Approximation-based Control Research Articles

Szasz-Favard-Mirakyan operators for Chaos synchronization: An Observer-based approach

In this paper, a Szasz-Favard-Mirakyan operator is exploited for chaos synchronization of a master-slave system. The universal approximation feature asset lets in Szasz-Favard-Mirakyan operators for approximating uncertainties, including un-modeled dynamics and disturbances. The above subject is considered in detail in this investigation. It is confirmed that the synchronization/approximation errors are uniformly bounded and stable if the Szasz-Favard-Mirakyan operators are applied as the regressors. Additionally, it has been presumed that the synchronization’s error rate is not available, and an observer will be designed for its estimation. The Duffing–Holmes oscillator is examined as the computer-generated chaotic framework with the target of examining the functioning of the recommended synchronization observer-based controller. The outcomes are compared with an effective approximation-based control strategy. Unlike Chebyshev Neural Network approximators in which the system’s inputs states are needed to define the regressor vector and approximate uncertainties, the suggested Szasz-Favard-Mirakyan operators–based strategy is independent of the system states for constructing the regressor vector.

Chaos synchronization using q-Chlodowsky operators as uncertainty approximator

In this article, q-Chlodowsky operators are utilized for synchronizing a chaotic master-slave system. The universal approximation property enables q-Chlodowsky operators to approximate uncertainties, which consist of disturbances and unmodeled dynamics. It is verified that uniformly ultimately bounded stability of the tracking/approximation errors can be assured if the q-Chlodowsky operators are utilized as regressors and the polynomial coefficients are tuned by the adaptive laws calculated in the stability analysis. Moreover, it has been assumed that the rate of synchronization error is not at hand and a filter-based strategy has been adopted instead of designing an observer. The Duffing-Holmes oscillator is studied and the results are also compared with three powerful approximation-based and robust control methods. Unlike neural network/fuzzy estimators in which system states are necessary to define the regressor vector, the proposed uncertainty estimator does not require them in the regressor vector. Furthermore, in comparison with radial basis functions neural networks (RBFNNs), we are involved in fewer adaptive coefficients in the regressor vector of the proposed uncertainty estimator. As a result, the tedious and time-consuming procedure of trial and error for tuning the uncertainty estimator is eliminated.

Adaptive Fuzzy Control With Global Stability Guarantees for Unknown Strict-Feedback Systems Using Novel Integral Barrier Lyapunov Functions

In this article, the adaptive fuzzy tracking control problem for a class of uncertain strict-feedback systems with unknown nonlinearities is investigated with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">particular emphasis on global stability</i> . The proposed control scheme is designed by integrating the barrier Lyapunov functions (BLFs) with the techniques of fuzzy approximation and backstepping. The novel integral BLFs (iBLFs) are introduced to overcome the design difficulties induced by the virtual control coefficients and determine <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> the compact set for guaranteeing the validity of fuzzy approximation. Compared with existing approximation-based control results, the developed controller not only guarantees global stability without requiring prior information of system nonlinearities and assumptions on the time derivatives of virtual control coefficients, but also prevents the “explosion of complexity” issue without attaching additional filters. The simulation results further confirm the effectiveness of the theoretical findings.

Finite-Time Approximation-Free Attitude Control of Quadrotors: Theory and Experiments

In this article, a novel finite-time approximation-free control scheme is proposed for the attitude tracking of quadrotor unmanned aerial vehicles. Prescribed performance functions are employed to transform the original attitude tracking problem into an alternative system stabilization problem. By incorporating a finite-time error compensation mechanism into the recursive control design, the finite-time error convergence, and singularity-free property can be guaranteed simultaneously. Compared with the existing approximation-based control schemes, the presented controller has a simple cascade proportional-like structure and less computational burden, and the coupling among the roll, pitch, and yaw dynamics can be successfully handled without requiring any model information or function approximations. With the proposed control scheme, the attitude tracking error can be retained within a prescribed boundary and converge into a sufficiently small region around origin in finite time. Extensive comparative experiments on a three degree-of-freedom (3-DOF) quadrotor platform are performed to validate the effectiveness of the proposed control scheme.

Adaptive Interval Type-2 Fuzzy Tracking Control of PV Grid-Connected Inverters

Fuzzy logic systems with approximation capabilities provide effective control for nonlinear and uncertain systems. Due to the characteristics of photovoltaic (PV) and the PWM method, a grid-connected PV system is a considerably nonlinear system with unpredictable parameters. In this study, a new adaptive interval type-2 fuzzy approximation-based controller (AIT2FAC) was developed to control a three-phase grid-connected PV system. The proposed controller can be implemented without any prior knowledge of the system mathematical model. In the presence of both parametric and modeling uncertainty, the developed controller can achieve the control objectives. The proposed controller utilizes the principle of input-output feedback linearization and the approximation capability of fuzzy systems to the control inverter current components to track prescribed reference values. The proposed AIT2FAC controller is capable of handling system uncertainties due to the interval type-2 fuzzy logic system capability to cope with a high level of uncertainty. Lyapunov analysis is used to determine the closed-loop system stability and the updating laws of the proposed controller parameters. The effectiveness of the designed controller to achieve the required tracking is validated for different operating cases, including system disturbances, modelling, and parameter uncertainties. For evaluation, the proposed type-2 fuzzy controller is compared to a type-1 fuzzy controller in terms of some performance measures. The comparison results demonstrate that the proposed type-2 fuzzy controller has better tracking performance than the type-1 fuzzy controller in terms of the settling time, the maximum overshoot, the integral absolute error (IAE) and the integral time of absolute error (ITAE).

Moving least squares approximation-based online control optimised by the team game algorithm for Duffing-Holmes chaotic problems

ABSTRACT In this paper, an online optimal adaptive robust fuzzy controller based on the Moving Least Squares (MLS) and Team Game Algorithm (TGA) is introduced to control uncertain chaotic nonlinear systems. At first, a robust supervisory stabiliser and a fuzzy adaptive PID controller are designed and combined to handle a Duffing-Holmes chaotic oscillator. Next, the TGA is utilised to optimise the parameters of the designed fuzzy adaptive robust controller for different values of system disturbances. Then, to adapt the optimum design parameters of the controller with the uncertain values of the external disturbances, the MLS approximation is implemented. In order to evaluate the performance of the proposed technique, it is applied on a Duffing-Holmes chaotic nonlinear oscillator with different initial conditions. The results and the analysis prove the efficiency of the proposed controller with regard to uncertain systems’ challenging external disturbances that impair the stability, accuracy and optimality of the system.

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 Adaptive Neural Consensus Control for Stochastic Nonlinear Multiagent Systems with Whole State Delays and Multiple Constraints

This paper presents a distributed adaptive neural tracking consensus control strategy for a class of stochastic nonlinear multiagent systems with whole state time delays, input and output constrains. The considered systems are involved in the existence of whole state delays and stochastic disturbances, which makes the controller design more difficult and complex. Firstly, time delays are related to unknown dynamic interactions with the whole states of the agent systems, and novel Lyapunov-Krasovskii functionals are constructed. Secondly, the smooth asymmetric saturation nonlinearity is given based on Gaussian error function, output constraints are achieved via barrier Lyapunov functions, and neural networks are utilized to deal with the completely unknown nonlinearities and stochastic disturbances. Then, based on Lyapunov stability theory, a delay-independent adaptive controller is developed via Lyapunov-Krasovskii functionals and backstepping technique, and it reduces the complexity of learning parameters. It is proved that the proposed approximation-based controller can guarantee that all closed-loop signals are cooperatively semi-globally uniformly ultimately bounded (CSGUUB), and the tracking errors between the followers and the leaders eventually converge to a small neighbourhood around the origin. Finally, simulation studies are carried out, and the simulation results verify the correctness and effectiveness of the proposed strategy.

Globally Exponentially Stable Tracking Control of Self-Restructuring Nonlinear Systems

This article proposes a neural networks (NNs)-based tracking control approach for a class of uncertain high-order self-restructuring nonaffine dynamic systems. Unlike most existing NN-based works that normally ignore the precondition on the functionality and reliability of NN unit and thus can only ensure semiglobal stability, the proposed method explicitly addresses the issue of reliable in-loop operation of NN approximation-based control unit, resulting in a safeguarded NN-based control solution capable of ensuring globally stable tracking. Furthermore, the control method proposed guarantees exponentially globally stable tracking for systems with self-restructuring nonlinearities and uncertainties, distinguishing itself from those that only yield uniformly ultimately bounded (UUB) regulation/tracking results for nonlinear systems with fixed structures. All of these features are achieved by the proposed strategy consisting of two cooperative control units: 1) safeguard control and 2) NN-based control. The role of the safeguard control is to force the states (starting from any initial condition) to enter a stable region, so that the NN-based control can be activated trustworthily and safely. It is such cooperation of the two units that not only ensures the tracking error entering the stable region first within a prespecified finite time but also guarantees the tracking error converging to zero exponentially thereafter, resulting in global zero-error tracking. Both the theoretical analysis and numerical simulation authenticate the effectiveness of the proposed method.

Neural-Network-Based Distributed Formation Tracking Control of Marine Vessels With Heterogeneous Hydrodynamics

In this paper, a robust scheme is presented for distributed formation tracking control of marine vessels with heterogeneous hydrodynamics. Provided that multiple ships sail in proximity at sea, the hydrodynamic forces and moments would be changed greatly due to the ship-ship interactions, thereby causing the heterogeneous hydrodynamics; nevertheless, it is usually ignored in formation control of marine vessels. The heterogeneous hydrodynamics is very difficult to model and induces the uncertainty; therefore, it is treated as unknown dynamics in this paper. The neural network (NN), which is well-known for approximation-based control, is employed to handle the unknown dynamics. Note that the NN is employed to approximate the unknown dynamics collectively for reducing the learning parameters. Furthermore, unlike traditional formation control of coordination of positions, the consensus mechanism of velocities is also included into the control design to ensure the performance of formation maintenance. The presented distributed controller for each ship only utilizes the information from itself and its neighbors, aiming to prevent single-point failure in harsh marine environment. All the closed-loop signals are proved to be stable based on Lyapunov theory. Various simulations are conducted to validate the effectiveness of proposed control design.

A Hybrid Adaptive Control Strategy for Industrial Robotic Joints

This paper presents a hybrid adaptive approximation-based control (HAAC) strategy for a class of uncertain robotic joints' system. The proposed control structure consists of a robust sliding mode controller and an adaptive approximation-based controller. The robust sliding mode controller is designed by using the super-twisting algorithm, which is a particularly effective method to decrease the chattering caused by the traditional sliding mode control (SMC) and compensate the disturbances. Another improvement of the robust sliding mode controller is that the robust control parameters only subject to the upper bound of the derivative of the external disturbances, rather than choosing a relatively large value. Moreover, the designed adaptive approximation-based controller has the following two distinctive features: 1) the control parameters are designed to be adjusted in real time and 2) the prior knowledge of actual robotic model is not required to be known. These features contribute to compensating the uncertainties. The stability of the closed-loop system is proved by using the Lyapunov theory, and the simulation results demonstrate the effectiveness of the proposed control method. Finally, the proposed HAAC could apply in the experiments of industrial robotic joints' system.

Nussbaum gain adaptive neural control for stochastic pure-feedback nonlinear time-delay systems with full-state constraints

Abstract In this paper, the problem concerned with adaptive approximation-based control is discussed for a class of stochastic pure-feedback nonlinear time-delay systems with unknown direction control gains and full-state constraints. In the controller design process, the approximation capability of neural networks is utilized to identify the unknown nonlinearities, the appropriate Lyapunov–Krasovskii functionals are constructed to compensate the unknown time-delay terms, barrier Lyapunov functions (BLFs) are designed to ensure that the state variables are constrained, and the Nussbaum-type gain function is used to solve the difficulties caused by the unknown virtual control gains. Then, based on adaptive backstepping technique and Lyapunov stability theory, a robust control scheme is presented, and the developed controller decreases the number of learning parameters and thus reduces the computational burden. It is shown that the proposed controller can guarantee that all the signals in the closed-loop system are 4-Moment (or 2 Moment) semi-globally uniformly ultimately bounded (SGUUB) and the tracking error converges to a compact set of the origin. Finally, two simulation examples are included to validate the effectiveness of the proposed approach.

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.

Parameterization and Adaptive Control of Multivariable Noncanonical T--S Fuzzy Systems

This paper conducts a new study for adaptive Takagi-Sugeno (T-S) fuzzy approximation-based control of multi-input and multi-output (MIMO) noncanonical-form nonlinear systems. Canonical-form nonlinear systems have explicit relative degree structures, whose approximation models can be directly used to derive desired parameterized controllers. Noncanonical-form nonlinear systems usually do not have such a feature, nor do their approximation models, which are also in noncanonical forms. This paper shows that it is desirable to reparameterize noncanonical-form T-S fuzzy system models with smooth membership functions for adaptive control, and such system reparameterization can be realized using relative degrees, a concept yet to be studied for MIMO noncanonical-form T-S fuzzy systems. This paper develops an adaptive feedback linearization scheme for control of such general system models with uncertain parameters, by first deriving various relative degree structures and normal forms for such systems. Then, a reparameterization procedure is developed for such system models, based on which adaptive control designs are derived, with desired stability and tracking properties analyzed. A detailed example is presented with simulation results to show the new control design procedure and desired control system performance.

Hybrid feedback feedforward: An efficient design of adaptive neural network control

This paper presents an efficient hybrid feedback feedforward (HFF) adaptive approximation-based control (AAC) strategy for a class of uncertain Euler–Lagrange systems. The control structure includes a proportional–derivative (PD) control term in the feedback loop and a radial-basis-function (RBF) neural network (NN) in the feedforward loop, which mimics the human motor learning control mechanism. At the presence of discontinuous friction, a sigmoid-jump-function NN is incorporated to improve control performance. The major difference of the proposed HFF-AAC design from the traditional feedback AAC (FB-AAC) design is that only desired outputs, rather than both tracking errors and desired outputs, are applied as RBF-NN inputs. Yet, such a slight modification leads to several attractive properties of HFF-AAC, including the convenient choice of an approximation domain, the decrease of the number of RBF-NN inputs, and semiglobal practical asymptotic stability dominated by control gains. Compared with previous HFF-AAC approaches, the proposed approach possesses the following two distinctive features: (i) all above attractive properties are achieved by a much simpler control scheme; (ii) the bounds of plant uncertainties are not required to be known. Consequently, the proposed approach guarantees a minimum configuration of the control structure and a minimum requirement of plant knowledge for the AAC design, which leads to a sharp decrease of implementation cost in terms of hardware selection, algorithm realization and system debugging. Simulation results have demonstrated that the proposed HFF-AAC can perform as good as or even better than the traditional FB-AAC under much simpler control synthesis and much lower computational cost.

Approximation‐based adaptive control of uncertain non‐linear pure‐feedback systems with full state constraints

This study proposes an adaptive approximation-based control approach for non-linear pure-feedback systems in the presence of full state constraints. Completely non-affine non-linear functions are considered and assumed to be unknown. The dynamic surface design based on integral barrier Lyapunov functionals is provided to achieve both the desired tracking performance and the constraints satisfaction, in consideration of the full-state-constrained non-affine non-linearities. In this design procedure, simple sufficient conditions for choosing control gains, which can be checked off-line, are established to guarantee the feasibility of the controller. The function approximation technique is employed to estimate unknown non-linearities induced from the controller design procedure where the adaptive laws using the projection operator are designed to ensure the boundedness of the function approximators in the feasibility conditions. It is shown that all the signals in the closed-loop system are uniformly ultimately bounded and the tracking error converges to an adjustable neighbourhood of the origin while all state variables always remain in the constrained state space.

Approximation-based Control of an Uncertain Robot with Output Constraints

In this paper, we consider the problem of tracking a desired trajectory for an uncertain robot in the presence of constraints and uncertainties. The dynamics of the uncertain robot are represented by an n-link rigid robotic manipulator. To deal with the system uncertainties and disturbances, the adaptive neural networks are used to approximate the unknown model of the robot and handle the unknown disturbance. Uniform ultimate boundedness of the closed loop system is guaranteed by using a Barrier Lyapunov Function (BLF). Extensive simulations for a robot with constraints are carried out to illustrate the effectiveness of the proposed control.

Tracking Control of a Closed-Chain Five-Bar Robot With Two Degrees of Freedom by Integration of an Approximation-Based Approach and Mechanical Design

The trajectory tracking problem of a closed-chain five-bar robot is studied in this paper. Based on an error transformation function and the backstepping technique, an approximation-based tracking algorithm is proposed, which can guarantee the control performance of the robotic system in both the stable and transient phases. In particular, the overshoot, settling time, and final tracking error of the robotic system can be all adjusted by properly setting the parameters in the error transformation function. The radial basis function neural network (RBFNN) is used to compensate the complicated nonlinear terms in the closed-loop dynamics of the robotic system. The approximation error of the RBFNN is only required to be bounded, which simplifies the initial "trail-and-error" configuration of the neural network. Illustrative examples are given to verify the theoretical analysis and illustrate the effectiveness of the proposed algorithm. Finally, it is also shown that the proposed approximation-based controller can be simplified by a smart mechanical design of the closed-chain robot, which demonstrates the promise of the integrated design and control philosophy.

Tracking control for nonaffine systems: a self-organizing approximation approach.

This paper considers tracking control for single-input, single-output nonaffine dynamic systems. A performance-dependent self-organizing approximation-based approach is proposed. The designer specifies a positive tracking error criterion. The self-organizing approximation-based controller then monitors the tracking performance and adds basis elements only as needed to achieve the tracking specification. Even though the system is not affine, the approach is defined such that the approximated function is independent of the control variable u. Stability is proved and the self-organization is derived in a Lyapunov-based methodology. To illustrate certain novel aspects of the proposed controller, a numerical example is included.

Leader–follower formation control of underactuated autonomous underwater vehicles

This paper is concerned with the leader–follower formation control of multiple underactuated autonomous underwater vehicles (AUVs). In the proposed leader–follower control, the follower tracks a reference trajectory based on the leader position and predetermined formation without the need for leader's velocity and dynamics. This is desirable in marine robotics due to weak underwater communication and low bandwidth. A virtual vehicle is constructed such that its trajectory converges to the reference trajectory of the follower. Position tracking control is designed for the follower to track the virtual vehicle using Lyapunov and backstepping synthesis. Approximation-based control technique is employed to handle the model parametric uncertainties and unknown disturbances for the follower. The residual error between vehicles within the formation is proven to converge to a bounded compact set and control performance is guaranteed by suitably choosing the design parameters. Extensive simulations are provided to demonstrate the effectiveness of the approaches presented.