Systems In Strict-feedback Form Research Articles

Robust adaptive fuzzy control for a class of uncertain nonaffine nonlinear systems with unknown control directions

In this paper, the adaptive fuzzy backstepping control problem is considered for a class of single-input single-output (SISO) unknown uncertain nonaffine nonlinear systems in strict-feedback form. Within this approach, Nussbaum gain functions are introduced to solve the problem of unknown control directions. The unknown nonlinear functions are approximated by employing adaptive fuzzy systems. The stability analysis of the closed-loop system in the sense of Lyapunov guarantees the global boundedness property for all the signals and states, and at the same time, steers the tracking error to a small neighborhood of the origin. The feasibility of the developed control approach is illustrated by numerical simulation.

Adaptive Event-Triggered Fuzzy Tracking Control of Uncertain Stochastic Nonlinear Systems With Unmeasurable States

This article is concerned with the problem of adaptive event-triggered tracking control for a class of uncertain stochastic nonlinear systems in strict-feedback form. First, with the help of fuzzy logic systems to approximate the unknown nonlinear functions, a robust fuzzy state observer is constructed to estimate all the unmeasurable states. Next, an adaptive output feedback controller, which can adjust the variables online is designed by using the backstepping scheme. Concomitantly, in order to reduce the computation of communication process, a new event triggering condition involving the decreasing function of tracking errors is introduced. Moreover, the desired closed-loop stability of the resulting systems can be achieved by exploiting Lyapunov function analysis. Finally, simulation results verify the effectiveness of the proposed method.

A novel adaptive control design method for stochastic nonlinear systems using neural network

This paper presents a novel method for designing an adaptive control system using radial basis function neural network. The method is capable of dealing with nonlinear stochastic systems in strict-feedback form with any unknown dynamics. The proposed neural network allows the method not only to approximate any unknown dynamic of stochastic nonlinear systems, but also to compensate actuator nonlinearity. By employing dynamic surface control method, a common problem that intrinsically exists in the back-stepping design, called “explosion of complexity”, is resolved. The proposed method is applied to the control systems comprising various types of the actuator nonlinearities such as Prandtl–Ishlinskii (PI) hysteresis, and dead-zone nonlinearity. The performance of the proposed method is compared to two different baseline methods: a direct form of backstepping method, and an adaptation of the proposed method, named APIC-DSC, in which the neural network is not contributed in compensating the actuator nonlinearity. It is observed that the proposed method improves the failure-free tracking performance in terms of the Integrated Mean Square Error (IMSE) by 25%/11% as compared to the backstepping/APIC-DSC method. This depression in IMSE is further improved by 76%/38% and 32%/49%, when it comes with the actuator nonlinearity of PI hysteresis and dead-zone, respectively. The proposed method also demands shorter adaptation period compared with the baseline methods.

Robust Neural Tracking Controller Design for a Class of Discrete-Time Nonlinear Systems Under Arbitrary Switching

In this article, the problem of robust tracking control is investigated for a class of discrete-time switched nonlinear uncertain system in strict-feedback form, and a novel robust neural tracking control scheme through backstepping technique is proposed for the first time. In order to handle the problem of causality contradiction, the original discrete-time nonlinear switched system is transformed into a special maximum n-step ahead predictor which suits the application of backstepping technique. The radial basis function(RBF) neural networks are adopted to approximate the unknown lumped function at the last step of backstepping procedure, not necessary to approximate all the virtual control laws at each intermediate step. Thus, the process of controller design is significantly simplified in contrast with the existing results. The stability analysis proves the proposed approach guarantees the semi-globally uniformly ultimately bounded(SGUUB) of the closed-loop system, and the tracking error asymptotically converges to an arbitrarily small neighborhood of origin by appropriately designing control parameters. Finally, simulation results are provided to demonstrate the correctness and effectiveness of the proposed scheme.

Fuzzy adaptive dynamic surface control for strict-feedback nonlinear systems with unknown control gain functions

In this work, the fuzzy adaptive output feedback control is investigated for single-input single-output (SISO) uncertain nonlinear systems in strict-feedback form. The controlled systems under consideration of this work contain the immeasurable states and unknown control gain functions. The immeasurable states are estimated by constructing a fuzzy state observer, and the uncertain nonlinear functions are approximated by fuzzy logic systems. In order to settle the issue of ‘explosion of complexity’ inherent in the conventional backstepping design process, the ‘dynamic surface control’ (DSC) method is introduced. An observer-based fuzzy adaptive control algorithm is proposed by employing the adaptive backstepping control design technique and constructing the Logarithm Lyapunov functions. The designed fuzzy adaptive control scheme can settle the complexity problem of control scheme and insure that the closed-loop system is semi-globally uniformly ultimately boundedness (SGUUB), a simulation example is considered to illustrate the availability of the designed controller.

Robust Intelligent Control of SISO Nonlinear Systems Using Switching Mechanism.

In this article, a robust adaptive learning control strategy for uncertain single-input-single-output systems in strict-feedback form and controllability canonical form (CCF) is studied. For the strict-feedback system, the dynamic surface control is introduced while for the controllability canonical system, sliding-mode control is further constructed. The finite-time design is introduced for fast convergence. Under the switching mechanism, the intelligent design and the robust technique work together to obtain robust tracking performance. Once the states run out of the domain of intelligent control, the robust item will pull the states back while inside the neural working domain, the composite learning is developed to achieve higher approximation precision by building the prediction error for the weight update. The closed-loop system stability is analyzed via the Lyapunov approach. Especially for the CCF, the finite-time convergence is achieved while the system signals are globally uniformly ultimately bounded. Simulation studies on the general nonlinear systems and the flight dynamics show that the new design scheme obtains better tracking performance with higher precision and stronger robustness.

Observer-Based Fuzzy Adaptive Inverse Optimal Output Feedback Control for Uncertain Nonlinear Systems

In this article, an observer-based fuzzy adaptive inverse optimal output feedback control problem is studied for a class of nonlinear systems in strict-feedback form. The considered nonlinear systems contain unknown nonlinear dynamics and their states are not measured directly. Fuzzy logic systems are applied to identify the unknown nonlinear dynamics and an auxiliary nonlinear system is constructed. Based on this auxiliary system, a fuzzy state observer is first designed to estimate the immeasurable states. By using the inverse optimal principle and adaptive backstepping design theory, an observer-based fuzzy adaptive inverse optimal output feedback control scheme is then developed. The proposed inverse optimal control scheme need not assume that the states are measurable. It also guarantees that the closed-loop system is semiglobally uniformly ultimately bounded, and achieves the optimal control objective as well. Finally, two simulation examples are provided to check the validity of the presented control method.

Finite-Time Backstepping of a Nonlinear System in Strict-Feedback Form: Proved by Bernoulli Inequality

In this paper, we propose a novel state-feedback backstepping control design approach for a single-input single-output (SISO) nonlinear system in strict-feedback form. Rational-exponent Lyapunov functions (ReLFs) are employed in the backstepping design, and the Bernoulli inequality is primarily adopted in the stability proof. Semiglobal practical finite-time stability, or global asymptotically stability, is guaranteed by a continuous control law using a commonly used recursive backstepping-like approach. Unlike the inductive design of typical finite-time backstepping controllers, the proposed method has the advantage of reduced design complexity. The virtual control laws are designed by directly canceling the nonlinear terms in the derivative of the specific Lyapunov functions. The terms with exponents are transformed into linear forms as their bases. The stability proof is simplified by applying several inequalities in the final proof, instead of in each step. Furthermore, the singularity problem no longer exists. The weakness of the concept of practical finite-time stability is discussed. The method can be applied to smoothly extend numerous design methodologies with asymptotic stability with a higher convergence rate near the equilibrium. Two numerical case studies are provided to present the performance of the proposed control.

Fault Tolerant Control Scheme for a Class of Interconnected Nonlinear Time Delay Systems Using Event-Triggered Approach

In this study, the event-triggered decentralized fault tolerant control (FTC) problem is investigated for a class of interconnected nonlinear time delay systems in strict feedback form with input quantization and actuator faults. The model of interconnected nonlinear time delay systems includes unstructured uncertainties and unmeasurable states. Firstly, the radial basis function neural networks (RBFNNs) are introduced to identify the unstructured uncertainties and time delay functions. An adaptive RBFNNs state observer is constructed to identify the unmeasurable states, which will be used to the design of decentralized fault tolerant controller. By utilizing the adaptive backstepping technique with low pass filter, an event-based decentralized FTC strategy is developed for the considered interconnected nonlinear systems, such that the output signals could track the reference signals, and the stability of the closed loop systems is guaranteed. Finally, the effectiveness of the designed control strategy is illustrated by a practical simulation example.

A Fuzzy Adaptive Tracking Control for MIMO Switched Uncertain Nonlinear Systems in Strict-Feedback Form

An adaptive fuzzy tracking control is investigated for a category of multiple-input multiple-output switched uncertain nonlinear systems in strict-feedback form. The unknown nonlinear function and switching signals with average dwell time are contained in the systems. The control method is designed by generalized fuzzy hyperbolic model and backstepping technique. Through adding a first-order filter into conventional backstepping method, the phenomenon of “explosion of complexity” is eliminated. In the controller design process of each subsystem, only one adaptive parameter is needed to adjust online. Compared with other existing control methods, the proposed control design is more simple and calculated amount is greatly decreased. It is guaranteed that the whole closed-loop system is stable by using Lyapunov function and average dwell-time methods. The effectiveness of the proposed control method can be demonstrated in the simulation part.

Decentralized Adaptive Event-triggered Control for Nonlinear Interconnected Systems in Strict-feedback Form

The decentralized event triggered control problem is investigated for nonlinear interconnected systems in strict-feedback form subjected to parametric uncertainty. For each subsystem in the interconnected systems, the decentralized adaptive backstepping controller is designed to guarantee that the tracking error is semi-globally ultimately bounded. The control update and parameter estimate action are aperiodical executed only when the desired control specifications cannot be ensured, drastically reducing the computational burden and the transmission cost. It can be proved that zeno phenomenon is avoided as a positive lower bound on the minimal inter-sample time exists. The impulsive dynamical systems tools and Lyapunov analysis are introduced to prove the stability property for closed-loop system. Finally, a numerical simulation example is included to validate the effectiveness of the control scheme.

Global Fixed-Time Stabilization of Switched Nonlinear Systems: A Time-Varying Scaling Transformation Approach

This brief is concerned with the problem of global fixed-time stabilization for a class of switched nonlinear systems in strict-feedback form. A novel time-varying scaling transformation is introduced to convert the original fixed-time stabilization problem into the asymptotic stabilization problem of transformed systems. Based on this, a new framework to address fixed-time state feedback for switched nonlinear systems is developed by employing the common Lyapunov function method and the adding a power integrator technique. It is proved that the proposed controller renders the states of the closed-loop system to zero in any prescribed finite time under arbitrary switchings. Finally, simulation results of a practical example are provided to confirm the efficacy of the proposed method.

Neuro-Fuzzy-Based Adaptive Dynamic Surface Control for Fractional-Order Nonlinear Strict-Feedback Systems With Input Constraint

This article investigates the issue of neuro-fuzzy-based adaptive dynamic surface control (DSC) for uncertain fractional-order (FO) nonlinear systems in strict-feedback form where input constraint is considered in the systems. In the recursive steps, the neuro-fuzzy network systems are employed to deal with the unknown nonlinear terms existing in systems. Furthermore, based on a DSC scheme, a modified FO filter is constructed to overcome the problem of explosion of complexity caused by the traditional backstepping design. Moreover, according to the FO Lyapunov stability theory, a neuro-fuzzy-based adaptive controller is designed to guarantee all the signals of FO closed-loop systems tend to be bounded. Finally, the three examples are provided to verify the validity and superiority of the presented control scheme.

Fuzzy Adaptive Distributed Event-Triggered Consensus Control of Uncertain Nonlinear Multiagent Systems

This paper studies the fuzzy adaptive distributed event-based control scheme for a class of uncertain nonlinear multiagent systems in strict feedback form. A novel distributed adaptive event-trigger condition and event-triggered controller are designed simultaneously. The main advantage with respect to event-sampling states is that the controllers were updated in an aperiodic manner at the event sampled instants, saving the computation resource and transmission load. Simulation results have shown satisfactory tracking performance of the developed control scheme.

Linear output-feedback-based semi-global stabilization for switched nonlinear time-delay systems

This paper focuses on the problem of semi-global output-feedback stabilization for a class of switched nonlinear time-delay systems in strict-feedback form. A switched state observer is first constructed, then switched linear output-feedback controllers for individual subsystems are designed. By skillfully constructing multiple Lyapunov–Krasovskii functionals and successfully solving several troublesome obstacles, such as time-varying delay and switching signals and nonlinearity in the design procedure, the switched linear output-feedback controllers designed can render the resulting closed-loop switched system semi-globally stabilizable under a class of switching signals with average dwell time. Furthermore, under some milder conditions on nonlinearities, the semi-global output-feedback stabilization problem for switched nonlinear time-delay systems is also studied. Simulation studies on two examples, which include a continuous stirred tank reactor, are carried out to demonstrate the effectiveness of the proposed approach.

Robust Adaptive Dynamic Surface Control of Nonlinear Time-varying Systems in Strict-feedback Form

In this paper, the problem of robust adaptive control for nonlinear systems with unknown time-varying parameters and disturbance is considered. For this purpose, an indirect adaptive controller based on the Dynamic Surface Control (DSC) is proposed where an adaptive scheme is presented to provide estimations of unknown time-varying parameters. First, the parameters are approximated in terms of polynomials with unknown coefficients using the Taylor series expansion. Then, these unknown coefficients are estimated using a novel adaptive law. It is shown that the proposed control scheme guarantees the stability of the overall system and the tracking error can converge to a desired small residual. Finally, simulation results are given to demonstrate the effectiveness of the proposed method.

Adaptive fuzzy backstepping control for uncertain nonlinear systems with tracking error constraints

This article deals with the design of adaptive fuzzy backstepping control for uncertain nonlinear systems in strict-feedback form with tracking error constraints. In this article, a fuzzy system is used to approximate the unknown nonlinear functions and the differential of virtual control law of each subsystem. In order to satisfy the limitation of tracking error constraints, the barrier Lyapunov function is introduced. Moreover, by applying the minimal learning parameters technique, the number of online parameters update for each subsystem is reduced to only 1. The control scheme not only ensures the tracking error is not to transgress the constraint bounds but also solves the problem of “explosion of complexity” and greatly reduces the initial control input and the number of the adaptive parameters; this provides the conditions for the practical application. The simulation results show the effectiveness of the proposed method.

Adaptive Fuzzy Sliding Mode Control for a Class of Uncertain Nonaffine Nonlinear Strict-Feedback Systems

In this paper, an adaptive fuzzy sliding mode scheme is proposed for a class of single-input, single-output (SISO) unknown uncertain nonaffine nonlinear systems in strict-feedback form. In this scheme, the backstepping algorithm is used to obtain the virtual controllers. The real control law is designed by using sliding mode approach. The both matched and mismatched unknown uncertainties are counteracted by using adaptive robust terms. The unknown nonlinear functions are approximated by employing adaptive fuzzy logic systems. The adaptation laws of the adjustable parameters are deduced from the stability analysis of the closed-loop system in the sense of Lyapunov. The derived adaptive fuzzy sliding mode control approach guarantees the global boundedness property for all the signals and states, and at the same time, steers the tracking error to a very small neighborhood of the origin. Numerical simulation results are provided to show the effectiveness of the proposed control scheme.

Adaptive asymptotic tracking using barrier functions

This paper studies the global output tracking problem for a class of unknown time-varying nonlinear systems in strict-feedback form. By utilizing the barrier functions, a universal adaptive state-feedback control strategy is proposed that achieves asymptotic tracking performance. Unlike the existing results in the literature, the proposed control scheme utilizes the barrier functions to ensure the unknown system nonlinearities to be the bounded “disturbance-like” terms, which are adaptively compensated at each step, this enables any approximation structures are not needed. Furthermore, the “explosion of complexity” issue in backstepping-like approaches is avoided without using additional filtering. Simulation results are presented to demonstrate the effectiveness of the proposed methodology.

Adaptive optimal fault-tolerant control scheme for a class of strict-feedback nonlinear systems

An active optimal fault-tolerant control (FTC) scheme for a class of nonlinear systems in strict-feedback form in the presence of partial loss of actuator effectiveness faults is proposed, using backstepping design technique and adaptive dynamic programming (ADP) algorithm to compensate the effects of failure. The proposed FTC scheme consists of feedforward controller that achieve the objective of fault-tolerant and feedback optimal controller, which can guarantee the performance index function is minimized. Since fault estimation and control law parameters are updated online, the control system has an adaptive failure compensation capability so as to reconfigure the control law in real time in response to failure indications. Based on Lyapunov stability theory, the whole closed-loop system is guaranteed to be ultimately uniformly bounded. Finally, the effectiveness of the proposed method is demonstrated by simulation.