Boundedness Of Tracking Error Research Articles

Polynomial filter-based adaptive fault-tolerant control for a class of nonlinear systems with multiplicative faults

In this article, the problem of fault-tolerant control is investigated for polynomial nonlinear systems (PNSs) subject to multiplicative faults and measurement noise. Firstly, a matrix Taylor expansion technique is applied to transform the PNS into a linear parameter varying system. Then the polynomial filter is constructed to get estimations of states and unmatched disturbances, which reduces the effect of measurement noise. A novel adaptive fault-tolerant controller is proposed to compensate for multiplicative faults and unmatched disturbances. Furthermore, the condition of the parameter-dependent linear matrix inequality (PDLMI) is deduced, which guarantees the boundedness of tracking errors and estimation errors, and the sum of squares (SOS) decomposition technique is utilized to solve PDLMI. Finally, the simulation of a rotary steerable drilling tool system confirms the feasibility of the proposed adaptive fault-tolerant control framework.

Adaptive Fuzzy Tracking Control With Global Prescribed-Time Prescribed Performance for Uncertain Strict-Feedback Nonlinear Systems.

For strict-feedback systems with mismatched uncertainties, adaptive fuzzy control techniques are developed to provide global prescribed performance with prescribed-time convergence. First, a class of prescribed-time prescribed performance functions are designed to quantify the performance constraints of the tracking error. Additionally, a novel error transformation function is provided to eliminate the initial value limitations and resolve the singularity issue in previous research. To ensure the convergence of the tracking error into a prescribed bounded region within a prescribed time and satisfactory transient performance, controllers with or without approximating structures are established. Notably, the settling time and initial condition of the prescribed performance function are completely independent of the initial tracking error and system parameters, thereby improving upon existing results. Furthermore, the disadvantage of the semi-global boundedness of tracking error induced by dynamic surface control can be eliminated through the use of a novel Lyapunov-like energy function. Finally, the effectiveness of the proposed strategies is validated through numerical simulations performed on practical examples.

Adaptive Nussbaum design using composite triggering condition for stochastic nonlinear systems with multiple unknown control directions

AbstractThe traditional Nussbaum‐type functions encounter challenges when extended to systems with multiple unknown control directions, as the coupling between the Nussbaum‐type function and Nussbaum‐type gain may lead to mutual elimination. To address this issue, a novel Nussbaum‐type function is introduced to handle multiple unknown control directions for stochastic systems. The stability of the proposed Nussbaum‐based design can be guaranteed using the Lyapunov analysis. To conserve computational resources, a static composite event‐triggered mechanism is integrated with intermittent feedback in the Nussbaum‐based design. In contrast to conventional event‐triggering mechanisms that depend solely on control signals, the presented design incorporates tracking errors into the event‐triggering conditions. Moreover, to further alleviate computational burdens, we further develop a dynamic composite event‐triggered mechanism. With the overall design scheme, bounded tracking errors and reduced computational burden can be ensured. Simulation examples validate the efficacy of the proposed adaptive Nussbaum composite event‐based robust control design.

Adaptive fuzzy backstepping secure control for incommensurate fractional order cyber–physical power systems under intermittent denial of service attacks

This paper proposes a fuzzy adaptive control strategy, based on backstepping, for incommensurate fractional-order power systems under denial-of-service (DoS) attacks. To address a wider range of fractional-order systems, this study assumes that the agents’ dynamics where each state have non-identical fractional derivative orders. This assumption brings up additional complexity to the design of the secure control system compared to existing approaches. Our approach tackles the challenge of maintaining system stability during both normal and attack periods by integrating adaptive control technique. The method includes an attack compensator to manage unavailable output measurements during attacks and a fuzzy estimator to approximate unobservable states. It also employs a fractional-order adaptive law and a backstepping controller to ensure performance and security, maintaining bounded tracking errors. MATLAB simulations confirm the effectiveness and robustness of the proposed method, demonstrating its ability to protect power systems from cyber threats while ensuring operational integrity and stability. Compared to prior studies, the presence of an attack compensator reduces both overshoots and convergence time following each attack.

Robust iterative learning control for discrete-time systems with random initial state shifts

This paper presents the sufficient conditions of the stability for discrete-time systems with random initial state shifts when applying iterative learning control. Different sufficient conditions have been developed for systems with different characteristics to ensure boundedness of tracking errors and convergence of the learning process under random initial condition. First, the relationship between the initial shifts of two adjacent iterations is addressed with the help of the transition matrix. And the different analysis models are established for each type of systems: one-dimensional matrix analysis model and 2-D Roesser model. Second, the new and sufficient conditions are established for the proposed learning control law to ensure systems convergence by matrix theory and two-dimensional system theory. Third, solving the linear matrix inequality (LMI) yields the control gains. Finally, digital simulations illustrate the effectiveness and sufficiency of the presented conditions.

Adaptive finite-time neural control of nonstrict-feedback nonlinear systems with input dead-zone and output hysteresis

This paper explores the adaptive finite-time neural control issue for nonlinear systems with input dead zone and output hysteresis in nonstrict-feedback form. The unknown functions are estimated by employing the radial basis function neural networks (RBFNN) approach. A systematic adaptive finite-time control method is introduced using the backstepping technique and neural network approximation properties. The stability of the system is also examined by using semi-global practical finite-time stability theory. The established control approach guarantees the boundedness of all signals within the closed-loop system, enabling the system output to accurately follow the desired signal within a finite time framework while maintaining a small and bounded tracking error. Finally, simulation results are shown to demonstrate the efficacy of the suggested strategy.

Event‐triggered obstacle avoidance control for autonomous surface vehicles with actuator faults

AbstractThe article addresses the event‐triggered obstacle avoidance control problem for autonomous surface vehicles subject to actuator faults. In order to tackle the challenges presented by unknown actuator faults, internal model uncertainties, and external disturbances, we propose a fault‐tolerant control scheme that relies on an extended state observer to estimate the unknown parameters. In addition, considering the occurrence of actuator faults, a novel event‐triggered mechanism is developed to reduce wear of actuator and ensures the performance of system. On this basis, we employ backstepping technique and an improved artificial potential function methods to develop an event‐triggered obstacle avoidance control scheme with the capability of fault tolerance. This proposed control strategy ensures the uniform ultimate boundedness of tracking errors. In contrast to existing results, the presented control strategy simultaneously holds the performance of obstacle avoidance and fault tolerance and reduce the update frequency of actuators. The effectiveness of the provided control scheme has been confirmed through simulation.

Learning-Based Model-Free Adaptive Control for Nonlinear Discrete-Time Networked Control Systems Under Hybrid Cyber Attacks.

A novel learning-based model-free adaptive control (LMFAC) approach is presented in this article for a class of unknown nonaffine nonlinear discrete-time networked control systems (NCSs) subject to hybrid cyber attacks. The aperiodic denial-of-service (DoS) attacks and persistent deception attacks are assumed to arise in feedback channels, which could result in the absence or authenticity lackness of system signals sent to the controller. With the aid of dynamic linearizaton technology, the equivalent dynamic linearized data models of considered NCSs are first established only based on I/O information instead of the knowledge of mathematical models that are commonly used under the model-based control framework. Then, an LMFAC scheme is designed on the basis of occurred maximum DoS attacks interval to adaptively tune the attenuation coefficient of the input signal for improving system performance during the next DoS attacks interval. Finally, the boundedness of tracking error is rigorously proved through the contraction mapping principle and the effectiveness of the proposed pure data-driven LMFAC method is demonstrated via simulations.

Adaptive Tracking Control of Robotic Manipulator Subjected to Actuator Saturation and Partial Loss of Effectiveness

Abstract This paper introduces an adaptive control design tailored for robotic systems described by Euler–Lagrange equations under actuator saturation and partial loss of effectiveness. The adaptive law put forth not only retains conventional control properties but also extends its scope to effectively address challenges posed by actuator saturation and partial loss of effectiveness. The framework’s primary focus is on bolstering system robustness, thereby ensuring the achievement of uniformly ultimate bounded tracking errors. The stability and convergence of the system’s behavior are rigorously established through the application of the Lyapunov analysis technique. Moreover, the effectiveness and superiority of the introduced framework are compellingly demonstrated through a series of practical simulations and experimental instances.

Adaptive Position Control for Two-Mass Drives with Nonlinear Flexible Joints

We consider a two-mass drive with a flexible joint with a nonlinear characteristic of the transmitted torque as a function of the torsion angle. We propose a new, nonlinear, adaptive position-tracking controller, taking this nonlinearity of stiffness into account. The derivation of the controller is based on nonlinear adaptive control theory, incorporates several non-standard mathematical techniques and provides a proof of the uniform ultimate boundedness of tracking errors. As the result, we present a controller that solves the position tracking problem, attenuates dangerous tortional oscillations in the shaft and operates correctly in the presence of unknown torques acting on both sides of the joint, even if all plant parameters are unknown. We demonstrate experimentally that using some materials indeed introduces a nonlinear characteristic of the joint. We prove via real plant experiments that the proposed control algorithm is easily implementable with a DSP controller in real-world applications.

Safe learning-based control for an aerial robot with manipulator arms

Aerial manipulation aims at combining the versatility and the agility of some aerial platforms as multirotor unmanned aerial vehicles (UAVs) with the manipulation capabilities of robotic arms. Trajectory tracking control of these vehicles is a substantial aspect for an increasing range of application domains. However, external disturbances and parts of the internal dynamics are often unknown or very time-consuming to model. To overcome this issue, we present a tracking control law using an learning-based oracle based on Gaussian process models. The presented approach guarantees a bounded tracking error with high probability where the bound is explicitly given. A numerical example highlights the effectiveness of the proposed control law.

Adaptive fuzzy prescribed time tracking control for nonlinear systems with input saturation

This article investigates the adaptive fuzzy prescribed time tracking control problem for a class of strict-feedback systems simultaneously considering the user-defined asymmetric tracking performance, input saturation, and external disturbances. From the perspective of ensuring the reliability for control implementation, a saturation-based fixed-time funnel boundary is constructed by embedding the modification signals related to input saturation errors into a funnel function, which is capable of automatically enlarging or recovering itself when input saturation occurs or disappears, thereby reducing the risk of system singularity. Subsequently, by constructing a fixed-time tracking performance function, any known bounded tracking error is recast into a new variable with a zero initial value. With such a treatment, funnel boundaries are no longer redeveloped for different initial tracking errors, and meanwhile the behavior of the tracking error is pre-specified as needed over a finite time interval. Also, auxiliary systems are devised to generate the aforesaid modification signals, while compensating for the adverse impact resulting from input saturation. Remarkably, by feat of the backstepping design based on the fuzzy approximation, it is proven that the tracking error converges to a user-defined region within a prescribed time (known as the practically prescribed time tracking), which is achieved without the fractional power feedback of system states. Finally, two simulation examples are presented to confirm the feasibility and effectiveness of the developed approach.

Feedback higher-order iterative learning control for nonlinear systems with non-uniform iteration lengths and random initial state shifts

For nonlinear discrete-time systems with non-uniform iteration lengths and random initial state shifts, this paper developed a feedback higher-order iterative learning control (ILC) approach. To compensate the absent information of last iteration caused by non-uniform iteration lengths, the tracking information in both iteration domain and time domain is included in ILC design with the help of higher-order control and feedback control, respectively, while the general ILC schemes just adopt the information in iteration domain. A sufficient condition based on the higher-order ILC gains is derived. It is guaranteed that as the iteration number goes to infinity, the asymptotic bound of tracking error is proportional to random initial state shifts in mathematical expectation sense. Specifically, as the expectation of initial state shifts is zero, the ILC tracking error can be controlled to zero along the iteration direction. Two examples with different initial conditions are provided to validate the proposed ILC approach.

Layered Time-Delay Robust Control Strategy for Yaw Stability of SbW Vehicles

The development of intelligent driving and X-by-wire chassis technology increases the computational complexity and the number of CAN bus-mounted nodes of steering-by-wire vehicles, which leads to non-negligible large random time delay (TD) of the SbW system's control input. TD has a great impact on SbW system's tracking error, which will inevitably deteriorate the safety and yaw stability of vehicles and even cause sideslip accidents. In order to ensure the safety of SbW vehicles, a layered time-delay robust control strategy (LTDRCS) consisting an upper and a lower controller is proposed. Specifically, a novel Lyapunov-Krasovskii (L-K) TD H∞ controller (H∞C) is designed as the lower controller to reduce tracking error, and the system's stability and convergence are synchronously guaranteed by an L-K function and an H∞ norm constraint. As for the upper controller, a novel terminal sliding mode controller (NTSMC) is established to control the yaw rate index and sideslip angle index. In order to restrain the serious influence of the SbW system's inevitable, unknown, and bounded tracking error caused by TD on vehicle yaw stability and ensure the rapid convergence of the system, a novel integral term is introduced. Simulation and experiments show that the proposed strategy efficiently improves the tracking accuracy of the SbW system and the vehicle yaw stability under random TD condition, which benefits from the synergy of the upper and lower layers.

Kernel-based regularized iterative learning control of repetitive linear time-varying systems

For data-driven iterative learning control (ILC) methods, both the model estimation and controller design problems are converted to parameter estimation problems for some chosen model structures. It is well-known that if the model order is not chosen carefully, models with either large variance or large bias would be resulted, which is one of the obstacles to further improve the modeling and tracking performances of data-driven ILC in practice. An emerging trend in the system identification community to deal with this issue is using regularization instead of the statistical tests, e.g., AIC, BIC, and one of the representatives is the so-called kernel-based regularization method (KRM). In this paper, we integrate KRM into data-driven ILC to handle a class of repetitive linear time-varying systems, and moreover, we show that the proposed method has ultimately bounded tracking error in the iteration domain. The numerical simulation results show that in contrast with the least squares method and some existing data-driven ILC methods, the proposed one can give faster convergence speed, better accuracy and robustness in terms of the tracking performance.

A novel asymmetric barrier Lyapunov function-based fixed-time ship berthing control under multiple state constraints

The construction of an asymmetric barrier function with internal security set, while ensuring its invariance in the presence of interference, is a challenging task to achieve stability and security. In this paper, a translation transformation is used to construct a novel asymmetric Barrier Lyapunov Function. To the best of our knowledge, this is the first designed continuously differentiable barrier Lyapunov function (BLF) of this type. A new adaptive sliding mode control (SMC) method based on asymmetrical BLF is proposed for uncertain external disturbances. The robust control method can ensure the safety performance in the presence of model uncertainty with bounded tracking error of the ship trajectory and velocity. Its fixed-time convergence is then proved by the Lyapunov theory. The results show that it has stronger anti-interference ability in bad sea conditions, better robustness, and concise form. In addition, the system can deal with unknown disturbances, ensure the safe navigation, and keep the good tracking of the route under wind and current disturbance due to random sea conditions.

Dynamic average consensus control based on event-triggered cloud access

A dynamic average consensus (DAC) control scheme is proposed in the presence of constrained communication, where event-triggered cloud access is presented with a cloud repository for agents. With this framework, agents upload and download information from the cloud repository, in which they exchange information with neighbors indirectly. We first propose a triggered condition with a deviation of control signals for reduction of the frequency of communication with the cloud repository. Due to the fact that the neighbors’ information in the condition can’t be obtained in the cloud framework, cloud access scheduling rules, with an adaptive threshold for control signal updating, are then introduced via available information at current instants, and agents access the cloud repository in an asynchronous manner to avoid congestion. Available information from the cloud is employed, as well as their own, into the triggered condition for prediction of next access instants, and the triggered consequence is generated. Stability analysis presents that the control scheme in this paper steers the MAS to achieve DAC with an ultimately bounded tracking error and rules out Zeno behaviors. The effectiveness of the control scheme is verified via an example.

Adaptive Fault-Tolerant Tracking Control for Uncertain Nonlinear Systems With Unknown Control Directions and Limited Resolution

This article addresses the adaptive fault-tolerant tracking control (FTTC) problem for a family of strict-feedback uncertain nonlinear systems subject to limited sensor resolution and unknown control directions. Both partial loss-of-effectiveness (LOE) and lock-in-place (LIP) faults of actuators are investigated. An adaptive control strategy based on a Nussbaum-type function and neural networks is presented by introducing a backstepping approach to make the system output track a desired reference output signal with bounded tracking error in the case of faulty actuators. The effect of the limited resolution is decoupled from the nonlinear system and is approximated using a neural network. The impact of disturbances is effectively compensated by utilizing adaptive parameter estimation terms in the backstepping procedure. It is proven that the proposed FTTC strategy can ensure the boundedness of all signals and guarantee that the output tracking error can converge into a small neighborhood of the origin. Two simulation examples are given to illustrate the effectiveness of the proposed FTTC strategy.

Reinforcement Learning-Based Prescribed Performance Motion Control of Pneumatic Muscle Actuated Robotic Arms With Measurement Noises

Featured with high power density, excellent flexibility, shock absorption capacity, etc., pneumatic muscles (PMs) promote the development of exoskeleton robots and rehabilitation equipment. However, the complex nonlinearities of PMs limit efficiency optimization in closed-loop control, while the force-displacement coupling, soft materials, deficient workspace, etc., make it more difficult to simultaneously increase motion speeds and ensure the safety of multiple PM-actuated (PMA) robots. Although force sensors can currently be replaced by applying state estimation techniques, the amplification effects of measurement noises still compromise control accuracy and stability in practice. To this end, this article proposes a reinforcement learning-based robust motion control method with the prescribed performance, which achieves efficient and satisfactory tracking control for PMA robotic arms. In particular, by elaborately incorporating an integral term, a robust generalized proportional integral observer is used to eliminate measurement noises. Meanwhile, by using an actor–critic network to optimize control performance, an error-transformation-based continuous controller is designed to guarantee the uniformly ultimately boundedness of tracking errors. Compared with most existing methods, this article provides the first solution to restrict the entire transient and steady-state performance of PMA robotic arms, improve the noise suppression capability, and optimize the control efficiency simultaneously. Finally, complete stability analysis based on Lyapunov techniques is provided, and several groups of hardware experiments demonstrate the practicability and robustness of the proposed method.

Anti-Disturbance Compensation for Quadrotor Close Crossing Flight Based on Deep Reinforcement Learning

The aim of this article is the design of a feedforward compensator based on deep reinforcement learning (DRL) for cooperative quadrotors in close crossing flight. Quadrotors are described by state-space models that include shearing airflow disturbance from other quadrotors. This disturbance is compensated in a feedforward way using DRL. Both value based compensator and policy based compensator algorithms are proposed for training purposes. Then, Lyapunov stability criteria are used to prove that the reference trajectory can be tracked boundedly even during the training process of the proposed algorithms, and that a smaller bound of tracking error can be achieved when the compensator converges. An indoor experimental system for online training has been developed for validation purposes. Both simulation and experimental results are provided to demonstrate the effectiveness and advantages of the proposed compensator.