Non-affine Pure-feedback Systems Research Articles

Adaptive fuzzy command filtered backstepping control for uncertain pure-feedback systems

This paper considers tracking control of non-affine pure-feedback systems with uncertain disturbances; a new adaptive fuzzy command filtered backstepping controller is presented. First, the non-affine difficulty of pure-feedback systems is solved with the help of the mean-value theorem, and the fuzzy logic systems (FLSs) are adopted to estimate the system uncertainties which contain external disturbances. Next, a novel command-filtered technology is proposed to avoid the so-called “explosion of complexity” in the backstepping design process. The compensation mechanism is employed to eliminate the shortcoming of the traditional dynamic surface control. The presented control scheme ensures that all closed-loop signals are bounded, and the system output can track the given reference signal. The main contribution of this paper is that the developed approach is not only generalized to uncertain non-affine pure-feedback systems, which extends the practical applications of classical command filtered schemes, but also has an error compensation system, which improves tracking performance in comparison with existing dynamic surface control proposals. Finally, several simulation experiments validate the effectiveness and superiority of the proposed control scheme.

Adaptive Neural Control for Nonaffine Pure-Feedback System Based on Extreme Learning Machine

For nonaffine pure-feedback systems, an adaptive neural control method based on extreme learning machine (ELM) is proposed in this paper. Different from the existing methods, this scheme firstly converts the original system into a nonaffine system containing only one unknown term by equivalent transformation, thus avoiding the cumbersome and complex indirect design process of traditional backstepping methods. Secondly, a high-performance finite-time-convergence-differentiator (FD) is designed, through which the system state variables and their derivatives are accurately estimated to ensure the control effect. Thirdly, based on the implicit function theorem, the ELM neural network is introduced to approximate the uncertain items of the system, which simplifies the repeated adjustment process of the network training parameters. Meanwhile, the minimum learning parameter algorithm (MLP) is adopted to design the adaptive law for the norm of the network weight vector, which significantly reduces calculations. And it is theoretically proved that the closed-loop control system is stable and the tracking error is bounded. Finally, the effectiveness of the designed controller is verified by simulation.

Fuzzy Adaptive DSC Design for an Extended Class of MIMO Pure-Feedback Non-Affine Nonlinear Systems in the Presence of Input Constraints

A novel adaptive fuzzy dynamic surface control (DSC) scheme is for the first time constructed for a larger class of (multi-input multi-output) MIMO non-affine pure-feedback systems in the presence of input saturation nonlinearity. First of all, the restrictive differentiability assumption on non-affine functions has been canceled after using the piecewise functions to reconstruct the model for non-affine nonlinear functions. Then, a novel auxiliary system with bounded compensation term is firstly introduced to deal with input saturation, and the dynamic system employed in this work designs a bounded compensation term of tangent function. Thus, we successfully relax the strictly bounded assumption of the dynamic system. Additionally, the fuzzy logic systems (FLSs) are used to approximate unknown continuous systems functions, and the minimal learning parameter (MLP) technique is exploited to simplify control design and reduce the number of adaptive parameters. Finally, two simulation examples with input saturation are given to validate the effectiveness of the developed method.

Distributed Containment Control for Multi-agent Systems in Pure-feedback Form under Switching Topologies

The distributed containment control problem is investigated for nonlinear multi-agent systems in pure-feedback form subjected to switching directed communication topologies. Based on command filtered back-stepping technology and singular-perturbation theorem, a containment controller is addressed to guarantee that the followers’ outputs can converge to the convex hull spanned by multi-dynamic leaders, and the containment tracking errors between leaders and followers is semi-global practical exponential stable in the closed-loop system. A nonlinear anti-tangent filter is adopted to estimate the virtual control, avoiding the analytical deduction inherent in traditional back-stepping method and overcoming the ?circular construction problem? in non-affine pure-feedback systems. Differing from the existing back-stepping method, the proposed control scheme only requires leaders’ output signals and their first derivative signals, does not need the prior knowledge of their nth order derivative signals. The stability analysis is completed via the singular perturbation theory, revealing some novel features of the underlying back-stepping method. Finally, two simulation examples are provided to illustrate the effectiveness of the control scheme.

Adaptive neural novel prescribed performance control for non-affine pure-feedback systems with input saturation

An error constraint control problem is considered for pure-feedback systems with non-affine functions being possibly in-differentiable. A new constraint variable is used to construct virtual control that guarantees the tracking error within the transient and steady-state performance envelopment. The new error transformation avoids non-differentiable problems and complex deductions caused by traditional error constraint approaches. A locally semi-bounded and continuous condition for non-affine functions is employed to ensure the controllability and transform the closed-loop system into a pseudo-affine form. An auxiliary system with bounded compensation term is proposed for nonlinear systems with input saturation. On the basis of backstepping technique, an adaptive neural controller is designed to handle unknown terms and circumvent repeated differentiations of virtual controls. The boundedness and convergence of the closed-loop system are proved by Lyapunov theory. Asymptotic tracking is achieved without violating control input constraint and error constraint. Two examples are performed to verify the theoretical findings.

Adaptive Fuzzy Control for Pure-Feedback Nonlinear Systems With Nonaffine Functions Being Semibounded and Indifferentiable

An adaptive fuzzy control approach is proposed for nonaffine pure-feedback systems with nonaffine functions being semibounded and possibly indifferentiable. A semibounded and continuous condition for nonaffine function is presented to guarantee the controllability of system. To overcome the difficulty from this mild condition, a compact set is introduced, and the nonaffine nonlinear functions are modeled in a new way based on this compact set, which is proved to be invariant set eventually. By using the dynamic surface control (DSC) technique, the problem of explosion of complexity inherent in backstepping method is avoided in the proposed adaptive fuzzy control scheme. Robust compensators are used to minify the influence of uncertainties and disturbances. Furthermore, it is proved that all the closed-loop signals are bounded and the tracking error converges to a small residual set asymptotically. Finally, simulation examples are provided to demonstrate the effectiveness of the designed method.

Adaptive control of pure-feedback systems in the presence of parametric uncertainties

The purpose of this study is to investigate an adaptive control approach for completely nonaffine pure-feedback systems with linear/nonlinear parameterization. In this approach, the parameter separation technique and the idea of the positive function of linearly connected parameters are coupled effectively with the combination of backstepping and time-scale separation. Subsequently, a fast dynamical equation is derived from the original subsystem, where the solution is sought to approximate the corresponding ideal virtual/actual control inputs. Furthermore, the adaptation law of unknown parameters can be derived based on Lyapunov theory in the backstepping technique and there is no need to design a state predictor for this purpose. Therefore, it results in higher accuracy and avoids complexity. The closed-loop stability and the state regulation of these systems are proved. Finally, the simulation results are provided to demonstrate the effectiveness of the proposed approach.

Adaptive control of pure-feedback systems with nonlinear parameterization via time-scale separation

In this paper an adaptive control approach for completely non-affine pure-feedback systems with nonlinear parameterization is proposed. By using the parameter separation technique and coupling it effectively with combination of backstepping and time scale separation, a fast dynamical equation is derived from the original subsystem, where the solution is sought to approximate the corresponding ideal virtual/actual control input. In this approach, since designing state predictor to derive adaptation law of unknown parameters is omitted, our design is more accurate and less complex. The closed loop stability and the state regulation of nonlinear parameterization pure-feedback systems are all proved by new theorem in singular perturbation theory. Finally the simulation results are provided to demonstrate the effectiveness of the proposed approach.

Adaptive NN dynamic surface control for a class of uncertain non-affine pure-feedback systems with unknown time-delay

Adaptive neural network (NN) dynamic surface control (DSC) is developed for a class of non-affine pure-feedback systems with unknown time-delay. The problems of "explosion of complexity" and circular construction of the practical controller in the traditional backstepping algorithm are avoided by using this controller design method. For removing the requirements on the sign of the derivative of function fi, Nussbaum control gain technique is used in control design procedure. The effects of unknown time-delays are eliminated by using appropriate Lyapunov-Krasovskii functionals. Proposed control scheme guarantees that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded. Two simulation examples are presented to demonstrate the method.

Approximation-based adaptive tracking control of nonlinear pure-feedback systems with time-varying output constraints

An adaptive neural network control problem of completely non-affine pure-feedback systems with a time-varying output constraint and external disturbances is investigated. For the controller design, we presents an appropriate Barrier Lyapunov Function (BLF) considering both the time-varying output constraint and the control direction nonlinearities induced from the implicit function theorem and mean value theorem. From an error transformation, the BLF dependent on the time-varying constraint is transformed into the explicitly time-independent BLF. Based on the explicitly time-independent BLF, an adaptive dynamic surface control scheme using the function approximation technique is designed to ensure both the constraint satisfaction and the desired tracking ability. It is shown that all signals in the closed-loop system are semi-globally uniformly ultimately bounded and the tracking error converges to an adjustable neighborhood of the origin while the time-varying output constraint is never violated.

Adaptive control of nonlinear pure-feedback systems with output constraints: Integral barrier Lyapunov functional approach

This paper studies an adaptive control problem of completely non-affine pure-feedback systems with an output constraint. The implicit function theorem and the mean value theorem are employed to deal with unknown output-constrained non-affine nonlinearities. Then, a dynamic surface design approach based on an appropriate Integral Barrier Lyapunov Functional is presented to design an adaptive controller ensuring both the constraint satisfaction and the desired tracking ability. For the controller design, the function approximation technique using neural networks is used for estimating unknown nonlinear terms with control direction nonlinearities. It is shown that all the signals in the closed-loop system are semi-globally uniformly ultimately bounded and the tracking error converges to an adjustable neighborhood of the origin while the output constraint is never violated.

Distributed containment control for uncertain nonlinear multi-agent systems in non-affine pure-feedback form under switching topologies

This paper considers the containment control problem of uncertain nonlinear multi-agent systems with multiple dynamic leaders under switching directed topologies. The followers are governed by a class of non-affine pure-feedback systems with arbitrary uncertainty. Implicit function and mean value theorems are employed to overcome the difficulty in controlling the non-affine pure-feedback systems and fuzzy logic systems are utilized to approximate the unknown nonlinear functions. Distributed adaptive containment controllers are proposed to guarantee that the outputs of all followers converge to the convex hull spanned by the multiple dynamic leaders. In addition, by incorporating the distributed dynamic surface control (DSC) technique, the developed containment controllers are able to eliminate the problem of “explosion of complexity” inherent in backstepping design. Based on Lyapunov stability theory, it is proved that all signals in the closed-loop systems are cooperatively semiglobally uniformly ultimately bounded (CSUUB), and the containment errors converge to a small neighborhood of the origin. An example is provided to show the effectiveness of the control approach.

출력 제약된 Pure-Feedback 시스템의 적응 신경망 제어

This paper investigates an adaptive approximation design problem for the tracking control of output-constrained non-affine pure-feedback systems. To satisfy the desired performance without constraint violation, we employ a barrier Lyapunov function which grows to infinity whenever its argument approaches some limits. The main difficulty in dealing with pure-feedback systems considering output constraints is that the system has a non-affine appearance of the constrained variable to be used as a virtual control. To overcome this difficulty, the implicit function theorem and mean value theorem are exploited to assert the existence of the desired virtual and actual controls. The function approximation technique based on adaptive neural networks is used to estimate the desired control inputs. It is shown that all signals in the closed-loop system are uniformly ultimately bounded.

Adaptive Neural Control of Pure-Feedback Nonlinear Time-Delay Systems via Dynamic Surface Technique

This paper is concerned with robust stabilization problem for a class of nonaffine pure-feedback systems with unknown time-delay functions and perturbed uncertainties. Novel continuous packaged functions are introduced in advance to remove unknown nonlinear terms deduced from perturbed uncertainties and unknown time-delay functions, which avoids the functions with control law to be approximated by radial basis function (RBF) neural networks. This technique combining implicit function and mean value theorems overcomes the difficulty in controlling the nonaffine pure-feedback systems. Dynamic surface control (DSC) is used to avoid "the explosion of complexity" in the backstepping design. Design difficulties from unknown time-delay functions are overcome using the function separation technique, the Lyapunov-Krasovskii functionals, and the desirable property of hyperbolic tangent functions. RBF neural networks are employed to approximate desired virtual controls and desired practical control. Under the proposed adaptive neural DSC, the number of adaptive parameters required is reduced significantly, and semiglobal uniform ultimate boundedness of all of the signals in the closed-loop system is guaranteed. Simulation studies are given to demonstrate the effectiveness of the proposed design scheme.

Approximation-Based Adaptive Tracking Control of Pure-Feedback Nonlinear Systems With Multiple Unknown Time-Varying Delays

This paper presents adaptive neural tracking control for a class of non-affine pure-feedback systems with multiple unknown state time-varying delays. To overcome the design difficulty from non-affine structure of pure-feedback system, mean value theorem is exploited to deduce affine appearance of state variables x(i) as virtual controls α(i), and of the actual control u. The separation technique is introduced to decompose unknown functions of all time-varying delayed states into a series of continuous functions of each delayed state. The novel Lyapunov-Krasovskii functionals are employed to compensate for the unknown functions of current delayed state, which is effectively free from any restriction on unknown time-delay functions and overcomes the circular construction of controller caused by the neural approximation of a function of u and [Formula: see text] . Novel continuous functions are introduced to overcome the design difficulty deduced from the use of one adaptive parameter. To achieve uniformly ultimate boundedness of all the signals in the closed-loop system and tracking performance, control gains are effectively modified as a dynamic form with a class of even function, which makes stability analysis be carried out at the present of multiple time-varying delays. Simulation studies are provided to demonstrate the effectiveness of the proposed scheme.

An ISS-modular approach for adaptive neural control of pure-feedback systems

Controlling non-affine non-linear systems is a challenging problem in control theory. In this paper, we consider adaptive neural control of a completely non-affine pure-feedback system using radial basis function (RBF) neural networks (NN). An ISS-modular approach is presented by combining adaptive neural design with the backstepping method, input-to-state stability (ISS) analysis and the small-gain theorem. The difficulty in controlling the non-affine pure-feedback system is overcome by achieving the so-called “ISS-modularity” of the controller-estimator. Specifically, a neural controller is designed to achieve ISS for the state error subsystem with respect to the neural weight estimation errors, and a neural weight estimator is designed to achieve ISS for the weight estimation subsystem with respect to the system state errors. The stability of the entire closed-loop system is guaranteed by the small-gain theorem. The ISS-modular approach provides an effective way for controlling non-affine non-linear systems. Simulation studies are included to demonstrate the effectiveness of the proposed approach.