Uniform Ultimate Boundedness Research Articles

Continuous‐discrete high gain observer design with PI output predictor

SummaryIn this article, a continuous‐discrete high gain observer with a proportional integral (PI) output predictor structure is designed for a class of nonlinear systems subject to sampled‐delayed data measurements. To counteract the effect of the well‐known sensitivity of the high gain observer to the noise measurements, a novel PI output predictor structure is proposed. The uniform ultimate boundedness (UUB) of the proposed observer is demonstrated by means of Lyapunov function and small gain method. The effectiveness of our proposed observer is demonstrated through numerical simulations and performances comparison with previous observer design methods in the literature.

Robust approximate constraint following control for SCARA robots system with uncertainty and experimental validation

In the context of mechanical systems characterized by nonlinearity, uncertainty, and unknown external disturbances, this paper presents a novel and practical robust control algorithm based on the Udwadia–Kalaba (UK) equation that incorporates the ideal system trajectory as a servo constraint. Firstly, the constraint force of the system under servo constraints can be written in analytic form, which is the first part of the proposed control algorithm. The remaining parts deal with the initial condition incompatibility and the uncertainty within the system, respectively. Secondly, it is proved that the controlled system with uniform boundedness (UB) and uniform ultimate boundedness (UUB) by using the Lyapunov method. Finally, based on the SCARA robot platform, numerical simulations and experiments are performed using the rapid control prototype CSPACE. The results indicate that the proposed robust control method provides excellent dynamic performance under the constraints of the ideal trajectory. The effectiveness and practicality of the control method are further demonstrated.

Cooperative Game-Based Approximate Optimal Control of Modular Robot Manipulators for Human-Robot Collaboration.

Major challenges of controlling human-robot collaboration (HRC)-oriented modular robot manipulators (MRMs) include the estimation of human motion intention while cooperating with a robot and performance optimization. This article proposes a cooperative game-based approximate optimal control method of MRMs for HRC tasks. A harmonic drive compliance model-based human motion intention estimation method is developed using robot position measurements only, which forms the basis of the MRM dynamic model. Based on the cooperative differential game strategy, the optimal control problem of HRC-oriented MRM systems is transformed into a cooperative game problem of multiple subsystems. By taking advantage of the adaptive dynamic programming (ADP) algorithm, a joint cost function identifier is developed via the critic neural networks, which is implemented for solving the parametric Hamilton-Jacobi-Bellman (HJB) equation and Pareto optimal solutions. The trajectory tracking error under the HRC task of the closed-loop MRM system is proved to be ultimately uniformly bounded (UUB) by the Lyapunov theory. Finally, experiment results are presented, which reveal the advantage of the proposed method.

Adaptive Robust Control for Uncertain Underactuated Mechanical Systems

AbstractAn adaptive robust control method is proposed for underactuated mechanical systems (UMSs) with uncertainty in the paper. UMSs are supposed to follow prescribed constraints when there is uncertainty. Inspired by the Udwadia–Kalaba modeling approach, those constraints are considered as control objectives realized by robust control. The uncertainty considered in UMSs is characterized as time‐varying and bounded, albeit with unknown bounds. In order to estimate the boundary information, a novel adaptive law is developed. By using the Lyapunov function, the UMS is guaranteed to be uniform boundedness and uniform ultimate boundedness. The simulation is validated by using a two‐wheeled bicycle as an example, compared to conventional PD control methods. The simulation comparison results show that the proposed control method exhibits notable characteristics, including fast transient response, small overshoots, and good robust performance.

Adaptive self-triggered control-based cooperative output regulation of heterogeneous multi-agent systems under sensor and actuator attack

The cooperative output regulation problem for heterogeneous multi-agent systems under sensor and actuator attacks with an adaptive self-triggering strategy is considered in this article. The attack considered is the false data injection attack. To resolve the system, a dynamic internal model is introduced for each agent, and a distributed adaptive self-triggering control scheme is proposed. The results show that intermittent communication is possible for all agents using the proposed control mechanism and no agent shows Zeno behavior. The adaptive controller is used to guarantee that the regulated output for the heterogeneous multi-agent systemss is achieved with cooperative uniform ultimate boundedness under both actuator and sensor attacks. Given the whole state of information is unavailable, the adaptive dynamic system is designed based on receivable compromised or uncompromised output information. The effectiveness of the given control scheme is illustrated by two examples.

The cascade backstepping controller design for supercavitating vehicle based on RBFNN observer

Aiming at the model uncertainty in design process of longitudinal motion control system of supercavitating vehicle, a backstepping controller is designed by combining with Radial Basis Function neural network (RBFNN). Considering that the environment of the supercavitating vehicle is complex during its traversal, it is not possible to measure all states directly. The calculation of planing force with nonlinear characteristics is related to vertical velocity, therefore, the accuracy of vertical velocity has impact on the value of planing force directly. A state observer is designed to estimate vertical velocity. In this paper, based on the backstepping method, the uncertain part of matrix coefficient in cascade control model of supercavitating vehicle is approximated by RBFNN. The output of the observer approximates the state variables, then, a depth tracking control law of the vehicle is obtained according to the observed values and the weight of neural network is calculated by Lyapunov function. Finally, it is proved that the control law can ensure the uniform ultimate boundedness of the closed-loop system.

Decentralized adaptive PI fault tolerant tracking control for strong interconnected nonlinear systems subject to unmodeled dynamics and actuator faults

In this study, a novel decentralized adaptive proportional–integral (PI) fault tolerant tracking control strategy is proposed for a class of strong interconnection nonlinear systems with unmodeled dynamics and actuator faults. The adverse effects generated by unmodeled dynamics could be eliminated by introducing a dynamic signal and the multiplicative actuator faults could be dealt with by using adaptive control technique. In addition, broad learning system (BLS) scheme is used to approximate the unknown nonlinear functions and additive actuator faults, the minimal-learning-parameter (MLP) scheme is used in this study to reduce the number of adaptive update parameters, which resltus in less computation overhead. The decentralized adaptive PI fault tolerant tracking controller including a switching compensation mechanism is designed for the considered interconnected systems, such that all the signals in the closed-loop systems are uniformly ultimately bounded (UUB) and the tracking errors converge to a small residual set. In the end, a chemical reactor example is utilized to illustrate the feasibility and significance of the designed control scheme.

OPTIMAL TRACKING CONTROL FOR ROBOT MANIPULATORS WITH INPUT CONSTRAINT BASED REINFORCEMENT LEARNING

This paper introduces an optimal tracking controller for robot manipulators with saturation torques. The robot model is presented as a strict-feedback nonlinear system. Firstly, the position tracking control problem is transformed into the optimal tracking control problem. Subsequently, the saturated optimal control law is designed. The optimal control law is determined through the solution of the Hamilton-Jacobi-Bellman (HJB) equation. We use a reinforcement learning algorithm with only one neural network (NN) to approximate the solution of the equation HJB. The technique of experience replay is used to relax a persistent citation condition. By Lyapunov analysis, the tracking and the approximation errors are uniformly ultimately bounded (UUB). Finally, the simulation on a robot manipulator with saturation torques is performed to verify the efficiency of the proposed controller.

Adaptive control for stochastic nonlinear systems with time‐varying delays via multidimensional Taylor network

AbstractControl of stochastic nonlinear systems turns out to be notoriously difficult when stochastic uncertainties and time‐varying delays occur simultaneously. This article presents a tractable adaptive control scheme for the stochastic nonlinear system with time‐varying delays. To mitigate the effects of stochastic uncertainties, an adaptive embedded cubature Kalman filter is developed to realize the robust estimation of the state. Unlike the conventional cubature Kalman filter with fixed construction, a semi‐definite programming is designed to adjust the weights of cubature points dynamically. Such programming guarantees the positive definiteness of the error covariance matrix, which enhances the reliability of the filtering procedure. Based on more accurate state estimations, the multidimensional Taylor network (MTN) is utilized to evaluate the dynamic performance under time‐varying delays and approximate the optimal policy in the deterministic policy gradient framework. Adaptive tracking control with high computational efficiency is achieved due to the concise topological structure of MTN. The exponential convergence of the state estimation error and the semi‐globally uniform ultimate boundness of the tracking error are verified theoretically. The effectiveness of the proposed method is confirmed by a numerical simulation based on a practical electric industrial system.

Attitude constraint-based recovery for under-actuated AUVs under vertical plane control during the capture stage

The landed docking method is one of the most widely used underwater recovery methods because it imposes fewer restrictions on AUV configuration and applies to a wide range of applications. During the capture stage, the AUV and the recovery device are in close proximity, and if the attitude of the AUV fluctuates significantly, a collision may occur and cause damage. Existing studies for the common underactuated AUV do not effectively constrain the non-directly controlled degrees of freedom, posing a significant safety risk to the underwater recovery in the capture stage. To address this issue, a vertical plane control strategy that considers the safety interval of non-direct controlled degrees of freedom is proposed to keep the attitude angle within the safety interval during the docking process. To begin, the non-directly controlled AUV's pitching motion is analyzed and simplified to demonstrate that it has input-to-state stability during the capture stage, and an estimate of its uniform ultimate bound is provided so that the motion boundary can be estimated. Then, based on the backstepping method, a thrust-gain-constrained vertical surface control strategy is proposed: for the directly controlled heaving motion and rolling motion, precise control is achieved through a reasonable configuration of control parameters; for the non-directly controlled pitching motion, the thrust-constrained gain is calculated through the ultimate bound so that the force applied to the pitch degrees of freedom of the thruster is constrained and the purpose of indirect control is achieved. Using this strategy, the pitch motion is guaranteed to be inside the safe interval, resulting in accurate, safe, and quick recovery control during the capture stage.

Event‐triggered adaptive fuzzy optimal control of modular robot manipulators using zero‐sum differential game through value iteration

SummaryIn this article, a novel event‐triggered zero‐sum differential game through value iteration (VI)‐based adaptive fuzzy control is proposed for modular robot manipulators (MRMs) with uncertain disturbance. The subsystem dynamic of MRMs is established by joint torque feedback (JTF) technology. An identifier based on fuzzy logic model (FLM) is established to identify unknown system dynamics. Then, the trajectory tracking control issue of MRMs with uncertain disturbance is transformed into a two‐player zero‐sum differential game. On the basis of event‐triggered mechanism and VI algorithm, the event‐triggered Hamilton–Jacobi–Issacs (HJI) equation is approximately solved through establishment of actor‐critic structure. Different from the traditional adaptive dynamic programming (ADP) algorithm, this scheme is based on adaptive fuzzy optimal control through an actor‐critic structure, which refers to adaptation of both optimal cost function and optimal control policy. Moreover, tracking errors of closed‐loop MRM system are demonstrated to be uniformly ultimately bounded (UUB) under Lyapunov stability theorem. At last, experimental examples and comparisons show the reliability and effectiveness of this method.

Discrete-time prediction based event-triggered controller design: An application to networked multi-area power system with time delays

This work proposes a solution for power system stability by utilizing prediction-based event-triggered control (ETC) in the discrete-time domain. The proposed control method can handle a large sampling period, and the event-triggered mechanism (ETM) is applied in both controller and actuator loops to reduce the network’s computational and communication burden. Input and output (I/O) quantizers are used to avoid the quantization error that arises due to sampling and are also included in the stability analysis. The proposed control strategy is evaluated under various load scenarios using three-area interconnected power systems. The results demonstrate that the proposed approach saves 25.5%, 22%, and 23.5% of channel bandwidth in each area, as compared to the conventional time-triggered control approach. A comparative study shows that the proposed work outperforms recently reported works in terms of better event triggering number, average inter-event time, and performance indices. The effectiveness of the proposed control schemes is further validated by considering uncertainty in system parameters and typical power system nonlinearities. The study also illustrates the integration of renewable energy resources (RERs) and electric vehicles (EVs). The closed-loop system stability is proved theoretically using uniform ultimate boundedness and validated through simulations in MATLAB R2018a.

Distributed Adaptive Human-in-the-Loop Event-Triggered Formation Control for QUAVs With Quantized Communication

To improve the safety and reliability of quadrotor unmanned aerial vehicles (QUAVs) with limited communication, system uncertainties, and unknown external disturbances in a highly uncertain and safety-critical environment, a distributed adaptive human-in-the-loop (HiTL) event triggered (ET) formation controller is proposed. The non-autonomous leader are controlled by receiving control commands decided by the gesture recognition system, and the followers are controlled indirectly via the connected communication network. Thus, the safety and flexibility of the closed-loop system are improved. The designed controller is quantized and then sent to actuator only at the ET instants to further reduce the network burden. The radial basis function neural network (RBFNN) is used to approximate the system uncertainties. The unknown approximation error and the unknown external disturbance are viewed as a compound disturbance compensated by the high order disturbance observer (HODO). In addition, the uniformly ultimately bounded (UUB) stability of the closed-loop system is achieved through the Lyapunov method. Finally, comparative experiments are implemented on the QUAVs to demonstrate the validity of the presented control scheme.

Approximate Optimal Control for Nonlinear Systems With Periodic Event-Triggered Mechanism.

This article investigates the approximate optimal control problem for nonlinear affine systems under the periodic event triggered control (PETC) strategy. In terms of optimal control, a theoretical comparison of continuous control, traditional event-based control (ETC), and PETC from the perspective of stability convergence, concluding that PETC does not significantly affect the convergence rate than ETC. It is the first time to present PETC for optimal control target of nonlinear systems. A critic network is introduced to approximate the optimal value function based on the idea of reinforcement learning (RL). It is proven that the discrete updating time series from PETC can also be utilized to determine the updating time of the learning network. In this way, the gradient-based weight estimation for continuous systems is developed in discrete form. Then, the uniformly ultimately bounded (UUB) condition of controlled systems is analyzed to ensure the stability of the designed method. Finally, two illustrative examples are given to show the effectiveness of the method.

On uniform-ultimate boundedness and periodicity results of solutions to certain second order non-linear vector differential equations

In this paper, we employ the second method of Lyapunov to examine sufficient conditions for the uniform-ultimate boundedness of solutions and existence of at least one periodic solution to the following second order vector differential equation: Ẍ+ F(X, Ẋ ) Ẋ + H(X) = P(t, X, Ẋ ), when the non-linear term H(X) is: (i) differentiable, (ii) non-necessarily differentiable. The results contain in this paper are new and complement related ones in the literature.

Adaptive control of cooperative robots in the presence of disturbances and uncertainties: A Bernstein–Chlodowsky approach

The function approximation technique (FAT) is a powerful mathematical tool recently utilized to design model‐free controllers for robots. However, some FAT‐based controllers depend on joint velocities, which may not be available in many real‐world applications. This problem is solved in this paper by proposing an output feedback tracking control for cooperative robotic arms using the Bernstein–Chlodowsky polynomial as an uncertainty approximator. In other words, the Bernstein–Chlodowsky approach is adopted to approximate the lumped uncertainties consisting of disturbances and unmodeled dynamics. An adaptive rule is then suggested to update the approximator's coefficients matrix. Moreover, it is assured that controlled system error signals are uniformly ultimately bounded (UUB) utilizing the Lyapunov lemma. Finally, the designed Bernstein–Chlodowsky controller is applied to a cooperative system with two arms handling a load. Besides, the results of applying the designed technique are compared with the outcomes of the Chebyshev neural network (CNN) as a state‐of‐the‐art approximation method. The simulation outcomes indicate the capability of the designed method.

Neural network-based adaptive controller design for robotic manipulator subject to varying loads and unknown dead-zone

In this article, aiming at handling the trajectory tracking issue of industrial manipulator system (IMS) with modeling uncertainty, varying loads (VL) and unknown dead-zone characteristic, a compensation-based adaptive switching controller synthesis is proposed. In this scheme, the dynamic model of the IMS under VL is regarded as a switched system (SS) with a specified modal set. The nonlinear term related to plant model in each subsystem is approximated by radial basis function neural network (RBFNN) so as to avoid the reliance of the controller on the accurate model, and the unknown dead-zone is estimated and compensated by NN, from which the corresponding NN robust compensation term is developed to eliminate the potential perturbations and estimated errors. The designed controller with switching mechanism effectively solves the problem of degradation of the tracking accuracy caused by VL. Finally, the uniform ultimate boundedness of error signals is analyzed by the average dwell time (ADT) approach, multi-Lyapunov function method and the synthesized adaptive control law, and the effectiveness of the developed scheme is verified by simulation.

Observer‐based adaptive backstepping control and the application for a class of multi‐input multi‐output nonlinear systems with structural uncertainties and perturbations

AbstractIn this study, the adaptive backstepping control of a class of multi‐input multi‐output nonlinear systems with immeasurable states, structural uncertainties and periodic perturbations is researched using neural network (NN) based nonlinear observer. Firstly, a series of harmonic components obtained from Fourier series expansion (FSE) are employed to estimate the unknown periodic perturbations. Then treating the estimated perturbations as inputs, a radial basis function‐based neural network (RBFNN) is constructed as a component of the observer, and the observer is proved to be uniformly ultimately bounded (UUB) in estimating the immeasurable system states with the negative‐gradient adaptive laws of NN parameters. Subsequently, the adaptive backstepping control is designed to track reference signals at the outputs of system based on the FSE‐RBFNN observer, and the stability of the closed‐loop system is proved on a finite set of system states. Finally, the proposed methods are applied to a triangular tethered satellite formation model to test the stability of the observer and control, and the effect of FSE in estimating perturbations is comparatively tested as well.

Finite time course keeping control for unmanned surface vehicles with command filter and rudder saturation

This paper studies the problem of course keeping control for unmanned surface vehicle (USV) with servo dynamic characteristics, environmental disturbances and model’s uncertainties, and rudder saturation. Considering the characteristics of the rudder servo system (RSS), a nonlinear Norrbin model with second-order rudder is reconstructed, and proposes a finite time simple and practical course controller by the finite time dynamic surface control (FTDSC) technology, which achieve excellent finite time command filter. Then, the nonlinear disturbance observer (NDOB) is applied to estimate the lumped disturbances, which is easy to implement in engineering. Meanwhile, the bandwidth between the controller and the observer is studied, which affects the convergence speed of the observer. Besides, finite time dynamic auxiliary systems with smooth switching function (FTADS-SSF) is designed to compensate for rudder physical constraints with finite time convergence. Further practical finite time stability analysis indicates that the entire system is uniformly ultimately bounded (UUB). Finally, simulation comparison results have been demonstrated the proposed strategy.

Event‐triggered neural experience replay learning for nonzero‐sum tracking games of unknown continuous‐time nonlinear systems

AbstractIn this paper, an online event‐triggered integral reinforcement learning is proposed to solve the nonzero‐sum tracking games of the two‐player nonlinear system with unknown dynamics and constrained input. Firstly, an augmented system of the nonzero‐sum game is constructed to describe the tracking problem. Thus, the optimal tracking problem is treated as solving the coupled Hamilton‐Jacobi (HJ) equations of the augmented system. To restrict the number of sampling states, a novel triggering threshold containing the augmented states is designed, and it avoids the existence of Zeno behavior. Secondly, a single‐critic network is utilized to approximate the solution of the coupled HJ equations. The critic network turning law with experience replay technology relaxes the dependence on the persistence of excitation (PE) conditions in traditional integral reinforcement learning (IRL) by using historical data in the stack. Moreover, the augmented system states and the critic NN weight errors are uniformly ultimate boundedness (UUB) by Lyapunov theory. Simulation examples are provided to demonstrate the availability of the proposed algorithm.