Step Of Backstepping Research Articles

Neural Network Backstepping Controller Design for Uncertain Permanent Magnet Synchronous Motor Drive Chaotic Systems via Command Filter

In this study, an adaptive neural network (NN) command filtered control (CFC) method is proposed for permanent magnet synchronous motor (PMSM) system with system uncertainties and external disturbance by means of backstepping technique. At every backstepping step, a novel command filter is proposed, and the complicated virtual input and its derivative together can be approximated by this filter. The ``explosion of complexity" problem in conventional backstepping design can be avoided because we do not need to calculate the derivative of the virtual input repeatedly. NNs are used to model system uncertainties and disturbances. Finally, an adaptive NN CFC is designed, and the convergence of the tracking error and the boundedness of all signals involved can be guaranteed. Finally, a simulation study is presented to verify the theoretical results.

A hybrid adaptive synchronization protocol for nondeterministic perturbed fractional-order chaotic nonlinear systems

In this paper, we investigate hybrid adaptive synchronization issue for a class of perturbed fractional-order chaotic systems with nondeterministic nonlinear terms. On the basis of fractional-order extended version of Lyapunov stability criterion, a novel fuzzy adaptive synchronization control protocol coupled with backstepping-based method is constructed, ensuring that the synchronization errors converge to a sufficiently small region of the origin. In order to avert the occurrence of “explosion of complexity”, we take advantage of a fuzzy logic system to estimate the unknown systematic term approximately in every backstepping step. Finally, some numerical simulations are given to exemplify the effectiveness of the proposed approach.

Adaptive neural network control for time-varying state constrained nonlinear stochastic systems with input saturation

This paper investigates the tracking control issue of nonlinear stochastic systems subject to time-varying full state constraints and input saturation. By employing both neural network-based approximator and backstepping technique, an adaptive neural network (NN) control approach is presented on the basis of the time-varying barrier Lyapunov function. To surmount the influence of saturation nonlinearity, a Gaussian error function-based continuous differentiable saturation model is introduced such that the actual control in the final backstepping step can be achieved. The designed controller can not only achieve the tracking control objective, but also surmount the impact of input saturation to stochastic system performance. Meanwhile, the norm of NN weight vector is taken as estimated parameter, and it can alleviate computation burden. The presented controller can ensure that all the signals in the closed-loop system are bounded in probability and all state variables are restricted the predefined regions. Finally, simulation results are given to illustrate the effectiveness of the established controller.

Composite Learning Adaptive Dynamic Surface Control of Fractional-Order Nonlinear Systems.

Adaptive dynamic surface control (ADSC) is effective for solving the complexity problem in adaptive backstepping control of integer-order nonlinear systems. This article focuses on the ADSC design for parametric uncertain fractional-order nonlinear systems (FONSs). In each backstepping step, the virtual controller is driven to pass through a fractional dynamic surface whose fractional-order derivative can be calculated easily. An ADSC law that ensure tracking error convergence is designed. The proposed ADSC requires a stringent condition called persistent excitation (PE) to achieve parameter convergence. To relax this limitation, a prediction error is defined by using online recorded data and instantaneous data, and a composite learning law is proposed to utilize both the prediction error and the tracking error. Then, a composite learning ADSC (CLADSC) method is developed to guarantee tracking error convergence and accurate parameter estimation under an interval excitation condition that is weaker than the PE one. Finally, an illustrative example is presented to show the performance of our methods.

Adaptive neural network backstepping control of fractional-order Chua\u2013Hartley chaotic system

In this paper, the control of uncertain fractional-order Chua–Hartley (FOCH) chaotic systems by means of adaptive neural network backstepping control is considered. Neural network is utilized as a universal approximator to estimate the unknown nonlinear function. By using the fractional Lyapunov stability criterion and the backstepping technique, an adaptive neural network control (ANNC) method is implemented. In each backstepping step, the unknown nonlinear functions are approximated by neural networks, and a virtual control input is designed. The proposed method guarantees that all the closed-loop signals keep bounded and tracking error converges to an arbitrary small region of zero. Finally, numerical simulation is given to confirm the effectiveness and the good control performance of the proposed method.

Neuroadaptive containment control of nonlinear multiagent systems with input saturations

SummaryThis paper investigates the output containment tracking problem of nonlinear multiagent systems with mismatched uncertain dynamics and input saturations. A neural network–based distributed adaptive command filtered backstepping (CFB) scheme is given, which can guarantee that the containment tracking errors reach to the desired neighborhood of origin and all signals in the closed‐loop system are bounded. Note that error compensation system and virtual control laws established in CFB only use local information, so the given scheme is completely distributed. Moreover, the applied sliding mode differentiator (SMD) can make the outputs of SMD fast approximate the virtual signal and its derivative at each step of backstepping, which can further improve the control quality. Finally, a simulation example is given to show the effectiveness of the proposed scheme.

Anti-disturbance backstepping control for air-breathing hypersonic vehicles based on extended state observer

In this paper, we propose an anti-disturbance backstepping control approach with extended state observer (ESO) for tracking control of air-breathing hypersonic vehicles. Considering the large uncertainties, the external disturbances, and especially the lack of aerodynamic knowledge, several ESOs are introduced in the backstepping controller. With the total disturbance estimation ability of ESOs, almost no aerodynamic knowledge is needed for the controller design. Meanwhile, ESOs are also used to estimate the derivatives of the virtual control signals. The problem of “explosion of terms” is avoided. A key strategy of the controller is that each step of backstepping is activated successively. Consequently, the closed-loop system has time-scale structure. Rigorous stability proof can be obtained. At last, compared simulation results verify the superior tracking performance of the proposed controller.

One Dimensional Non-Linear Control of ST Fusion Reactor with D3He Fuel Using Back-Stepping Method

In this article a model for calculating energy and particles densities is presented in order to control burning plasma in the case of D3He fuel in spherical tokamak (ST). A spherical tokamak is a possible candidate for a D3He fusion reactor due to its high-beta value. In this model a one-dimensional approximation of the transport equation for energy as well as the density of deuterium-helium3 fuel ions and alpha particles is represented in cylindrical coordinates by a system of partial differential equation. By applying these simulation equations to control particles and energy densities profiles, the system is discretized in space using a finite difference method and a back-stepping design for D3He aneutronic fuel in tokamaks. Results obtained for boundary control conditions shows that initial values are approaching to equilibrium profiles and also the system achieves the stable equilibrium. It is also seen that with the controller modulation of the alpha particles, deuterium-helium3 and energy densities at the edge of the plasma, they are approaching to stable equilibrium points. The results of boundary control law show that the profiles can be successfully controlled with just one step of back-stepping.

Extended state observer based control for generic hypersonic vehicles with nonaffine-in-control character

This paper investigates the flight control problem of generic hypersonic vehicles subject to nonaffine-in-control character. Considering the large uncertainties and external disturbance, the disturbance observer based control strategy is incorporated in the control scheme. Firstly, an extended state observer is used to estimate the system states and the total disturbance. Then, based on the output of the extended state observer, we follow the backstepping design procedure. The dynamic inversion method is involved in the last step of backstepping to solve the nonaffine-in-control problem. The proposed control scheme ensures that the hypersonic vehicle tracks the command signal with almost no aerodynamic knowledge. Rigorous stability proof is given based on the separated time-scale structure of the extended state observer and the dynamic inversion method. At last, numerical simulations are presented in different conditions to demonstrate the effectiveness and good tracking performance of the proposed control scheme.

Optimized Backstepping for Tracking Control of Strict-Feedback Systems

In this paper, a control technique named optimized backstepping is first proposed by implementing tracking control for a class of strict-feedback systems, which considers optimization as a design philosophy of the high-order system control. The basic idea is that designing the actual and virtual controls of backstepping is the optimized solutions of the corresponding subsystems so that overall control of the high-order system is optimized. In general, optimization control is designed based on the solution of Hamilton-Jacobi-Bellman equation, but solving the equation is very difficult due to the inherent nonlinearity and intractability. In order to overcome the difficulty, the neural network (NN)-based reinforcement learning strategy of actor-critic architecture is used. In every backstepping step, the actor and critic NNs are constructed for executing control behavior and evaluating control performance, respectively. According to the Lyapunov stability theorem, it is proven that the desired control performance can be obtained. Finally, a simulation example is carried out to further demonstrate the effectiveness of the proposed control approach.

Generation of stable oscillations in uncertain nonlinear systems with matched and unmatched uncertainties

ABSTRACTIn this paper, a robust limit cycle control technique is proposed for generation of stable oscillations in a class of uncertain nonlinear systems with both matched and unmatched uncertainties. For this purpose, first, the modified Lyapunov function is introduced which is appropriate for stability analysis of invariant sets (instead of equilibrium points). The structure of the proposed Lyapunov function is related to the shape of the desirable limit cycle. Next, in order to design the robust limit cycle control input, the backstepping and Lyapunov redesign methods are employed, simultaneously. The The classical Lyapunov redesign controller is discontinuous and robust with respect to matched uncertainties. To overcome unmatched uncertainties, a modified version of the Lyapunov redesign controller is suggested in each step of backstepping which results in a continuous robust control law. Furthermore, the convergence of the phase trajectories of the uncertain closed-loop system to the target limit cycle is proved using the extended Lyapunov stability theorem. Finally, computer simulations are performed to show the applicability of the given approach. In this regard, two uncertain nonlinear practical systems are considered and robust stable oscillations are generated in these systems via the proposed controller. Simulation results confirm the effectiveness of the proposed technique.

Prescribed Performance Switched Adaptive Dynamic Surface Control of Switched Nonlinear Systems With Average Dwell Time

In this paper, the problem of adaptive fuzzy tracking control is investigated for a class of switched nonlinear systems. A new switched adaptive output feedback control method is presented where the changes of plant can be considered explicitly. Mode-dependent fuzzy logic systems are employed to approximate the switching nonlinear functions in system. To reduce the conservativeness caused by adoption of a common adaptive law for all subsystems, the switching optimal weight vectors are directly estimated via switched adaptive laws at each step of backstepping. However, the difficulties are how to ensure the estimation performance subject to persistent switchings, and how to design common virtual controls based on switching parameters. Further, based on the estimation of optimal weight vectors, the switched fuzzy state observer can be also applied to obtain the unmeasured states. By designing a novel Lyapunov function, the improved dynamic surface control incorporated by prescribed performance technique guarantees all the total state tracking errors, not partial ones, within predefined performance bounds. Simulation results are provided to demonstrate the effectiveness of the proposed method.

Robust backstepping control for a class of nonlinear systems using generalized disturbance observer

This paper presents a new composite robust controller for a class of uncertain nonlinear systems through backstepping method and disturbance observer (DOB) technique. By designing a generalized disturbance observer (GDOB), the mismatched/matched uncertainty is estimated and compensated at each backstepping step. By virtue of the proposed GDOB, the proposed controller only requires the lumped uncertainties are differentiable and the higher-order derivatives are bounded. With the help of GDOB, the high-gain backstepping controller is also avoided completely. Asymptotical stability of the closed-loop system is established using direct Lyapunov method. An illustrative example of missile autopilot design is given and simulations are carried out to validate the proposed method.

Control Chaos for Permanent Magnet Synchronous Motor Base on Adaptive Backstepping of Error Compensation

Permanent magnet synchronous motor (PMSM) can appear chaos phenomenon when PMSM in turn on or turn off. When control chaotic PMSM to zero, the state variables of the system can appear fluctuate which is not the ideal control result. In order to overcome fluctuate in control chaos. This paper propose adaptive backstepping of error compensation to control chaotic PMSM, an error compensation item is developed in the virtual control design of each backstepping step to compensate the effect of unknown error dynamics on system for obtaining smooth process of chaos control. This scheme can eliminate oscillation in course of control chaos. Numerical simulations further test the effectiveness of the theoretical analysis.

Backstepping-Based Lyapunov Function Construction Using Approximate Dynamic Programming and Sum of Square Techniques

In this paper, backstepping for a class of block strict-feedback nonlinear systems is considered. Since the input function could be zero for each backstepping step, the backstepping technique cannot be applied directly. Based on the assumption that nonlinear systems are polynomials, for each backstepping step, Lypunov function can be constructed in a polynomial form by sum of square (SOS) technique. The virtual control can be obtained by the Sontag feedback formula, which is equivalent to an optimal control-the solution of a Hamilton-Jacobi-Bellman equation. Thus, approximate dynamic programming (ADP) could be used to estimate value functions (Lyapunov functions) instead of SOS. Through backstepping technique, the control Lyapunov function (CLF) of the full system is constructed finally making use of the strict-feedback structure and a stabilizable controller can be obtained through the constructed CLF. The contributions of the proposed method are twofold. On one hand, introducing ADP into backstepping can broaden the application of the backstepping technique. A class of block strict-feedback systems can be dealt by the proposed method and the requirement of nonzero input function for each backstepping step can be relaxed. On the other hand, backstepping with surface dynamic control actually reduces the computation complexity of ADP through constructing one part of the CLF by solving semidefinite programming using SOS. Simulation results verify contributions of the proposed method.

Adaptive finite-time stabilization of a class of switched nonlinear systems using neural networks

This paper investigates the adaptive finite-time stabilization of a class of switched nonlinear systems via backstepping technique, where unknown nonlinear functions are only required to be continuous. At each step of backstepping, neural network is used to appropriate the unknown continuous function, an adaptive law and a virtual controller are constructed simultaneously. By using the Lyapunov function method, it is shown that, under the designed controller and adaptive laws, the state of the closed-loop system is semi-globally uniformly finite-time bounded. An example is provided to show the effectiveness of the proposed method.

Robust terminal angle constraint guidance law with autopilot lag for intercepting maneuvering targets

In this paper, a robust continuous guidance law with terminal angle constraint for intercepting maneuvering targets in the presence of autopilot lag is proposed and its finite-time stability is proved. First, assuming that the missile autopilot is sufficiently fast, a composite fast nonsingular terminal sliding mode guidance (CFNTSMG) law is presented. The proposed guidance law is obtained through a combination of fast nonsingular terminal sliding mode control theory and generalized disturbance observer (GDOB). The presented guidance law requires no priori information on target maneuver, which is estimated and compensated online by GDOB. Moreover, no discontinuous term exists in CFNTSMG law and therefore chattering is eliminated effectively. Next, viewing the missile autopilot as an uncertain second-order system, an integrated guidance and control dynamics is formulated and the systematic step-by-step backstepping technique is used to derive a new robust guidance law, which not only holds the advantages of CFNTSMG law, but also is insensitive to autopilot lag. At each backstepping step, novel continuous virtual/real control laws using finite-time control approach are designed and tracking differentiator is used to overcome the ‘explosion of complexity’ problem encountered with traditional backstepping method. Theoretical analysis and numerical simulations demonstrate the effectiveness of the proposed method.

Adaptive backstepping repetitive learning control design for nonlinear discrete‐time systems with periodic uncertainties

SummaryThis paper addresses a tracking problem for uncertain nonlinear discrete‐time systems in which the uncertainties, including parametric uncertainty and external disturbance, are periodic with known periodicity. Repetitive learning control (RLC) is an effective tool to deal with periodic unknown components. By using the backstepping procedures, an adaptive RLC law with periodic parameter estimation is designed. The overparameterization problem is overcome by postponing the parameter estimation to the last backstepping step, which could not be easily solved in robust adaptive control. It is shown that the proposed adaptive RLC law without overparameterization can guarantee the perfect tracking and boundedness of the states of the whole closed‐loop systems in presence of periodic uncertainties. In addition, the effectiveness of the developed controller is demonstrated by an implementation example on a single‐link flexible‐joint robot. Copyright © 2014 John Wiley & Sons, Ltd.

Robust Force Control of an SRM-Based Electromechanical Brake and Experimental Results

In this paper, we propose robust nonlinear force controllers for a switched-reluctance-motor (SRM) electromechanical brake system which is a promising replacement for hydraulic brakes in the automotive industry. A torque-level control law is first designed using robust backstepping. The backstepping proceeds via the force and the velocity states. The voltage-level control laws are obtained from the virtual control law for the torque using either an additional step of backstepping incorporating a novel voltage-commutation scheme or a torque-ripple-minimizing algorithm based on a design of turn-on/turn-off angles and torque factors. The controllers do not require knowledge of the motor mechanical parameters and the functional forms of the relationships among the motor position, the brake force, and the motor load torque. A detailed model of the motor including current-dependent inductance coefficients is used. The load exerted on the motor by the caliper may be modeled as a spring; however, the actual load model is taken to be an unknown nonlinear function of position to allow for uncertainties in the model. Hence, the developed controllers work for a wide variety of loads including brake systems. Moreover, the controllers provide significant robustness to uncertainty in the inductances and address practical current and voltage constraints. The performance of the proposed controllers is demonstrated through both simulation and experimental studies.

Architecture Induced by Distributed Backstepping Design

An important problem in the distributed control of large-scale and infinite dimensional systems is related to the choice of the appropriate controller architecture. We utilize backstepping as a tool for distributed control of nonlinear infinite dimensional systems on lattices, and provide the answer to the following question: what is the worst case controller architecture induced by distributed backstepping design? We demonstrate that distributed backstepping design yields controllers that are intrinsically decentralized, with a strong similarity between plant and controller architectures. In particular, we study the 'worst case' control design in which all interactions are cancelled at each step of backstepping. Any other backstepping strategy yields controllers with better information transmission properties. For this 'worst case' situation we quantify the number of control induced interactions necessary to guarantee desired dynamical behavior of the infinite dimensional system. We also provide an example of systems on lattices and show how the controllers with favorable architectures can be designed.