Semi-strict Feedback Form Research Articles

Nonlinear Differential Braking Control for Collision Avoidance During Lane Change

In this paper, a nonlinear differential braking control method is developed to avoid collision during lane change under driver torque. The lateral dynamics consist of lateral offset error and yaw error dynamics and can be interpreted as a semi-strict feedback form. In the differential braking control problem under the driver torque, a matching condition does not satisfy, and the system is not in the form of, the strict feedback form. Thus, a general backstepping control method cannot be applied. To overcome this problem, the proposed method is designed via the combination of the sliding mode control and backstepping. Two sliding surfaces are designed for differential braking control. One of the surfaces is designed considering the lateral offset error, and the other sliding surface is designed using the combination of the yaw and yaw rate errors as the virtual input of the lateral offset error dynamics. A brake steer force input is developed to regulate the two sliding surfaces using a backstepping procedure under the driver torque. Integral action and a super twisting algorithm are used in the lateral controller to ensure the robustness of the system. The proposed method, which is designed via the combination of the sliding mode control and backstepping, can improve the lateral control performance using differential braking. The proposed method is validated through simulations.

Active disturbance rejection adaptive output feedback control of uncertain nonlinear systems

AbstractThis article concerns the output feedback tracking control problem for a class of single‐input‐single‐output (SISO) nonlinear systems in a semistrict feedback form, which possesses both matched and mismatched uncertainties, including parametric uncertainties and unknown nonlinear disturbances. An active disturbance rejection adaptive output feedback control scheme is proposed via backstepping method, in which parametric uncertainties are handled by adaptation law and disturbances are estimated by a series of linear extended state observers (LESOs). The learning burden of LESOs is much alleviated due to active compensation of parametric uncertainties via adaptation. In addition, both matched and mismatched disturbances are estimated in the construction of LESO for each channel of the considered nonlinear plant. Hence, all various of uncertainties can be mostly compensated. Consequently, prescribed transient performance and steady‐state tracking accuracy can be guaranteed. Theoretical analysis reveals the tracking error can be made arbitrarily small. Furthermore, exponentially asymptotic output tracking can be achieved when unknown disturbances are constants. Finally, comparative simulation studies are presented to illustrate the effectiveness of the proposed scheme.

Robust Output Regulation Via Sliding Mode Control and Disturbance Observer: Application in a Forced Van Der Pol Chaotic Oscillator

This paper considers the problem of robust output regulation of nonlinear systems in semi strict-feedback form in the presence of model uncertainties and nonvanishing disturbances. In the proposed procedure, two exosystems are considered to generate the disturbance and reference signals. In order to reduce both the conservatism of the control law and the chattering phenomena, a disturbance observer is designed for disturbance estimation instead of assuming the known upper bound for the disturbance. Moreover, a novel sliding surface is designed based on the tracking error to guarantee that the output of the system tracks the output of the exosystem. In this regard, some theorems are given and according to the Lyapunov approach, it is proved that the robust output regulation is guaranteed in the presence of model uncertainties and external disturbances. Finally, in order to show the applicability of the proposed controller, it is applied to the Van der Pol chaotic oscillator. Computer simulations verify the theoretical results and also show the effective performance of the proposed controller.

Robust adaptive dynamic surface control of uncertain non‐linear systems with output constraints

This study presents a novel constraints transformation-based robust adaptive dynamic surface control (DSC) scheme for tracking control of a class of uncertain non-linear systems with output constraints. The considered systems are in semi-strict feedback form that contain not only parametric uncertainties, but also general non-linear function uncertainties as well as unknown input gains. In the proposed approach, a constraints transformation technique is firstly introduced to convert the original constrained system into an equivalent one without constraints. By ensuring the stability of the transformed system, it suffices to guarantee the constraints are not transgressed. To attain this, a decoupled DSC design methodology is then developed to stabilise the transformed system and a systematic design procedure is established. Through rigorous analysis, it is shown that the proposed control scheme can achieve output tracking objective and guarantee that all the closed-loop signals remain bounded while simultaneously preventing the prescribed output constraints from being violated. Simulation studies illustrate the efficiency and feasibility of the proposed approach.

Adaptive Dynamic Surface Control of Bouc–Wen Hysteretic Systems

This paper develops an adaptive dynamic surface algorithm for designing the control law for uncertain hysteretic structural systems with seismic disturbances that can be converted to a semi strict feedback form. Hysteretic behavior is usually described by Bouc–Wen model for hysteretic structural systems like base isolation systems. Adaptive sliding mode and adaptive backstepping algorithms are also studied and simulated for comparison purposes. The presented simulation results indicate the effectiveness of the proposed control law in reducing displacement, velocity and acceleration responses of the structural system with acceptable control force. Moreover, using dynamic surface control (DSC), the study analyzes the stability of the controlled system based on the Lyapunov theory.

Design of adaptive fuzzy backstepping sliding mode control for MIMO uncertain discrete-time nonlinear systems based on noisy measurements

This paper presents an adaptive fuzzy backstepping sliding mode control for multi-input and multi-output uncertain nonlinear systems in semi-strict feedback form. The systems are described by a discrete-time state equation with uncertainties viewed as the modeling errors and the unknown external disturbances, and the observation of the states is taken with independent measurement noises. Combining the adaptive fuzzy backstepping control with the sliding mode control approach for the comprehensive improvement in the stability and the robustness, the adaptive fuzzy backstepping sliding mode control is approximately designed where the design parameters are selected using an appropriate Lyapunov function. The uncertainities are approximated as fuzzy logic systems using the fuzzy inference approach based on the extended single input rule modules to reduce the number of the fuzzy IF-THEN rules. The estimates for the un-measurable states and the adjustable parameters are taken by the proposed simplified weighted least squares estimator. It is proved that the trajectory of the tracking error and the sliding surface is uniformly ultimately bounded. The effectiveness of the proposed approach is indicated through the simulation experiment of a simple numerical system.

Nonlinear Disturbance Observer-Based Dynamic Surface Control of Mobile Wheeled Inverted Pendulum

In this brief, a dynamic model of a mobile wheeled inverted pendulum (MWIP) system is improved considering friction forces, and a nonlinear disturbance observer (NDO)-based dynamic surface controller is investigated to control the MWIP system. Using a coordinate transformation, this non-Class-I type underactuated system is presented as a semistrict feedback form, which is convenient for dynamic surface controller design. A dynamic surface controller together with an NDO is designed to stabilize the underactuated plant. The proposed approach can compensate the external disturbances and the model uncertainties to improve the system performance significantly. The stability of the closed-loop MWIP system is proved by Lyapunov theorem. Experiment results are presented to illustrate the feasibility and efficiency of the proposed method.

Design of terminal block backstepping controllers for perturbed systems in semi-strict feedback form

Based on finite-time stability approach, a terminal block backstepping control with perturbation estimation scheme is proposed in this paper for a class of multi-input systems with matched and mismatched perturbations to solve regulation problems. The design procedure of the proposed control scheme mainly contains two parts. The first part is the design of pseudo-control inputs, the second part is the design of control input functions. In order to eliminate the problem of ‘explosion of complexity’ and overcome the problem of singularity, a derivative estimation algorithm embedded in the controller is utilised to estimate the perturbations. The control efforts are capable of suppressing the mismatched perturbations in most cases effectively so that the controlled states are able to reach zero within finite time. A numerical example is also given for demonstrating the feasibility of the proposed control scheme.

Output Tracking via Adaptive Backstepping Higher Order Integral Sliding Mode for Uncertain Nonlinear Systems

The authors propose a new tracking control design strategy for uncertain non-linear systems which are convertible to Semi-Strict Feedback Form (SSFF). The system in SSFF is first converted into new variables via existing adaptive backstepping control techniques. The control law is obtained by combining adaptive backstepping procedure and higher order integral sliding mode. The component of control law designed via backstepping is continuous which shows robustness against parametric uncertainties where as the discontinuous control component provides robustness against unmodeled dynamics and external disturbances. Since, this strategy relies on an integral manifold of the adaptively developed variables, therefore, the reaching phase is eliminated in this approach, which is an advantage in term of robustness. Furthermore, the parameters update law correctly provides the estimation of parameters which is again results in enhanced robustness of the strategy. The stability of proposed method is analysed theoretically and validated through a numerical example.

Dynamic Surface Control of Mobile Wheeled Inverted Pendulum Systems with Nonlinear Disturbance Observer

In this study, a dynamic surface controller based on a nonlinear disturbance observer is investigated to control mobile wheeled inverted pendulum system. By using a coordinate transformation, the underactuated system is presented as a semi-strict feedback form which is convenient for controller design. A dynamic surface controller together with a nonlinear disturbance observer is designed to stabilize the underactuated plant. The proposed dynamic surface controller with a nonlinear disturbance observer can compensate the external disturbances and the model uncertainties to improve the system performance significantly. The stability of the closed-loop mobile wheeled inverted pendulum system is proved by Lyapunov theorem. Simulation results show that the dynamic surface controller with a nonlinear disturbance observer can suppress the effects of external disturbances and model uncertainties effectively.

Distributed adaptive controller for the output-synchronization of networked systems in semi-strict feedback form

In this paper, we investigate the output synchronization of networked SISO nonlinear systems that can be transformed into semi-strict feedback form. Due to parameter uncertainty, the agents have heterogeneous dynamics. Combined backstepping method together with graph theory, we construct an augmented Laplacian potential function for analysis and a distributed controller is designed recursively for each agent such that its output can be synchronized to its neighbors' outputs. The distributed controller of each agent has three parts: state feedback of itself, neighborhood information transmitted through the network and adaptive parameter updaters both for itself and its neighbors. Moreover, distributed tuning function is designed to minimize the order of the parameter updater. It is proved that when the undirected graph is connected, all agents’ outputs in the network can be synchronized, i.e., cooperative output synchronization of the network is realized. Simulation results are presented to verify the effectiveness of the proposed controllers.

Adaptive Dynamic Surface Based Nonsingular Fast Terminal Sliding Mode Control for Semistrict Feedback System

This paper focuses on an adaptive dynamic surface based nonsingular fast terminal sliding mode control (ADS-NFTSMC) for a class of nth-order uncertain nonlinear systems in semistrict feedback form. A simple and effective controller has been obtained by introducing dynamic surface control (DSC) technique on the basis of second-order filters that the “explosion of terms” problem caused by backstepping method can be avoided. The nonsingular fast terminal sliding mode control is adopted in the last step of the controller design, and the error convergence rate is improved. An composite adaptive law is used to gain fast and accurate parameter estimation. Finally, simulation results are presented to illustrate the effectiveness of the proposed method.

Stabilising tracking of uncertain switched non-linear systems in semi-strict feedback form

This study concentrates on the stabilising tracking problem of a class of switched non-linear systems in semi-strict feedback form with parametric and non-linear uncertainties. The switching signal is arbitrary and occurs on the known non-linear functions. To solve the tracking problem, a switched adaptive robust dynamic surface control approach is proposed. The continuous error signals are constructed using the dynamic surface control approach, so the switched error dynamics is obtained. Then, semi-global stability analysis is discussed by using the common Lyapunov function method, which shows that the solutions of the closed-loop system are uniformly ultimate bounded and the tracking error can be prescribed arbitrarily small. A numerical example is provided to illustrate the effectiveness of the proposed control method.

Adaptive backstepping sliding mode control for feedforward uncertain systems

Output tracking backstepping sliding mode control for feedforward uncertain systems is considered in this article. Feedforward systems are not usually transformable to the parametric semi-strict feedback form, and they may include unmatched uncertainties consisting of disturbances and unmodelled dynamics terms. The backstepping method presented in this article, even without uncertainties differs from that of Ríos-Bolívar and Zinober [Ríos-Bolívar, M. and Zinober, A.S.I. (1999), ‘Dynamical Adaptive Sliding Mode Control of Observable Minimum Phase Uncertain Nonlinear Systems’, in Variable Structure Systems: Variable Structure Systems, Sliding Mode and Nonlinear Control, eds., K.D. Young and Ü. Özgüner. Ozguner, London, Springer-Verlag, pp. 211–236; Ríos-Bolívar, M., and Zinober, A.S.I. (1997a), ‘Dynamical Adaptive Backstepping Control Design via Symbolic Computation’, in Proceedings of the 3rd European Control Conference, Brussels]. In this article, the backstepping is not a dynamical method as in Ríos-Bolívar and Zinober (1997a, 1999), since at each step, the control and map input remain intact, and the differentiations of the control are not used. Therefore, the method can be introduced as static backstepping. Two different controllers are designed based upon the backstepping approach with and without sliding mode. The dynamic and static backstepping methods are applied to a gravity-flow/pipeline system to compare two methods.

Output Feedback Adaptive Robust Learning Control of a Class of Nonlinear Systems with Periodic Disturbances

AbstractIn this paper, a discontinuous projection-based output feedback adaptive robust learning control (OARLC) scheme is constructed for a class of nonlinear systems in a semi-strict feedback form by incorporating an observer and a dynamic normalization signal. Since only output signal is available for measurement, an observer is firstly designed to provide exponentially convergent estimates of the unmeasurable states. Using certain known basis functions to capture the characteristics of unknown general periodic disturbances, the discontinuous projection type adaptation law can then be used to tune the amplitudes of those basis functions on-line to recover the unknown general periodic disturbances asymptotically. The estimation errors due to the unknown initial states, uncompensated disturbances, and the uncertain nonlinearities are also effectively dealt with via certain robust feedback at each step of the proposed OARLC backstepping design. The resulting controller achieves a guaranteed transient and a prescribed final tracking accuracy for output tracking performance. In addition, when the general periodic disturbances fall within the approximation ranges of the periodic basis functions, asymptotic output tracking performance is achieved as well.

Adaptive robust control of a class of nonlinear systems in semi‐strict feedback form with non‐uniformly detectable unmeasured internal states

AbstractThis paper proposes a novel control method for a special class of nonlinear systems in semi‐strict feedback form. The main characteristic of this class of systems is that the unmeasured internal states are non‐uniformly detectable, which means that no observer for these states can be designed to make the observation error exponentially converge to zero. In view of this, a projection‐based adaptive robust control law is developed in this paper for this kind of system. This method uses a projection‐type adaptation algorithm for the estimation of both the unknown parameters and the internal states. Robust feedback term is synthesized to make the system robust to uncertain nonlinearities and disturbances. Although the estimation error for both the unknown parameters and the internal states may not converge to zero, the tracking error of the closed‐loop system is proved to converge to zero asymptotically if the system has only parametric uncertainties. Furthermore, it is theoretically proved that all the signals are bounded, and the control algorithm is robust to bounded disturbances and uncertain nonlinearities with guaranteed output tracking transient performance and steady‐state accuracy in general. The class of system considered here has wide engineering applications, and a practical example—control of mechanical systems with dynamic friction—is used as a case study. Simulation results are obtained to demonstrate the applicability of the proposed control methodology. Copyright © 2010 John Wiley & Sons, Ltd.

Dynamic surface control approach to adaptive robust control of nonlinear systems in semi-strict feedback form

This article considers the adaptive robust control of a class of single-input-single-output nonlinear systems in semi-strict feedback form using radial basis function (RBF) networks. It is well known that the standard backstepping design may suffer from “explosion of terms”. To overcome this problem, the recently developed dynamic surface control technique which employs a first-order low-pass filter at each step of the backstepping design procedure is generalized to the nonlinear system under study. Our attention is paid to achieve guaranteed transient performance of the adaptive controller. At each step of design, a feedback controller strengthened by nonlinear damping terms to counteract nonlinear uncertainties is designed to guarantee input-to-state practical stability of the corresponding subsystem, and then parameter adaptations are introduced to reduce the ultimate error bound. Furthermore, for the output trajectory tracking problem, it is recommended to adopt the partial adaptation policy to reduce the computational burden due to “curse of dimension” of the RBF networks. Finally, numerical examples are included to verify the results of theoretical analysis.

DYNAMIC SURFACE CONTROL APPROACH TO ADAPTIVE ROBUST CONTROL OF NONLINEAR SYSTEMS IN SEMI-STRICT FEEDBACK FORM

This paper consider the adaptive robust control of a class of single-input-single-output (SISO) nonlinear systems in semi-strict feedback form, where nonlinearities can exist in the input channel of each subsystem. To overcome the problem of “explosion of terms”, the recently developed dynamic surface control technique is generalized to the nonlinear systems under study. At each step of design, a feedback controller strengthened by nonlinear damping terms to counteract modelling errors is designed to guarantee input-to-state practical stability of the corresponding subsystem, and then parameter adaptions are introduced to reduce the ultimate error bound. Finally, simulational examples are included to verify the results of theoretical analysis.

ADAPTIVE SLIDING MODE BACKSTEPPING CONTROL OF NONLINEAR SYSTEMS WITH UNMATCHED UNCERTAINTY

ABSTRACTThis paper considers an adaptive backstepping algorithm for designing the control for a class of nonlinear continuous uncertain processes with disturbances that can be converted to a parametric semi‐strict feedback form. Sliding mode control using a combined adaptive backstepping sliding mode control (SMC) algorithm, is also studied. The algorithm follows a systematic procedure for the design of adaptive control laws for the output tracking of nonlinear systems with matched and unmatched uncertainty.

Neural network adaptive robust control of nonlinear systems in semi-strict feedback form

In this paper, the recently proposed neural network adaptive robust control (NNARC) design is generalized to synthesize performance oriented control laws for a class of nonlinear systems transformable to the semi-strict feedback forms through the incorporation of backstepping design techniques. Both repeatable (or state dependent) unknown nonlinearities and nonrepeatable unknown nonlinearities such as external disturbances are considered. In addition, unknown nonlinearities can exist in the virtual control input channel of each intermediate step as well. All unknown but repeatable nonlinear functions are approximated by outputs of multi-layer neural networks to achieve a better model compensation for an improved performance. All NN weights are tuned on-line with no prior training needed. In order to avoid the possible divergence of the on-line tuning of neural networks, discontinuous projections with fictitious bounds are used in the NN weight adjusting law to make sure that all NN weights are tuned within a prescribed range. By doing so, even in the presence of approximation error and nonrepeatable nonlinearities such as disturbances, a controlled learning is achieved and the possible destabilizing effect of on-line tuning of NN weights could be avoided. Certain robust control terms can then be constructed to attenuate various model uncertainties effectively for a guaranteed output tracking transient performance and a guaranteed final tracking accuracy in general. The precision motion control of a linear motor drive system with slow electrical dynamics is used as a case study to illustrate the proposed NNARC methodology. Experimental results with a simulated slow electrical dynamics are presented.