Research on Improved Adaptive Control Methods for High-Order Nonlinear System Using Finite-Time Techniques

Abstract

This study develops four enhanced adaptive finite-time control strategies for high-order nonlinear maritime systems, addressing critical limitations in conventional backstepping controllers when applied to vehicles operating under uncertain oceanic conditions. The research proposes four advanced control strategies: multilayer adaptive control, neural network-based adaptive control, model predictive control (MPC) combined with backstepping, and event-triggered control employing barrier Lyapunov functions. These methods have been validated through simulations on a third-order nonlinear system and a cart-pendulum system. Comprehensive MATLAB simulations on third-order nonlinear systems demonstrate substantial performance improvements: the neural network controller achieves 88% faster convergence with 93.4% higher tracking accuracy, the MPC-backstepping approach reduces control energy by 32.7%, and the event-triggered method cuts computational load by 93% while maintaining strict state constraints. Quantitative analysis reveals steady-state error reductions from 0.085 to 0.014 (83.5% improvement) and settling time decreases from 1.2 s to 0.144 s (88% improvement) compared to conventional finite-time backstepping controllers. Furthermore, the proposed controllers were experimentally validated on practical applications, including robotic manipulators, quadrotor UAVs, and industrial hydraulic systems, exhibiting outstanding performance in highly nonlinear and noisy environments.

Keywords

adaptive control nonlinear systems finite-time control neural networks model predictive control event-triggered control

1. Introduction

High-order nonlinear maritime systems present substantial control challenges due to parametric uncertainties, unmodeled dynamics, and external oceanic disturbances including waves, currents, and wind loads (Sheng et al., 2023; Yao et al., 2024; Zhao et al., 2015). These complexities necessitate advanced control methodologies that can ensure robust performance while maintaining computational efficiency for onboard systems (Bhat & Bernstein, 2000; Chen & Liao, 2002; Hwang & Jung, 2023; Krstić et al., 1995). The inherent nonlinear characteristics, parameter uncertainties, and external disturbances complicate the design of effective controllers. Backstepping control (Vaidyanathan & Azar, 2021; Zhou et al., 2008) combined with finite-time techniques (Sarkar & Rakhlin, 2019; Zhao and Jia, 2015) has been widely investigated to ensure system state convergence within predetermined time intervals (Chen et al., 2000; Ge & Wang, 2004; Mayne et al., 2000). However, conventional approaches still exhibit several notable limitations.

Traditional finite-time backstepping controllers face various challenges, including sensitivity to disturbances, computational inefficiency, difficulty in maintaining state constraints, and limited performance in handling complex nonlinearities. The sensitivity issue arises from using time-barrier functions (Ahmadi et al., 2019) such as $\frac{1}{T_{p} - t}$ , which significantly increases control gain as time approaches $T_{p^{'}}$ , making systems highly sensitive to noise (Di et al., 2022; Li & Shi, 2014). Additionally, continuous control updates required by conventional methods lead to considerable computational resource consumption (Ames et al., 2019; Chen et al., 2020). Traditional methods also struggle to guarantee that system states remain within predefined safety boundaries (Sun et al., 2021; Wang et al., 2022). Furthermore, conventional controllers often underperform when handling highly nonlinear or incompletely modeled systems (Pham et al., 2024; Yu et al., 2005).

While existing finite-time control approaches (Chen et al., 2000; Ge and Wang, 2004; Mayne et al., 2000; Sarkar & Rakhlin, 2019; Zhao & Jia, 2015) demonstrate theoretical convergence properties, they exhibit significant practical limitations when implemented on platforms. Specifically, conventional time-barrier functions cause excessive control gains near convergence time, leading to actuator saturation and noise amplification in harsh environments. Furthermore, existing methods lack integrated mechanisms for simultaneous parameter estimation, disturbance compensation, and computational optimization—critical requirements for autonomous vehicles with limited onboard processing capabilities.

This research addresses these gaps through four novel contributions: (1) A multilayer adaptive architecture incorporating anti-windup mechanisms and adaptive disturbance estimation specifically designed for parameter uncertainties, (2) An RBF neural network controller with online learning algorithms optimized for maritime nonlinearities, (3) A hybrid MPC-backstepping framework with recursive system identification for energy-efficient operations, and (4) An event-triggered control strategy utilizing barrier Lyapunov functions to ensure state constraints while reducing computational demands for real-time applications.

In recent years, several improved techniques, such as neural network-based approaches (Nascimento et al., 2000; Pham & Han, 2022; Pham & Han, 2023a; Pham & Han, 2023b), model predictive control (MPC) (Afram and Janabi-Sharifi, 2014; Köhler et al., 2024; Kouvaritakis & Cannon, 2016; Schwenzer et al., 2021; Serale et al., 2018), and event-triggered control (Heemels et al., 2012; Nakayama et al., 2024; Sunil et al., 2024; Zhang et al., 2024), have been proposed. However, these studies typically focus on isolated aspects without offering comprehensive comparisons among various enhancement methods.

This research addresses these gaps by proposing and evaluating four improved finite-time adaptive control methods aimed at enhancing overall performance. Key contributions include the development of a multilayer adaptive controller incorporating anti-windup mechanisms, adaptive disturbance estimation, adaptive gain tuning, and integral actions to significantly improve baseline performance. Additionally, an adaptive controller integrated with a Radial Basis Function (RBF) neural network is designed with online learning mechanisms to approximate and compensate for unmodeled nonlinearities, thereby enhancing accuracy and disturbance rejection capabilities. A hybrid MPC-backstepping control strategy is proposed, integrating online model identification to optimize energy efficiency and predict future trajectories. An event-triggered control approach employing Lyapunov, and barrier functions is developed to substantially reduce computational demands and ensure state constraints. A comprehensive evaluation of these methods through simulations and experimental tests across multiple platforms is provided, along with rigorous mathematical proofs of stability and detailed guidelines for practical implementation.

Existing finite-time control approaches can be categorized into three main streams:

Time-Barrier Function Methods (Ge & Wang, 2004; Sarkar & Rakhlin, 2019; Zhao & Jia, 2015): Zhao et al. (Zhao and Jia, 2015) pioneered time-barrier functions for finite-time convergence but suffered from excessive control gains near terminal time. Their approach lacks robustness to parameter uncertainties—a critical limitation for applications where model parameters vary significantly with loading conditions and environmental factors.

Sliding Mode Approaches (Ahmadi et al., 2019; Yu et al., 2005): Yu et al. (2005) developed terminal sliding mode controllers achieving finite-time convergence. However, these methods exhibit chattering phenomena that can damage actuators and excite unmodeled high-frequency dynamics. The discontinuous control nature is particularly problematic for vehicles requiring smooth actuation for passenger comfort and equipment longevity.

Neural Network-Based Methods (Nascimento et al., 2000; Pham & Han, 2022; Pham & Han, 2023a; Pham & Han, 2023b): Recent works by Nascimento et al. (2000) and our previous studies (Pham & Han, 2022; Pham & Han, 2023a; Pham & Han, 2023b) demonstrate neural networks’ effectiveness in handling nonlinearities (Gautam, 2024). However, existing approaches lack integration with finite-time convergence guarantees and fail to address computational constraints of embedded systems.

Research Gaps Identified:

Lack of integrated parameter estimation and disturbance rejection in finite-time controllers.

Insufficient consideration of actuator constraints and computational limitations.

Absence of event-triggered mechanisms for bandwidth-limited communication systems.

Limited comparative analysis of different enhancement approaches under realistic conditions.

Recent Advancements Integration:

Recent developments in resilient consensus control (Ali et al., 2017; Sharafian et al., 2025) and energy-aware algorithms provide valuable insights for cooperative control scenarios. These concepts inspire our event-triggered approach and energy-efficient MPC integration.”

Recent finite-time adaptive control research, such as the work by (Min et al., 2019; Pham et al., 2025), has made significant contributions to ensuring finite-time convergence for uncertain nonlinear systems through adaptive parameter estimation and Lyapunov-based stability analysis. While these foundational approaches establish theoretical convergence guarantees, several practical limitations remain unaddressed.

Our work differs from existing finite-time adaptive control methods in three key aspects:

Multi-Method Integration: Unlike single-technique approaches, we present four distinct enhancement strategies (multilayer adaptive, neural network-based, MPC-backstepping, event-triggered) providing comprehensive solutions for different application requirements.

Computational Efficiency Focus: While traditional finite-time methods prioritize convergence speed, our event-triggered approach achieves 93% computational reduction while maintaining performance—critical for embedded systems with limited processing capabilities.

Practical Implementation Analysis: Beyond theoretical convergence proofs, we provide detailed computational profiling, simulation-to-reality gap analysis, and specific implementation guidelines addressing real-world deployment challenges.

These enhancements advance beyond conventional finite-time adaptive control by integrating multiple advanced techniques while maintaining rigorous stability guarantees and practical feasibility.

The paper is structured as follows: Section 2 presents theoretical backgrounds and system models; Section 3 details the proposed control methods; Section 4 presents simulation results and analyses; Section 5 provides stability analysis; Section 6 discusses practical implementations and applications; Section 7 analyzes sensitivity and robustness; Section 8 addresses parameter estimation and adaptability analysis; Section 9 evaluates overall performance comprehensively; and Section 10 concludes the study and suggests future research directions.

2. Theoretical Background and System Models

This study examines two representative nonlinear systems widely encountered in practical applications.

2.1. Mathematical Models

2.1.1 Third-Order Nonlinear System

The mathematical model of the third-order nonlinear system is expressed as follows:

\begin{aligned} {\begin{matrix} {\dot{x}}_{1} = x_{2} + θ x_{1} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = u + θ x_{3}^{2} + d (t) \end{matrix} \end{aligned}

(1)

where

x = [x_{1}, x_{2}, x_{3}]^{T} \in R^{3}

denotes the state vector,

u \in R

is the control input,

θ

represents the uncertain parameter requiring estimation, and

d (t)

denotes external disturbances. This model represents various practical systems such as robotic manipulators, mechatronic systems, and industrial processes.

2.1.2 Cart-pendulum System

The second-order mathematical model of the cart-pendulum system is described by:

\begin{aligned} {\begin{matrix} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = u - θ x_{2} - \frac{1}{2} m g l \sin (\frac{x_{1}}{M}) \end{matrix} \end{aligned}

(2)

where

x_{1}

represents the cart position,

x_{2}

is the cart velocity, M denotes cart mass, m is the pendulum mass, l is the pendulum length, g is gravitational acceleration, and

θ = \frac{ω}{M}

represents the uncertain parameter. This model captures the dynamics of mechanical systems with flexible joints and mobile robotic applications.

2.2. Finite-Time Backstepping Control Method

The traditional finite-time backstepping control method utilizes a time-barrier function $\frac{1}{T_{p} - t}$ to ensure state convergence within a specified finite time $T_{p}$ . For a third-order system, the control design involves several steps:

2.2.1 Backstepping Design for Third-Order Systems

Define the first tracking error variable and virtual control law:

\begin{aligned} z_{1} = x_{1}, \; α_{1} = - σ_{1} \frac{z_{1}}{T_{p} - t} - \hat{θ} f_{1} \end{aligned}

(3)

where $σ_{1} > 0$ is a design parameter, $\hat{θ}$ is the parameter estimate, and $f_{1} = x_{1}$ is a nonlinear function.

Define the second tracking error variable and second virtual control:

\begin{aligned} z_{2} = x_{2} - α_{1}, \; α_{2} = - σ_{2} \frac{z_{2}}{T_{p} - t} - z_{1} - \hat{θ} f_{2} + \frac{\partial α_{1}}{\partial \hat{θ}} τ + \frac{\partial α_{1}}{\partial x_{1}} (x_{2} + \hat{θ} f_{1}) + \frac{\partial α_{1}}{\partial t} \end{aligned}

(4)

where

σ_{2} > 0

is a design parameter,

f_{2} = 0

for this system, and the partial derivatives are calculated as:

\begin{aligned} \frac{\partial α_{1}}{\partial \hat{θ}} = - f_{1}, \; \frac{\partial α_{1}}{\partial x_{1}} = - σ_{1} \frac{1}{T_{p} - t} - \hat{θ}, \; \frac{\partial α_{1}}{\partial t} = - σ_{1} z_{1} \frac{1}{{(T_{p} - t)}^{2}} \end{aligned}

(5)

Define the third tracking error variable and the final control law:

\begin{aligned} z_{3} = x_{3} - α_{2}, \; u = - σ_{3} \frac{z_{3}}{T_{p} - t} - z_{2} - \hat{θ} f_{3} + ϕ_{2} \end{aligned}

(6)

where

σ_{3} > 0

is a design parameter,

f_{3} = x_{3}^{2}

, and

ϕ_{2}

represents a complex nonlinear function determined by the backstepping procedure. The adaptive parameter estimation law is given as:

\begin{aligned} \overset{\cdot}{\hat{θ}} = τ = z_{1} f_{1} + z_{3} f_{3} \end{aligned}

(7)

2.2.2 Analysis of Limitations of the Traditional Method

The traditional finite-time backstepping control method has several critical limitations. The first significant issue is the chattering phenomenon. As t approaches $T_{p}$ , the control gain $\frac{1}{T_{p} - t}$ increases excessively, causing chattering in the control signal. Furthermore, this high gain makes the system highly sensitive to disturbances, especially near the final convergence time $T_{p}$ . Additionally, the traditional approach lacks explicit mechanisms for enforcing state constraints, making it difficult to ensure that states always remain within predefined safety limits. Finally, traditional methods exhibit limited adaptability to unmodeled nonlinearities, significantly affecting their performance in practical scenarios. These limitations motivate the development of improved control methodologies, which are presented in the subsequent sections.

3. Theoretical Background and System Models

3.1. Multilayer Adaptive Controller

The multilayer adaptive controller has been enhanced by integrating the following advanced features:

3.1.1 Anti-Windup Mechanism

To prevent unlimited parameter accumulation, we propose a constrained adaptive law:

\begin{aligned} \overset{\cdot}{\hat{θ}} = γ_{1} sat\; (\frac{z_{1} f_{1} + z_{2} (f_{2} - \frac{\partial α_{1}}{\partial x_{1}} f_{1}) + z_{3} f_{3}}{α_{max}}) α_{max} \end{aligned}

(8)

where

γ_{1} > 0

is the adaptive learning gain,

α_{max}

is the parameter estimation bound, and the saturation function

sat (\cdot)

is defined as follows:

\begin{aligned} sat (s) = {\begin{array}{ll} s, & \; if\; | s | \leq 1 \\ sign (s), & \; if\; | s | > 1 \end{array} \end{aligned}

(9)

The anti-windup structure offers several advantages, particularly in applications. One key benefit is parameter bound enforcement, where the saturation function ensures that the estimated parameters, ∣θ $^{\land}$ ∣, do not exceed the maximum value of αmax, preventing unrealistic estimates that could destabilize the system, especially during initialization or under severe disturbances. The structure also enhances robustness by preventing excessive parameter drift during large transient errors, which are common during wave impacts. Additionally, the anti-windup mechanism helps protect actuators by limiting adaptive gains and preventing control signal saturation, which could otherwise damage the actuators. It also ensures consistent performance by maintaining repeatable controller behavior across various operating conditions and mission scenarios.

However, there are also some drawbacks. The parameter saturation can slow down the adaptation speed, particularly when the true parameters are near the saturation boundaries, potentially extending convergence time by 15–25% in the worst-case scenarios. Moreover, the fixed bounds (αmax) may be too restrictive for certain operating conditions, which limits the system's adaptability when rapid parameter changes are necessary. Additionally, if the true parameters exceed the saturation bounds, persistent steady-state errors may arise, necessitating careful selection of bounds based on system identification studies. Finally, tuning the bounds requires prior knowledge of the system's parameter ranges, which adds complexity to the design process.

For applications, it is recommended to set αmax = 1.5θnominal, where θnominal is derived from system identification under nominal loading conditions. This provides a 50% adaptation margin while preventing excessive parameter drift.

The saturation function in equation (9) exhibits non-differentiability at ∣s∣ = 1, which can lead to several practical issues. One of the primary problems is gradient discontinuity, as Lyapunov stability analysis requires smooth functions, but the derivative of the saturation function, ddssat(s), is undefined at switching points. Additionally, numerical integration can be problematic; MATLAB's ode45 solver may experience step-size reductions near these switching points, resulting in increased computation time. Another issue is control signal chattering, where rapid switching between the linear and saturated regions causes high-frequency oscillations in the control signal.

To address these issues, we propose a smooth saturation approximation using the function satsmooth(s) = tanh(βs), where β = 10. This approximation offers continuous differentiability, with the derivative given by ddssatsmooth(s) = β(1−tanh²(βs)). It also provides a close approximation to the original saturation function, with the error satisfying ∣sat(s) - satsmooth(s)∣ < 0.01 for ∣s∣ ≤ 2. The smooth approximation effectively eliminates chattering by ensuring smooth transitions between the linear and saturated regions.

Simulations using the smooth saturation approximation demonstrate several advantages. There is a 12% reduction in control signal variations and an 8% improvement in numerical integration speed, as fewer ODE solver steps are required. Additionally, the performance degradation is minimal, with less than a 2% difference in convergence time. Importantly, the smooth approximation maintains the essential boundedness property of the original saturation function, enabling rigorous Lyapunov analysis with continuous derivatives and ensuring the stability of the system.

3.1.2 Adaptive Disturbance Estimation

To address unmeasurable disturbances, an adaptive disturbance estimator is designed as:

\begin{aligned} \overset{\cdot}{\hat{d}} = γ_{2} \frac{z_{1} + z_{2} + z_{3}}{\sqrt{1 + z_{1}^{2} + z_{2}^{2} + z_{3}^{2}}} \end{aligned}

(10)

where

γ_{2} > 0

is the adaptive learning coefficient for disturbance estimation. The normalization in the denominator helps prevent excessive accumulation when the state variables become large.

3.1.3 Adaptive Gain Adjustment

To improve the trade-off between response speed and noise sensitivity, an adaptive gain adjustment mechanism is introduced:

\begin{aligned} \dot{λ} = 0.5 (z_{1}^{2} + z_{2}^{2} + z_{3}^{2} - 0.1 λ) \end{aligned}

(11)

where λ is the adaptive gain parameter. This mechanism allows the adaptive gain to increase when the error is large and decrease as the system approaches equilibrium.

3.1.4 Integral Action

To eliminate steady-state errors and enhance disturbance rejection capabilities, an integral action term is incorporated into the control law as follows:

\begin{aligned} u = - σ_{3} \frac{z_{3}}{T_{p} - t} - z_{2} - \hat{θ} f_{3} - κ_{3} sat (\frac{z_{3}}{0.1}) - \hat{d} - β \int z_{1} d t \end{aligned}

(12)

where

κ_{3} > 0

is the sliding-mode control parameter enhancing robustness, and

β > 0

is the integral control gain.

3.2. Neural Network-Based Adaptive Controller

To effectively manage uncertain nonlinearities, we integrate a Radial Basis Function (RBF) neural network into the adaptive controller.

3.2.1 RBF Neural Network Structure

The RBF neural network approximation function is defined as:

\begin{aligned} \hat{f} (x) = \sum_{i = 1}^{N} w_{i} ϕ_{i} (x) = W^{T} Φ (x) \end{aligned}

(13)

where

W = [w_{1}, w_{2}, \dots, w_{N}]^{T}

is the weight vector, and

Φ (x) = [ϕ_{1} (x), ϕ_{2} (x), \dots, ϕ_{N} (x)]^{T}

is the vector of basis functions. Each Gaussian radial basis function is defined as:

\begin{aligned} ϕ_{i} (x) = exp (- \frac{‖ x - {c_{i}}^{2} ‖}{b_{i}^{2}}) \end{aligned}

(14)

where

c_{i} \in R^{n}

represents the center of the

i^{t h}

neuron, and

b_{i} > 0

denotes the Gaussian function width.

3.2.2 Weight Update Law

To ensure neural network weights convergence, we propose the weight update law:

\begin{aligned} \dot{W} = γ_{3} Φ (x) z_{1} - β W \end{aligned}

(15)

where

γ_{3} > 0

is the learning rate, and

β > 0

is the weight damping factor to prevent excessive accumulation.

\begin{aligned} u = - σ_{2} \frac{z_{2}}{T_{p} - t} - z_{1} - \hat{θ} x_{2} - \hat{f} (x) - κ tanh (10 z_{2}) λ - \hat{d} \end{aligned}

(16)

where

κ > 0

is the sliding control parameter, and

λ

is the adaptive gain parameter. The hyperbolic tangent function

tanh (\cdot)

is used instead of the sign function to reduce chattering.

3.2.3 Neural Network Architecture Justification

The comparison between Shallow and Deep Network Analysis reveals that the Shallow Radial Basis Function (RBF) Network, with a 3-15-1 architecture, offers several advantages in applications. It achieves fast convergence, taking only 0.087 s per step, and guarantees universal approximation for continuous functions, with interpretable Gaussian basis functions. For specific uses, the shallow RBF network is robust to sensor noise and provides smooth approximation surfaces suitable for control applications.

In contrast, the deep network alternative, tested as a 3-10-10-1 multi-layer perceptron, requires a significantly longer training time of 0.296 s per step, which is 340% longer than the shallow network. Despite this, the accuracy improvement is marginal, at just 1.2%. Thus, the shallow RBF network provides an optimal balance between complexity and performance for control applications.

For the RBF network optimization, the center selection strategy involves uniformly distributing the centers within the expected state space, specifically within the range ci∈[−2,2]³, based on the analysis of the vehicle's operating envelope. The width parameter is set to bi = 0.5, selected through grid search optimization, ensuring a balance between approximation accuracy and generalization capability. Additionally, the online learning mechanism uses a weight adaptation rate of γ₃ = 5 and a forgetting factor of λ = 0.99, preventing weight drift while maintaining the network's adaptation capability.

Performance metrics for the RBF network show an approximation accuracy of 98% for polynomial nonlinearities and 94% for trigonometric functions. The convergence time for weight stabilization is 0.15 s, with memory requirements of 4.5MB for weight storage and computation.

3.3. MPC-Backstepping Hybrid Controller

This approach combines advantages of backstepping control and Model Predictive Control (MPC) to optimize future trajectories.

3.3.1. Online Model Identification

The Recursive Least Squares (RLS) method is employed for online system identification:

\begin{aligned} {\hat{θ}}_{M P C}^{\cdot} = P ϕ \frac{x_{3} - ϕ^{T} {\hat{θ}}_{M P C}}{α + ϕ^{T} P ϕ}, \; \dot{P} = \frac{1}{α} P - \frac{1}{α} \frac{P ϕ ϕ^{T} P}{α + ϕ^{T} P ϕ} \end{aligned}

(17)

where

{\hat{θ}}_{M P C} \in R^{3}

is the MPC model parameter vector,

ϕ = [x_{1}, x_{2}, x_{3}^{2}]^{T}

is the regression vector,

P \in R^{3 \times 3}

is the covariance matrix, and

α \in (0, 1)

is the forgetting factor enhancing adaptability.

3.3.2. Future State Prediction

Using the identified model, future states are predicted as:

\begin{aligned} \hat{x} (k + i ∣ k) = A_{M P C} \hat{x} (k + i - 1 ∣ k) + B_{M P C} u (k + i - 1) \end{aligned}

(18)

where

\hat{x} (k + i ∣ k)

represents the predicted state at time

k + i

based on information at time

k . A_{M P C}

and

B_{M P C}

are system matrices constructed from

{\hat{θ}}_{M P C}

3.3.3. MPC Cost Function Calculation

Following the constrained MPC formulation (Mayne et al., 2000), the optimization problem incorporates both control and state constraints. The MPC cost function is defined with exponentially decreasing weights along the prediction horizon:

\begin{aligned} J = \sum_{i = 1}^{N_{p}} w_{i} (\hat{x} {(k + i ∣ k)}^{T} Q \hat{x} (k + i ∣ k) + u {(k + i - 1)}^{T} R u (k + i - 1)) \end{aligned}

(19)

where

N_{p}

is the prediction horizon,

w_{i} = exp (- 0.2 (i - 1))

provides higher priority to nearer predictions, and

Q \in R^{3 \times 3}, R \in R

are weighting matrices for state and control inputs.

The selection of weighting matrices $Q$ and $R$ in Equation (19) is critical for achieving the desired closed-loop dynamics and ensuring system stability. The matrices are designed based on the following principles:

State Weighting Matrix Q:

\begin{aligned} Q = diag (q_{1}, q_{2}, q_{3}) \end{aligned}

(19a)

The diagonal structure ensures decoupled state penalties, where:

$q_{1}$ : High penalty on position tracking error $x_{1}$ , ensuring precise trajectory following.

$q_{2}$ : Medium penalty on velocity error $x_{2}$ , balancing responsiveness and smoothness.

$q_{3}$ : Lower penalty on acceleration $x_{3}$ , preventing excessive control activity.

Control Weighting Matrix R:

\begin{aligned} R \approx 0.1 \end{aligned}

(19b)

The small control weight emphasizes tracking performance over control effort, suitable for applications where high precision is prioritized over energy consumption.

Stability Analysis with Respect to Q and R Selection:

The closed-loop stability is guaranteed when the matrices satisfy the following conditions:

Positive Definiteness:

\begin{aligned} Q ≻ 0, \; R > 0 \end{aligned}

(19c)

ensure the cost function is strictly convex, guaranteeing unique optimal solutions.

Detectability Condition:

The pair ( $A_{MPC}, Q^{1 / 2}$ ) must be detectable, which is satisfied since:

\begin{aligned} rank [\begin{matrix} Q^{1 / 2} \\ A_{MPC} - λ I \end{matrix}] = 3, \; \forall | λ | \geq 1 \end{aligned}

(19d)

Stabilizability Condition:

The pair ( $A_{MPC}, B_{MPC}$ ) must be stabilizable, verified through:

\begin{aligned} rank [B_{MPC}, A_{MPC} B_{MPC}, A_{MPC}^{2} B_{MPC}] = 3 \end{aligned}

(19e)

Dynamic Response Shaping:

The eigenvalues of the closed-loop system ( $A_{MPC} - K_{MPC} C_{MPC}$ ) can be explicitly designed through Q and R selection:

\begin{aligned} λ_{cl} = [- 2.1 \pm 1.8 j, - 3.2] \end{aligned}

(19f)

These eigenvalues provide:

Damping ratio $ζ = 0.76$ (well-damped response)

Natural frequency $ω_{n} = 2.7 rad / s$ (appropriate bandwidth for maritime applications)

Real pole at −3.2 ensures fast disturbance rejection.

Sensitivity Analysis:

The controller maintains stability margins when Q and R vary within the following bounds:

$Q_{i i} \in [0.1 q_{i i}, 10 q_{i i}]$ maintains gain margin $> 6 \; dB$

$R \in [0.01, 1.0]$ maintains phase margin $> 45^{\circ}$

This robust stability ensures practical implementation reliability across varying operating conditions.

The gradient of the MPC cost function is computed as:

\begin{aligned} \nabla J = \sum_{i = 1}^{N_{p}} w_{i} (2 \hat{x} {(k + i ∣ k)}^{T} Q B_{M P C} + 2 R u (k + i)) \end{aligned}

(20)

3.3.4. Combined Backstepping and MPC Control Law

A combined control law is proposed to exploit benefits from both methods:

\begin{aligned} u = 0.7 u_{B S} + 0.2 u_{M P C} + 0.1 u_{smooth\;} \end{aligned}

(21)

where

u_{B S}

is the traditional backstepping control input,

u_{M P C} = - 0.2 \nabla J

represents the MPC-based control component (with 0.2 being the learning factor for gradient-based adjustment), and

u_{smooth\;} = 0.2 (u_{prev\;} - u_{B S})

is the smoothing term designed to reduce fluctuations in the control signal.

3.4. Event-Triggered Control with Barrier Lyapunov Functions

This method significantly reduces computational load and enforces state constraints by updating control inputs only when necessary.

3.4.1. Barrier Lyapunov Functions

To ensure state constraints are strictly enforced, we use the barrier Lyapunov functions defined as:

\begin{aligned} B_{i} = log (\frac{x_{i, max}^{2} - x_{i}^{2}}{x_{i, max}^{2}}), \; B = γ_{B} (B_{1} + B_{2}) \end{aligned}

(22)

where

x_{i, max}

is the safe boundary for the state

x_{i}

, and

γ_{B} > 0

is a design parameter for the barrier function.

The derivative of the barrier function with respect to the states is given by:

\begin{aligned} \frac{\partial B_{i}}{\partial x_{i}} = - \frac{2 x_{i}}{x_{i, max}^{2} - x_{i}^{2}} \end{aligned}

(23)

3.4.2. Event-Triggered Mechanism

To reduce computational burden, we define the event-triggering variable and triggering condition as follows:

\begin{aligned} e_{event\;} = ‖ x - x (t_{last}) ‖ \end{aligned}

(24)

The event-triggering condition is:

\begin{aligned} e_{event\;} > η V (x) + ϵ \; \; and\; \; t - t_{last\;} \geq δ \end{aligned}

(25)

where

x (t_{last\;})

is the state at the last control update,

V (x)

is the system's Lyapunov function,

η > 0

and

ε > 0

are design parameters for the triggering condition, and

δ > 0

is the minimum interval between updates to prevent Zeno behavior. This approach extends the distributed event-triggered consensus methods of Zhang et al. (2024) to single-agent nonlinear systems, while incorporating the adaptive fuzzy principles demonstrated by Mercorelli (2018) for enhanced robustness in uncertain environments.

3.4.3. Event-Triggered Control Law

The control law is updated only when the triggering condition is satisfied:

\begin{aligned} u = {\begin{matrix} u_{f u l l}, if\; the\; triggering\; condition\; is\; satisfied \\ u_{last\;}, otherwise \end{matrix} \end{aligned}

(26)

with the full control law defined as:

\begin{aligned} u_{f u l l} = - σ_{2} \frac{z_{2}}{T_{p} - t} - z_{1} - \hat{θ} x_{2} - f_{nonlinear\;} - κ_{B} tanh (10 z_{2}) - γ_{B} \frac{\partial B}{\partial x_{2}} \end{aligned}

(27)

where

κ_{B} > 0

is the sliding control parameter, and the term

γ_{B} \frac{\partial B}{\partial x_{2}}

enforces the state constraints via the barrier Lyapunov function.

3.4.4. Mitigation of Switching Effects

To reduce potential transient effects due to switching between control values, we propose a smoothing mechanism as follows:

\begin{aligned} u_{smooth\;} (t) = u_{last\;} + (u_{full\;} - u_{last\;}) (1 - e^{- α (t - t_{trigger\;})}) \end{aligned}

(28)

where

α > 0

controls the rate of transition between control values, and

t_{trigger\;}

is the most recent triggering time. This smoothing mechanism ensures a smooth transition, minimizing excitation of highorder dynamic modes.

3.5. Implementation Details and Computational Analysis

3.5.1. Neural Network Training Specifications

The RBF neural network implementation utilizes:

Network architecture: 3 input nodes → 15 hidden neurons → 1 output node

Gaussian RBF centers: Uniformly distributed in $[- 2, 2]^{3}$

Width parameters: $b_{i} = 0.5$ for all neurons

Learning rate: $γ_{3} = 5$ with exponential decay factor 0.99

Weight initialization: Random normal distribution $(0, 0.1)$

Training Time Comparison (MATLAB R2024a, Intel i7-13700KF):

Multilayer Adaptive: 0.023 s average per simulation step

Neural Network: 0.087 s average per simulation step (including weight updates)

MPC-Backstepping: 0.156 s average per simulation step

Event-Triggered: 0.008 s average per simulation step (during non-triggering periods)

Traditional Backstepping: 0.019 s average per simulation step

3.5.2. MPC Implementation Details

The MPC optimization problem is solved using MATLAB's quadratic programming solver with:

Prediction horizon: $N_{p} = 10$ steps

Control horizon: $N_{c} = 5$ steps

Sampling time: $T_{s} = 0.01$ seconds

Convergence tolerance: $10^{- 6}$

Maximum iterations: 50 per optimization cycle

3.6. Implementation Details and Computational Analysis

3.6.1. Decision Logic for Controller Selection

The proposed controllers are designed to address different operational scenarios rather than real-time switching. The Multilayer Adaptive Controller is intended for general-purpose vehicle control, where it requires moderate computational resources and must balance performance requirements. Optimal conditions for this controller are when the disturbance ∥d(t)∥ is less than or equal to 0.5, ensuring steady-state operation.

The Neural Network Controller is suited for high-precision operations, such as dynamic positioning and docking, where high computational capacity and maximum accuracy are essential. This controller is most effective when handling complex nonlinearities, particularly when the function Δf(x) exceeds 0.2, indicating a need for precise control under dynamic conditions.

The MPC-Backstepping Controller is designed for energy-constrained missions, such as those involving autonomous underwater vehicles, where the environment is predictable, and energy optimization is a priority. This controller is ideal for long-duration missions, where trajectory references are known in advance.

The Event-Triggered Controller is used in networked systems with limited communication bandwidth, typically in scenarios involving the coordination of multiple vehicles. It is suitable when communication bandwidth is below 10 Hz and distributed control is necessary to manage multiple vehicles effectively.

3.6.2. Implementation Strategy

Rather than online switching, we recommend:

Mission Planning Phase: Select controller based on mission requirements and available resources

Parameter Tuning: Adjust selected controller for specific platform characteristics

Backup Controller: Implement simplified event-triggered controller as failsafe mode

This approach avoids switching-induced transients while providing flexibility for different applications.

4. Simulation Results and Analysis

4.1. Simulation Setup

To evaluate the proposed controllers, simulations were conducted on a third-order nonlinear system and a cart-pendulum system with the following parameters:

- Third-order nonlinear system:

- Uncertain parameter: $θ = 0.5$

- Finite-time horizon: $T_{p} = 5$

- Control parameters: $σ_{1} = σ_{2} = σ_{3} = 5 - 8$ (depending on methods)

- Adaptive parameters: $γ_{1} = 2, γ_{2} = 1, γ_{3} = 5$

- Event-triggered parameters: $η = 0.1, ϵ = 0.01, δ = 0.02$

Cart-pendulum system:

- Cart mass: $M = 1 \; kg$

- Pendulum mass: $m = 0.5 \; kg$

- Pendulum length: $l = 1 \; m$

- Gravity: $g = 9.81 \; m / s^{2}$

- Uncertain parameter: $θ = 1$

- Finite-time horizon: $T_{p} = 0.5$

Initial conditions were selected to assess robustness:

- Third-order system: $x_{1} (0) \in {1, - 1}, x_{2} (0) \in {- 2, 2}, x_{3} (0) \in {1, - 1}$

- Cart-pendulum: $(x_{1} (0), x_{2} (0)) \in {(5, - 5), (3, - 3), (1, - 1)}$

Simulations were implemented in MATLAB R2024a using the ode45 solver with relative and absolute tolerances set to $1 e^{- 6}$ .

4.2. Results for the Nonlinear System

4.2.1 Multilayer Adaptive Controller Performance

Figure 1 presents state responses, control signals, and parameter estimation performance of the multilayer adaptive controller compared to the traditional method. The multilayer adaptive controller significantly improved convergence time by approximately sometimes up to 53.1%. Overshoot was reduced, steady-state error decreased by approximately sometimes up to 88.5%, and control energy consumption was lowered by approximately sometimes up to 8.5%. Parameter estimation accuracy improved notably, reducing parameter estimation errors by approximately sometimes up to 86.2% and decreasing parameter convergence time by approximately sometimes up to 56.5%. The adaptive disturbance estimation effectively compensated external disturbances.

Figure 1.

Comprehensive Performance Comparison Between Traditional Finite-Time Backstepping Controller (Blue Lines) and Proposed Multilayer Adaptive Controller (Red Lines) for Third-Order Nonlinear System with Initial Conditions x₁(0) = 1, x₂(0) = −2, x₃(0) = 1 and Disturbance d(t) = 0.1sin(2t).

A detailed performance analysis of the system presented in Figure 1 reveals significant improvements with the multilayer adaptive controller compared to the traditional controller in several key areas.

For the state trajectory, the x₁ response shows that the traditional controller experiences an initial transient overshoot of 23%, peaking at 1.23 at t = 0.18 s, while the multilayer adaptive controller achieves a more controlled approach with only 8% overshoot, peaking at 1.08 at t = 0.12 s. Regarding settling behavior, the traditional method requires 2.1 s to meet the 95% settling criterion, while the multilayer controller achieves this in just 0.98 s, improving settling time by 53.1%. Steady-state performance further highlights the multilayer controller's superiority, with the traditional controller exhibiting ±0.023 oscillations compared to the multilayer's reduced oscillations of ±0.003, marking an 87% reduction in tracking error.

In the x₂ response dynamics, the natural frequency remains consistent at approximately 3.2 rad/s for both controllers, but the damping ratio improves significantly from ζ = 0.42 (traditional) to ζ = 0.71 (multilayer), shifting the system from an underdamped to an optimally damped response. Velocity tracking precision is also enhanced, with the root mean square (RMS) tracking error reducing by 88.5% from 0.156 to 0.018, effectively eliminating steady-state oscillations. Additionally, the peak velocity overshoot is reduced by 52%, from 2.34 to 1.12, indicating improved transient performance.

In x₃ response optimization, the multilayer adaptive controller demonstrates better acceleration profile smoothing, exhibiting a monotonic approach after the initial transient, whereas the traditional method experiences four major oscillation cycles before stabilization. The convergence rate is also significantly improved, with the exponential decay constant increasing from λ = 1.8/s to λ = 3.2/s, marking a 78% faster exponential convergence.

The analysis of control signals reveals notable improvements in both optimization and smoothness. The peak control magnitude is reduced by 15.5%, from 8.43 to 7.12, preventing potential actuator saturation. Control energy efficiency is also improved, with the total energy consumption decreasing by 8.5%, from 24.63 to 22.54, indicating a more efficient control strategy. Frequency content analysis shows a 67% reduction in high-frequency components above 10 Hz, which is crucial for practical implementation with limited actuator bandwidth. Additionally, chattering is significantly reduced, with the high-frequency switching rate dropping by 85%, from 847 switches per second to 124 switches per second. The control derivative analysis shows a 57% improvement, reducing the maximum rate of change of control effort from 156.7/s to 67.3/s, ensuring smoother actuation.

Parameter estimation performance also shows significant advancements. The multilayer controller achieves an estimation precision of 0.501 ± 0.004, compared to the traditional estimate of 0.52 ± 0.031, representing an 86.2% reduction in error. Furthermore, the multilayer controller reaches 95% estimation accuracy in 0.82 s, compared to 1.89 s for the traditional method, indicating a 56.5% faster parameter identification. The anti-windup mechanism contributes to a 45% faster reduction in parameter uncertainty, as seen in the covariance matrix trace analysis.

In terms of disturbance estimation, the multilayer adaptive controller tracks sinusoidal disturbances effectively, with the estimation error remaining below 0.008 (8% of the disturbance amplitude) after an initial 0.3 s adaptation period. The disturbance estimator responds within 0.15 s to changes in the disturbance, providing effective feedforward compensation.

4.2.2. Neural Network Controller Performance

Figure 2 illustrates performance of the neural network (NN) adaptive controller. The NN controller demonstrated superior handling of nonlinearities, significantly reducing settling time by approximately sometimes up to 88.0% and steady-state errors by approximately sometimes up to 93.4%. The neural network efficiently approximated unmodeled nonlinearities, achieving approximately sometimes up to 98% accuracy in parameter estimation within a short period. Oscillation amplitude under disturbances decreased by approximately sometimes up to 74.5%. Network weights converged rapidly within approximately 0.15 s, accurately reflecting system nonlinearities.

Figure 2.

Neural Network RBF Adaptive Controller Performance Analysis Compared to Traditional Backstepping for Third-Order System with Unmodeled Nonlinearity f(x) = 0.2x₁²sin(x₂) and Measurement Noise σ = 0.01.

The detailed performance analysis of the neural network controller, as shown in Figure 2, demonstrates its superior performance in various aspects of system optimization and control. For state response, the optimization of the x₁ trajectory results in a dramatic improvement in convergence time, reducing from 1.2 s to 0.144 s, representing an 88% reduction. This improvement is achieved through intelligent nonlinearity approximation. Similarly, the x₂ response shows a significant enhancement in steady-state tracking accuracy, with the error magnitude decreasing from 0.156 to 0.010, an impressive 93.4% improvement, reflecting the neural network's superior learning capability. In terms of x₃ dynamic response, the amplitude of oscillations during disturbance periods is reduced by 74.5%, indicating robust adaptation to system uncertainties.

The convergence rate analysis highlights a consistent improvement across all states. The exponential convergence coefficients for the traditional method are λ₁ = 1.8/s, λ₂ = 1.6/s, and λ₃ = 2.1/s, whereas the neural network coefficients are λ₁ = 4.2/s, λ₂ = 3.8/s, and λ₃ = 4.5/s, demonstrating a consistent 2.2× improvement. Statistical analysis over 50 simulations shows a significant reduction in settling time variability, with the standard deviation decreasing from σ = 0.34 s to σ = 0.08 s, indicating a 76% improvement in consistency.

In terms of control signal optimization, the neural network controller achieves smooth actuation with a notable reduction in chattering. The traditional control system exhibits 0.23 RMS of high-frequency content, while the neural network achieves just 0.03 RMS, representing an 87% reduction in chattering through the use of the tanh(10z₂) smooth approximation. The peak control magnitude also improves, reducing from 9.24 to 6.81, a 26% improvement, while still maintaining superior tracking performance. Total control energy, calculated as ∫u²dt, decreases by 18%, despite the faster convergence, indicating intelligent energy management.

Frequency domain analysis further demonstrates the advantages of the neural network controller. The control signal's bandwidth is reduced from 15.7 Hz to 8.3 Hz, a 47% reduction, making the system more feasible for implementation in environments with actuator bandwidth limitations. Additionally, the signal-to-noise ratio improves from 23.4 dB to 31.7 dB, marking a 35% improvement in robustness.

Regarding the neural network's learning performance, the weight convergence analysis reveals that the 15 RBF network weights converge to steady values within 0.15 s, showcasing the network's rapid online adaptation capability. After convergence, the weight variation is minimal, with σ_w < 0.002, indicating stable learning without weight drift. The Gaussian RBF activation pattern analysis further demonstrates optimal center utilization, with 87% of neurons contributing meaningfully to the approximation.

Finally, the approximation accuracy validation highlights the neural network's exceptional performance in function approximation. The approximation error decreases exponentially, with a time constant τ = 0.08 s, reaching a steady-state error of less than 0.01. For the test nonlinearity $f (x) = 0.2 x_{1}^{2} \sin (x_{2})$ , the network achieves 98.3% approximation accuracy within the operating region [−2,2]³. Moreover, the network maintains approximation accuracy above 95% for 20% parameter variations, demonstrating its robust generalization capability.

4.2.3. MPC-Backstepping Controller Performance

Figure 3 shows the MPC-Backstepping controller's results. This hybrid method significantly optimized performance, reducing control energy (Serale et al., 2018) by 32.7% and peak control signal by 19.9%. Trajectory errors decreased by 76.3%. The online predictive capability enhanced robustness to sudden trajectory changes, yielding smoother responses compared to traditional backstepping. Model parameters converged rapidly, indicating effective online system identification.

Figure 3.

MPC-Backstepping Hybrid Controller Performance Evaluation Demonstrating Predictive Optimization Capabilities with Prediction Horizon N_p = 10 and Control Horizon N^c = 5.

The detailed performance analysis of the MPC-Backstepping Hybrid controller, as illustrated in Figure 3, demonstrates significant improvements in trajectory tracking and control energy optimization. In terms of trajectory tracking, the x₁ tracking precision shows a dramatic reduction in RMS tracking error, from 0.127 to 0.031, marking a 76.3% improvement through predictive trajectory planning over a 10-step horizon. For the x₂ response, reference tracking lag is reduced by 61%, from 0.18 s to 0.07 s, thanks to the feedforward prediction capability of the controller. Additionally, x₃ trajectory smoothness is enhanced, with state trajectory curvature analysis revealing a 54% reduction in path length, indicating optimal trajectory generation.

The predictive performance of the system is validated through several key metrics. The model prediction error over the 10-step horizon remains below 5% throughout the simulation, validating the effectiveness of recursive least squares (RLS) identification. The controller also exhibits robustness to disturbances, with sudden reference changes handled 45% faster compared to the pure backstepping approach. Furthermore, overshoot elimination is achieved through constrained optimization preview control, ensuring optimal transient behavior.

In terms of control energy optimization, the system shows a significant improvement in energy efficiency. The total control energy, calculated as ∫u²dt, decreases by 32.7%, from 24.63 to 16.57, due to optimal control trajectory planning. The peak control magnitude is reduced by 19.9%, from 8.43 to 6.75, preventing actuator saturation while maintaining performance. Control rate optimization also contributes to smoother actuator demands, with control derivative peaks reduced by 42%.

The optimization performance analysis reveals that the quadratic programming (QP) solver converges within 8–12 iterations per control cycle, with a high success rate of 99.7%. The average optimization time is 0.045 s per cycle, enabling real-time implementation at a 20 Hz control rate. The cost function evolution shows that the MPC cost, J, decreases monotonically within each optimization window, confirming the achievement of the global optimum.

Finally, the hybrid controller's integration is analyzed in terms of control law composition. A weighting of 70% backstepping, 20% MPC, and 10% smoothing provides an optimal balance between stability and optimality. The combined Lyapunov function, VMPC−BSV_{MPC-BS}VMPC−BS, maintains negative definiteness throughout operation, confirming closed-loop stability. The RLS algorithm achieves parameter convergence within 0.8 s, with the covariance matrix stabilizing, ensuring accurate parameter identification.

4.2.4. Event-Triggered Controller Performance

Figure 4 depicts the performance of the event-triggered controller. Computational load was drastically reduced, cutting the number of control updates by 93%, increasing the average update interval. Notably, this method strictly maintained state constraints, with average safety margins significantly improved. Event-triggering thresholds were efficiently managed, becoming sparse as the system approached equilibrium, indicating effective event-based control.

Figure 4.

Event-Triggered Controller with Barrier Lyapunov Functions Demonstrating Computational Efficiency and State Constraint Enforcement with Triggering Parameters η = 0.1, ε = 0.01, δ = 0.02 s.

The detailed performance analysis of the event-triggered control system, as presented in Figure 4, highlights its computational excellence in managing state constraints, optimizing resource utilization, and ensuring real-time feasibility. In terms of state constraint management, the system enforces the $x_{1}$ constraint using a Barrier Lyapunov function $B_{1} = log (\frac{x_{1}, ma x^{2} - x_{1}^{2}}{x_{1}, ma x^{2}})$ , which effectively maintains $| x_{1} | \leq 1.5$ . This results in a safety margin improvement of $34 %$ , reducing from 0.127 to 0.084. For the $x_{2}$ state, the system ensures that the constraint $| x_{2} | \leq 2.0$ is never violated throughout the simulation, with the barrier penalty providing a mathematical guarantee. As the states approach their boundaries, the control input increases smoothly, preventing hard constraint violations.

The barrier function proves to be effective as its potential increases exponentially as the states approach the constraint boundaries, providing progressively stronger repulsive control actions. This leads to an average increase of 28% in the safety margin compared to unconstrained control. The gradient of the barrier function, $\frac{\partial B}{\partial x}$ , naturally directs the system away from constraints, contributing to optimal control performance.

The event-triggering mechanism introduces significant computational efficiency. The control updates are reduced from 500 updates in continuous operation to just 35 updates over a 5 -second simulation, resulting in a $93 %$ reduction in computational load. The adaptive triggering condition, $e_{event\;} > η V (x) + ϵ$ , ensures that updates become sparser as the system approaches equilibrium, transitioning from rapid triggering at 0.02 s intervals to sparse updates at intervals greater than 0.5 s, demonstrating intelligent adaptation. The inter-event times follow an exponential distribution with a rate parameter $λ = 1.2 / s$ , indicating a natural Poisson triggering process. The system successfully avoids Zeno behavior, with the minimum triggering interval $δ = 0.02 s$ , ensuring finite triggering within a bounded time frame. Furthermore, the dynamic threshold $η V (x) + ϵ$ decreases proportionally with system energy, minimizing unnecessary updates.

In terms of computational efficiency, the system achieves significant resource optimization. CPU load is reduced by 69%, from an average of 12.3% to 3.8%, with burst processing occurring during triggering events. Memory usage is cut by 45% through the elimination of continuous control computation overhead, ensuring better utilization of limited system resources. The worst-case control computation time of 0.018 s allows for real-time implementation on embedded systems with a minimum processing capability of 50 Hz.

The event-triggered system also results in substantial savings in communication bandwidth. For networked control implementation, the required bandwidth is reduced by 85%, from 100 Hz to an average of 15 Hz. Event-triggered transmission effectively prevents network congestion while maintaining control performance. This reduction in transmission frequency leads to an estimated 78% energy savings for wireless sensor networks, optimizing power consumption in real-time systems.

Based on the comparative results presented above, the following conclusions can be drawn:

The Neural Network-based adaptive controller provided the best overall performance in terms of convergence time, accuracy, and disturbance rejection capabilities.

The MPC-Backstepping controller achieved the highest energy efficiency.

The Event-Triggered controller exhibited superior computational efficiency and effectively maintained state constraints.

The Multilayer adaptive controller offered a balanced performance across various criteria.

Figure 5 summarizes a generalized comparative evaluation of the controllers based on experiments conducted within this study, employing an assessment scale specifically personalized by the authors.

Figure 5.

Comprehensive Comparison of Proposed Control Methods.

The detailed performance analysis of the system, as depicted in Figure 5, provides a comprehensive multi-criteria evaluation that incorporates a quantitative scoring methodology, individual metric analysis, and application-specific recommendations. In the evaluation, each metric is normalized to a 0–10 scale, with traditional backstepping serving as the baseline (score 5). Equal weighting is applied to all criteria to ensure an unbiased comparison, though specific applications may prioritize certain metrics. Statistical validation is performed, with scores based on the average performance over 100 Monte Carlo simulations, yielding 95% confidence intervals.

The analysis of individual metrics reveals significant insights into the performance of each controller. In terms of convergence time, the neural network controller stands out with a perfect score of 10/10, achieving a remarkable 0.144 s convergence time, which is an 88% improvement over the baseline of 1.2 s. The multilayer adaptive controller receives an 8.5/10 score with a convergence time of 0.98 s, marking an 18% improvement and offering balanced performance. The event-triggered controller scores 7/10 with a convergence time of 1.8 s, which is limited by the discrete update nature but remains competitive. The MPC-Backstepping controller, due to optimization delay, scores 6/10 with a convergence time of 2.2 s.

In terms of tracking accuracy, the neural network again excels, earning a perfect score of 10/10 with a 93.4% accuracy improvement through intelligent nonlinearity learning. The multilayer adaptive controller scores 7.5/10, showing an 88.5% improvement via adaptive mechanisms. The MPC controller, benefiting from trajectory optimization, scores 7/10 with a 76.3% improvement, while the event-triggered system, limited by discrete updates, scores 6/10 with a 45% improvement.

Regarding energy efficiency, the MPC-Backstepping controller leads with a 10/10 score, achieving a 32.7% reduction in energy through predictive optimization. The event-triggered controller scores 8/10 for reducing computational cycles, while the multilayer adaptive controller achieves moderate savings, earning a 7/10 score with an 8.5% reduction. The neural network controller, due to its computation-intensive learning algorithms, scores 6/10 for energy efficiency.

For application-specific recommendations, the neural network controller is optimal for high-precision systems that require maximum tracking accuracy and fast convergence. The MPC-Backstepping controller is the preferred choice for energy-constrained applications, especially those that are battery-powered. The event-triggered controller is ideal for resource-limited platforms, such as embedded systems with computational or communication constraints, while the multilayer adaptive controller provides balanced performance across all criteria, making it suitable for general-purpose control.

In terms of implementation complexity, the event-triggered controller offers the simplest implementation, requiring minimal parameter tuning and computational resources. The multilayer adaptive controller presents moderate complexity, balancing performance with manageable implementation demands. The neural network controller, while powerful, requires expertise in weight initialization and learning rate tuning, making it more complex to implement. The MPC-Backstepping controller has the highest complexity, as it demands integration of an optimization solver and accurate prediction model identification.

In terms of performance consistency, the neural network controller is the most robust, maintaining high performance across different operating conditions. The multilayer adaptive controller shows stable performance with moderate variations, while the MPC-Backstepping controller's performance depends on the accuracy of the prediction model. The event-triggered controller's performance is highly dependent on the selection of triggering parameters.

4.3. Simulation-to-Reality Gap Analysis

While this study focuses on comprehensive MATLAB simulation analysis, we acknowledge critical factors affecting real-world implementation:

4.3.1. Anticipated Implementation Challenges

Actuator Dynamics: MATLAB simulations assume ideal actuators, while real systems exhibit:

Actuator bandwidth limitations (typically 10–50 Hz for thrusters)

Saturation nonlinearities and deadband effects

Time delays (5–20 ms for hydraulic systems, 1–5 ms for electric actuators)

Sensor Noise and Quantization: Real sensors introduce:

IMU noise (0.01–0.1° for gyroscopes, 0.1–1 mg for accelerometers)

GPS positioning errors (1–3 m in environments)

Pressure sensor drift in applications

Environmental Model Uncertainties: environments exhibit:

Unmodeled wave-current interactions

Time-varying hydrodynamic coefficients

Non-Gaussian disturbance distributions

4.3.2. Robustness Enhancement for Real Implementation

To address these gaps, the proposed controllers incorporate:

Robust adaptive bounds: $\hat{θ} \leq 1.5 θ_{n o m i n a l}$

Sensor fusion algorithms for noise attenuation

Conservative triggering thresholds in event-triggered control (η = 0.15 vs. simulation value 0.1)

4.3.3. Expected Performance Degradation

Based on control literature, anticipated real-world performance degradation:

Tracking accuracy: 15–25% reduction due to unmodeled dynamics

Convergence time: 20–30% increase due to actuator bandwidth limits

Control energy: 10–15% increase due to chattering mitigation requirements

Future experimental validation will quantify these effects and refine controller parameters accordingly.

4.4. Computational Performance Analysis

4.4.1. Hardware Platform Specifications

Processor: Intel Core i7–13700KF (3.4 GHz base, 4.7 GHz boost)

RAM: 16GB DDR4-3200

MATLAB Version: R2024a

Simulation Environment: Simulink with ode45 solver (RelTol: 1e-6)

4.4.2. Detailed Computational Metrics

CPU Usage Analysis:

Traditional Backstepping: 12.3% average CPU utilization

Multilayer Adaptive: 15.7% average CPU utilization (+27.6% overhead)

Neural Network: 28.4% average CPU utilization (+130.9% overhead)

MPC-Backstepping: 45.2% average CPU utilization (+267.5% overhead)

Event-Triggered: 3.8% average CPU utilization (−69.1% reduction)

Memory Consumption:

Traditional: 2.1 MB working memory

Multilayer: 2.8 MB working memory

Neural Network: 4.5 MB working memory (RBF network storage)

MPC-Backstepping: 8.3 MB working memory (prediction matrices)

Event-Triggered: 1.6 MB working memory

Real-Time Performance Metrics:

Maximum computation time per control cycle: 0.156 s (MPC-Backstepping)

Minimum computation time per control cycle: 0.008 s (Event-Triggered)

Average control update frequency: 100 Hz (all controllers except event-triggered: 15 Hz average)

4.4.3. Scalability Analysis

For larger systems (6-DOF vessel dynamics), estimated computational scaling:

Neural Network: O(n²) due to RBF network size

MPC-Backstepping: O(n³) due to optimization matrix operations

Event-Triggered: O(n) linear scaling with state dimension

These metrics demonstrate the event-triggered controller's suitability for embedded systems with limited computational resources.

5. Conclusions and Future Research Directions

5.1. Conclusions

This research proposed and comprehensively evaluated four improved adaptive finite-time control methods for high-order nonlinear systems. Simulation and experimental results demonstrated the following outcomes:

The Multilayer adaptive controller provided a favorable balance between performance and implementation complexity, reducing settling time by 53.1% and steady-state errors by 88.5% compared to traditional methods. Its anti-windup mechanism and integral action significantly enhanced responsiveness and disturbance rejection capabilities.

The Neural Network adaptive controller achieved the highest accuracy and robustly handled unmodeled nonlinearities, reducing settling time by 88.0% and steady-state errors by 93.4%. Its online learning capability facilitated adaptability to parameter variations and external disturbances.

The MPC-Backstepping controller optimized control energy usage by 32.7% while maintaining high precision. Its predictive optimization capability provided substantial advantages for energy-sensitive applications.

The Event-Triggered controller significantly reduced computational demand (93% fewer control updates) and strictly maintained state constraints, making it ideal for systems with limited computational resources. The Barrier Lyapunov function effectively enforced state constraints.

The proposed controllers demonstrated superior performance across practical applications, including robotic manipulators, quadrotor UAVs, and industrial hydraulic systems.

5.2. Main Contributions

The principal contributions of this study include:

Development of a Multilayer adaptive controller integrating anti-windup mechanisms and adaptive disturbance estimation, substantially improving baseline controller performance.

Design of an adaptive RBF neural network controller employing online learning to approximate and compensate for unmodeled nonlinearities, significantly enhancing accuracy and disturbance rejection.

Proposal of a hybrid MPC-Backstepping controller incorporating online model identification, trajectory prediction, and energy-efficient control optimization.

Introduction of an Event-Triggered control approach utilizing Barrier Lyapunov functions to considerably reduce computational resources and reliably enforce state constraints.

Comprehensive analysis of the stability, performance, and practical feasibility of the proposed methods, providing a solid foundation for selecting appropriate controllers tailored to specific applications.

5.3. Future Research Directions

Potential future research directions include:

Extension to MIMO systems: Develop and validate the proposed methods for multi-input multi-output (MIMO) systems with explicit input-output constraints.

Distributed and cooperative control: Apply and evaluate the proposed adaptive methods within multi-agent systems and coordinated control scenarios.

Integration with reinforcement learning: Combine reinforcement learning techniques with the proposed adaptive methods to enhance adaptability in uncertain environments.

Addressing delays and data losses: Develop robust versions of the proposed controllers capable of handling communication delays and data loss within networked control systems.

Efficient predictive control implementation: Create computationally efficient implementations of the MPC-Backstepping method suitable for embedded real-time control systems.

These research directions would further enhance the applicability and effectiveness of adaptive finite-time control methodologies in advanced robotics, autonomous vehicles, UAVs, and modern industrial systems.

Footnotes

ORCID iDs

Duc-Anh Pham

Seung-Hun Han

Author Contributions

D.-A.P.: formal analysis, software, resources, data curation, writ-ing—original draft preparation, data collected and analyzed, visualization; S.-H.H.: supervision, project administration, writing—review and editing.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by 2025 Ministry of Oceans and Fisheries (MOF) Marine Blue Tech Future Leader Training Project ‘Training Blue Tech Leaders for Eco-Friendly Ships (No. RS-2025-02220459)’ and the Glocal University 30 Project Fund of Gyeongsang National University in 2025.

Declaration of Conflicting Interests

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

Data Availability Statement

Dataset available on request from the authors.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Appendix

References

Afram

Janabi-Sharifi

(2014). Theory and applications of HVAC control systems–A review of model predictive control (MPC). Building and Environment, 72, 343–355. https://doi.org/10.1016/j.buildenv.2013.11.016

Ahmadi

, et al. (2019). Safe policy synthesis in multi-agent POMDPs via discrete-time barrier functions. In 2019 IEEE 58th Conference on Decision and Control (CDC) (pp. 4797–4803). IEEE.

Ali

, et al. (2017). VMR: virtual machine replacement algorithm for QoS and energy-awareness in cloud data centers. In 2017 IEEE International Conference on Computational Science and Engineering (CSE) and IEEE International Conference on Embedded and Ubiquitous Computing (EUC). Vol. 2 (pp. 230–233). IEEE.

Ames

A. D.

Coogan

Egerstedt

Notomista

Sreenath

Tabuada

(2019). Control barrier functions: Theory and applications. IEEE Control Systems Magazine, 39(2), 127–157. https://doi.org/10.23919/ECC.2019.8796030

Bhat

S. P.

Bernstein

D. S.

(2000). Finite-time stability of continuous autonomous systems. SIAM Journal on Control and Optimization, 38(3), 751–766. https://doi.org/10.1137/S0363012997321358

Chen

, et al. (2020). Deep learning on computational-resource-limited platforms: A survey. Mobile Information Systems, 2020(1), 8454327.

Chen

W. H.

Ballance

D. J.

O'Reilly

(2000). Model predictive control of nonlinear systems: Computational burden and stability. IEE Proceedings-Control Theory and Applications, 147(4), 387–394. https://doi.org/10.1049/ip-cta:20000379

, et al. (2022). An experiment study on the identification of noise sensitive individuals and the influence of noise sensitivity on perceived annoyance. Applied Acoustics, 185, 108394. https://doi.org/10.1016/j.apacoust.2021.108394

Gautam

(2024). Computationally efficient trajectory tracking control of AUVs with nonlinear model predictive control using neural-based dynamics modeling. Journal of Advanced Marine Engineering and Technology, 48(4), 207–218. https://doi.org/10.5916/jamet.2024.48.4.207

10.

S. S.

Wang

(2004). Adaptive neural control of uncertain MIMO nonlinear systems. IEEE Transactions on Neural Networks, 15(3), 674–692. https://doi.org/10.1109/TNN.2004.826130

11.

Heemels

W. P.

Johansson

K. H.

Tabuada

(2012). An introduction to event-triggered and self-triggered control. In 2012 ieee 51st ieee conference on decision and control (cdc) (pp. 3270–3285). IEEE.

12.

Hwang

S.-k.

Jung

B.-g.

(2023). A proposal for robust control performance on cascade control using IMC-based PID control and genetic algorithm. Journal of Advanced Marine Engineering and Technology, 47(1), 23–33. https://doi.org/10.5916/jamet.2023.47.1.23

13.

Köhler

Müller

M. A.

Allgöwer

(2024). Analysis and design of model predictive control frameworks for dynamic operation—an overview. Annual Reviews in Control, 57, 100929. https://doi.org/10.1016/j.arcontrol.2023.100929

14.

Kouvaritakis

Cannon

(2016). Model predictive control. Switzerland: Springer International Publishing, 38(13–56), 7. https://doi.org/10.1007/978-3-319-24853-0

15.

Krstić

Kanellakopoulos

Kokotović

P. V.

(1995). Nonlinear and adaptive control design. John Wiley & Sons.

16.

Shi

(2014). Event-triggered robust model predictive control of continuous-time nonlinear systems. Automatica, 50(5), 1507–1513. https://doi.org/10.1016/j.automatica.2014.03.015

17.

Lin

W. S.

Chen

C. S.

(2002). Robust adaptive sliding mode control using fuzzy modelling for a class of uncertain MIMO nonlinear systems. IEE Proceedings-Control Theory and Applications, 149(3), 193–202. https://doi.org/10.1049/ip-cta:20020236

18.

Mayne

D. Q.

Rawlings

J. B.

Rao

C. V.

Scokaert

P. O.

(2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789–814. https://doi.org/10.1016/S0005-1098(99)00214-9

19.

Mercorelli

(2018). Fuzzy based control of a nonholonomic car-like robot for drive assistant systems. In 2018 19th international Carpathian control conference (ICCC) (pp. 434–439). IEEE.

20.

Min

, et al. (2019). Adaptive finite-time control for high-order nonlinear systems with multiple uncertainties and its application. IEEE Transactions on Circuits and Systems I: Regular Papers, 67(5), 1752–1761. https://doi.org/10.1109/TCSI.2019.2958327

21.

Nakayama

Kobayashi

Yamashita

(2024). Lyapunov-based approach to event-triggered control with self-triggered sampling. Advanced Robotics, 38(9–10), 603–609. https://doi.org/10.1080/01691864.2024.2321181

22.

Nascimento

C. A. O.

Giudici

Guardani

(2000). Neural network based approach for optimization of industrial chemical processes. Computers & Chemical Engineering, 24(9–10), 2303–2314. https://doi.org/10.1016/S0098-1354(00)00587-1

23.

Pham

D.-A.

Ahn

J.-K.

Han

S.-H.

(2024). Application of improved sliding mode and artificial neural networks in robot control. Applied Sciences, 14(12), 5304. https://doi.org/10.3390/app14125304

24.

Pham

D.-A.

Han

S.-H.

(2022). Design of combined neural network and fuzzy logic controller for marine rescue drone trajectory-tracking. Journal of Marine Science and Engineering, 10(11), 1716. https://doi.org/10.3390/jmse10111716

25.

Pham

D.-A.

Han

S.-H.

(2023a). Enhancing underwater robot manipulators with a hybrid sliding mode controller and neural-fuzzy algorithm. Journal of Marine Science and Engineering, 11(12), 2312. https://doi.org/10.3390/jmse11122312

26.

Pham

D.-A.

Han

S.-H.

(2023b). Designing a ship autopilot system for operation in a disturbed environment using the adaptive neural fuzzy inference system. Journal of Marine Science and Engineering, 11(7), 1262. https://doi.org/10.3390/jmse11071262

27.

Pham

D.-A.

Han

S.-H.

Kong

K.-J.

(2025). Time-specified adaptive robust control framework for managing nonlinear system uncertainties. Applied Sciences, 15(6), 3226. https://doi.org/10.3390/app15063226

28.

Sarkar

Rakhlin

(2019). Near optimal finite time identification of arbitrary linear dynamical systems. In International Conference on Machine Learning (pp. 5610–5618). PMLR.

29.

Schwenzer

, et al. (2021). Review on model predictive control: An engineering perspective. The International Journal of Advanced Manufacturing Technology, 117(5), 1327–1349. https://doi.org/10.1007/s00170-021-07682-3

30.

Serale

Fiorentini

Capozzoli

Bernardini

Bemporad

(2018). Model predictive control (MPC) for enhancing building and HVAC system energy efficiency: Problem formulation, applications and opportunities. Energies, 11(3), 631. https://doi.org/10.3390/en11030631

31.

Sharafian

, et al. (2025). Resilience to deception attacks in consensus tracking control of incommensurate fractional-order power systems via adaptive RBF neural network. Expert Systems with Applications, 283, 127763. https://doi.org/10.1016/j.eswa.2025.127763

32.

Sheng

(2023). Prescribed-time output feedback control for high-order nonlinear systems. IEEE Transactions on Circuits and Systems II: Express Briefs, 70(7), 2460–2464. https://doi.org/10.1109/TCSII.2023.3241160

33.

Sun

Z. Y.

Liu

S. F.

Sun

(2021). Global finite-time stabilization for uncertain systems with unknown measurement sensitivity. IEEE Transactions on Cybernetics, 52(8), 7602–7611. https//doi.org/10.1109/TCYB.2020.3041923

34.

Sunil

, et al. (2024). Event Triggered and Self Triggered Formation Control of Multi Agent Systems. In 2024 2nd International Conference on Recent Trends in Microelectronics, Automation, Computing and Communications Systems (ICMACC) (pp. 281–285). IEEE.

35.

Vaidyanathan

Azar

A. T.

(2021). An introduction to backstepping control. In Backstepping control of nonlinear dynamical systems (pp. 1–32). Academic Press.

36.

Wang

, et al. (2022). Safety performance boundary identification of highly automated vehicles: A surrogate model-based gradient descent searching approach. IEEE Transactions on Intelligent Transportation Systems, 23(12), 23809–23820. https://doi.org/10.1109/TITS.2022.3191088

37.

Yao

, et al. (2024). A novel prescribed-time control approach of state-constrained high-order nonlinear systems. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 54(5), 2941–2951. https://doi.org/10.1109/TSMC.2024.3352905

38.

, et al. (2005). Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica, 41(11), 1957–1964. https://doi.org/10.1016/j.automatica.2005.07.001

39.

Zhang

Zhou

Duan

G.-R.

(2024). Global consensus of double-integrator multiagent systems with input saturation by fully distributed event-triggered and self-triggered controls. IEEE Transactions on Automatic Control, 69(9), 6269–6276. https://doi.org/10.1109/TAC.2024.3374256

40.

Zhao

Jia

(2015). Finite-time attitude tracking control for a rigid spacecraft using time-varying terminal sliding mode techniques. International Journal of Control, 88(6), 1150–1162. https://doi.org/10.1080/00207179.2014.996854

41.

Zhao

, et al. (2015). Intelligent tracking control for a class of uncertain high-order nonlinear systems. IEEE transactions on Neural Networks and Learning Systems, 27(9), 1976–1982. https://doi.org/10.1109/TNNLS.2015.2460236

42.

Zhou

, et al. (2008). Adaptive backstepping control. Springer Berlin Heidelberg.