Backstepping Control For Nonlinear Systems Research Articles

Practical finite-/fixed-time improved command-filtered backstepping control for nonlinear systems via immersion and invariance

This paper presents a practical finite-/fixed-time adaptive tracking control scheme based on improved command filtered backstepping (ICFBS) and immersion and invariance (I&I) for a class of single-input single-output (SISO) strict-feedback nonlinear systems with uncertainty parameterization. The conventional command-filtered backstepping method (CFBS) is improved by introducing any bounded function such that ICFBS relaxes the boundedness assumption of the control direction function and elevates the tunable space of the design scheme, with almost no increase in computational complexity. The higher-power composite term of a parameter estimation is designed in the parameter update law based on I&I and the inequality constructed by us proves that this form of the adaptive update law can be augmented to suit for not only uniform ultimate bounded (UUB) and practical finite-time stability but also practical fixed-time stability of the whole closed-loop system. The error compensation system is also extended to realize the practical finite-/fixed-time convergence of the error compensation signals. The proposed controller guarantees practical finite-/fixed-time stability of the closed-loop system. Therefore, our work solves two unsolved problems: both practical fixed-time and practical finite-/fixed-time ICFBS and I&I adaptive tracking control. Finally, the proposed control scheme is validated by a practical electromechanical system and a nonlinear system with unbounded control direction functions.

Output feedback backstepping control for non-linear systems using an adaptive finite time sliding mode observer

Abstract In this paper, a class of non-linear systems in normal form is considered, which is composed of internal and external dynamics. An adaptive finite time sliding mode observer is first designed so that the system states, unmatched uncertain parameters and matched uncertainties can all be observed in finite time. Then, the systematic backstepping design procedure is employed to develop a novel output feedback backstepping control (OFBC). The proposed OFBC method can stabilize the considered non-linear systems despite the presence of non-linear internal dynamics and unmatched uncertainties. A Lyapunov method is used to ensure that the closed-loop system is asymptotically stable. Two MATLAB simulation examples are used to demonstrate the method.

Adaptive neural network observer‐based filtered backstepping control for nonlinear systems with fuzzy dead‐zone and uncertainty

SummaryThis study proposes a high gain observer‐based filtered backstepping control scheme for nonlinear systems with unknown dynamics and dead‐zone constraint. Majority of the previously papers describe dead‐zone by the conventional certain models, assume that all state variables are available and invoke the backstepping or conventional dynamic surface control (CDSC) approaches for controller design. This work describes the dead‐zone by the fuzzy model, invokes radial basis function neural network (RBFNN) to model the unknown functions and proposes a RBFNN‐based high gain observer to estimate unmeasurable states. To avoid the complexity explosion in the backstepping approach and eliminate the sensitivity to the time‐constant of the first‐order filters in the CDSC, a nonlinear tracking differentiator (NTD) with finite time convergence property is used instead of the first‐order filters in the CDSC to obtain derivative of the virtual inputs. Moreover, the error compensation mechanism and auxiliary signal are proposed to eliminate the influence of the filtering error and input nonlinearity, respectively. Despite the fuzzy dead‐zone, the proposed scheme makes the closed‐loop signals uniformly ultimately bounded. In comparison with the existing results, (i) some assumptions like certain model of dead‐zone and full state measurement are removed, (ii) “explosion of complexity” and sensitivity to the time constant of the first‐order filters are eliminated, (iii) effect of the filtering error and input nonlinearity are compensated, (iv) number of adjustable parameters and online computational burden are decreased effectively. Simulation results on the two well‐known examples verify the effectiveness and applicability of the proposed new design technique.

Transients Improvement in the Output-Feedback Adaptive Backstepping Control of Nonlinear Systems with Input Constraints

The paper addresses the problem of transient performance improvement in the output-feedback adaptive backstepping control of parametrically uncertain nonlinear system represented in the cascaded output-feedback form. The system state is inaccessible for measurement, the system input is constrained, and the control gain is assumed unknown (its sign is known). The problem is solved by applying specially modified K-filters with adjustable parameter, using the backstepping procedure taking into account the input constraints, and involving the Kreisselmeier-like adaptation algorithm with a modified swapping scheme. The algorithm enables us to improve the closed-loop transient performance without increasing initial “swings” in the control error transients. The latter property is achieved by including the high-order time derivatives of the adjustable parameters, calculated by the adaptation algorithm, in the control law. The properties of the closed-loop system are illustrated via simulation.

Finite-time command-filtered backstepping control for nonlinear systems with input delay and time-varying full-state constraints

This paper examines a finite-time command-filtered backstepping control proposed for a class of strict-feedback nonlinear systems with input delay and time-varying asymmetric full-state constraints. The existing control method either ignores the effect of input delay or converges asymptotically in infinite time. A command-filtered backstepping design is used to decrease computational burden. A first-order Levant differentiator is employed to replace the command filter for estimating the virtual control signals in finite time. The time-varying asymmetric barrier Lyapunov function (TVABLF) and command-filtered backstepping design are applied to alleviate the significant challenges caused by the backstepping approach and full-state constraints. A novel finite-time delay compensation mechanism is proposed to remove the impact of input delay. The closed-loop signals are proved to be practical bounded. Simulation results are provided to demonstrate the effectiveness of the proposed control scheme.

Error-Based Gain-Varying Finite-Time Command Filtered Backstepping Control for Nonlinear Systems With Disturbances

In this brief, a novel error-based gain-varying backstepping control strategy is proposed for nonlinear systems with disturbances. The error-based gain-varying functions are constructed to substitute the static gains to enhance the anti-disturbance capability and improve the transient performance. The command filter is applied to avoid the differential explosion problem and the filtering errors are reduced by using gain-varying error compensation mechanism. The error-based gain-varying backstepping strategy ensures that the tracking error converges into a small neighborhood of origin in finite time and all signals in the closed-loop system are bounded in finite time. An example of single-link manipulator is employed to illustrate the effectiveness of the proposed strategy.

Adaptive neural output feedback finite‐time command filtered backstepping control for nonlinear systems with full‐state constraints

AbstractIn this paper, an adaptive neural finite‐time control method via barrier Lyapunov function, command filtered backstepping, and output feedback is proposed to solve the tracking problem of uncertain high‐order nonlinear systems with full‐state constraints and input saturation. By utilizing the neural network (NN) to approximate unknown nonlinear functions, the finite‐time command filters are used to filtering the virtual control signals and get the intermediate control signals in a finite time in the backstepping process. Because there are errors between the output of finite‐time command filters and the virtual control signals, the error compensation signals are added to eliminate the influence of filtering errors. Based on the proposed control scheme, the states of the system can be constrained in the predetermined region, all signals in the system are bounded in finite time, and the tracking error can converge to the desired region in finite time. At last, a simulation example is given to show the effectiveness of the proposed control method.

Prescribed Adaptive Backstepping Control of Nonlinear Systems Preceded by Hysteresis in Piezoelectric Actuators

Prescribed Adaptive Backstepping Control of Nonlinear Systems Preceded by Hysteresis in Piezoelectric Actuators

Adaptive NN Cross Backstepping Control for Nonlinear Systems With Partial Time-Varying State Constraints and Its Applications to Hyper-Chaotic Systems

This paper proposes a novel cross backstepping technique-based controller design for a class of nonlinear constrained systems with its applications to hyper-chaotic systems. Considering the strict feedback systems with partial time-varying state constraints, we divide the special systems into constrained subsystems and unconstrained subsystems. However, the normal backstepping method is only an effective method structured to control lower-triangular systems. Since the method is strictly limited to the system structure, the previous works cannot solve the problem considered. Thus, we employ the cross backstepping method to solve the problem of partial and alternate time-varying state constraints. For the constrained subsystems, the time-varying barrier Lyapunov function (TVBLF) is employed to ensure that the violation of any time-varying constraint does not occur. Besides, the radial basis function neural network (RBFNN) is used to approximate the uncertainties. Then, based on the stability analysis, it is concluded that the output follows the desired trajectory as closely as possible, all the signals in closed-loop systems are bounded, and the alternate time-varying state constraints are never violated. Finally, the simulation results demonstrate the effectiveness of the proposed strategy.

Barrier Lyapunov Function Based Adaptive Cross Backstepping Control for Nonlinear Systems with Time-varying Partial State Constraints

This paper focuses on an adaptive cross-backstepping control for a class of nonstrict-feedback nonlinear systems affected by time-varying partial state constraints. The nonstrict-feedback nonlinear systems considered in this paper are divided into two strict-feedback nonlinear subsystems. One constrained subsystems and another unconstrained subsystems. In view of the normal backstepping method is only an effective method to control lower-triangular systems, cross-backstepping control is introduced which successfully solves the problem of alternate time-varying state constraints. For the constrained subsystems, novel time-varying tan-type barrier Lyapunov function (TBLF) is employed in each step of backstepping design to guarantee the boundedness of the fictitious or actual state tracking errors. Besides, proper adaptive laws are designed for the upper bound of uncertain parameters, which successfully neutralize the influence of parametric uncertainties. Based on the stability analysis, it is concluded that the states of the closed-loop system maintain in the predefined time-varying compact sets and the output can be guaranteed to be as close to the desired trajectory as possible. The effectiveness of the proposed technique is illustrated by a constrained hyperchaotic system in two cases.

Asymptotic adaptive control of nonlinear systems with elimination of overparametrization in a Nussbaum-like design

Making a trade-off between the control accuracy and computational reduction is a problem frequently encountered in the system control design. This is especially difficult when one designs adaptive fuzzy (or neural network) controls for nonlinear systems, in which fuzzy controls have to consume many computational resources to tune a sufficiently large number of adaptive parameters, meanwhile nonlinear uncertainties block the high demanding control accuracy. Current works usually face a dilemma that, either the computation is reduced but the control accuracy is degraded due to the use of the norm estimation, or the asymptotic control is resulted but the computation is increased due to the extra compensation controls. To address such dilemma, we propose an asymptotic adaptive fuzzy tracking control algorithm, whose main feature is that only two adaptive laws are needed throughout the control scheme. In particular, we allocate one adaptive law to achieve adaptive fuzzy backstepping control for nonlinear systems with a focus on stabilizing the closed-loop system. We then allocate the other adaptive law not only to asymptotically drive the stabilization error to the zero, but also to achieve the elimination of overparametrization in a Nussbaum-like design, which is inspired by the tuning function-based approach.

Uncertainty and Disturbance Estimator-Based Backstepping Control for Nonlinear Systems With Mismatched Uncertainties and Disturbances

This paper proposes an uncertainty and disturbance estimator (UDE)-based controller for nonlinear systems with mismatched uncertainties and disturbances, integrating the UDE-based control and the conventional backstepping scheme. The adoption of the backstepping scheme helps to relax the structural constraint of the UDE-based control. Moreover, the reference model design in the UDE-based control offers a solution to address the “complexity explosion” problem of the backstepping approach. Furthermore, the strict-feedback form condition in the conventional backstepping approach is also relaxed by using the UDE-based control to estimate and compensate “disturbance-like” terms including nonstrict-feedback terms and intermediate system errors. The uniformly ultimate boundedness of the closed-loop system is analyzed. Both numerical and experimental studies are provided.

Robust mixed order backstepping control of non‐linear systems

Robust backstepping control of non-linear systems with derivation orders (commensurate or non-commensurate) lying at interval (0, 2) is proposed in this study. The stability and robustness properties are proved using a linearisation procedure in contrast to the classic recursive Lyapunov approach to backstepping. Vector, adaptive and robustness extensions are then developed to solve general tracking and stabilisation control problems. The adaptive extension is proved with a non-Lyapunov approach based on inequalities on non-negative fractional integrals. The mixed-order nature enables to control integer order systems using fractional operators. Through simulations, it is shown that the performance of the controllers depends on their derivation orders. In particular, for transient behaviour and the root mean square value of the control signal, the fractional controllers proposed exhibit improved responses as compared with the integer ones.

Robust and adaptive backstepping control for nonlinear systems using RBF neural networks.

In this paper, two different backstepping neural network (NN) control approaches are presented for a class of affine nonlinear systems in the strict-feedback form with unknown nonlinearities. By a special design scheme, the controller singularity problem is avoided perfectly in both approaches. Furthermore, the closed loop signals are guaranteed to be semiglobally uniformly ultimately bounded and the outputs of the system are proved to converge to a small neighborhood of the desired trajectory. The control performances of the closed-loop systems can be shaped as desired by suitably choosing the design parameters. Simulation results obtained demonstrate the effectiveness of the approaches proposed. The differences observed between the inputs of the two controllers are analyzed briefly.

Robust backstepping control of nonlinear systems using neural networks

A controller is proposed for the robust backstepping control of a class of general nonlinear systems using neural networks (NNs). A tuning scheme is proposed which can guarantee the boundedness of tracking error and weight updates. Compared with adaptive backstepping control schemes, we do not require the unknown parameters to be linear parametrizable. No regression matrices are needed, so no preliminary dynamical analysis is needed. One salient feature of our NN approach is that there is no need for the off-line learning phase. Three nonlinear systems, including a one-link robot, an induction motor, and a rigid-link flexible-joint robot, were used to demonstrate the effectiveness of the proposed scheme.

Adaptive dynamic surface control: A simplified algorithm for adaptive backstepping control of nonlinear systems

In this paper, we propose a new algorithm for adaptive backstepping control of non-linear uncertain systems. Current backstepping algorithms require repeated differentiations of the modelled non-linearities. The addition of n first order low pass filters allows the algorithms to be implemented without differentiating any model non-linearities. The uncertainties are assumed to be linear in the unknown constant parameters. The combined adaptive backstepping/first order filter system is proven to be semi-globally stable for sufficiently fast filters by a singular perturbation approach.