Filtered Tracking Error Research Articles

Trajectory Tracking Control for a QUAV With Performance Constraints

In this paper, a trajectory tracking control scheme is developed for a quadrotor unmanned aerial vehicle (QUAV) with unknown inertial moments and unknown interconnections among every attitude subsystem. In this scheme, the overall control system is decoupled into a position subsystem and an attitude subsystem. For the position subsystem, a position constraints controller is proposed, while a prescribed performance constraints controller is developed for the attitude subsystem. It is proved that all the signals in the closed-loop system are bounded and that both the position constraints and the prescribed performance constraints on attitude tracking errors can be achieved by guaranteeing the boundedness of the filtered tracking errors. The simulation results demonstrate the effectiveness of the proposed scheme.

Composite learning robot control with guaranteed parameter convergence

Parameter convergence is desirable in adaptive control as it enhances the overall stability and robustness properties of the closed-loop system. However, a stringent condition termed persistent excitation (PE) must be satisfied to guarantee parameter convergence in the conventional adaptive control. This paper provides the first result of parameter convergence without the PE condition for adaptive control of a general class of robotic systems. More specifically, we develop a composite learning robot control (CLRC) strategy to achieve fast and accurate parameter estimation under a condition termed interval excitation (IE) which is much weaker than the PE condition. In the composite learning, a time-interval integral of a filtered regressor is utilized to construct a prediction error such that the time derivation of plant states is not necessary, and both the prediction error and a filtered tracking error are employed to update the parameter estimate. The closed-loop system is proven to be globally exponentially stable under the IE condition. Robustness against external disturbances of the CLRC is analyzed in the Lyapunov sense. An illustrative example shows the effectiveness and superiority of the proposed approach.

Asymptotical Cooperative Tracking Control for Unknown High-Order Multi-Agent Systems via Distributed Adaptive Critic Design

In this paper, a distributed adaptive critic design-based asymptotical cooperative tracking control scheme is established for a class of unknown high-order multi-agent systems with disturbances on undirected graphs. The cooperative tracking problem of high-order multi-agent systems is first transformed to the stabilizing problem of filtered tracking errors dynamic systems, which only depend on the information from neighboring agents. Then, a novel distributed adaptive critic design-based controller is developed for the newly obtained filtered tracking errors dynamic systems. The special local filtered tracking errors and local critic signals are defined for facilitating the implementation of distributed adaptive critic design. Novel distributed weights updating laws for critic neural networks, actor neural networks, and adaptive parameters are proposed. Furthermore, the RISE technique is introduced to suppress the external disturbances and neural network approximation errors. The asymptotical stability of a closed-loop system is demonstrated by using Lyapunov stability theory. Finally, a simulation example justifying the effectiveness of the proposed control scheme is given.

Adaptive fuzzy control for feedback linearizable MIMO nonlinear systems with prescribed performance

In this paper, a new adaptive fuzzy control scheme with prescribed performance is developed for a class of feedback linearizable uncertain MIMO nonlinear systems with unknown control direction and external disturbances. The fuzzy systems are employed to approximate the unknown nonlinear functions, and a Nussbaum-type function is applied to resolve the unknown control direction problems. By using the prescribed performance bounds, an adaptive fuzzy controller equipped with the Nussbaum-type gain function is developed. The proposed design scheme guarantees that all the signals in the closed-loop systems are bounded and that the tracking errors converge to a prescribed performance bounds by guaranteeing the convergence of the filtered tracking error to a predefine performance bounds. Two simulation examples are used to demonstrate the effectiveness of the proposed scheme.

Uniformly Ultimately Bounded Tracking for Uncertain Euler-Lagrange Systems with Unknown Time-Varying Input Delay

This paper proposes a robust compensator for a class of uncertain nonlinear systems subjected to unknown time-varying input delay. The proposed control law is based on the integral of past values of control and a novel filtered tracking error, which can compensate for large input delays. Sufficient gain conditions dependent on the known bound of the delay are derived using a Lyapunov based stability analysis, where suitable Lyapunov-Krasovskii (L-K) functionals are used to achieve a uniformly ultimately bounded (UUB) result. Simulation on an example of Euler-Lagrange (E-L) system is illustrated to evaluate the performance and robustness of controller for different values of time-varying input delay.

Adaptive sliding mode control of non‐linear non‐minimum phase system with input delay

This study proposes an adaptive sliding mode control (SMC) design method for multi-input multi-output non-linear non-minimum phase system with input delay. The problem is challenging, not only because of the unstable internal dynamics of the non-minimum phase system, but also the existence of input delay. The partially linearised model of the original non-linear system is obtained through input/output linearisation, and a state tracking model is constructed based on the computed ideal internal dynamics. A filtered tracking error is introduced to handle the input delay. A SMC strategy is proposed for the stability of filtered tracking error system, and an adaptive law is designed to estimate the upper bound of perturbations. Finally, comparison simulation results on vertical takeoff and landing aircraft are provided to verify the efficiency of the proposed method.

Design of three exponentially convergent robust controllers for the trajectory tracking of autonomous underwater vehicles

This paper deals with the trajectory tracking problem of autonomous underwater vehicles (AUVs) in the presence of dynamic uncertainties and time-varying external disturbances. Three exponentially convergent robust controllers, namely, the min-max type controller, the saturation type controller and the smooth transition type controller are proposed to drive an AUV to track a predefined trajectory. It is shown that the filtered tracking errors, position tracking errors and velocity tracking errors for the three proposed controllers are exponentially convergent. Moreover, all the above tracking errors for the three proposed controllers can be shaped by specific analytic expressions and such expressions illustrate how the transient responses of the above tracking errors can be modified by adjusting the control parameters. The characteristics of the three proposed controllers are summarized and demonstrated with numerical simulations. Theoretical comparison analysis and comparative simulations with the existing RISE-based controller of AUV are presented to show the effectiveness of the three proposed exponentially convergent robust controllers.

Design of Adaptive Fuzzy Control for a Class of Networked Nonlinear Systems

This paper presents an adaptive fuzzy controller for a class of unknown nonlinear systems over network. The network-induced delays can degrade the performance of the networked control systems (NCSs) and also can destabilize the system. Moreover, the seriousness of the delay problem is aggravated when packet losses occur during a transmission of data. The proposed controller uses a filtered tracking error to cope the time-varying network-induced delays. It is also robust enough to cope some packet losses in the system. Fuzzy logic systems (FLSs) are used to approximate the unknown nonlinear functions that appear in the tracking controller. Based on Lyapunov stability theory, the constructed controller is proved to be asymptotically stable. Stability of the adaptive fuzzy controller is guaranteed in the presence of bounded external disturbance, time-varying delays, and data packet dropouts. Simulated application of the inverted pendulum tracking illustrates the effectiveness of the proposed technique with comparative results.

Adaptive variable structure control of input delay non-minimum phase hypersonic flight vehicles

An adaptive variable structure control strategy is proposed for the output tracking control of input delay non-minimum hypersonic flight vehicles. The problem is challenging because of the complex nonlinearity of hypersonic flight vehicles and the existence of input delay. The nonlinear model of hypersonic flight vehicles is partially linearized, and a state tracking model is constructed based on the ideal internal dynamics of hypersonic flight vehicles. A filtered tracking error is introduced to handle the input delay. A variable structure control strategy is proposed for the stability of filtered tracking error system, and an adaptive law is established for the unknown perturbations. Finally, the effectiveness of the proposed control method is shown by the simulation results.

Trajectory tracking with obstacle avoidance of redundant manipulator based on fuzzy inference systems

In presence of the uncertainties in robot dynamics, model-based control law can easily fail. In our paper, this important question is tackled by using fuzzy adaptive control to drive with obstacle avoidance an industrial redundant manipulator under the hypothesis of uncertain dynamics. This hypothesis makes our controller an original one, since the problem of the tracking trajectory in the presence of obstacles and assuming that the robot dynamics are unknown has not been discussed before in the literature. Besides, the proposed obstacle avoidance is achieved by generating a self-motion. Furthermore, the usual procedure related to the obstacle avoidance based on Euclidean distance in 3D space is made easier since this one includes only the parameters of 2D space. This self-motion is directly incorporated into the adaptive fuzzy control scheme via the filtered tracking errors. The proposed method is tested by simulations in the case of redundant robot (PUMA 560) operating in 3D space in presence of obstacles. Despite that the robot dynamics are assumed to be uncertain, the proposed control performance is satisfactory. Indeed, the trajectory tracking control, incorporating self-motion obstacle avoidance, operates effectively with weak tracking control errors and bounded actuator torques evolving in admissible range.

Neural Learning Control of Marine Surface Vessels With Guaranteed Transient Tracking Performance

This paper studies neural learning control with predefined tracking error bound for a marine surface vessel whose accurate dynamics could not be obtained a priori . With the introduction of an error transformation function, the constrained tracking control of the original vessel is transformed into the stabilization of an unconstrained system. A filtered tracking error is introduced based on the error transformation, and radial basis function (RBF) neural networks (NNs) are employed to approximate unknown vessel dynamics. Subsequently, stable adaptive NN control is proposed to ensure ultimate boundedness of all the signals in the closed-loop system and to guarantee prescribed tracking performances. Under persistent excitation (PE) condition, the proposed adaptive NN control is shown to be capable of acquiring knowledge on the vessel dynamics and of storing the learned knowledge in memory. The stored knowledge is reused to develop neural learning control such that the improved control performance with faster tracking convergence rate and less computational burden could be achieved, while prescribed transient and steady-state tracking control performances are guaranteed. Simulation studies are performed to demonstrate the effectiveness of the proposed design techniques.

Lyapunov-Based Control of an Uncertain Euler-Lagrange System with Uncertain Time-Varying Input Delays without Delay Rate Constraints

A tracking controller is developed for a class of uncertain Euler-Lagrange systems with bounded external disturbances and an uncertain time-varying input delay. A novel filtered tracking error signal is designed to obtain a delay-free input signal in the closed-loop error system to facilitate the control design and analysis. The designed novel filtered tracking error consists of a finite integral over a constant estimated delay interval of the past control inputs as a means to compensate for the input delay. The maximum tolerable error between unknown time-varying delay and a constant estimate of the delay can be determined based on the selection of the control gains to provide uniformly ultimately bounded convergence of the tracking error to the origin. Lyapunov-Krasovskii functionals are used in the Lyapunov-based stability analysis

Neural control of hypersonic flight dynamics with actuator fault and constraint

This paper deals with the control problem of actuator fault and saturation for hypersonic flightvehicles. Different from previous back-stepping design, the scheme is on transforming the dynamics into the“prediction function”. The controller is constructed with high gain observer, while the effect of fault andsaturation is compensated by neural networks. For the input saturation, the auxiliary dynamics is included todesign the adaptive learning law. The neural weights and filtered tracking error are guaranteed to be boundedvia Lyapunov approach. The effectiveness of the proposed method is verified by simulation of winged-conemodel.

Minimal-learning-parameter technique based adaptive neural control of hypersonic flight dynamics without back-stepping

This paper investigates one robust adaptive controller for hypersonic flight dynamics. The altitude tracking is transformed into the control problem of the attitude subsystem, which is composed of flight path angle, pitch angle and pitch rate. Different from previous design using back-stepping related technique, this paper analyzed the tracking control without back-stepping where the controller is synthesized with high gain observer and minimal-learning-parameter technique. The highlight is that the design procedure is greatly simplified and the computation burden of parameter updating is reduced. It is proved that the filtered tracking error is guaranteed in the semiglobal sense. Simulation results are presented to demonstrate the effectiveness of the design.

Control strategy and application of hysteretic chaotic neuron and neural network

In order to remain the structure of the neural network in the process of the optimization unchanged, taking the hysteretic chaotic neuron and the hysteretic chaotic neural network as controlled plants, a novel control strategy based on the filtered tracking error is proposed to perform the stability control for the single hysteretic chaotic neuron or the hysteretic chaotic neural network. Especially, the hysteretic chaotic neuron and the hysteretic chaotic neural network can be used to solve the optimization problem through using the control strategy on condition that the generation mechanisms of the nonlinear characteristics, hysteresis and chaos, are unchanged. The control law is composed of two terms: one is the equivalent control term in the ideal filtered tracking error surface, and the other is the control term which can make the system reach the filtered tracking error surface quickly. Lyapunov stability method is used to prove the stability of the control strategy for the single hysteretic chaotic neuron and hysteretic chaotic neural network. The control laws of hysteretic chaotic neurons can be obtained according to the optimization function. The state of the single hysteretic chaotic neuron or the hysteretic chaotic neural network can converge to an extreme point of the optimization function gradually by the control law. In this way, the optimization problem can be solved effectively. Simulation results prove the feasibility and validity of the control strategy for optimization problem.

Design of adaptive fuzzy-based tracking control of input time delay nonlinear systems

This paper proposes an adaptive fuzzy logic tracking control for unknown nonlinear systems having input time delay. In the controller design, a filtered tracking error is introduced to facilitate handling of the time delay. Approximation capability of fuzzy logic systems are exploited to provide approximations of unknown nonlinear functions that appear in the tracking controller. Lyapunov stability analysis shows that the designed tracking control yields bounded signals of the closed-loop system. Simulation examples are provided to validate the effectiveness of the proposed controller in achieving desired tracking with bounded control signals.

Adaptive fuzzy optimal control using direct heuristic dynamic programming for chaotic discrete-time system

In this paper, we aim to solve the optimal tracking control problem for the Henon Mapping chaotic system using the direct heuristic dynamic programming (DHDP) setting with filtered tracking error. The fuzzy logic system is used to approximate the long-term utility function. Compared with the results for chaotic discrete-time system, the cost of the controller is reduced. The Lyapunov analysis approach is utilized to prove the stability of the chaotic system. It is shown that the tracking error, the adaptation law and the control input retain the property of uniformly ultimate boundedness. A simulation example is given to demonstrate the effectiveness of the proposed approach.

Dynamic learning from adaptive neural network control of a class of nonaffine nonlinear systems.

This paper studies the problem of learning from adaptive neural network (NN) control of a class of nonaffine nonlinear systems in uncertain dynamic environments. In the control design process, a stable adaptive NN tracking control design technique is proposed for the nonaffine nonlinear systems with a mild assumption by combining a filtered tracking error with the implicit function theorem, input-to-state stability, and the small-gain theorem. The proposed stable control design technique not only overcomes the difficulty in controlling nonaffine nonlinear systems but also relaxes constraint conditions of the considered systems. In the learning process, the partial persistent excitation (PE) condition of radial basis function NNs is satisfied during tracking control to a recurrent reference trajectory. Under the PE condition and an appropriate state transformation, the proposed adaptive NN control is shown to be capable of acquiring knowledge on the implicit desired control input dynamics in the stable control process and of storing the learned knowledge in memory. Subsequently, an NN learning control design technique that effectively exploits the learned knowledge without re-adapting to the controller parameters is proposed to achieve closed-loop stability and improved control performance. Simulation studies are performed to demonstrate the effectiveness of the proposed design techniques.

ADAPTIVE FUZZY-BASED TRACKING CONTROL FOR A CLASS OF STRICT-FEEDBACK SISO NONLINEAR TIME-DELAY SYSTEMS WITHOUT BACKSTEPPING

In this paper, an adaptive fuzzy tracking control is presented for a class of SISO nonlinear strict feedback systems with unknown time delays. The proposed algorithm does not use the backstepping scheme rather it converts the strict-feedback time-delayed system to the normal form. The Mamdani-type fuzzy system is employed to approximate on-line the lumped uncertain system nonlinearity. The developed controller guarantees uniform ultimate boundedness of all signals in the closed-loop system. The designed control law is independent of the time delays and has a simple form with only one adaptive parameter vector needed to be updated on-line. As a result, the proposed control algorithm is considerably simpler than the previous ones based on backstepping. The Lyapunov stability of the fuzzy system parameters and the filtered tracking error is employed to guarantee semiglobal uniform boundedness for the closed loop system. Simulation results are presented to verify the effectiveness of the proposed approach.

Adaptive neural control based on HGO for hypersonic flight vehicles

This paper describes the design of adaptive neural controller for the longitudinal dynamics of a generic hypersonic flight vehicle (HFV) which are decomposed into two functional systems, namely the altitude subsystem and the velocity subsystem. For each subsystem, one adaptive neural controller is investigated based on the normal output-feedback formulation. For the altitude subsystem, the high gain observer (HGO) is taken to estimate the unknown newly defined states. Only one neural network (NN) is employed to approximate the lumped uncertain system nonlinearity during the controller design which is considerably simpler than the ones based on back-stepping scheme with the strict-feedback form. The Lyapunov stability of the NN weights and filtered tracking error are guaranteed in the semiglobal sense. Numerical simulation study of step response demonstrates the effectiveness of the proposed strategy in spite of system uncertainty.