Ultimate Boundedness Research Articles

Precise Tracking Control for Articulating Crane: Prescribed Performance, Adaptation, and Fuzzy Optimality by Nash Game.

Articulating crane (AC) is used in various industrial activities. The articulated multisection arm exacerbates nonlinearities and uncertainties, making the precise tracking control challenging. This study proposes an adaptive prescribed performance tracking control (APPTC) for AC to robustly fulfill the task of precise tracking control, with adaptation to resist time-variant uncertainties, whose bounds are unknown but lie in prescribed fuzzy sets. Particularly, a state transformation is applied to simultaneously track the desired trajectory and satisfy the prescribed performance. Adopting the fuzzy set theory to describe uncertainties, APPTC does not invoke any IF-THEN fuzzy rules. There is no linearizations, or nonlinear cancelation for APPTC, thus making it approximation free. The performance of the controlled AC is twofold. First, deterministic performance in fulfilling the control task is ensured by the Lyapunov analysis using uniform boundedness and uniform ultimate boundedness. Second, fuzzy-based performance is further improved by an optimal design, which seeks the optima of control parameters by formulating a two-player Nash game. The existence of Nash equilibrium is theoretically proved, and its acquisition process is given. The simulation results are provided for validations. This is the first endeavor that explores the precise tracking control for fuzzy AC.

Analysis of a stochastic Leslie-Gower three-species food chain system with Holling-II functional response and Ornstein-Uhlenbeck process

<abstract><p>This paper studies a stochastic Leslie-Gower model with a Holling-II functional response that is driven by the Ornstein-Uhlenbeck process. Some asymptotic properties of the solution of the stochastic Leslie-Gower model are introduced: The existence and uniqueness of the global solution of the model are given; the ultimate boundedness of the model is proven; by constructing the Lyapunov function and applying Ito's formula, the existence of the stationary distribution of the model is demonstrated; and the conditions for system extinction are discussed. Finally, numerical simulations are used to validate our conclusion.</p></abstract>

Uniform Ultimate Boundedness of Lur’e Systems with Switchings and Delays

Uniform Ultimate Boundedness of Lur’e Systems with Switchings and Delays

Event-Triggered Robust Adaptive Dynamic Programming for Multiplayer Stackelberg-Nash Games of Uncertain Nonlinear Systems.

In this article, an event-triggered robust adaptive dynamic programming (ETRADP) algorithm is developed to solve a class of multiplayer Stackelberg-Nash games (MSNGs) for uncertain nonlinear continuous-time systems. Considering the different roles of players in the MSNG, the hierarchical decision-making process is described as the designed value functions for the leader and all followers, which assist to transform the robust control problem of the uncertain nonlinear system into an optimal regulation problem of the nominal system. Then, an online policy iteration algorithm is formulated to solve the derived coupled Hamilton-Jacobi equation. Meanwhile, an event-triggered mechanism is designed to alleviate computational and communication burdens. Moreover, critic neural networks (NNs) are constructed to obtain the event-triggered approximate optimal control polices for all players, which constitute the Stackelberg-Nash equilibrium of the MSNG. By using Lyapunov's direct method, the stability of the closed-loop uncertain nonlinear system is guaranteed under the ETRADP-based control scheme in the sense of uniform ultimate boundedness. Finally, a numerical simulation is provided to demonstrate the effectiveness of the present ETRADP-based control scheme.

Adaptive Neural Sliding-Mode Control for Fuzzy Singularly Perturbed Systems: Sojourn-Probability-Based Stochastic Communication Protocol.

This article addresses an adaptive neural network (NN) sliding-mode control (SMC) strategy for fuzzy singularly perturbed systems against unrestricted deception attacks and stochastic communication protocol (SCP). Instead of relying on the traditional transition probability, a sojourn-probability-based SCP is efficiently established to characterize the stochastic nature more precisely. In response to unrestricted deception attacks, an NN-based technique is deployed to estimate and counteract their detrimental impacts on system performance. Moreover, the design of the sliding-mode controller integrates the singular perturbation parameter and fuzzy rules, addressing the challenge of imperfect premise matching. The proposed controller guarantees exponential ultimate boundedness in the mean square sense and ensures the reachability of the specified sliding surface for the closed-loop system. Finally, the efficacy of the proposed theoretical framework is validated through two illustrative examples, confirming its practical applicability and robustness.

Control of Uncertain High-Order Fully Actuated Strict-Feedback Systems: A Backstepping Approach With High-Gain Observer-Based Derivative Approximation.

In this article, a high-gain observer (HGO)-based differentiator is proposed to approximate the derivatives of the virtual control to ease the "explosion of complexity" problem in backstepping for the second-order strict-feedback systems (SOSFSs) and high-order strict-feedback systems (HOSFSs). Unlike the existing high-order command-filtered backstepping or extended dynamic surface control that needs to tune a large number of parameters, this proposed high-order HGO-based backstepping (HOHGOB) scheme can improve the derivative approximation performance by only tuning a single parameter, i.e., the observer gain. A further advantage of the proposed HOHGOB scheme, in addition to its simplicity and ease of implementation, is that the estimation error of the derivatives shrinks to zero as the observer gain grows to infinity. We have also rigorously established that the states of the closed-loop system achieve uniform ultimate boundedness under the designed high-order backstepping controller. Additionally, the output tracking error can be made arbitrarily small by the designer. The efficacy of the proposed scheme is numerically validated through a benchmark application to a single-link robot arm.

On the boundedness of solutions of some fuzzy dynamical control systems

<abstract> <p>The asymptotic behavior of solutions of fuzzy control systems is a component of the study of fuzzy control theory. The study of stability for T-S (Takagi-Sugeno) fuzzy systems, which process qualitative data through linguistic expressions, is the subject of this paper. Asymptotic stability is conservative in many real-world applications due to measurement noise and other disruptions. The ultimate limit, which indicates that the mistakes stay in a specific area close to the origin after a long enough amount of time, is a crucial characteristic that is frequently defined for such systems. We are interested with the problem of the state feedback controller for T-S fuzzy models with uncertainties where the global exponential ultimate boundedness of solutions is studied for certain fuzzy control systems. We use common quadratic Lyapunov function and parallel distributed compensation controller techniques to study the asymptotic behavior of the solutions of fuzzy control system in presence of perturbations. An example demonstrating the validity of the main result is discussed.</p> </abstract>

Ultimate Boundedness of Solutions of Some System of Third-Order Nonlinear Differential Equations

Ultimate Boundedness of Solutions of Some System of Third-Order Nonlinear Differential Equations

Adaptive Fuzzy Tracking Control for Switched Nonlinear Systems Under FDI Attacks and Input Saturation: A Flexible Transient Performance Approach.

The present study proposes an adaptive fuzzy tracking control strategy for switched nonlinear systems, capable of effectively addressing false data injection (FDI) attacks and input saturation, while achieving flexible prescribed performance control as well as semi-global uniform ultimate boundness for the resultant system. Compared to the previous work, the provided control strategy exhibits two notable strengths: 1) it introduces a novel modified fixed-time pregiven performance function to effectively balance input saturation and output constraint and 2) the detrimental impacts resulting from FDI attacks are successfully mitigated by implementing the fuzzy logic systems approximation technique in the backstepping procedure. A set of switching fuzzy observers are established to estimate the unobservable states while a first-order differential filter is utilized to handle the complexity explosion problem. Finally, the mass-spring-damper system is given to substantiate the developed approach.

Advanced robust control design and experimental verification for trajectory tracking of model-based uncertain collaborative robots

At the core of this research is the pursuit of enhancing the trajectory tracking performance of six-degree-of-freedom collaborative robots, with a particular focus on addressing the challenges posed by uncertainties in real-world applications. One of the primary issues encountered with existing methods is the susceptibility of trajectory tracking to uncertainties, which can significantly hinder the performance of robotic systems. To address these challenges, we propose an advanced control method, known as the model-based proportional-derivative controller, or MPDP controller for short, which represents an innovative fusion of model-based PD control principles with a robust control algorithm. This amalgamation is driven by the need to mitigate the impact of uncertainties and external disturbances on trajectory tracking. A comprehensive assessment grounded in Lyapunov theory has been undertaken to validate the effectiveness of our approach. The analysis has firmly established that our method ensures not only the ultimate boundedness but also the uniform boundedness of the robotic system, which is critical for its operational stability. Both experimental and simulation studies have been meticulously conducted to benchmark the performance of the MPDP controller against the conventional proportional-integral-derivative controller, which serves as a widely adopted baseline in the field. The results unequivocally demonstrate the superiority of the MPDP controller across multiple dimensions. It exhibits exceptional robustness, resulting in a smaller steady-state tracking error, a critical advantage when addressing inherent uncertainties and external disturbances that can perturb the robot system. This translates to a more stable trajectory tracking performance. Furthermore, the MPDP controller empowers the robot with the capability to precisely follow predefined trajectories, thus ensuring high-precision and reliable execution of tasks. This feature significantly contributes to an overall enhancement of system performance and productivity.

Nonsingleton Gaussian type-3 fuzzy system with fractional order NTSMC for path tracking of autonomous cars

Path-tracking and lane-keeping tasks are critical to guarantee safety and navigation performance considerations for deploying autonomous cars. This paper presents a novel control framework for the path-tracking control of high-speed autonomous cars with structured uncertainties. This study introduces a nonlinear adaptive control system based on a fractional-order terminal sliding mode system while incorporating a novel Gaussian Nonsingleton type-3 fuzzy system (FOTSM-NT3FS). Therefore, the proposed controller is independent of the information about the ego vehicle’s dynamic information, and instead, the dynamics are approximated through a developed NT3FLS. The developed control system exhibits robustness to measurement errors and faulty sensors, because the inputs to the NT3FS are uncertain. In order to guarantee the boundedness of the adaptation parameters, the σ−mod approach is employed. The Lyapunov stability theorem and Barbalat’s lemma are used to ensure the uniform ultimate boundedness of the closed loop system and the convergence of tracking errors to the origin in finite time. High-fidelity co-simulations with CarSim and MATLAB are performed to verify the effectiveness of the proposed control scheme and are also compared to other reported methods in the literature. Based on the obtained results, the schemed controller exhibits competitive effectiveness in path-tracking tasks and strong efficiency under various road conditions, parametric uncertainties, and unknown disturbances.

Neural Network-Based Adaptive Height Tracking Control of Active Air Suspension System with Magnetorheological Fluid Damper Subject to Uncertain Mass and Input Delay.

In this paper, we present a novel robust adaptive neural network-based control framework to address the ride height tracking control problem of active air suspension systems with magnetorheological fluid damper (MRD-AAS) subject to uncertain mass and time-varying input delay. First, a radial basis function neural network (RBFNN) approximator is designed to compensate for unmodeled dynamics of the MRD. Then, a projector-based estimator is developed to estimate uncertain parameter variation (sprung mass). Additionally, to deal with the effect of input delay, a time-delay compensator is integrated in the adaptive control law to enhance the transient response of MRD-AAS system. By introducing a Lyapunov-Krasovskii (LK) functional, both ride height tracking and estimator errors can robustly converge towards the neighborhood of the desired values, achieving uniform ultimate boundness. Finally, comparative simulation results based on a dynamic co-simulator built in AMESim 2021.2 and Matlab/Simulink 2019(b) are given to illustrate the validity of the proposed control framework, showing its effectiveness to operate ride height regulation with MRD-AAS systems accurately and reliably under random road excitations.

A novel saturated PID‐type observer‐based controller for wheeled mobile robots with a guaranteed performance considering path curvature

AbstractThis article proposes a novel saturated PID‐type formation controller for a class of nonholonomic wheeled mobile robots (WMRs) of type (m, s) with mobility m and steerability s. Since the path curvature causes a severe deviation of the followers from the leaders' trajectory in the corners, a new definition of output equations is suggested to construct the coordinates of a virtual reference point in the front of each follower based on the concept of the extended look‐ahead control that utilizes a corrective angle to compensate the trajectory curvature efficiently. A nonlinear transformation and the robot's kinematic and dynamic equations are heuristically combined to develop a novel second‐order Euler–Lagrange formulation of unconstrained errors from constrained errors. Then, a novel saturated PID‐type controller is proposed for WMRs to build a desirable formation with a prescribed performance. An innovative nonlinear observer is presented to estimate followers' velocities. An efficient mixture of a radial basis function neural network (RBFNN) and an adaptive mechanism is proposed to compensate for model uncertainties, external disturbances, and network approximation errors. Lyapunov's direct method verifies semi‐global uniform ultimate boundedness stability by estimating a gain‐dependent region of attraction. Numerical simulations finally show the efficiency of the proposed controller.

Boundedness analysis of neutral stochastic systems driven by [formula omitted]-Brownian motion

This paper is concerned with exponential ultimate boundedness problem for a class of neutral stochastic differential systems driven by G-Brownian motion. By developing a method that combines the comparison principle and the reduction to absurdity, we provide sufficient conditions for exponential ultimate boundedness in the mean square of neutral stochastic differential systems driven by G-Brownian motion. The obtained results improve greatly some previous works given in the literature. Finally, some comparisons with existing results and examples are provided to show the effectiveness and superiority of the theoretical results.

Online accelerated data‐driven learning for optimal feedback control of discrete‐time partially uncertain systems

SummaryIn this paper, we develop an online learning algorithm for solving the Bellman equation for affine in the control discrete‐time nonlinear uncertain dynamical systems. To ensure accelerated learning of our algorithm in generating optimal control policies, we use an actor‐critic structure predicated on higher‐order tuner laws. More specifically, we construct a Nesterov‐like architecture involving momentum‐based learning laws leading to an accelerated convergence of the optimal control policy. The proposed online learning‐based optimal control framework guarantees uniform ultimate boundedness of the closed‐loop system under the assumption that the system is persistently excited. Finally, two illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.

A deterministic robust control with parameter optimization for uncertain two-wheel driven mobile robot

The uncertainty in mobile robot greatly affects control accuracy. This makes it difficult to apply to more rigorous high-precision engineering fields. Therefore, the fuzzy set theory is introduced to describe the uncertainty. Based on that, the fuzzy mobile robot system is established. The virtual speed controller using backstepping method is designed. Then, a robust control method is proposed to guarantee the uniform boundedness and uniform ultimate boundedness of the controlled system. Furthermore, the balance optimization problem of the performance and cost of the controlled system is explored. By minimizing the performance index containing fuzzy numbers, the optimal control parameter is obtained. Compared with the linear quadratic regulator algorithm, which is the representative optimal robust controller, the proposed control method and optimization strategy based on fuzzy set theory are verified to be effective. The control accuracy is further improved.

Finite‐time tracking control for time‐delayed pure‐feedback systems with guaranteed performance

AbstractThis article investigates the finite‐time tracking control problem with guaranteed performance for a class of nonlinear pure‐feedback systems subject to both time‐varying state delays and control input delays. Both the state delays and control input delays are assumed to be unknown yet bounded by some known constants. A new performance function is introduced that ensures the finite‐time convergence of tracking errors and guaranteed performance. The key to handling both the states and input delays lies in a novel construction of the Lyapunov–Krasovskii (LK) function, which also avoids the reconstruction of the LK function in stability analysis. In addition, a continuous package function is introduced, which allows the unknown nonlinear terms arising from the uncertain perturbation and the derivation of the LK function to be eliminated. Under the proposed method, the semi‐global uniform ultimate boundedness of all signals in the closed‐loop system can also be ensured. Simulation examples are provided to demonstrate the effectiveness of the proposed approach in this article.

Robust Bearing-Based Formation Tracking Control of Underactuated Surface Vessels: An Output Regulation Approach

This article is concerned with the robust formation tracking control of a group of underactuated surface vessels whose formation is characterized by the relative bearings between the vessels. Control protocols are proposed to asymptotically steer certain points on the vessels, called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hand points</i> , which lie ahead of their centers of mass along the longitudinal axes, to track the desired moving formation; such a control method is called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">an output regulation approach</i> . The moving formation is guided by the motions of two or more agents, which are referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">leaders</i> , and the other <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">follower</i> agents track the leaders by maintaining bearing constraints to neighbors. The proposed control schemes are designed based on a backstepping technique and sliding mode control. The latter is employed to attenuate known bounded disturbances present in the vessels' dynamics. When the bounds of the disturbances are unknown, an adaptive control protocol is subsequently investigated, under which the system is shown to converge to a small neighborhood of the target formation. Further, sufficient conditions on the upper bound of the hand points' desired velocities are provided for the uniform ultimate boundedness of the vessels' internal sway-yaw dynamics. Simulation results are provided to support the theoretical analysis.

Geometric ergodicity and ultimate boundedness of a stochastic chemostat model with general nutrient uptake function

The ergodicity of solutions for stochastic systems has been widely concerned by many scholars in recent years. Thus, in this paper, we further analyze the geometric ergodicity of a stochastic chemostat model with general nutrient uptake function. By verifying minorization condition and Lyapunov condition, geometric ergodicity of stochastic chemostat system is presented, which means the transfer probability of Markov process will converge to its limiting distribution at an exponentially rate. Meanwhile, we prove that the solution of stochastic system is stochastically ultimately bounded.

Uniform ultimate boundedness and approximate synchronization of coupled reaction-diffusion systems with parameter mismatches

We present a framework for studying the uniform ultimate boundedness and approximate synchronization of coupled systems. The individual subsystems of the coupled systems are described by general reaction-diffusion (RD) systems, and we account for parameter mismatches between individual subsystems while considering a general coupling scheme. We derive several criteria for the uniform ultimate boundedness of coupled RD systems under different assumptions regarding the properties of the reaction terms and coupling schemes. Using M-matrix theory and iterative methods, we establish an approximate synchronization criterion for the coupled RD systems. This criterion allows the coupled systems to synchronize with synchronization errors arising from parameter mismatches, and perfect synchronization is achieved when there are no parameter mismatches. Moreover, the criterion provides an estimated bound for the synchronization error, which decreases as the degree of inhomogeneity caused by parameter mismatches reduces. To demonstrate the effectiveness of our approaches, we apply our theories to the FitzHugh-Nagumo (FHN) neuron model.