}^{- \frac{a}{b}} {α (x, y)}^{\frac{a + b}{b}} m^{- \frac{a}{b}} {|y|}^{a + b},

(3)

where α(x, y) > 0 and a, b, m are positive constants.

Lemma 2

Qian and Lin (2002) Let two real variables x ≥ 0, y > 0, there holds $x \leq y + {(\frac{x}{n})}^{n} {(\frac{n - 1}{y})}^{n - 1}$ , where n ≥ 1 is a real constant.

Neural networks theory

In the following procedure, we will use the radial basis function neural network to approximate nonlinear terms. For the unknown continuous nonlinear vector function $\bar{f} (x) = {[{\bar{f}}_{1} (x), {\bar{f}}_{2} (x), \dots, {\bar{f}}_{n} (x)]}^{T}$ , there exists the following approximation equation Si et al., (2017)

{\bar{f}}_{i} (x) = W^{* T} J (x) + δ (x),

j_{i} (x) = \exp \{- \frac{{(x - Z_{i})}^{T} (x - Z_{i})}{γ_{i}^{2}}\}, i = 1, \dots, l,

where x ∈ Ω_x ⊂ R^q represents the input vector, and q denotes the dimensionality of the neural network input.

J (x) = {[j_{1} (x), j_{2} (x), \dots, j_{l} (x)]}^{T} \in R^{l}

is the vector of Gaussian basis functions, where l > 1 denotes the number of nodes in the neural network.

Z_{i} = {[Z_{i 1}, Z_{i 2}, \dots, Z_{i q}]}^{T}

is the center vector and γ_i represents the width of the Gaussian function.

W = {[W_{1}, W_{2}, \dots W_{l}]}^{T} \in R^{l}

is the weight vector, where

W^{*} = \arg \min_{W \in R^{l}} \{\sup_{x \in Ω_{x}} |\bar{f} (x) - W^{T} J (x)|\}

δ (x)

is defined as the approximation error.

Problem formulation

Consider a stochastic nonholonomic system with partial input saturation and full-state constraints in the following form

\{\begin{matrix} d x_{0} = u_{0} d t, & (4 a) \\ d x_{i} = u_{0} x_{i + 1} d t + f_{i} (x_{0}, {\bar{x}}_{i}) d t + g_{i}^{T} (x_{0}, {\bar{x}}_{i}) d ω, i = 1, \dots, n - 1, \\ d x_{n} = u d t + f_{n} (x_{0}, x) d t + g_{n}^{T} (x_{0}, x) d ω, & (4 b) \end{matrix}

where

{(x_{0}, x^{T})}^{T} = {(x_{0}, x_{1}, \dots, x_{n})}^{T} \in R^{n + 1}

are the states,

{\bar{x}}_{i} = {(x_{1}, \dots, x_{i})}^{T}

and

{\bar{x}}_{n} = x

. The input signals u₀ and u satisfy |u₀(x₀)| ≤ K, where K is a positive constant.

f_{i} (x_{0}, {\bar{x}}_{i}) : R^{i + 1} \to R

and

g_{i} (x_{0}, {\bar{x}}_{i}) : R^{i + 1} \to R^{r}

are, respectively, the drift and diffusion terms, both being smooth unknown functions that satisfy

f_{i} (0,0) = 0, g_{i} (0,0) = 0, i = 1, \dots, n

. ω represents an r-dimensional standard Wiener process. Moreover, the states x₀, …, x_n are all limited within the following sets

Ω_{x_{i}} = {x_{i} \in R | - {\underset{̲}{s}}_{i} < x_{i} < {\bar{s}}_{i}},

where

{\underset{̲}{s}}_{i}

and

{\bar{s}}_{i}

are predetermined positive constants. The control objective of this study is to design an adaptive neural network controller that ensures the UUB of all signals in the closed-loop system and guarantees convergence to a small neighborhood around zero, without violating the AFSC and overcoming the influence of input saturation on the system. To achieve this objective, the following assumption is required:

Assumption 1

Here exist the smooth functions ${\bar{φ}}_{i} (x_{0}, {\bar{x}}_{i})$ and ${\bar{ν}}_{i} (x_{0}, {\bar{x}}_{i})$ , such that

|f_{i} (x_{0}, {\bar{x}}_{i})| \leq (|x_{1}| + \dots + |x_{i}|) {\bar{φ}}_{i} (x_{0}, {\bar{x}}_{i}), |g_{i} (x_{0}, {\bar{x}}_{i})| \leq (|x_{1}| + \dots + |x_{i}|) {\bar{ν}}_{i} (x_{0}, {\bar{x}}_{i}) .

Remark 1

Assumption 1 requires the uncertain nonlinear terms $f_{i} (x_{0}, {\bar{x}}_{i})$ and $g_{i} (x_{0}, {\bar{x}}_{i})$ to follow a triangular condition, while ${\bar{φ}}_{i} (x_{0}, {\bar{x}}_{i})$ and ${\bar{ν}}_{i} (x_{0}, {\bar{x}}_{i})$ are considered genuinely nonlinear. This assumption is common in adaptive and robust nonlinear control frameworks Krstic et al., (1995) and Marino and Tomei (1996). In Lv et al., (2015), ${\bar{φ}}_{i} (x_{0}, {\bar{x}}_{i})$ and ${\bar{ν}}_{i} (x_{0}, {\bar{x}}_{i})$ are set as nonnegative constants. However, Assumption 1 relaxes some constraints by treating ${\bar{φ}}_{i} (x_{0}, {\bar{x}}_{i})$ and ${\bar{ν}}_{i} (x_{0}, {\bar{x}}_{i})$ as functions related to the system state, thus increasing its applicability to a wider range of cases.

Controller design

In this part, we present a design scheme for an adaptive neural network controller. To ensure clarity, we first consider the situation where the initial condition $x_{0} (t_{0}) \neq 0$ . Subsequently, we consider another situation where $x_{0} (t_{0}) = 0$ . Due to the specific structure of the system (4), we will split the design of control inputs u₀ and u into two separate sections.

Design u₀ for x₀ − subsystem

Select the control input for the x₀ − subsystem

u_{0} (x_{0}) = - η_{0} ρ (x_{0}),

(5)

with η₀ is a determined positive constant, and

ρ (x_{0})

is designed as

ρ (x_{0}) = \{\begin{cases} sign (x_{0}) ς, & |x_{0}| > ς \\ x_{0}, & |x_{0}| \leq ς \end{cases}

(6)

where ς is a small positive constant.

Theorem 1

For any initial condition $x_{0} (t_{0}) \neq 0$ , where t₀ ≥ 0, the corresponding solution $x_{0} (t)$ converges globally and asymptotically to zero without crossing it. This solution also satisfies the condition $x_{0} (t) \leq |x_{0} (t_{0})|$ . Furthermore, from (5), the controller u₀ exists and remains not pass zero.

Proof

Selecting function $V_{0} (x_{0}) = \frac{1}{2} x_{0}^{2}$ , we can get

\dot{V} (x_{0}) = x_{0} {\dot{x}}_{0} = \{\begin{cases} - η_{0} ς x_{0}, & x_{0} > ς > 0 \\ η_{0} ς x_{0}, & x_{0} < - ς < 0 \\ - η_{0} x_{0}^{2}, & |x_{0}| < ς \end{cases}

so, we can infer that

V (t) \leq V (t_{0})

and

x_{0} (t) \leq |x_{0} (t_{0})|

. In the given situation, the solution

x_{0} (t)

exists and

x_{0} (t) \to 0

as t → ∞ in the situation when

|x_{0}| > ς

. Additionally, x₀(t) does not pass zero. Under the influence of (5), the x₀ − subsystem has become

{\dot{x}}_{0} = - η_{0} x_{0}

. Thus, we obtain the solution for the x₀ − subsystem represented in terms of

x_{0} (t) = x_{0} (t_{0}) e^{- η_{0} (t - t_{0})}

. Given that

x_{0} (t_{0}) \neq 0

, it follows that

x_{0} (t)

does not intersect zero for

\forall t \in (t_{0}, \infty)

. Based on (5), it can be concluded that u₀ never crosses zero for

\forall t \in (t_{0}, \infty)

and satisfies

\lim_{t \to \infty} u_{0} (t) = 0

. Thus, the proof is complete. □

Remark 2

By selecting design parameters η₀ and ς such that η₀ς < K, we ensure that the control input $|u_{0} (x_{0})| \leq K$ . Thus, the control input u₀ is limited by constant η₀ς. This section addresses a more comprehensive system that incorporates full-state constraints and partial input saturation, making it better suited for practical applications.

Design u for x − subsystem

Next, we consider the subsystem (4). By introducing a transformation of the state-input scaling coordinates

z_{i} (t) = \frac{x_{i} (t)}{u_{0}^{n - i}},

(7)

the x − subsystem (4) can be transformed into the following form

\{\begin{aligned} d z_{i} = z_{i + 1} d t + F_{i} (x_{0}, {\bar{z}}_{i}) d t + G_{i}^{T} (x_{0}, {\bar{z}}_{i}) d ω, i = 1, \dots, n - 1, \\ d z_{n} = u d t + F_{n} (x_{0}, z_{n}) d t + G_{n}^{T} (x_{0}, z_{n}) d ω, \end{aligned}

(8)

where

F_{i} (x_{0}, {\bar{z}}_{i}) = \frac{f_{i} (x_{0}, {\bar{x}}_{i})}{u_{0}^{n - i}} + (n - i) z_{i} |\frac{{\dot{u}}_{0}}{u_{0}}|, G_{i}^{T} (x_{0}, {\bar{z}}_{i}) = \frac{g_{i}^{T} (x_{0}, {\bar{x}}_{i})}{u_{0}^{n - i}}, 1 \leq i \leq n .

(9)

Next, we apply a state coordinate transformation in the following form

\{\begin{aligned} ε_{1} = z_{1}, \\ ε_{i} = z_{i} - α_{i - 1}, i = 2, \dots, n, \end{aligned}

(10)

where α₁, …, α_n−1 represent virtual controllers to be determined in subsequent discussions. Define the following form of adaptive law

Θ ≜ \max \{N_{i} {‖W_{i}^{*}‖}^{2}, i = 1,2, \dots, n\},

(11)

where N₁, N₂, …, N_n represent the node numbers of the radial basis function neural network and

W_{1}^{*}, W_{2}^{*}, \dots, W_{n}^{*}

are ideal weight vectors.

Therefore, based on (8)–(11) and Itô’s formula in Mao (2007), we can obtain

\{\begin{aligned} d ε_{1} = [(ε_{2} + α_{1}) + \frac{f_{1}}{u_{0}^{n - 1}} + (n - 1) z_{1} |\frac{{\dot{u}}_{0}}{u_{0}}|] d t + \frac{g_{1}^{T}}{u_{0}^{n - 1}} d ω, \\ d ε_{i} = (ε_{i + 1} + α_{i}) d t + Φ_{i} (x_{0}, {\bar{z}}_{i}) d t + {Ψ_{i}}^{T} (x_{0}, {\bar{z}}_{i}) d ω, i = 2, \dots, n - 1, \\ d ε_{n} = u d t + Φ_{n} (x_{0}, {\bar{z}}_{n}) d t + {Ψ_{n}}^{T} (x_{0}, {\bar{z}}_{n}) d ω, \end{aligned}

(12)

where

{\bar{z}}_{i} = (z_{1}, z_{2}, \dots z_{i}), i = 1, \dots, n

\begin{align} Φ_{i} (x_{0}, {\bar{z}}_{i}) = & \frac{f_{i}}{u_{0}^{n - i}} + (n - i) z_{i} |\frac{{\dot{u}}_{0}}{u_{0}}| - \frac{\partial α_{i}}{\partial x_{0}} u_{0} - \sum_{k = 1}^{i - 1} \frac{\partial α_{i}}{\partial z_{k}} \{z_{k + 1} + \frac{f_{k}}{u_{0}^{n - k}} + (n - k) z_{k} |\frac{{\dot{u}}_{0}}{u_{0}}|\} \\ - \frac{1}{2} \sum_{j, k = 1}^{i - 1} \frac{\partial^{2} α_{i}}{\partial z_{j} \partial z_{k}} (\frac{g_{j}^{T}}{u_{0}^{n - j}} \cdot \frac{g_{k}^{T}}{u_{0}^{n - k}}) + Γ_{i}, \\ Ψ_{i} (x_{0}, {\bar{z}}_{i}) = & \frac{g_{i}^{T}}{u_{0}^{n - i}} - \sum_{k = 1}^{i - 1} \frac{\partial α_{i}}{\partial z_{k}} \frac{g_{k}^{T}}{u_{0}^{n - k}}, Γ_{i} = - \frac{\partial α_{i}}{\partial \hat{Θ}} \dot{\hat{Θ}} . \end{align}

(13)

Then, the following lemma is introduced for the required design procedure.

Lemma 3

For i = 1, …, n, there exist positive smooth functions $φ_{i j} (x_{0}, {\bar{x}}_{i}), j = 1, \dots, 5$ , such that the following inequalities hold

\begin{array}{l} |\frac{f_{i} (x_{0}, {\bar{x}}_{i})}{u_{0}^{n - i}}| \leq (\sum_{l = 1}^{i} |ε_{l}|) φ_{i 1} (x_{0}, {\bar{z}}_{i}), \\ |\frac{g_{i} (x_{0}, {\bar{x}}_{i})}{u_{0}^{n - i}}| \leq (\sum_{l = 1}^{i} |ε_{l}|) φ_{i 2} (x_{0}, {\bar{z}}_{i}), \end{array}

\begin{array}{l} |\frac{f_{i}}{u_{0}^{n - i}} + (n - i) z_{i} |\frac{{\dot{u}}_{0}}{u_{0}}|| \leq (\sum_{l = 1}^{i} |ε_{l}|) φ_{i 3} (x_{0}, {\bar{z}}_{i}), \\ |\frac{g_{i}^{T}}{u_{0}^{n - i}} - \sum_{l = 1}^{i - 1} \frac{\partial α_{i}}{\partial z_{l}} \frac{g_{l}^{T}}{u_{0}^{n - l}}| \leq (\sum_{l = 1}^{i} |ε_{l}|) φ_{i 4} (x_{0}, {\bar{z}}_{i}), \end{array}

|- \frac{1}{2} \sum_{j, l = 1}^{i - 1} \frac{\partial^{2} α_{i}}{\partial z_{j} \partial z_{l}} (\frac{g_{j}^{T}}{u_{0}^{n - j}} \cdot \frac{g_{l}^{T}}{u_{0}^{n - l}})| \leq (\sum_{l = 1}^{i} | ε_{l} |) φ_{i 5} (x_{0}, {\bar{z}}_{i}) .

(14)

Proof

Regarding Lemma 3, we direct attention to the third inequality since the remaining four can be similarly obtained based on Assumption 1. Through the integration of equations (5)–(7), and (13) with Assumption 1, and building upon the theoretical framework established in (Du and Ma, 2024a; Du and Wang, 2024b), we derive the following key results

\begin{array}{l} |\frac{f_{i}}{u_{0}^{n - i}} + (n - i) z_{i} |\frac{{\dot{u}}_{0}}{u_{0}}|| \\ \leq \frac{(| x_{1} | + \dots + | x_{i} |)}{| u_{0}^{n - i} |} {\bar{φ}}_{i} (x_{0}, {\bar{x}}_{i}) + (n - i) | z_{i} | |\frac{{\dot{u}}_{0}}{u_{0}}| \\ \leq \frac{(| z_{1} u_{0}^{(n - 1)} | + \dots + | z_{i} u_{0}^{(n - i)} |)}{| u_{0}^{n - i} |} {\bar{φ}}_{i} (x_{0}, {\bar{x}}_{i}) + (n - i) | z_{i} | \times \{\begin{cases} 0, & | x_{0} | > ς \\ η_{0}, & | x_{0} | \leq ς \end{cases} \\ \leq (\sum_{l = 1}^{i} |ε_{l}|) φ_{i 3} (x_{0}, {\bar{z}}_{i}) . \end{array}

Then, we define a new barrier Lyapunov function as following

V_{e_{i}} (ε_{i}) = \frac{{({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ε_{i}^{4}}{4 {({\bar{l}}_{i} - ε_{i})}^{4} {({\underset{̲}{l}}_{i} + ε_{i})}^{4}},

(15)

where

ε_{i} \in Ω_{ε_{i}} : = {ε_{i} \in R, | - {\underset{̲}{l}}_{i} < ε_{i} < {\bar{l}}_{i}}

with

{\underset{̲}{l}}_{i}

and

{\bar{l}}_{i}

are positive constants. It is noteworthy that the function

V_{e_{i}} (ε_{i})

is effective within the set

Ω_{ε_{i}} : = {ε_{i} \in R, | - {\underset{̲}{l}}_{i} < ε_{i} < {\bar{l}}_{i}}

. Furthermore,

V_{e_{i}} (ε_{i}) \to \infty

only as ɛ_i approaches

- {\underset{̲}{l}}_{i}

{\bar{l}}_{i}

. Hence, we need to ensure that ɛ_i belongs to a compact set

Ω_{ε_{i}} : = {ε_{i} \in R, | - {\underset{̲}{l}}_{i} < ε_{i} < {\bar{l}}_{i}}

. By performing the calculation on (15), it is easy to obtain the following result

\begin{align} \frac{\partial V_{e_{i}} (ε_{i})}{\partial ε_{i}} & = \frac{{({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4}}{4} \frac{ε_{i}^{3} {({\bar{l}}_{i} - ε_{i})}^{4} {({\underset{̲}{l}}_{i} + ε_{i})}^{4} + 4 ε_{i}^{4} {({\bar{l}}_{i} - ε_{i})}^{3} {({\underset{̲}{l}}_{i} + ε_{i})}^{4} - 4 ε_{i}^{4} {({\bar{l}}_{i} - ε_{i})}^{4} {({\underset{̲}{l}}_{i} + ε_{i})}^{3}}{{({\bar{l}}_{i} - ε_{i})}^{8} {({\underset{̲}{l}}_{i} + ε_{i})}^{8}} \\ = \frac{{({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ({\underset{̲}{l}}_{i} {\bar{l}}_{i} + ε_{i}^{2})}{{({\bar{l}}_{i} - ε_{i})}^{5} {({\underset{̲}{l}}_{i} + ε_{i})}^{5}} ε_{i}^{3} \\ = H_{i} (ε_{i}) ε_{i}^{3}, \end{align}

(16)

\begin{align} \frac{\partial^{2} V_{e_{i}} (ε_{i})}{\partial ε_{i}^{2}} = & 3 H_{i} (ε_{i}) ε_{i}^{2} + \frac{2 {({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ε_{i}^{4} {({\bar{l}}_{i} - ε_{i})}^{5} {({\underset{̲}{l}}_{i} + ε_{i})}^{5} + 5 {({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ({\underset{̲}{l}}_{i} {\bar{l}}_{i} + ε_{i}^{2}) ε_{i}^{3} {({\bar{l}}_{i} - ε_{i})}^{4} {({\underset{̲}{l}}_{i} + ε_{i})}^{5}}{{({\bar{l}}_{i} - ε_{i})}^{10} {({\underset{̲}{l}}_{i} + ε_{i})}^{10}} \\ - \frac{5 {({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ({\underset{̲}{l}}_{i} {\bar{l}}_{i} + ε_{i}^{2}) ε_{i}^{3} {({\bar{l}}_{i} - ε_{i})}^{5} {({\underset{̲}{l}}_{i} + ε_{i})}^{4}}{{({\bar{l}}_{i} - ε_{i})}^{10} {({\underset{̲}{l}}_{i} + ε_{i})}^{10}} \\ = & 3 H_{i} (ε_{i}) ε_{i}^{2} + \frac{2 {({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ε_{i}^{4}}{{({\bar{l}}_{i} - ε_{i})}^{5} {({\underset{̲}{l}}_{i} + ε_{i})}^{5}} + \frac{5 ({\underset{̲}{l}}_{i} - {\bar{l}}_{i} + 2 ε_{i})}{({\bar{l}}_{i} - ε_{i}) ({\underset{̲}{l}}_{i} + ε_{i})} H_{i} (ε_{i}) ε_{i}^{3}, \end{align}

(17)

where $H_{i} (ε_{i}) = [{({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ({\underset{̲}{l}}_{i} {\bar{l}}_{i} + ε_{i}^{2})] / [{({\bar{l}}_{i} - ε_{i})}^{5} {({\underset{̲}{l}}_{i} + ε_{i})}^{5}]$ is a smooth positive function on $Ω_{ε_{i}}$ .

Remark 3

Based on the results from (Liu et al., 2018; Hua et al., 2020; Peng et al., 2024; He et al., 2015; Song and Zhou, 2018) and related literature, it is evident that the majority of existing results are targeted at symmetrically constrained systems. Based on these insights, Liu et al., (2018) proposed a segmented barrier Lyapunov function to handle systems with AFSC, thereby increasing the complexity of the control strategy. Moreover, leveraging the constrained states as demonstrated in Peng et al., (2024) and Song and Zhou (2018), a barrier Lyapunov function capable of directly handling AFSC was proposed. This paper extends these findings to systems (4) with AFSC, further simplifying the design of controllers and the process of stability analysis.

Remark 4

It is important to highlight that the barrier Lyapunov function (6) is applicable for solving systems with or without AFSC. When the system state has no constraint requirements, that is, ${\underset{̲}{s}}_{i} = {\bar{s}}_{i} = s_{i} \to \infty$ , it follows from (10) that ${\underset{̲}{l}}_{i} = {\bar{l}}_{i} = l_{i} \to \infty$ . Therefore, we can get

\lim_{l_{i} \to \infty} \frac{{({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ε_{i}^{4}}{4 {({\bar{l}}_{i} - ε_{i})}^{4} {({\underset{̲}{l}}_{i} + ε_{i})}^{4}} = \frac{ε_{i}^{4}}{4} .

(18)

However, according to Du et al., (2015) and Gao and Yuan (2013), it is clear that $\frac{ε_{i}^{4}}{4}$ is commonly used to stabilize system (4) in the absence of constraints. Therefore, the asymmetric barrier Lyapunov function proposed in this paper offers a more versatile approach to addressing constraint-related issues.

Next, we will apply barrier Lyapunov function to the design of the controller and present the specific process.

Theorem 2

For the stochastic nonholonomic system (4) subject to AFSC under Assumption 1, the combined implementation of controllers (5), (44) and adaptive law (45) ensures uniform ultimate boundedness of all closed-loop signals while strictly enforcing all state constraints.

Proof

The proof is constructed using the backstepping approach with the following steps.

Step 1 Select a positive definite barrier Lyapunov function

V_{1} = \frac{{({\underset{̲}{l}}_{1} {\bar{l}}_{1})}^{4} ε_{1}^{4}}{4 {({\bar{l}}_{1} - ε_{1})}^{4} {({\underset{̲}{l}}_{1} + ε_{1})}^{4}} + \frac{1}{2 ϒ} {\tilde{Θ}}^{2},

(19)

where ϒ > 0 is an adaptive gain,

\tilde{Θ} = Θ - \hat{Θ}

is the estimation error. Within set

ε_{1} \in Ω_{ε_{1}} : = {ε_{1} \in R, | - {\underset{̲}{l}}_{1} < ε_{1} < {\bar{l}}_{1}}

, it is evident that V₁ possesses continuous differentiability. From Lemma 3, there exist nonnegative smooth functions

φ_{11} (x_{0}, {\bar{z}}_{1})

φ_{12} (x_{0}, {\bar{z}}_{1})

φ_{13} (x_{0}, {\bar{z}}_{1})

, such that

|\frac{f_{1}}{u_{0}^{n - 1}}| \leq |ε_{1}| φ_{11} (x_{0}, z_{1}),

|\frac{g_{1}}{u_{0}^{n - 1}}| \leq |ε_{1}| φ_{12} (x_{0}, z_{1}),

|\frac{f_{1}}{u_{0}^{n - 1}} + (n - 1) z_{1} |\frac{{\dot{u}}_{0}}{u_{0}}|| \leq |ε_{1}| φ_{13} (x_{0}, z_{1}) .

According to Definition 1 and the coordinate transformation (10), we can obtain

\begin{align} L V_{1} \leq & H_{1} ε_{1}^{3} [(ε_{2} + α_{1}) + ε_{1} φ_{13}] + \frac{1}{2} \{3 H_{1} ε_{1}^{2} + \frac{2 {({\underset{̲}{l}}_{1} {\bar{l}}_{1})}^{4} ε_{1}^{4}}{{({\bar{l}}_{1} - ε_{1})}^{5} {({\underset{̲}{l}}_{1} + ε_{1})}^{5}} \\ + \frac{5 ({\underset{̲}{l}}_{1} - {\bar{l}}_{1} + 2 ε_{1})}{({\bar{l}}_{1} - ε_{1}) ({\underset{̲}{l}}_{1} + ε_{1})} H_{1} ε_{1}^{3}\} Tr \{{Ψ_{1}}^{T} Ψ_{1}\} - \frac{\tilde{Θ}}{ϒ} \dot{\hat{Θ}} . \end{align}

(20)

Based on Lemmas 1 and 2, it can be obtained

H_{1} ε_{1}^{3} ε_{2} \leq ω_{1} ε_{2}^{4} + β_{1,1} (x_{1}) ε_{1}^{4},

(21)

\begin{align} \frac{3}{2} H_{1} ε_{1}^{2} Tr \{{Ψ_{1}}^{T} Ψ_{1}\} & \leq \frac{3 H_{1} ε_{1}^{2} φ_{12}^{2}}{2} ε_{1}^{2} \\ \leq μ_{11} + β_{1,2} (x_{1}) ε_{1}^{4}, \end{align}

(22)

\begin{align} \frac{{({\underset{̲}{l}}_{1} {\bar{l}}_{1})}^{4} ε_{1}^{4}}{{({\bar{l}}_{1} - ε_{1})}^{5} {({\underset{̲}{l}}_{1} + ε_{1})}^{5}} Tr \{{Ψ_{1}}^{T} Ψ_{1}\} & \leq \frac{{({\underset{̲}{l}}_{1} {\bar{l}}_{1})}^{4} ε_{1}^{4} φ_{12}^{2}}{{({\bar{l}}_{1} - ε_{1})}^{5} {({\underset{̲}{l}}_{1} + ε_{1})}^{5}} ε_{1}^{2} \\ \leq μ_{12} + β_{1,3} (x_{1}) ε_{1}^{4}, \end{align}

(23)

\begin{align} \frac{5 ({\underset{̲}{l}}_{1} - {\bar{l}}_{1} + 2 ε_{1})}{2 ({\bar{l}}_{1} - ε_{1}) ({\underset{̲}{l}}_{1} + ε_{1})} H_{1} ε_{1}^{3} Tr \{{Ψ_{1}}^{T} Ψ_{1}\} & \leq \frac{5 ({\underset{̲}{l}}_{1} - {\bar{l}}_{1} + 2 ε_{1}) H_{1} ε_{1}^{3} φ_{12}^{2}}{2 ({\bar{l}}_{1} - ε_{1}) ({\underset{̲}{l}}_{1} + ε_{1})} ε_{1}^{4} \\ \leq μ_{13} + β_{1,4} (x_{1}) ε_{1}^{4}, \end{align}

(24)

where, μ₁₁, μ₁₂, and μ₁₃ > 0 are constants,

β_{1,1} (x_{1}) = \frac{3}{4} {(\frac{1}{4 ω_{1}})}^{(1 / 3)} H_{1}^{(4 / 3)}

β_{1,2} (x_{1}) = \frac{1}{μ_{11}} {(\frac{3}{4} H_{1} ε_{1}^{2} φ_{12}^{2})}^{2}

β_{1,3} (x_{1}) = \frac{1}{μ_{12}} {(\frac{{({\underset{̲}{l}}_{1} {\bar{l}}_{1})}^{4} ε_{1}^{4} φ_{12}^{2}}{2 {({\bar{l}}_{1} - ε_{1})}^{5} {({\underset{̲}{l}}_{1} + ε_{1})}^{5}})}^{2}

and

β_{1,4} (x_{1}) = \frac{1}{μ_{13}} {(\frac{5 ({\underset{̲}{l}}_{1} - {\bar{l}}_{1} + 2 ε_{1}) H_{1} ε_{1}^{3} φ_{12}^{2}}{4 ({\bar{l}}_{1} - ε_{1}) ({\underset{̲}{l}}_{1} + ε_{1})})}^{2}

. Substituting (21)–(24) into (20), we can obtain

\begin{matrix} L V_{1} & \leq H_{1} ε_{1}^{3} (α_{1} + \bar{f_{1}}) + {ω_{1} ε}_{2}^{4} + S_{1} + (β_{1,1} + β_{1,2} + β_{1,3} + β_{1,4}) ε_{1}^{4} - \frac{\tilde{Θ}}{ϒ} \dot{\hat{Θ}}, \end{matrix}

(25)

where S₁ = μ₁₁ + μ₁₂ + μ₁₃,

\bar{f_{1}} = ε_{1} φ_{13}

and

\bar{f_{1}}

can be approximated by neural networks as

{\bar{f}}_{1} = W_{1}^{* T} J_{1} (Z_{1}) + δ_{1} (Z_{1}),

(26)

where

Z_{1} = {[x_{0}, z_{1}]}^{T}

and δ₁ is defined as the approximation error that satisfies |δ₁| ≤ k₁. From (11) and

J_{1}^{T} J_{1} \leq N_{1}

, we can obtain

{|W_{1}^{* T}|}^{2} {|J_{1}|}^{2} \leq {|W_{1}^{* T}|}^{2} N_{1} \leq Θ

, and by combining (26) and Lemma 1, the following results can be obtained

\begin{align} H_{1} ε_{1}^{3} {\bar{f}}_{1} & = H_{1} ε_{1}^{3} [W_{1}^{* T} J_{1} (Z_{1}) + δ_{1} (Z_{1})] \\ \leq H_{1} | ε_{1} |^{3} τ_{1} (\hat{Θ}) + \frac{H_{1} | ε_{1} |^{3}}{2} \tilde{Θ} \\ \leq ξ_{1} + η_{1} (\hat{Θ}) H_{1}^{\frac{4}{3}} ε_{1}^{4} + \frac{H_{1} | ε_{1} |^{3}}{2} \tilde{Θ}, \end{align}

(27)

where ξ₁ is a designed positive constant,

τ_{1} (\hat{Θ}) = 1 + \frac{\sqrt{1 + {\hat{Θ}}^{2}}}{2} + \frac{1}{2} k_{1}^{2}

, and

η_{1} (\hat{Θ}) = \frac{3}{4} {(\frac{1}{4 ξ_{1}})}^{(1 / 3)} τ_{1}^{(4 / 3)}

. Bringing (27) into (25) yields

\begin{align} L V_{1} \leq & H_{1} ε_{1}^{3} α_{1} + {ω_{1} ε}_{2}^{4} + ξ_{1} + η_{1} (\hat{Θ}) H_{1}^{\frac{4}{3}} ε_{1}^{4} + \frac{H_{1} | ε_{1} |^{3}}{2} \tilde{Θ} \\ + S_{1} + (β_{1,1} + β_{1,2} + β_{1,3} + β_{1,4}) ε_{1}^{4} - \frac{\tilde{Θ}}{ϒ} \dot{\hat{Θ}} . \end{align}

(28)

Then, choose the virtual controller as

α_{1} = \frac{1}{H_{1}} [- c_{1} ε_{1} - η_{1} H_{1}^{\frac{4}{3}} ε_{1} - (β_{1,1} + β_{1,2} + β_{1,3} + β_{1,4}) ε_{1}] .

(29)

Similarly, we can get

L V_{1} \leq - c_{1} ε_{1}^{4} + {ω_{1} ε}_{2}^{4} + Δ_{1} - \frac{1}{ϒ} \tilde{Θ} (\dot{\hat{Θ}} - π_{1}),

(30)

where Δ₁ = S₁ + ξ₁ and

π_{1} = \frac{ϒ}{2} H_{1} | ε_{1} |^{3}

Step i (2 ≤ i ≤ n − 1) Consider the i-th Lyapunov function candidate $V_{i} = \sum_{m = 1}^{i} \frac{{({\underset{̲}{l}}_{m} {\bar{l}}_{m})}^{4} ε_{m}^{4}}{4 {({\bar{l}}_{m} - ε_{m})}^{4} {({\underset{̲}{l}}_{m} + ε_{m})}^{4}} + \frac{1}{2 ϒ} {\tilde{Θ}}^{2}$ .

Following the approach in (30) and applying mathematical induction, we derive

L V_{i - 1} \leq - \sum_{j = 1}^{i - 1} c_{j} ε_{j}^{4} + {ω_{i - 1} ε}_{i}^{4} + Δ_{i - 1} - \frac{1}{ϒ} \tilde{Θ} (\dot{\hat{Θ}} - π_{i - 1}),

(31)

where c_j > 0,

π_{i - 1} = \sum_{j = 1}^{i - 1} \frac{ϒ}{2} H_{j} | ε_{j} |^{3}

, and

Δ_{i - 1} = \sum_{j = 1}^{i - 1} (S_{j} + ξ_{j})

are positive constants. From (10)–(13), we have

d ε_{i} = (ε_{i + 1} + α_{i} + Φ_{i} (x_{0}, {\bar{z}}_{i})) d t + {Ψ_{i}}^{T} (x_{0}, {\bar{z}}_{i}) d ω,

(32)

where

\begin{matrix} Φ_{i} (x_{0}, {\bar{z}}_{i}) = & \frac{f_{i}}{u_{0}^{n - i}} + (n - i) z_{i} |\frac{{\dot{u}}_{0}}{u_{0}}| - \frac{\partial α_{i}}{\partial x_{0}} u_{0} - \sum_{l = 1}^{i - 1} \frac{\partial α_{i}}{\partial z_{l}} \{z_{l + 1} + \frac{f_{l}}{u_{0}^{n - l}} + (n - l) z_{l} |\frac{{\dot{u}}_{0}}{u_{0}}|\} \\ - \frac{1}{2} \sum_{j, l = 1}^{i - 1} \frac{\partial^{2} α_{i}}{\partial z_{j} \partial z_{l}} (\frac{g_{j}^{T}}{u_{0}^{n - j}} \cdot \frac{g_{l}^{T}}{u_{0}^{n - l}}) + Γ_{i}, \\ Ψ_{i} (x_{0}, {\bar{z}}_{i}) = & \frac{g_{i}^{T}}{u_{0}^{n - i}} - \sum_{l = 1}^{i - 1} \frac{\partial α_{i}}{\partial z_{l}} \frac{g_{l}^{T}}{u_{0}^{n - l}}, Γ_{i} = - \frac{\partial α_{i}}{\partial \hat{Θ}} \dot{\hat{Θ}} . \end{matrix}

Combining Definition 1 and Lemma 3, we can get

\begin{align} L V_{i} \leq & - \sum_{j = 1}^{i - 1} c_{j} ε_{j}^{4} + ω_{i - 1} ε_{i}^{4} + Δ_{i - 1} + H_{i} ε_{i}^{3} \{\frac{f_{i}}{u_{0}^{n - i}} + ε_{i + 1} + α_{i} + (n - i) z_{i} |\frac{{\dot{u}}_{0}}{u_{0}}| \\ - \frac{\partial α_{i}}{\partial x_{0}} u_{0} - \sum_{k = 1}^{i - 1} \frac{\partial α_{i}}{\partial z_{k}} [z_{k + 1} + \frac{f_{k}}{u_{0}^{n - k}} + (n - k) z_{k} |\frac{{\dot{u}}_{0}}{u_{0}}|] - \frac{1}{2} \sum_{j, k = 1}^{i - 1} \frac{\partial^{2} α_{i}}{\partial z_{j} \partial z_{k}} (\frac{g_{j}^{T}}{u_{0}^{n - j}} \cdot \frac{g_{k}^{T}}{u_{0}^{n - k}}) \\ - \frac{\partial α_{i}}{\partial \hat{Θ}} \dot{\hat{Θ}}\} + \frac{1}{2} \{3 H_{i} ε_{i}^{2} + \frac{2 {({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ε_{i}^{4}}{{({\bar{l}}_{i} - ε_{i})}^{5} {({\underset{̲}{l}}_{i} + ε_{i})}^{5}} + \frac{5 ({\underset{̲}{l}}_{i} - {\bar{l}}_{i} + 2 ε_{i})}{({\bar{l}}_{i} - ε_{i}) ({\underset{̲}{l}}_{i} + ε_{i})} H_{i} ε_{i}^{3}\} Tr \{{Ψ_{i}}^{T} Ψ_{i}\} \\ - \frac{1}{ϒ} \tilde{Θ} (\dot{\hat{Θ}} - π_{i - 1}) . \end{align}

(33)

From Lemma 1, we can obtain the following inequalities

H_{i} ε_{i}^{3} ε_{i + 1} \leq ω_{i} ε_{i + 1}^{4} + β_{i, 1} (x_{i}) ε_{i}^{4},

(34)

\begin{align} \frac{3}{2} H_{i} ε_{i}^{2} Tr \{{Ψ_{i}}^{T} Ψ_{i}\} & \leq \frac{3 H_{i} \sum_{j = 1}^{i} ε_{j}^{2} φ_{i 4}^{2}}{2} ε_{i}^{2} \\ \leq μ_{i 1} + β_{i, 2} (x_{i}) ε_{i}^{4}, \end{align}

(35)

\begin{align} \frac{{({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ε_{i}^{4}}{{({\bar{l}}_{i} - ε_{i})}^{5} {({\underset{̲}{l}}_{i} + ε_{i})}^{5}} Tr \{{Ψ_{i}}^{T} Ψ_{i}\} & \leq \frac{{({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} \sum_{j = 1}^{i} ε_{j}^{2} φ_{i 4}^{2}}{{({\bar{l}}_{i} - ε_{i})}^{5} {({\underset{̲}{l}}_{i} + ε_{i})}^{5}} ε_{i}^{4} \\ \leq μ_{i 2} + β_{i, 3} (x_{i}) ε_{i}^{4}, \end{align}

(36)

\begin{align} \frac{5 ({\underset{̲}{l}}_{i} - {\bar{l}}_{i} + ε_{i})}{2 ({\bar{l}}_{i} - ε_{i}) ({\underset{̲}{l}}_{i} + ε_{i})} H_{i} ε_{i}^{3} Tr \{{Ψ_{i}}^{T} Ψ_{i}\} & \leq \frac{5 ({\underset{̲}{l}}_{i} - {\bar{l}}_{i} + 2 ε_{i}) H_{i} \sum_{j = 1}^{i} ε_{j}^{2} φ_{i 4}^{2}}{2 ({\bar{l}}_{i} - ε_{i}) ({\underset{̲}{l}}_{i} + ε_{i})} ε_{i}^{3} \\ \leq μ_{i 3} + β_{i, 4} (x_{i}) ε_{i}^{4}, \end{align}

(37)

where μ_i1, μ_i2, and μ_i3 are positive constants,

β_{i, 1} (x_{i}) = \frac{3}{4} {(\frac{1}{4 ω_{i}})}^{(1 / 3)} H_{i}^{(4 / 3)}

β_{i, 2} (x_{i}) = \frac{1}{μ_{i 1}} {(\frac{3}{4} H_{i} \sum_{j = 1}^{i} ε_{j}^{2} φ_{i 4}^{2})}^{2}

β_{i, 3} (x_{i}) = \frac{1}{μ_{i 2}} {(\frac{{({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ε_{i}^{2} \sum_{j = 1}^{i} ε_{j}^{2} φ_{i 4}^{2}}{2 {({\bar{l}}_{i} - ε_{i})}^{5} {({\underset{̲}{l}}_{i} + ε_{i})}^{5}})}^{2}

β_{i, 4} (x_{i}) = \frac{1}{μ_{i 3}} {(\frac{5 ({\underset{̲}{l}}_{i} - {\bar{l}}_{i} + 2 ε_{i}) H_{i} ε_{i} \sum_{j = 1}^{i} ε_{j}^{2} φ_{i 4}^{2}}{4 ({\bar{l}}_{i} - ε_{i}) ({\underset{̲}{l}}_{i} + ε_{i})})}^{2}

. Substituting (34)–(37) into (33), we get

\begin{align} L V_{i} \leq & - \sum_{j = 1}^{i - 1} c_{j} ε_{j}^{4} + Δ_{i - 1} + H_{i} ε_{i}^{3} (\frac{ω_{i - 1}}{H_{i}} ε_{i} + α_{i + 1} + \bar{f_{i}}) + ω_{i} ε_{i + 1}^{4} + S_{i} \\ + (β_{i, 1} + β_{i, 2} + β_{i, 3} + β_{i, 4}) ε_{i}^{4} - \frac{1}{ϒ} \tilde{Θ} (\dot{\hat{Θ}} - π_{i - 1}), \end{align}

(38)

where

\begin{matrix} \bar{f_{i}} = & ε_{i} φ_{i 3} - \frac{\partial α_{i}}{\partial x_{0}} u_{0} - \sum_{k = 1}^{i - 1} \frac{\partial α_{i}}{\partial z_{k}} [z_{k + 1} + \frac{f_{k}}{u_{0}^{n - k}} + (n - k) z_{k} |\frac{{\dot{u}}_{0}}{u_{0}}|] \\ - \frac{1}{2} \sum_{j, k = 1}^{i - 1} \frac{\partial^{2} α_{i}}{\partial z_{j} \partial z_{k}} (\frac{g_{j}^{T}}{u_{0}^{n - j}} \cdot \frac{g_{k}^{T}}{u_{0}^{n - k}}) - \frac{\partial α_{i}}{\partial \hat{Θ}} \dot{\hat{Θ}} . \end{matrix}

Seemingly, $\bar{f_{i}}$ is a smooth unknown term that can be approximated by neural networks

{\bar{f}}_{i} = W_{i}^{* T} J_{i} (Z_{i}) + δ_{i} (Z_{i}), | δ_{i} (Z_{i}) | \leq k_{i},

(39)

where

Z_{i} = {[x_{0}, z_{1}, z_{2}, \dots, z_{i}, \hat{Θ}]}^{T}

, k_i > 0, δ_i is defined as the approximation error and satisfies

\begin{align} H_{i} ε_{i}^{3} {\bar{f}}_{i} & = H_{i} ε_{i}^{3} [W_{i}^{* T} J_{i} (Z_{i}) + δ_{i} (Z_{i})], \\ \leq H_{i} | ε_{i} |^{3} τ_{i} (\hat{Θ}) + \frac{H_{i} | ε_{i} |^{3}}{2} \tilde{Θ}, \\ \leq ξ_{i} + η_{i} (\hat{Θ}) H_{i}^{\frac{4}{3}} ε_{i}^{4} + \frac{H_{i} | ε_{i} |^{3}}{2} \tilde{Θ}, \end{align}

(40)

where ξ_i is a designed nonnegative constant,

τ_{i} (\hat{Θ}) = 1 + \frac{\sqrt{1 + {\hat{Θ}}^{2}}}{2} + \frac{1}{2} k_{i}^{2}

and

η_{i} (\hat{Θ}) = \frac{3}{4} {(\frac{1}{4 ξ_{i}})}^{(1 / 3)} τ_{i}^{(4 / 3)}

. Then, we set up the virtual controller α_i as follows

α_{i} = \frac{1}{H_{i}} [- c_{i} ε_{i} - η_{i} H_{i}^{\frac{4}{3}} ε_{i} - (β_{i, 1} + β_{i, 2} + β_{i, 3} + β_{i, 4}) ε_{i} - ω_{i - 1} ε_{i}],

(41)

By substituting (40) and (41) into (38), we obtain

L V_{i} \leq - \sum_{j = 1}^{i} c_{j} ε_{j}^{4} + {ω_{i} ε}_{i}^{4} + Δ_{i} - \frac{1}{ϒ} \tilde{Θ} (\dot{\hat{Θ}} - π_{i}),

(42)

where c_i, S_i, and ξ_i are designed nonnegative constants,

π_{i} = π_{i - 1} + \frac{ϒ}{2} H_{i} ε_{i}^{3} = \sum_{j = 1}^{i} \frac{ϒ}{2} H_{j} ε_{j}^{3}

, and

Δ_{i} = Δ_{i - 1} + S_{i} + ξ_{i} = \sum_{j = 1}^{i} (S_{j} + ξ_{j})

Step n According (10)–(13), we can get

d ε_{n} = (u + Φ_{n} (x_{0}, {\bar{z}}_{n})) d t + {Ψ_{n}}^{T} (x_{0}, {\bar{z}}_{n}) d ω,

(43)

where

\begin{matrix} Φ_{n} (x_{0}, {\bar{z}}_{n}) = & f_{n} - \frac{\partial α_{n}}{\partial x_{0}} u_{0} - \sum_{k = 1}^{n - 1} \frac{\partial α_{n}}{\partial z_{k}} \{z_{k + 1} + \frac{f_{k}}{u_{0}^{n - k}} + η_{0} (n - k) z_{k}\} \\ - \frac{1}{2} \sum_{j, k = 1}^{n - 1} \frac{\partial^{2} α_{n}}{\partial z_{j} \partial z_{k}} (\frac{g_{j}^{T}}{u_{0}^{n - j}} \cdot \frac{g_{k}^{T}}{u_{0}^{n - k}}) + Γ_{n}, \\ Ψ_{n} (x_{0}, {\bar{z}}_{n}) & = g_{n}^{T} - \sum_{k = 1}^{n - 1} \frac{\partial α_{n}}{\partial z_{k}} \frac{g_{k}^{T}}{u_{0}^{n - k}}, Γ_{n} = - \frac{\partial α_{n}}{\partial \hat{Θ}} \dot{\hat{Θ}} . \end{matrix}

By designing control input and adaptive law in the following way

u = \frac{1}{H_{n}} [- c_{n} ε_{n} - ω_{n - 1} ε_{n} - η_{n} H_{n}^{\frac{4}{3}} ε_{n} - (β_{n, 1} + β_{n, 2} + β_{n, 3} + β_{n, 4}) ε_{n}],

(44)

\dot{\hat{Θ}} = π_{n} = \sum_{j = 1}^{n} \frac{ϒ H_{j} | ε_{j} |^{3}}{2} - ϖ_{0} \hat{Θ},

(45)

we can make

V_{n} = \sum_{m = 1}^{n} \frac{{({\underset{̲}{l}}_{m} {\bar{l}}_{m})}^{4} ε_{m}^{4}}{4 {({\bar{l}}_{m} - ε_{m})}^{4} {({\underset{̲}{l}}_{m} + ε_{m})}^{4}} + \frac{1}{2 ϒ} {\tilde{Θ}}^{2},

(46)

satisfy

L V_{n} \leq - \sum_{j = 1}^{n} c_{j} ε_{j}^{4} + Δ_{n} + \frac{ϖ_{0}}{ϒ} \tilde{Θ} \hat{Θ},

(47)

where

ε_{i} \in Ω_{ε_{i}} : = {ε_{i} \in R, | - {\underset{̲}{l}}_{i} < ε_{i} < {\bar{l}}_{i}}

, c_n, ω_n−1, ξ_n, and ϖ₀ are non-negative constants that have been designed,

Δ_{n} = \sum_{j = 1}^{n} (S_{j} + ξ_{j})

. From

\hat{Θ} = Θ - \tilde{Θ}

, we derive the subsequent inequality

\frac{ϖ_{0}}{ϒ} \tilde{Θ} \hat{Θ} \leq - \frac{ϖ_{0}}{2 ϒ} {\tilde{Θ}}^{2} + \frac{ϖ_{0}}{2 ϒ} Θ^{2} .

(48)

However, due to the positivity and continuity of $V_{e_{i}}$ , there exist sets $Λ_{i} : = {ε_{i} \in Ω_{ε_{i}} | V_{e_{i}} \leq Q_{i}} \subseteq Ω_{ε_{i}}$ with some constants Q_i > 0 such that $- {\underset{̲}{l}}_{i} / 2 < ε_{i} < {\bar{l}}_{i} / 2$ . For all instances of ɛ_i ∈ Λ_i, one can derive

V_{e_{i}} = \frac{{({\underset{̲}{l}}_{i} {\bar{l}}_{i})}^{4} ε_{i}^{4}}{4 {({\bar{l}}_{i} - ε_{i})}^{4} {({\underset{̲}{l}}_{i} + ε_{i})}^{4}} \leq \frac{ε_{i}^{4}}{{(1 - \frac{ε_{i}}{{\bar{l}}_{i}})}^{4} {(1 - \frac{ε_{i}}{{\underset{̲}{l}}_{i}})}^{4}} \leq 2^{8} ε_{i}^{4} .

(49)

Combining (48) and (49) into (47), we obtain

\begin{align} L V_{n} & \leq - \sum_{j = 1}^{n} c_{j} ε_{j}^{4} + Δ_{n} - \frac{ϖ_{0}}{2 ϒ} {\tilde{Θ}}^{2} + \frac{ϖ_{0}}{2 ϒ} Θ^{2} \\ \leq - \sum_{j = 1}^{n} 2^{- 8} c_{j} \frac{{({\underset{̲}{l}}_{j} {\bar{l}}_{j})}^{4} ε_{j}^{4}}{4 {({\bar{l}}_{j} - ε_{j})}^{4} {({\underset{̲}{l}}_{j} + ε_{j})}^{4}} - \frac{ϖ_{0}}{2 ϒ} {\tilde{Θ}}^{2} + \frac{ϖ_{0}}{2 ϒ} Θ^{2} + Δ_{n} \\ \leq - c \sum_{j = 1}^{n} (\frac{{({\underset{̲}{l}}_{j} {\bar{l}}_{j})}^{4} ε_{j}^{4}}{4 {({\bar{l}}_{j} - ε_{j})}^{4} {({\underset{̲}{l}}_{j} + ε_{j})}^{4}} + \frac{1}{2 ϒ} {\tilde{Θ}}^{2}) + ρ \\ \leq - c V_{n} + ρ, \end{align}

(50)

where c = min{2⁻⁸c_i, ϖ₀|1 ≤ i ≤ n} and

ρ = Δ_{n} + \frac{ϖ_{0}}{2 ϒ} Θ^{2}

. Then, by taking expectations on both sides of (50) and multiplying by e^bt, we can obtain

d (e^{b t} E (V_{n})) \leq ρ e^{b t}

. For all t ≥ 0, integrating yields

E (V_{n}) \leq E (V_{n} (0)) e^{- b t} + \frac{ρ}{b} \leq E (V_{n} (0)) + \frac{ρ}{b} .

(51)

Therefore, considering (47), we obtain $ε_{i}, \hat{Θ}$ and $\tilde{Θ}$ are UUB. From x₁ = ɛ₁, we know that x₁ is UUB. Incorporating (29), (41), and (44), these conditions ensure that α₁, …, α_n−1, u are UUB. Furthermore, we can obtain all states x₁, …, x_n are UUB.

Based on z₁ = ɛ₁, we can get $- {\underset{̲}{l}}_{1} < z_{1} < {\bar{l}}_{1}$ . Therefore, from (5) and (7), we can conclude $- {\underset{̲}{l}}_{1} K^{n - 1} < x_{1} < K^{n - 1} {\bar{l}}_{1}$ , denote $- {\underset{̲}{l}}_{1} K^{n - 1} = - {\underset{̲}{s}}_{1}$ , $K^{n - 1} {\bar{l}}_{1} = {\bar{s}}_{1}$ , then $- {\underset{̲}{s}}_{1} < x_{1} < {\bar{s}}_{1}$ . It is evident that the continuous virtual controller α_i is bounded. Therefore, for i = 1, …, n − 1, there exist non-negative constants T_i such that |α_i| ≤ T_i. From ɛ_i = z_i − α_i−1, we have $- {\underset{̲}{l}}_{i} - T_{i} < z_{i} < {\bar{l}}_{i} + T_{i}$ . Thus, from (7), we can conclude $- K^{n - 1} ({\underset{̲}{l}}_{i} + T_{i}) < x_{i} < K^{n - 1} ({\bar{l}}_{i} + T_{i})$ . By denoting $- K^{n - 1} ({\underset{̲}{l}}_{i} + T_{i}) = - {\underset{̲}{s}}_{i}$ and $K^{n - 1} ({\bar{l}}_{i} + T_{i}) = {\bar{s}}_{i}$ , we obtain $- {\underset{̲}{s}}_{i} < x_{i} < {\bar{s}}_{i} .$ Thus, all states of the system (4) are within the constraint set.□

Remark 5

Through the use of asymmetric barrier Lyapunov function and appropriate adaptive parameters (11), we have designed controller (5) and (44) with only one adaptive law (45). This controller is not only applicable to a more general class of stochastic nonholonomic systems with AFSC but also significantly reduces computational burden. Therefore, compared with existing results, the results of this paper are more practical.

Stability analysis

Based on the above analysis, we have designed controllers (5) and (44) for u₀ and u of systems (4) when x₀(t₀) ≠ 0. Now, let’s consider how to select controllers u₀ and u when x₀(t₀) = 0. In such cases, we set u₀ as follows

u_{0} = u_{0}^{*},

(52)

where

u_{0}^{*} < K

is a nonnegative constant. Utilizing the previously described design scheme, we derive the controller u*(x₀, x_n) and the adaptation law

{\hat{Θ}}_{*}

. This ensures that the solution x₀ of the x₀ − subsystem remains stable throughout the interval [t₀, t_s]. Once t > t_s and given that x₀(t_s) ≠ 0, we can, respectively, switch controllers u₀ and u to (5) and (44).

The main conclusion is expressed in the upcoming theorem.

Theorem 3

To address the issue of uncontrollability, we can devise the control strategy $u = {(u_{0}, u)}^{T}$ for systems (4) in the following manner

u_{0} = \{\begin{cases} (5), if x_{0} (t_{0}) \neq 0, \\ \{\begin{cases} (52), & t_{0} \leq t < t_{s}, \\ (5), & t \geq t_{s}, \end{cases} if x_{0} (t_{0}) = 0, \end{cases}

u = \{\begin{cases} (44), if x_{0} (t_{0}) \neq 0, \\ \{\begin{cases} u^{*}, & t_{0} \leq t < t_{s}, \\ (44), & t \geq t_{s}, \end{cases} if x_{0} (t_{0}) = 0, u_{0} = u_{0}^{*} \end{cases}

and the adaption law

\dot{\hat{Θ}} = \{\begin{cases} (45), if x_{0} (t_{0}) \neq 0, \\ \{\begin{cases} {\dot{\hat{Θ}}}_{*}, & t_{0} \leq t < t_{s}, \\ (45), & t \geq t_{s} . \end{cases} if x_{0} (t_{0}) = 0, \end{cases}

Subsequently, under any initial conditions, all signals within the closed-loop systems (4) are UUB.

Proof

Initially, when x₀(t₀) ≠ 0, we can infer from Theorems 1 and 2 that signals x₀, α_i, ɛ_i, u, $\hat{Θ}$ are UUB. The emphasis is that x₀ will not cross zero. Based on (10), we can conclude that z_i are UUB. From (7), it is evident x₀, …, x_n are UUB.

Furthermore, if x₀(t₀) = 0, directly using state-input scaling coordinates may lead to singularity issues. To avoid singularity problems, given any finite t_s > 0, we select the controller u* as a nonzero constant in [t₀, t_s). This guarantees that the solution x₀ of the x₀-subsystem remains bounded during the interval [t₀, t_s). In this situation, the x-subsystem transforms into a standard strict feedback system that can be stabilized using a backstepping-based nonlinear feedback u₀ = u*(x₀, x) and the adaptive law ${\hat{Θ}}_{*}$ . Next, by employing a differential inequality similar to (50), we can establish that all states of system (4) remain bounded throughout the time interval [t₀, t_s). At t_s > 0, if x₀(t_s) ≠ 0, we have the option to transition the control inputs u₀ and u to (5) and (44). The proof is concluded.

Remark 6

Regardless of initial conditions, the controller and switching strategy ensure that system states converge to a small neighborhood near zero. Compared to (Du and Wang, 2024b; Liu et al., 2018; Hua et al., 2020; Peng et al., 2024), the controller designed in this paper is more general. By introducing a novel barrier Lyapunov function combined with an adaptive neural network, it can handle stochastic nonholonomic systems with AFSC and partial input saturation. Furthermore, it ensures that all signals of the closed-loop system are uniformly ultimately bounded and do not violate full-state constraints. However, the feasibility conditions still need to be further addressed.

Simulation example

To validate the controller design approach, we implement it on a tricycle mobile robot Hespanha et al., (1999) with the following dynamics

\{\begin{cases} {\dot{x}}_{a} = v \cos θ, \\ {\dot{y}}_{a} = v \sin θ, \\ \dot{θ} = ω, \end{cases}

(53)

where v and ω represent the linear and angular velocity control inputs, respectively.

When the angular velocity ω is subject to stochastic disturbances, modeled by

ω (x_{a}, y_{a}, θ) = ω_{1} (x_{a}, y_{a}, θ) + ω_{2} (x_{a}, y_{a}, θ) \dot{B} (t),

(54)

where B(t) is white noise. The dynamical system (53) can be transformed into

\{\begin{cases} d x_{a} = v \cos θ d t, \\ d y_{a} = v \sin θ d t, \\ d θ = ω_{1} d t + ω_{2} d B . \end{cases}

(55)

For system (55), we introduce the state and input transformation

x_{0} = x_{a}, x_{1} = y_{a}, x_{2} = \tan θ, u_{0} = v \cos θ, u = ω_{1} \sec^{2} θ,

(56)

which yields

\{\begin{cases} d x_{0} = u_{0} d t, \\ d x_{1} = u_{0} x_{2} d t, \\ d x_{2} = u d t + (1 + x_{2}^{2}) ω_{2} d B . \end{cases}

(57)

Assuming $ω_{2} = \frac{sin (x_{2})}{1 + x_{2}^{2}}$ , we define the system state as

\{\begin{aligned} d x_{0} = u_{0} d t, \\ d x_{1} = u_{0} x_{2} d t, \\ d x_{2} = u d t + sin (x_{2}) d ω . \end{aligned}

(58)

It is verifiable that Assumption 1 is satisfied when ${\bar{φ}}_{1} = 1, {\bar{φ}}_{2} = 1, {\bar{ν}}_{1} = 1, {\bar{ν}}_{2} = 1$ . Based on the above analysis, the design procedure for the adaptive neural network controller and the updating law can be presented as follows:

Construct the transformation of state-input scaling

z_{1} = \frac{x_{1}}{u_{0}}, z_{2} = x_{2} .

Introduce the transformations of coordinates

ε_{1} = z_{1}, ε_{2} = z_{2} - α_{1} .

Design the control law for partial input saturation

u_{0} (x_{0}) = \{\begin{cases} - 0.5 sign (x_{0}), & |x_{0}| > 0.5, \\ - x_{0}, & |x_{0}| \leq 0.5, \end{cases}

u = \frac{1}{H_{2}} [- c_{2} ε_{2} - ω_{1} ε_{2} - η_{2} H_{2}^{\frac{4}{3}} ε_{2} - (β_{2,1} + β_{2,2} + β_{2,3} + β_{2,4}) ε_{2}],

with

α_{1} = \frac{1}{H_{1}} [- c_{1} ε_{1} - η_{1} H_{1}^{\frac{4}{3}} ε_{1} - (β_{1,1} + β_{1,2} + β_{1,3} + β_{1,4}) ε_{1}],

τ_{1} (\hat{Θ}) = 1 + \frac{\sqrt{1 + {\hat{Θ}}^{2}}}{2} + \frac{1}{2} k_{1}^{2}, η_{1} (\hat{Θ}) = \frac{3}{4} {(\frac{1}{4 ξ_{1}})}^{(1 / 3)} τ_{1}^{(4 / 3)},

τ_{2} (\hat{Θ}) = 1 + \frac{\sqrt{1 + {\hat{Θ}}^{2}}}{2} + \frac{1}{2} k_{2}^{2}, η_{2} (\hat{Θ}) = \frac{3}{4} {(\frac{1}{4 ξ_{2}})}^{(1 / 3)} τ_{2}^{(4 / 3)},

β_{1,1} (x_{1}) = \frac{3}{4} {(\frac{1}{4 ω_{1}})}^{(1 / 3)} H_{1}^{(4 / 3)}, β_{1,2} (x_{1}) = {(\frac{3}{4} H_{1} ε_{1}^{2})}^{2},

β_{1,3} (x_{1}) = {(\frac{{({\underset{̲}{l}}_{1} {\bar{l}}_{1})}^{4} ε_{1}^{4}}{2 {({\bar{l}}_{1} - ε_{1})}^{5} {({\underset{̲}{l}}_{1} + ε_{1})}^{5}})}^{2}, β_{1,4} (x_{1}) = {(\frac{5 ({\underset{̲}{l}}_{1} - {\bar{l}}_{1} + 2 ε_{1}) H_{1} ε_{1}^{3}}{4 ({\bar{l}}_{1} - ε_{1}) ({\underset{̲}{l}}_{1} + ε_{1})})}^{2},

β_{2,1} (x_{2}) = \frac{3}{4} {(\frac{1}{4 ω_{2}})}^{(1 / 3)} H_{2}^{(4 / 3)}, β_{2,2} (x_{2}) = {(\frac{3}{4} H_{2} ε_{2}^{2})}^{2},

β_{2,3} (x_{2}) = {(\frac{{({\underset{̲}{l}}_{2} {\bar{l}}_{2})}^{4} ε_{2}^{4}}{2 {({\bar{l}}_{2} - ε_{2})}^{5} {({\underset{̲}{l}}_{2} + ε_{2})}^{5}})}^{2}, β_{2,4} (x_{2}) = {(\frac{5 ({\underset{̲}{l}}_{2} - {\bar{l}}_{2} + 2 ε_{2}) H_{2} ε_{2}^{3}}{4 ({\bar{l}}_{2} - ε_{2}) ({\underset{̲}{l}}_{2} + ε_{2})})}^{2} .

Design the adaptive update law as follows

\dot{\hat{Θ}} = \sum_{j = 1}^{2} \frac{ϒ H_{j} | ε_{j} |^{3}}{2} - ϖ_{0} \hat{Θ} .

Let n = 2, $V_{2} = \sum_{m = 1}^{2} \frac{{({\underset{̲}{l}}_{m} {\bar{l}}_{m})}^{4} ε_{m}^{4}}{4 {({\bar{l}}_{m} - ε_{m})}^{4} {({\underset{̲}{l}}_{m} + ε_{m})}^{4}} + \frac{1}{2 ϒ} {\tilde{Θ}}^{2}$ , then

\begin{array}{l} L V_{2} & \leq - \sum_{j = 1}^{2} c_{j} ε_{j}^{4} + Δ_{2} - \frac{ϖ_{0}}{2 ϒ} {\tilde{Θ}}^{2} + \frac{ϖ_{0}}{2 ϒ} Θ^{2} \\ \leq - \sum_{j = 1}^{2} 2^{- 8} c_{j} \frac{{({\underset{̲}{l}}_{j} {\bar{l}}_{j})}^{4} ε_{j}^{4}}{4 {({\bar{l}}_{j} - ε_{j})}^{4} {({\underset{̲}{l}}_{j} + ε_{j})}^{4}} - \frac{ϖ_{0}}{2 ϒ} {\tilde{Θ}}^{2} + \frac{ϖ_{0}}{2 ϒ} Θ^{2} + Δ_{2} \\ \leq - 2^{- 6} (\sum_{j = 1}^{2} \frac{{({\underset{̲}{l}}_{j} {\bar{l}}_{j})}^{4} ε_{j}^{4}}{4 {({\bar{l}}_{j} - ε_{j})}^{4} {({\underset{̲}{l}}_{j} + ε_{j})}^{4}} + \frac{1}{2 ϒ} {\tilde{Θ}}^{2}) + ρ \\ \leq - 2^{- 6} V_{2} + ρ . \end{array}

Therefore, (50) and (51) are satisfied for system (58) when c = 2⁻⁶, c₁ = 0.5, c₂ = 0.3, Γ = 10, ϖ₀ = 14.5, ϒ = 10. Moreover, taking k₁ = 2, k₂ = 1.75, η₀ = 1, ξ₁ = 0.8, ξ₂ = 0.6, μ₁₁ = 0.1, ω₁ = 0.1, and the initial values $x (0) = {(x_{0} (0), x_{1} (0), x_{2} (0))}^{T} = {(0.8, 0.2, 0.3)}^{T}$ , Theorems 1–3 can be satisfied. Therefore, the controller designed with an adaptive law for system (58) can ensure that the closed-loop signals are UUB.

From Figure 1, it can be observed that states x₀, x₁, x₂ are UUB. Clearly, this confines states x₀, x₁, x₂ within predetermined constraints. It is noteworthy that state x₀ globally asymptotically converges to zero and does not cross zero. In Figures 2 and 3, the control inputs u₀ and u are shown to be bounded. Figure 4 demonstrates the ultimate boundedness of the adaptive parameter.

Remark 7

In comparison with Peng et al., (2024), the proposed control framework addresses a more complex system dynamics while offering broader applicability. Simulation results demonstrate that our adaptive law and controller achieve significantly faster convergence and substantially reduced oscillation amplitude, confirming their superior stability and transient performance. These findings validate the enhanced robustness and control efficiency of the proposed strategy in handling system uncertainties.

Remark 8

It can be seen that, although Liu et al., (2018) considers the impact of AFSC on stochastic nonlinear systems, the use of piecewise barrier Lyapunov function complicates the design and stability analysis of the controller. In reference Gao et al., (2019), the authors address stochastic nonlinear systems with full state constraints, which require a greater number of adaptive parameters. In contrast to these approaches, the solution proposed in this paper is applicable to stochastic nonholonomic systems with AFSC and partial input saturation, utilizing only a single adaptive rate, thereby reducing the computational burden. In this paper, we highlight the specific advantages of this controller: Firstly, it accounts for systems with AFSC, ensuring that the system state strictly adheres to predefined boundaries, even with a relatively small range. Secondly, it employs only one adaptive rate, which reduces computational load and increases response speed. Thirdly, it considers the effects of partial input saturation, resulting in smaller control input amplitudes and reduced oscillations.

Remark 9

To verify the superiority of our method, we compare our control method with existing approaches for stochastic nonlinear systems, including both unconstrained methods (Sun et al., 2016; Zhao and Li 2025) and full-state constrained methods (Liu et al., 2018; Liu and Zhu, 2020). The simulation results confirm that the system states remain bounded under both our proposed controller and the reference methods (Sun et al., 2016; Liu et al., 2018; Zhao and Li, 2025; Liu and Zhu, 2020). However, it should be noted that the methods in (Sun et al., 2016; Zhao and Li, 2025) cannot guarantee the system states to evolve within predefined ranges, and none of the methods in (Sun et al., 2016; Liu et al., 2018; Zhao and Li, 2025; Liu and Zhu, 2020) consider partial input saturation issues. Furthermore, these existing approaches require more adaptive parameters for implementation.

Remark 10

To further demonstrate the superiority of our approach, we perform comparative simulations using Sun et al., (2016) as a benchmark, with the results clearly showing that our controller achieves better performance. Firstly, our method effectively handles strict state constraints within a compact range through novel constraint enforcement mechanisms. Secondly, we eliminate inherent system limitations by employing state coordinate transformation and a switching control strategy, thereby preventing uncontrollable phenomena. Third, the computational burden is significantly reduced through a simplified adaptive law structure. Moreover, by explicitly considering input saturation effects in system (58), our controller generates smoother control signals with reduced amplitude and minimal oscillations. These combined advantages lead to faster response times and superior overall control performance compared to conventional approaches.

Conclusion

This paper proposes an adaptive neural network control scheme based on a novel barrier Lyapunov function for stochastic nonholonomic systems subject to AFSC and partial input saturation. Theoretical analysis and numerical simulations demonstrate the effectiveness of the proposed method. The research findings hold significant potential for practical applications in intelligent mobile robotics, autonomous vehicles, and aerospace systems, offering new solutions for motion control in complex environments, path tracking, and attitude control problems. Future research will focus on experimental validation, extension to higher-order systems, handling of time-varying constraints, multi-agent cooperative control, and computational efficiency optimization. Furthermore, the integration of multi-modal perception and interactive learning mechanisms, such as the vision-language model for industrial defect detection proposed in Shen et al., (2025), could be explored to enhance situational awareness and decision-making. These investigations will further advance the practical implementation of the proposed method and provide more comprehensive solutions for nonlinear system control under complex constraints.

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Ningxia (2025AAC030133).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Adaptive neural network control for a class of stochastic nonholonomic systems with asymmetric full-state constraints

Abstract

Keywords

Introduction

Preliminaries

Stochastic theory

Neural networks theory

Problem formulation

Controller design

Design u0 for x0 − subsystem

Design u for x − subsystem

Stability analysis

Simulation example

Conclusion

Footnotes

Funding

Declaration of conflicting interests

References

Design u₀ for x₀ − subsystem