Integral-type Lyapunov Function Research Articles

Divergent Component of Motion Planning and Adaptive Repetitive Control for Wearable Walking Exoskeletons.

Wearable walking exoskeletons show great potentials in helping patients with neuro musculoskeletal stroke. Key to the successful applications is the design of effective walking trajectories that enable smooth walking for exoskeletons. This work proposes a walking planning method based on the divergent component of motion to obtain a stable joint angle trajectory. Since periodic and nonperiodic disturbances are ubiquitous in the repeating walking motion of an exoskeleton system, a major challenge in the walking control of wearable exoskeleton is the joint angle drift problem, that is, the joint angle motion trajectories are not necessarily periodic due to the presence of disturbance. To address this challenge, this work develops an adaptive repetitive control strategy to guarantee that the motion trajectories of joint angle are repetitive. In particular, by treating the disturbance as system uncertainties, an adaptive controller is designed to compensate for the uncertainties based on an integral-type Lyapunov function. A fully saturated learning approach is then developed to achieve asymptotic tracking of repetitive walking trajectories. Extensive experiments are carried out to demonstrate the effectiveness of the tracking performance.

Manipulator Control Based on Adaptive RBF Network Approximation

With the popularization of intelligent manufacturing, manipulator has found ever wider application in various industries. A manipulator requires a real-time and fast control algorithm in order to improve the accuracy in all kinds of precise operations. This paper proposes an algorithm based on adaptive radial basis function (RBF) for approximating the parameters of the manipulator, and the adaptive equations are designed to automatically adjust the weight of RBF. Proportional integral (PI) robust based on dynamic error tracking is used in controller to reduce the steady state errors and enhance the anti-interference performance of the system. The global asymptotic stability of the system is demonstrated by defining an integraltype Lyapunov function. Finally, MATLAB is used to simulate the angular positions tracking and angular velocities tracking of the double joints manipulator. The results show that the manipulator can track the ideal output signal quickly and accurately and has good anti-interference performance.

Fuzzy Adaptive Finite-Time Consensus Control for High-Order Nonlinear Multiagent Systems Based on Event-Triggered

This article studies the fuzzy adaptive finite-time consensus control problem for high-order nonlinear multiagent systems with unknown nonlinear dynamics. In control design,fuzzy logic systems (FLSs) are adopted to approximate the unknown nonlinear dynamics, and under the frameworks of adaptive backstepping recursive design and finite-time stability theory, an adaptive fuzzy finite-time consensus control method is developed. To save communication resources and reduce the numbers of controller execution times, a dynamic event-triggered mechanism with a relative threshold is established. Subsequently, an event-triggered-based finite-time fuzzy adaptive control scheme is formulated. Furthermore, by constructing novel integral-type Lyapunov functions and adding a power integrator technique, the finite-time stability of the closed-loop system and the convergence of consensus tracking errors are proved. Finally, a numerical simulation example is provided to verify the effectiveness of the proposed adaptive event-triggered consensus control method.

Fuzzy Enhanced Adaptive Admittance Control of a Wearable Walking Exoskeleton With Step Trajectory Shaping

The generation of motor adaptation in response to mechanical perturbation during human walking is seldom considered in an exoskeleton system. Reshaping step trajectory over consecutive gait cycles for a walking exoskeleton is investigated in this article. Step adjustment of a walking exoskeleton can adapt to human walking intention by shaping step trajectory. This work develops an admittance adaptive fuzzy control strategy for a walking exoskeleton robot to provide assistance for human lower limb movement. Considering human walking intention and utilizing an admittance model, it shapes a reference trajectory to ensure that the walking exoskeleton follows it according to the human–robot force produced by its wearer. Considering a nonlinear and dynamic model with uncertainties, this work designs an integral-type Lyapunov function controller to track a reference trajectory. A disturbance observer is integrated into the controller design to compensate for uncertain disturbance in order to achieve an effective tracking performance. Finally, this work conducts experiments on two healthy subjects with the proposed method on a walking exoskeleton to validate its effectiveness. The results show that it can be applied to walking exoskeletons to enhance human mobility.

Fuzzy adaptive finite-time consensus tracking control for nonlinear multi-agent systems

This paper focuses on the finite-time consensus tracking control problem of nonlinear multi-agent systems. Dynamics of each agent has completely unknown nonlinear terms that cannot be directly used for control design. Therefore, fuzzy logic systems are employed to approximate these nonlinear functions. Furthermore, a finite-time fuzzy adaptive consensus tracking protocol is proposed for a class of nonlinear multi-agent systems by using integral-type Lyapunov functions. The developed adaptive backstepping design scheme successfully avoids the singularity problem of the derivatives of virtual control signals. It is shown that with the presented control protocol, the consensus tracking errors converge to a small neighbourhood of the origin in finite time, and the other signals of multi-agent systems are bounded. Finally, a numerical example is used to verify the effectiveness of the proposed control protocol.

분산 PID 알고리즘을 이용한 입력 포화가 존재하는 다중 모터 시스템의 동기화

In this paper, we study the speed synchronization problem for a group of multiple motor systems by considering input saturations. To solve this problem, we propose the distributed PID algorithm based on the consensus technique. In addition, the integral controller may cause the integral windup effect due to input saturations. Therefore, we apply the anti-windup technique to overcome the windup problem. Then, we analyze the stability of the overall error system using an integral-type Lyapunov function. Finally, experiments are performed to demonstrate the validity of the proposed algorithm.

An exploration on adaptive iterative learning control for a class of commensurate high-order uncertain nonlinear fractional order systems

This paper explores the adaptive iterative learning control method in the control of fractional order systems for the first time. An adaptive iterative learning control U+0028 AILC U+0029 scheme is presented for a class of commensurate high-order uncertain nonlinear fractional order systems in the presence of disturbance. To facilitate the controller design, a sliding mode surface of tracking errors is designed by using sufficient conditions of linear fractional order systems. To relax the assumption of the identical initial condition in iterative learning control U+0028 ILC U+0029, a new boundary layer function is proposed by employing Mittag- Leffler function. The uncertainty in the system is compensated for by utilizing radial basis function neural network. Fractional order differential type updating laws and difference type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. The hyperbolic tangent function and a convergent series sequence are used to design robust control term for neural network approximation error and bounded disturbance, simultaneously guaranteeing the learning convergence along iteration. The system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapnov-like composite energy function U+0028 CEF U+0029 containing new integral type Lyapunov function, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

Adaptive neural dynamic surface control of strict-feedback nonlinear systems with full state constraints and unmodeled dynamics

In this paper, the problem of adaptive neural network (NN) dynamic surface control (DSC) is discussed for a class of strict-feedback nonlinear systems with full state constraints and unmodeled dynamics. By introducing a one to one nonlinear mapping, the strict-feedback system with full state constraints is transformed into a novel pure-feedback system without state constraints. Radial basis function (RBF) neural networks (NNs) are used to approximate unknown nonlinear continuous functions. Unmodeled dynamics is dealt with by introducing a dynamical signal. Using modified DSC and introducing integral-type Lyapunov function, adaptive NN DSC is developed. Using Young’s inequality, only one parameter is adjusted at each recursive step in the design. It is shown that all the signals in the closed-loop system are semi-global uniform ultimate boundedness (SGUUB), and the full state constraints are not violated. Simulation results are provided to verify the effectiveness of the proposed approach.

Distributed Consensus of Second-Order Multi-Agent Systems With Heterogeneous Unknown Inertias and Control Gains Under a Directed Graph

In this paper, we study the consensus problem for second-order multi-agent systems with heterogeneous unknown inertias and control gains under a general directed graph. Unlike the existing consensus algorithms for second-order multi-agent systems in which all agents are assumed to have common unit inertias or share common control gains, we allow the inertias and the control gains to be heterogeneous and time-varying for each agent. We propose fully distributed consensus algorithms over a general directed graph when there exist, respectively, absolute velocity damping and relative velocity damping. Novel integral-type Lyapunov functions are proposed to study the consensus convergence. Moreover, the adaptive $\sigma$ - modification schemes for the gain adaptation are proposed, which renders smaller control gains and thus requires smaller amplitude on the control input without sacrificing consensus convergence. Furthermore, we show that one proposed algorithm also works for consensus of agents with intrinsic Lipschitz nonlinear dynamics. The control gains are varying and updated by distributed adaptive laws. As a result, the proposed algorithms require no global information and thus can be implemented in a fully distributed manner.

Disturbance Observer-Based Fuzzy Control of Uncertain MIMO Mechanical Systems With Input Nonlinearities and its Application to Robotic Exoskeleton

We develop a novel disturbance observer-based adaptive fuzzy control approach in this paper for a class of uncertain multi-input-multi-output mechanical systems possessing unknown input nonlinearities, i.e., deadzone and saturation and time-varying external disturbance. It is shown that the input nonlinearities can be represented by a nominal part and a nonlinear disturbance term. High-dimensional integral-type Lyapunov function is used to construct the controller. Fuzzy logic system is employed to cancel model uncertainties, and disturbance observer is also integrated into control design to compensate the fuzzy approximation error, external disturbance, and nonlinear disturbance caused by the unknown input nonlinearities. Semiglobally uniformly ultimately boundness of the closed-loop control system is guaranteed with tracking errors keeping bounded. Experimental studies on a robotic exoskeleton using the proposed control demonstrate the effectiveness of the approach.

Continuous composite finite-time convergent guidance laws with autopilot dynamics compensation

This paper has proposed two continuous composite finite-time convergent guidance laws to intercept maneuvering targets in the presence of autopilot lag: one is for hit-to-kill and the other is for zeroing the line-of-sight (LOS) angular rate. More specifically, the nonlinear disturbance observer (NDOB) is used to estimate the lumped uncertainty online while the finite-time control technique is used to fulfill the design goal in finite time. The key feature in derivation of the proposed guidance law is that two integral-type Lyapunov functions are used to avoid analytic differentiation of virtual control law encountered with traditional backstepping. The finite-time stability of the closed-loop nonlinear observer-controller system is established using finite-time bounded (FTB) function and Lyapunov function methods. Numerical simulations with some comparisons are carried out to demonstrate the superiority of the proposed method.

Adaptive Control of Nonlinear Time-varying Delay Systems with Unknown Control Gain Signs

Abstract Based on the principle of sliding mode control and the property of Nussbaum-type functions, an adaptive neural control scheme is proposed for a class of MIMO nonlinear time delay systems with unknown function control gains. By utilizing the integral-type Lyapunov function and Young's inequality, the restriction of the control gains and the assumption of time-varying delay uncertainties are relaxed. By choosing appropriate Lyapunov–Krasovskii functionals, unknown time-varying delay uncertainties are compensated for. By theoretical analysis, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded.

Adaptive Tracking for Periodically Time-Varying and Nonlinearly Parameterized Systems Using Multilayer Neural Networks

This brief addresses the problem of designing adaptive neural network tracking control for a class of strict-feedback systems with unknown time-varying disturbances of known periods which nonlinearly appear in unknown functions. Multilayer neural network (MNN) and Fourier series expansion (FSE) are combined into a novel approximator to model each uncertainty in systems. Dynamic surface control (DSC) approach and integral-type Lyapunov function (ILF) technique are combined to design the control algorithm. The ultimate uniform boundedness of all closed-loop signals is guaranteed. The tracking error is proved to converge to a small residual set around the origin. Two simulation examples are provided to illustrate the feasibility of control scheme proposed in this brief.

Adaptive Neural Control for a Class of Nonlinear Systems With Uncertain Hysteresis Inputs and Time-Varying State Delays

In this paper, adaptive variable structure neural control is investigated for a class of nonlinear systems under the effects of time-varying state delays and uncertain hysteresis inputs. The unknown time-varying delay uncertainties are compensated for using appropriate Lyapunov-Krasovskii functionals in the design, and the effect of the uncertain hysteresis with the Prandtl-Ishlinskii (PI) model representation is also mitigated using the proposed control. By utilizing the integral-type Lyapunov function, the closed-loop control system is proved to be semiglobally uniformly ultimately bounded (SGUUB). Extensive simulation results demonstrate the effectiveness of the proposed approach.

Adaptive variable structure control of MIMO nonlinear systems with time-varying delays and unknown dead-zones

In this paper, adaptive variable structure neural control is presented for a class of uncertain multi-input multi-output (MIMO) nonlinear systems with state time-varying delays and unknown nonlinear dead-zones. The unknown time-varying delay uncertainties are compensated for using appropriate Lyapunov-Krasovskii functionals in the design. The approach removes the assumption of linear function outside the deadband without necessarily constructing a dead-zone inverse as an added contribution. By utilizing the integral-type Lyapunov function and introducing an adaptive compensation term for the upper bound of the residual and optimal approximation error as well as the dead-zone disturbance, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded. In addition, a modified adaptive control algorithm is given in order to avoid the high-frequency chattering phenomenon. Simulation results demonstrate the effectiveness of the approach.

Adaptive Neural Network Tracking Control of MIMO Nonlinear Systems With Unknown Dead Zones and Control Directions

In this paper, adaptive neural network (NN) tracking control is investigated for a class of uncertain multiple-input-multiple-output (MIMO) nonlinear systems in triangular control structure with unknown nonsymmetric dead zones and control directions. The design is based on the principle of sliding mode control and the use of Nussbaum-type functions in solving the problem of the completely unknown control directions. It is shown that the dead-zone output can be represented as a simple linear system with a static time-varying gain and bounded disturbance by introducing characteristic function. By utilizing the integral-type Lyapunov function and introducing an adaptive compensation term for the upper bound of the optimal approximation error and the dead-zone disturbance, the closed-loop control system is proved to be semiglobally uniformly ultimately bounded, with tracking errors converging to zero under the condition that the slopes of unknown dead zones are equal. Simulation results demonstrate the effectiveness of the approach.

Adaptive dynamic surface control of nonlinear systems with unknown dead zone in pure feedback form

In this paper, adaptive dynamic surface control (DSC) is developed for a class of pure-feedback nonlinear systems with unknown dead zone and perturbed uncertainties using neural networks. The explosion of complexity in traditional backstepping design is avoided by utilizing dynamic surface control and introducing integral-type Lyapunov function. It is proved that the proposed design method is able to guarantee semi-global uniform ultimate boundedness of all signals in the closed-loop system, with arbitrary small tracking error by appropriately choosing design constants. Simulation results demonstrate the effectiveness of the proposed approach.

Adaptive Neural Control of SISO Time-Delay Nonlinear Systems with Unknown Hysteresis Input

In this paper, adaptive variable structure neural control is investigated for a class of SISO nonlinear systems in a Brunovsky form with state time-varying delays and unknown hysteresis input. The unknown time-varying delay uncertainties are compensated for using appropriate Lyapunov-Krasovskii functionals in the design. The effect of the unknown hysteresis with the Prandtl-Ishlinskii model is mitigated using the proposed adaptive control. By utilizing the integral-type Lyapunov function, the closed-loop control system is proved to be semi-globally uniformly ultimately bounded. Simulation results demonstrate the effectiveness of the approach.

Adaptive Fuzzy Control for a Class of MIMO Nonlinear Systems with Unknown Dead-zones

A design scheme of adaptive fuzzy controller for a class of uncertain MIMO nonlinear systems with unknown dead-zones and a triangular control structure is proposed in this paper. The design is based on the principle of sliding mode control and the property of Nussbaum function. The approach does not require a priori knowledge of the signs of the control gains and the upper bounds and lower bounds of dead-zone parameters to be known a priori. By introducing the integral-type Lyapunov function and adopting the adaptive compensation term of the upper bound of the optimal approximation error and the dead-zone disturbance, the closed-loop control system is proved to be semi-globally stable in the sense that all signals involved are bounded, with tracking errors converging to zero.

Adaptive control of uncertain continuously stirred tank reactors with unknown actuator nonlinearities

Adaptive nonlinear control is investigated for continuously stirred tank reactor (CSTR) systems using neural networks. The CSTR plant under study belongs to a class of nonaffine nonlinear systems, and contains an unknown parameter that enters the model nonlinearly. Using adaptive backstepping and neural network (NN) approximation techniques, an alternative adaptive NN controller is developed that achieves asymptotic output tracking control. A novel integral-type Lyapunov function, which includes both system states and control input as its arguments, is constructed to solve the difficulty associated with the nonaffine control problem. Numerical simulation is performed to show the feasibility of the proposed approach for chemical process control.