Adaptive Function Approximation Research Articles

Cooperative multiple elastic‐joint arms control using the (p, q)‐analogue of Bernstein operators as the uncertainty approximator

AbstractThis paper presents the position‐force tracking control problem of cooperative multiple elastic‐joint robotic arms, encountering model nonlinearities, unknown perturbations, and dynamic uncertainties. To this mean, an adaptive function approximation technique (FAT) is proposed, authorizing the reference trajectory to be tracked by the object. This potential originates from the universal approximation feature of the FAT‐based methodologies. The (p, q)‐analogue of the Bernstein operators is utilized for this purpose. Since the systems' parameters are not precisely known, adaptive laws are employed for adjusting the uncertainty coefficients. The Lyapunov stability theorem assures that all signals in the error space remain uniformly ultimately bounded (UUB). In the end, two elastic‐joint arms carrying a rigid object are utilized for validation of the theoretical outcomes. The suggested strategy is also compared to the Chebyshev neural network (CNN). The results show the benefit of the existing method, managing the system even in the presence of disturbances and uncertainties with lower tuning parameters compared with CNN.

Adaptive piecewise Poly-Sinc methods for function approximation

We propose a new method of adaptive piecewise function approximation based on Sinc points. The adaptive method is a Lagrange interpolation scheme which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. We prove the exponential convergence of our algorithm. We derive an a priori error estimate for our adaptive method and show its exponential convergence. We use a statistical approach for partition refinement. The adaptive greedy piecewise Poly-Sinc algorithm is validated on continuous, singular, and piecewise functions. We demonstrate that Runge's phenomenon can be handled within a given accuracy goal.

Adaptive Control of Wire-Borne Underactuated Brachiating Robots Using Control Lyapunov and Barrier Functions

Executing safe brachiation maneuvers with a cable-suspended underactuated robot is a challenging problem due to the complications induced by the cable dynamics. We present and experimentally implement an online adaptive controller for a wire-borne brachiating robot swinging on a vibrating cable. An adaptive function approximation approach is proposed to estimate the unknown dynamics of the flexible cable as an external force applied to the robot. Robust control Lyapunov and barrier functions are designed and incorporated into quadratic programs to synthesize a unified adaptive control framework, which formally guarantees the stability and safety of the brachiating robot in the presence of dynamic uncertainties, actuator constraints, and obstacles in the environment. The stability analysis and derivation of the adaptation law are carried out through a Lyapunov analysis. We demonstrate and validate the proposed control framework using extensive hardware experiments with an underactuated brachiating robot operating on a flexible cable. Simulation results, hardware experiments, and comparisons with a baseline controller show that the proposed quadratic programming-based controller achieves reliable tracking performance and disturbance estimation in the presence of model uncertainties, actuator limits, and safety constraints.

Robust Adaptive Finite-Time Synergetic Tracking Control of Delta Robot Based on Radial Basis Function Neural Networks

This paper presents a robust proportional derivative adaptive nonsingular finite-time synergetic tracking control (PDAFS) for a parallel Delta robot system. First, a finite-time synergetic controller combined with a proportional derivative (PD) control is constructed based on an object-oriented model to fulfill the robust tracking control of the robot. Then, an adaptive radial basis function approximation neural network (RBF) is designed to compensate for the effects of uncertainty parameters and external disturbances. Second, a second-order sliding mode (SOSM) differentiator is implemented to reduce the chattering noises due to the low-resolution encoders. Third, the stability theorems of the proposed control scheme are provided, where the Lyapunov stability theory is used to prove the theorems. Then, simulations of the helix trajectory tracking and the pick-and-place task are demonstrated on the Delta robot to validate the advantages of the proposed control scheme. Based on the advances, an implementing control system of the proposed controller is performed to improve the Delta robot’s performance in the experiments.

Functional Feature Extraction for Hyperspectral Image Classification With Adaptive Rational Function Approximation

A functional feature extraction method based on rational function approximation for hyperspectral image (HSI) classification is proposed. In digital imagery, the spectral information of a pixel can be regarded as a 1-D signal. An HSI is composed of these 1-D signals arranged in a certain spatial structure. According to the functional characteristic of hyperspectral data, 1-D signals can be approximated by a linear combination of basis functions. Thus, a joint rational basis function system (JRBFS) based on class adaptivity is here first built for an HSI by adaptive Fourier decomposition (AFD). Second, the functional representations (FRs) and corresponding reconstructed spectral curves are obtained by decomposing the original spectral information in a JRBFS. Furthermore, the functional spectral-spatial features are extracted on the basis of FRs by an edge-preserving filtering method, FR-EPFs. Finally, the functional spectral-spatial features are used for HSI classification by SVM. Experimental results for five commonly used HSI data sets demonstrate the effectiveness and advantages of the proposed method FR-EPFs.

Approximation‐free design for distributed consensus tracking of networked uncertain non‐linear multi‐agent systems with heterogenous high powers

This study investigates a distributed approximation-free design problem for guaranteed consensus tracking quality of networked uncertain heterogenous power-chained multi-agent systems with completely unknown non-linearities. Compared with the existing studies in the presence of power-chained state variables, the main contribution of this study is to provide a distributed low-complexity consensus tracker design strategy using only the relative output information among agents, without the use of adaptive neural or fuzzy function approximators for estimating unknown non-linear functions. A novel recursive design strategy independent of the feedback domination technique is established to deal with power-chained state variables under directed networks. Using the Lyapunov stability theorem, it is shown that the preselected performance of the consensus tracking error is ensured.

Tracking Control Strategy Using Filter-Based Approximation for the Unknown Control Direction Problem of Uncertain Pure-Feedback Nonlinear Systems

A filter-based recursive tracker design approach is presented for the problem of unknown control directions of pure-feedback systems with completely unknown non-affine nonlinearities. In the controller design procedure, the first-order filters for error surfaces, a control input, and state variables are employed to design nonadaptive virtual and actual control laws independent of adaptive function approximators. In addition, for the unknown control direction problem, the filtering signals are incorporated with Nussbaum functions. Different from existing adaptive approximation-based control schemes in the presence of unknown control directions, the proposed approach does not require any adaptive technique regardless of completely unknown nonlinear functions. Therefore, a simplified tracking structure can be constructed. Using the Lyapunov stability analysis, it is shown that the tracking error is reduced within an adjustable neighborhood of the origin while ensuring all the closed-loop signals are bounded.

Distributed approximation-free design for ensuring network connectivity of uncertain nonholonomic multi-robot synchronized tracking systems with disturbances

A distributed approximation-free design for maintaining initial connectivity of uncertain nonholonomic multi-robot synchronized tracking systems is developed under limited communication ranges where the models of followers are regarded as the kinematics and dynamics of uncertain nonholonomic mobile robots. All nonlinearities and external disturbances in the followers’ dynamics are assumed to be unknown. The primary improvement in the current development is that local trackers for ensuring connectivity among robots are designed by using only relative posture errors and adaptive function approximators such as neural networks or fuzzy logic systems are not utilized despite unknown nonlinearities in the robot dynamics. This improvement has been done by transforming individual linear position errors among mobile robots to individual nonlinear position errors using connectivity-preserving functions with communication ranges. The predefined synchronized tracking performance and the guaranteed connectivity of the overall closed-loop system are proved via the Lyapunov stability theorem.

Decentralized low-complexity fault-tolerant tracking of a class of arbitrarily switched interconnected nonaffine nonlinear systems with unexpected faults

A low-complexity design strategy is proposed to ensure the decentralized fault-tolerant tracking quality of a class of uncertain switched interconnected nonaffine nonlinear systems. All system nonlinearities, nonlinear interaction faults, and actuator faults are arbitrarily switched and unknown. Compared with the existing decentralized tracking results, a universal decentralized tracking strategy is recursively provided to deal with asynchronously switched nonlinear interconnections and their faults, without requiring the signs of control coefficient functions and employing any fault compensators using adaptive nonlinear function approximators. Using the common Lyapunov function method, it is shown that the local tracking errors are maintained within preselected time-varying bounds regardless of arbitrary switched faults.

An Error Transformation Approach for Connectivity-Preserving and Collision-Avoiding Formation Tracking of Networked Uncertain Underactuated Surface Vessels.

This paper investigates a distributed connectivity-preserving and collision-avoiding formation tracking problem of networked uncertain underactuated surface vessels (USVs) with heterogeneous limited communication ranges. All nonlinearities in the dynamic model are assumed to be completely unknown. Compared with the existing formation tracking results for USVs, our primary contribution is to develop a new nonlinearly transformed formation error for achieving the initial connectivity preservation, the collision avoidance, and the distributed formation tracking without switching the desired formation pattern and using any additional potential functions. In other words, these three objectives can be achieved by using only one transformed formation error surface. The local tracker design strategy using the nonlinearly transformed error is established under the direct graph topology, where the adaptive function approximation technique and the auxiliary variables are employed to compensate for uncertain nonlinearities and to deal with the underactuated problem of USVs, respectively. Finally, the Lyapunov stability analysis and simulations are performed to verify the effectiveness of the proposed theoretic result.

Robust event-driven tracking control with preassigned performance for uncertain input-quantized nonlinear pure-feedback systems

This paper presents a simplified design methodology for robust event-driven tracking control of uncertain nonlinear pure-feedback systems with input quantization. All nonlinearities and quantization parameters are assumed to be completely unknown. Different from the existing event-driven control approaches for systems with completely unknown nonlinearities, the main contribution of this paper is to design a simple event-based tracking scheme with preassigned performance, without the use of adaptive function approximators and adaptive mirror models. It is shown in the Lyapunov sense that the proposed event-driven low-complexity tracker consisting of nonlinearly transformed error surfaces and a triggering condition can achieve the preselected transient and steady-state performance of control errors in the presence of the input quantization.

Filter-Driven-Approximation-Based Control for a Class of Pure-Feedback Systems With Unknown Nonlinearities by State and Output Feedback

This paper presents a new approximation-based control approach for uncertain nonlinear pure-feedback systems. The main idea of this paper is to estimate unknown continuous nonlinear functions through a linear combination of first-order filtered signals of state variables and a control input in the nonadaptive control framework, instead of using conventional adaptive neural or fuzzy function approximators. Based on the proposed filter-driven approximation technique, we first present a state-feedback control scheme for pure-feedback systems with unknown nonaffine nonlinearities and a dead-zone input. Then, a filter-driven-approximation-based output-feedback control scheme is proposed via a system transformation and an observer to estimate unmeasurable state variables. Based on the Lyapunov stability theorem, the control errors and the filter-driven approximation errors are considered to prove that the controlled closed-loop system is semi-globally uniformly ultimately bounded. Finally, simulation results are provided to show that the proposed filter-driven-approximation-based controller and the existing function-approximation-based adaptive controllers have similar control performance for nonlinear pure-feedback systems.

Direct Adaptive Function Approximation Techniques Based Control of Robot Manipulators

In this paper, a simple model-free controller for electrically driven robot manipulators is presented using function approximation techniques (FAT) such as Legendre polynomials (LP) and Fourier series (FS). According to the orthogonal functions theorem, LP and FS can approximate nonlinear functions with an arbitrary small approximation error. From this point of view, they are similar to fuzzy systems and can be used as controller to approximate the ideal control law. In comparison with fuzzy systems and neural networks, LP and FS are simpler and less computational. Moreover, there are very few tuning parameters in LP and FS. Consequently, the proposed controller is less computational in comparison with fuzzy and neural controllers. The case study is an articulated robot manipulator driven by permanent magnet direct current (DC) motors. Simulation results verify the effectiveness of the proposed control approach and its superiority over neuro-fuzzy controllers.

Fast smooth rank function approximation based on matrix tri-factorization

Recently, Smooth Rank Function (SRF) is proposed for matrix completion problem. The main idea of this algorithm is based on a continuous and differentiable approximation of the rank function. However, it need to deal with singular value decomposition of matrix in each iteration, which consumes much time for large matrix. In this paper, by utilizing the tri-factorization of matrix, a fast matrix completion method based on SRF is proposed. Then, based on our fast matrix completion method, a rank adaptive smooth rank function approximation is presented with appropriate rank estimation. We mathematically prove the convergence of the proposed method. Experimental results show that our proposed method improves the running time significantly. Furthermore, our proposed method outperforms other existing matrix completion approaches in most cases.

미지의 방향성을 갖는 불확실한 스위치드 순궤환 시스템의 추종 제어를 위한 강인 저 복잡성 설계

This paper investigates a robust low-complexity design problem for tracking control of uncertain switched pure-feedback systems in the presence of unknown control direction. The completely unknown non-affine nonlinearities are assumed to be arbitrarily switched. By combining the nonlinear error transformation technique and Nussbaum-type functions, a robust tracking controller is designed without using any adaptive function approximators. Thus, compared with existing results, the proposed control scheme has the low-complexity property. From Lyapunov stability theory, it is shown that the tracking error remains within the preassigned transient and steady-state error bounds.

Heavyweight airdrop flight control design using feedback linearization and adaptive sliding mode

This paper investigates the problem of designing a novel adaptive sliding-mode controller for heavyweight airdrop operations. The design objective is to guarantee asymptotic tracking performance of the aircraft states, in the presence of bounded nonlinear uncertainties without prior knowledge of the bounds. On the basis of feedback linearization of the aircraft–cargo model, a sliding-mode control method with projection-based adaptive function approximation is proposed. This method uses an adaptation strategy to achieve robustness against model uncertainties, and a knowledge of the bounds on the complex uncertainties is not required. Notably, the adaptation law with projection can bound the estimated function, and this avoids singularity of the control signal. Simulations are conducted under the condition that one transport aircraft performs a maximum load airdrop mission at a height of 25 m, using single-row single-platform mode. The results verify the good properties of the control method, which can meet the airdrop mission performance indexes well in the presence of ±20% aerodynamic data uncertainty and 20% actuator fault.

Robust fault-tolerant prescribed performance tracking for uncertain switched pure-feedback nonlinear systems under arbitrary switching

ABSTRACTThis paper presents a model-free prescribed performance design methodology for the robust fault-tolerant tracking (RFTT) of uncertain switched pure-feedback nonlinear systems under arbitrary switching. Unexpected faults in switched non-affine nonlinearities and in an actuator are considered. Using the prescribed performance design and the common Lyapunov function method, a common RFTT scheme is proposed to ensure that the tracking error remains within preassigned performance bounds and finally converges to a preselected neighbourhood of the origin, regardless of arbitrary switching and unexpected faults. Contrary to existing results in the literature, the proposed methodology does not require fault compensation mechanisms such as adaptive techniques and function approximators using neural networks or fuzzy systems. Thus, the structure of the proposed RFTT scheme is simpler than that of the existing control schemes. Moreover, the proposed approach can predesign the transient performance bounds at the instants when switching and faults occur. Finally, the simulation results are provided to demonstrate the effectiveness of the proposed theoretical approach.

Minimal-Approximation-Based Decentralized Backstepping Control of Interconnected Time-Delay Systems.

A decentralized adaptive backstepping control design using minimal function approximators is proposed for nonlinear large-scale systems with unknown unmatched time-varying delayed interactions and unknown backlash-like hysteresis nonlinearities. Compared with existing decentralized backstepping methods, the contribution of this paper is to design a simple local control law for each subsystem, consisting of an actual control with one adaptive function approximator, without requiring the use of multiple function approximators and regardless of the order of each subsystem. The virtual controllers for each subsystem are used as intermediate signals for designing a local actual control at the last step. For each subsystem, a lumped unknown function including the unknown nonlinear terms and the hysteresis nonlinearities is derived at the last step and is estimated by one function approximator. Thus, the proposed approach only uses one function approximator to implement each local controller, while existing decentralized backstepping control methods require the number of function approximators equal to the order of each subsystem and a calculation of virtual controllers to implement each local actual controller. The stability of the total controlled closed-loop system is analyzed using the Lyapunov stability theorem.

Adaptive fuzzy multi‐surface sliding control of multiple‐input and multiple‐output autonomous flight systems

In this study, we proposed an adaptive fuzzy multi-surface sliding control (AFMSSC) for trajectory tracking of 6 degrees of freedom inertia coupled aerial vehicles with multiple inputs and multiple outputs (MIMO). It is shown that an adaptive fuzzy logic-based function approximator can be used to estimate the system uncertainties and an iterative multi-surface sliding control design can be carried out to control flight. Using AFMSSC on MIMO autonomous flight systems creates confluent control that can account for both matched and mismatched uncertainties, system disturbances and excitation in internal dynamics. It is proved that the AFMSSC system guarantees asymptotic output tracking and ultimate uniform boundedness of the tracking error. Simulation results are presented to validate the analysis.

Projection-Based Adaptive Backstepping Control of a Transport Aircraft for Heavyweight Airdrop

An autopilot inner loop that combines backstepping control with adaptive function approximation is developed for airdrop operations. The complex nonlinear uncertainty of the aircraft-cargo model is factorized into a known matrix and an uncertainty function, and a projection-based adaptive approach is proposed to estimate this function. Using projection in the adaptation law bounds the estimated function and guarantees the robustness of the controller against time-varying external disturbances and uncertainties. The convergence properties and robustness of the control method are proved via Lyapunov theory. Simulations are conducted under the condition that one transport aircraft performs a maximum load airdrop task at a height of 82 ft, using single row single platform mode. The results show good performance and robust operation of the controller, and the airdrop mission performance indexes are satisfied, even in the presence of ±15% uncertainty in the aerodynamic coefficients, ±0.01 rad/s pitch rate disturbance, and 20% actuators faults.