Class Of Pure-feedback Nonlinear Systems Research Articles

Adaptive neural control for a class of non-affine pure-feedback nonlinear systems

ABSTRACTA novel adaptive neural control scheme is designed for a class of pure-feedback nonlinear systems with non-affine functions possibly being discontinuous. The non-affine function is not necessary to be continuous with respect to control variables or input, and the bounds of non-affine function are unknown functions. Some compact sets are constructively introduced to investigate the bounds of non-affine function so as to cope with the difficulty from these unknown bounds. Moreover, the dynamic surface control technique has been utilised for handling with the problem of ‘explosion of complexity’, and the minimal learning parameter technique is also employed to overcome the problem of excessive parameters. Furthermore, it is highly proved that all the variables will always stay in the introduced compact sets, and all the signals in the closed-loop control system are semi-globally uniformly ultimately bounded by choosing the appropriate design parameters. Finally, simulation examples are provided to demonstrate the effectiveness of the designed approach.

Adaptive Neural Dynamic Surface Control of Pure-Feedback Nonlinear Systems With Full State Constraints and Dynamic Uncertainties

In this paper, adaptive neural dynamic surface control (DSC) is developed using radial basis function neural networks (NNs) for a class of pure-feedback nonlinear systems with full state constraints and dynamic uncertainties. Based on a one-to-one nonlinear mapping, the pure-feedback system with full state constraints is transformed into a novel pure-feedback system without state constraints. The dynamic uncertainties are dealt with using a dynamic signal. Using modified DSC and mean value theorem as well as Nussbaum function, two adaptive NN control schemes are proposed based on the transformed system. The designed control strategy removes the conditions that the upper bound of the control gain is known, and the lower bounds and upper bounds of the virtual control coefficients are known. It is shown that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded, and the full state constraints are not violated. Two numerical examples are provided to illustrate the effectiveness of the proposed approach.

Adaptive fuzzy tracking control for a class of pure-feedback nonlinear systems with time-varying delay and unknown dead zone

This paper investigates the problem of adaptive fuzzy tracking control for a category of pure-feedback nonlinear systems with time-varying delay and unknown dead zone. Fuzzy logic systems are used to identify unknown functions existing in systems. Mean-value theorem is introduced to overcome the difficulty arising from the pure-feedback structure in designing controller. Based on the information of dead-zone slopes, dead zone is handled. Only one adaptive parameter needs updating online by considering the norm of membership function vectors rather than all sub-vectors. Moreover, utilizing the adaptive backstepping technique, a novel adaptive fuzzy tracking control scheme is developed to guarantee all signals of the closed-loop systems are semi-globally uniformly ultimately bounded, and the tracking error can be regulated to the origin with a small neighborhood. The stability of the closed-loop system is proved and simulation results are given to demonstrate the effectiveness of the proposed control approach.

Adaptive fuzzy output-feedback control for a class of nonlinear pure-feedback systems with time delays

ABSTRACTThis paper investigates observer-based adaptive fuzzy control for a class of delayed nonlinear systems in pure-feedback form. We first proposed a linear observer to get the estimation of the system's state. And then utilise fuzzy logic systems to model those nonlinearities. Further, adaptive fuzzy approach and backstepping technique are combined to develop the desired controller via Lyapunov analysis. The suggested adaptive fuzzy control scheme ensures that all the signals of the resulting nonlinear system converge to a small neighbourhood of the origin. Then a real simulation is used to illustrated the effectiveness of our results.

Barrier Lyapunov Functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints

In this study, an adaptive control technique is developed for a class of uncertain nonlinear parametric systems. The considered systems can be viewed as a class of nonlinear pure-feedback systems and the full state constraints are strictly required in the systems. One remarkable advantage is that only less adjustable parameters are used in the design. This advantage is first to take into account the pure-feedback systems with the full state constraints. The characteristics of the considered systems will lead to a difficult task for designing a stable controller. To this end, the mean value theorem is employed to transform the pure-feedback systems to a strict-feedback structure but non-affine terms still exist. For the transformed systems, a novel recursive design procedure is constructed to remove the difficulties for avoiding non-affine terms and guarantee that the full state constraints are not violated by introducing Barrier Lyapunov Function (BLF) with the error variables. Moreover, it is proved that all the signals in the closed-loop system are global uniformly bounded and the tracking error is remained in a bounded compact set. Two simulation studies are worked out to show the effectiveness of the proposed approach.

Adaptive Neural Control for a Class of Pure-Feedback Nonlinear Systems via Dynamic Surface Technique.

This brief addresses the adaptive control problem for a class of pure-feedback systems with nonaffine functions possibly being nondifferentiable. Without using the mean value theorem, the difficulty of the control design for pure-feedback systems is overcome by modeling the nonaffine functions appropriately. With the help of neural network approximators, an adaptive neural controller is developed by combining the dynamic surface control (DSC) and minimal learning parameter (MLP) techniques. The key features of our approach are that, first, the restrictive assumptions on the partial derivative of nonaffine functions are removed, second, the DSC technique is used to avoid "the explosion of complexity" in the backstepping design, and the number of adaptive parameters is reduced significantly using the MLP technique, third, smooth robust compensators are employed to circumvent the influences of approximation errors and disturbances. Furthermore, it is proved that all the signals in the closed-loop system are semiglobal uniformly ultimately bounded. Finally, the simulation results are provided to demonstrate the effectiveness of the designed method.

Observer-based adaptive neural control for a class of nonlinear pure-feedback systems

This paper focuses on the issue of adaptive neural output-feedback control for a class of nonlinear pure-feedback systems. A state observer is first constructed to estimate the unmeasurable state variables. Radial basis function (RBF) neural networks (NNs) are used to approximate the unknown nonlinear functions and backstepping adaptive technique is utilized to construct controller. The proposed control scheme guarantees semi-globally uniformly ultimate boundedness (SGUUB) of all the signals in the closed-loop system. At last, a simulation example is used to show the effectiveness of the proposed method.

Adaptive iterative learning control for nonlinear pure-feedback systems with initial state error based on fuzzy approximation

In this paper, an iterative learning control strategy is presented for a class of nonlinear pure-feedback systems with initial state error using fuzzy logic system. The proposed control scheme utilizes fuzzy logic systems to learn the behavior of the unknown plant dynamics. Filtered signals are employed to circumvent algebraic loop problems encountered in the implementation of the existing controllers. Backstepping design technique is applied to deal with system dynamics. Based on the Lyapunov-like synthesis, we show that all signals in the closed-loop system remain bounded over a pre-specified time interval [0,T]. There even exist initial state errors, the norm of tracking error vector will asymptotically converge to a tunable residual set as iteration goes to infinity and the learning speed can be easily improved if the learning gain is large enough. A time-varying boundary layer is introduced to solve the problem of initial state error. A typical series is introduced in order to deal with the unknown bound of the approximation errors. Finally, two simulation examples show the feasibility and effectiveness of the approach.

Adaptive neural tracking control of pure-feedback nonlinear systems with unknown gain signs and unmodeled dynamics

In this paper, robust adaptive control is proposed for a class of pure-feedback nonlinear systems with unmodeled dynamics and unknown gain signs using radial basis function neural networks (RBFNNs). Dynamic uncertainties are dealt with using a dynamic signal. The unknown virtual gain signs are solved using the Nussbaum functions. Using mean value theorem and Young's inequality, only one learning parameter needs to be tuned online at each step of recursion. It is proved that the proposed design scheme can guarantee semi-global uniform ultimate boundedness (SGUUB) of all signals in the closed-loop system. Simulation results demonstrate the effectiveness of the proposed approach.

Adaptive neural tracking control for a class of perturbed pure-feedback nonlinear systems

This paper focuses on the problem of the adaptive neural control for a class of a perturbed pure-feedback nonlinear system. Based on radial basis function (RBF) neural networks’ universal approximation capability, an adaptive neural controller is developed via the backstepping technique. The proposed controller guarantees that all the signals in the closed-loop system are bounded and the tracking error eventually converges to a small neighborhood around the origin. The main advantage of this note lies in that a control strategy is presented for a class of pure-feedback nonlinear systems with external disturbances being bounded by functions of all state variables. A numerical example is provided to illustrate the effectiveness of the suggested approach.

Adaptive tracking control for a class of pure-feedback non-linear systems including actuator hysteresis and dynamic uncertainties

In this study, an adaptive neural network dynamic surface control for a class of pure-feedback non-linear systems with dynamic uncertainties and unknown hysteresis is proposed. The hysteresis is described by a saturated-type Prandtl–Ishlinskii (PI) model, which is more applicable than the traditional PI model. The main advantages of the authors scheme are that by introducing an initialising technique, the ℒ∞ performance of the tracking error can be achieved, the assumption on dynamic uncertainties is relaxed, the explosion of the complexity problem when the hysteresis is fused with back-stepping design can be eliminated, which together with the estimation of vector norm of unknown parameters makes the control law be simplified and the computational burden be greatly reduced. Simulation results are presented to demonstrate the efficiency of the proposed scheme.

Dynamic inversion control for a class of pure-feedback systems

A dynamic inversion control method is proposed for a class of pure-feedback nonlinear systems. By combing the back-steeping control method with the singular perturbation theory, the virtual and the final actual control inputs are derived from the solutions of a series of fast dynamical equations. Stability analysis shows that the system output tracks the desired trajectory with bounded errors, which can be made arbitrarily small by choosing appropriate design parameters. Tracking performance is illustrated by simulation results. © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Adaptive Fuzzy Control of Nonlinear Systems in Pure Feedback Form Based on Input-to-State Stability

Using mean value theorem and backstepping technique, a robust adaptive fuzzy control scheme is proposed for a class of pure-feedback nonlinear systems with unknown dead zone and disturbances via input-to-state stability. Takagi-Sugeno (T--S) type fuzzy logic systems are used to approximate the uncertain nonlinear functions and fewer learning parameters need to be adjusted online. Based on small gain theorem, the closed-loop control system is proven to be semiglobally uniformly ultimately bounded, and the tracking error converges to a neighborhood of zero by choosing appropriate parameters. Simulation results demonstrate the effectiveness of the control scheme.

Adaptive Control of a Class of Nonlinear Pure-Feedback Systems Using Fuzzy Backstepping Approach

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A controller is proposed for the robust backstepping control of a class of nonlinear pure-feedback systems using fuzzy logic. The proposed control scheme utilizes fuzzy logic systems to learn the behavior of the unknown plant dynamics. Filtered signals are employed to circumvent algebraic loop problems encountered in the implementation of the usual controllers, and the approximation errors can be efficiently counteracted by employing smooth robust compensators. Most importantly, the uniform ultimate boundedness of all signals in the closed-loop system can be guaranteed, and <emphasis emphasistype="italic">a priori</emphasis> knowledge of the plant dynamics is no longer required. Furthermore, the proposed method can be used for adaptive control of a large class of single-input--single-output nonlinear systems in both strict-feedback and pure-feedback forms, and has great potential in many diverse applications. The performance of the proposed approach is demonstrated through three simulation examples, including one nonlinear pure-feedback and two nonlinear strict-feedback systems. </para>

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.