Backstepping Approach Research Articles

Neural-Based Decentralized Adaptive Finite-Time Control for Nonlinear Large-Scale Systems With Time-Varying Output Constraints

This paper addresses the adaptive finite-time decentralized control problem for time-varying output-constrained nonlinear large-scale systems preceded by input saturation. The intermediate control functions designed are approximated by neural networks. Time-varying barrier Lyapunov functions are used to ensure that the system output constraints are never breached. An adaptive finite-time decentralized control scheme is devised by combining the backstepping approach with Lyapunov function theory. Under the action of the proposed approach, the system stability and desired control performance can be obtained in finite time. The feasibility of this control strategy is demonstrated by using simulation results.

Folding Bilateral Backstepping Output-Feedback Control Design for an Unstable Parabolic PDE

We present a novel methodology for designing output-feedback backstepping bilateral boundary controllers for an unstable 1-D diffusion-reaction partial differential equation with spatially-varying reaction. Using folding transforms the parabolic PDE into a 2X2 coupled PDE system with coupling through compatibility conditions. We apply a two-tiered backstepping approach, where the invertibility of the transformations guarantees the state-feedback controllers exponentially stabilize the trivial solution of the PDE system. A state observer is also designed for two collocated measurements at an arbitrary interior point, generating exponentially stable state estimates. The output feedback control law is formulated by composing the independently designed state-feedback controller with the observer, and the resulting dynamic feedback is shown to stabilize the trivial solution. Some numerical analysis on how the selection of these points affect the responses of the controller and observer are discussed, with simulations illustrating various choices of folding points and their effect on the stabilization in different performance indexes.

Neuroadaptive control of saturated nonlinear systems with disturbance compensation

Extended state observer acting as a popular tool can estimate the system states and total disturbances simultaneously. However, for extended-state-observer-based control of high-order nonlinear systems, there are still some difficult issues to solve, such as how to simultaneously reject matched and mismatched model uncertainties with strict theoretical proof, especially in the case of output feedback, “explosion of complexity” and so on. Motivated by these reasons, different control schemes in full-state feedback and output feedback conditions respectively will be integrated via the filter-based backstepping approach for saturated nonlinear systems. For the full-state feedback condition, adaptive neural network and extended state observer will be combined for each dynamic to handle the unknown nonlinear dynamics and external disturbances, respectively. For the output feedback condition, nonlinear disturbance observer design will be incorporated into the neural-network-based extended state observer scheme to handle mismatched disturbances at the same time. In particular, an auxiliary system will be constructed to compensate for the saturation influence. Moreover, the anticipate control effects of the developed controllers have been demonstrated by contrastive results for a hydraulic servo system.

Adaptive stabilization for a reaction–diffusion equation with uncertain nonlinear actuator dynamics

This paper is devoted to the stabilization of a reaction–diffusion equation with uncertain nonlinear actuator dynamics described by a cascaded parabolic PDE–ODE system. Essentially different from the systems in the related literature, control input appears in the ODE subsystem rather than in the PDE one, and moreover, the actuator dynamics are nonlinear and possess uncertainties rather than linear without uncertainty. Note that the existing control schemes so far are ineffective to handle such systems, and hence the stabilization problem for these systems is still unsolved. For this, backstepping approaches in both infinite and finite-dimension are smartly combined with the adaptive dynamic compensation technique in order to solve the control problem. Specifically, by using an infinite-dimensional backstepping transformation and its inverse, the original PDE subsystem is first changed to a new one from which control design is much convenient. Then, based on the new system together with the actuator dynamics, an adaptive state-feedback controller is explicitly constructed by using a couple of finite-dimensional backstepping transformations, which guarantees that all the states of the resulting closed-loop systems are bounded. Moreover, both the control input and the states of the original system converge to zero ultimately. Finally, a simulation example is provided to validate the effectiveness of the established theoretical results.

Adaptive Control of Wind Turbine Generators for Power Capture Optimization by Using Integral Backstepping Approach During Partial-Load Operation

The paper proposes an effective nonlinear integral backstepping approach strategy (IBS) through the use of a new adaptive power control algorithm for variable-speed wind turbine generators (VS-WTGs) to optimize the power extracted from the WTGs. The suggested approach is applied to one of the most frequently used maximum power control methods called as tip speed ratio (TSR) during partial-load operation considering the effects of the variations in wind velocity profile and unmodeled system dynamics such as external disturbances. The proposed method of control is known to scale-back the mechanical stress at the level of the GT shafts (generator and turbine). Furthermore, the IBS approach is relatively simple, which significantly reduces the online computational time and cost. Some numerical simulation results prove, under different wind speed models, that the proposed AC-TSR-IBS method works well on the level of the improved efficiency and the rapid system response compared to conventional TSR-PI method.

Output feedback compensation to state and measurement delays for a first-order hyperbolic PIDE with recycle

This paper develops a novel observer-based output-feedback control law for a class of interconnected first-order hyperbolic partial integral differential equations (PIDEs) with in-domain, non-local coupling terms and measurement delay compensation. A transport PDE is introduced to account for in-domain recycle delay which results in an extended spatial domain with the PIDE and transport PDE being cascaded. Since standard backstepping approach cannot be applied, an affine Volterra integral transform is introduced which allows for the construction of a boundary controller for the PIDE and delay compensation. An observer backstepping transformation containing four kernels is realized, reconstructing the states throughout the delayed measurement. By the use of the backstepping method, a set of coupled kernel equations where one of the kernels has two boundary conditions is established. The existence and invertibility for the control and observer transformations are proved and the finite-time stability for the output system is established while results are illustrated by numerical simulations.

Event-triggered adaptive output-feedback neural-networks control for saturated strict-feedback nonlinear systems in the presence of external disturbance

An event-triggered (ET) neural-networks (NNs) adaptive output-feedback control approach is proposed for a class of input-saturated strict-feedback nonlinear systems with external disturbance. Compared with overall existing event-triggered control strategies, which are free from control input nonlinearities and suffer from the explosion of complexity, the proposed event-triggered-based NNs controller is able to handle system input saturation along with completely avoiding the complexity explosion problem. First, by introducing alternative state variables, and by implementing a low-pass filter, the difficulty arising from the cascading of the input-saturated strict-feedback system has been avoided. Thus, the system is converted to the normal canonical system, for which the controller synthesis is much simpler without resort to traditional back-stepping approach. Then, an observer is adopted to estimate the unknown states of the newly derived canonical system based on strictly positive real theory. In the design procedure, the unknown nonlinear functions are approximated by NNs to design a baseline controller, for which an additional robust term is embedded to deal with the input saturation nonlinearity, unknown disturbances and approximation errors using only two adaptive parameters. The proposed ET adaptive NNs control scheme is shown to guarantee the convergence of the output of the system to a small neighborhood of the origin along with the boundedness of all signals in the closed loop. Finally, simulation examples are presented to show the effectiveness of the proposed controller.

Decentralised state-feedback prescribed performance control for a class of interconnected nonlinear full-state time-delay systems with strong interconnection

This paper investigates the problem of the state-feedback control design for a class of interconnected nonlinear full-state time-delay systems with strong interconnection and prescribed performance. Based on the dynamic gain approach and Lyapunov-Krasovskii method, it overcomes the rigorous restriction that the interconnected time-delay functions should satisfy linear growth condition. Compared with the existing results, the main contribution is that the dynamic gain approach is extended to state-feedback control for interconnected time-delay systems, and it is strictly proved that the closed-loop system is globally bounded stable with the proposed decentralised control strategy. The problem of the strong interconnection terms is solved by combining the backstepping approach and graph theory. Furthermore, the expected transient and steady-state performance of the output can be guaranteed by considering prescribed performance control into such strongly interconnected systems. Finally, simulation is performed and the results show the effectiveness of the proposed method.

Disturbance compensation based asymptotic tracking control for nonlinear systems with mismatched modeling uncertainties

AbstractIn order to acquire high‐performance tracking control for a class of nonlinear systems with matched and mismatched modeling uncertainties, a disturbance‐compensation‐based nonlinear robust controller with a continuous control law is integrated via a novel asymptotic‐filter‐based backstepping approach. Specially, a set of novel asymptotic disturbance observers with exact estimation performance is constructed to estimate matched and mismatched total disturbances respectively in real time. Moreover, the integrated control algorithm just needs the state measurements only and does not require any internal dynamics and other physical parameter information generally. Furthermore, only the first derivative of the command to be tracked is demanded. It is noted that asymptotic output tracking performance under the closed‐loop operation can be achieved. Finally, the whole closed‐loop stability is proven strictly via the Lyapunov theoretical analysis and the feasibility of the integrated controller is demonstrated by contrastive application results.

Neuro‐adaptive distributed output‐feedback containment control for multiagent systems with nonstrict‐feedback nonlinear dynamics and input constraints

AbstractIn this article, distributed adaptive controllers are designed for containment control of multiagent systems (MASs) with high‐order nonlinear dynamics in nonstrict‐feedback form. To broaden the application horizon, the agents' dynamics are considered heterogeneous, and the control saturation is taken into account. Moreover, the agents' dynamics, as well as their full state information, are assumed unknown. An adaptive observer is designed in the first step for each agent to estimate the agents' states using available output measurements. An adaptive backstepping controller is then integrated with the observer to solve the containment control problem. The computational complexity of the controller is reduced by adopting the command filtered adaptive backstepping approach to the containment control of MASs. The effect of the input saturation is also compensated by employing an auxiliary system throughout the design procedure. The proposed control approach guarantees that the states of all agents are cooperatively semiglobally uniformly ultimately bounded and containment errors converge to a small neighborhood of the origin. Finally, the effectiveness of the presented control algorithm is confirmed by a simulation example.

Modelling, Simulation and Dynamic Sliding Mode Control of a MEMS Gyroscope.

An adaptive dynamic sliding mode control via a backstepping approach for a microelectro mechanical system (MEMS) vibratory z-axis gyroscope is presented in this paper. The time derivative of the control input of the dynamic sliding mode controller (DSMC) is treated as a new control variable for the augmented system which is composed of the original system and the integrator. This DSMC can transfer discontinuous terms to the first-order derivative of the control input, and effectively reduce the chattering. An adaptive dynamic sliding mode controller with the method of backstepping is derived to real-time estimate the angular velocity and the damping and stiffness coefficients and asymptotical stability of the designed systems can be guaranteed. Simulation examples are investigated to demonstrate the satisfactory performance of the proposed adaptive backstepping sliding mode control.

Robust adaptive command-filtered backstepping synchronization navigation control with prescribed performance for supply ship during underway replenishment

In the presence of unknown environmental disturbances, this paper develops a robust adaptive synchronization navigation control design with prescribed performance for the supply ship during underway replenishment using the adaptive technique, the prescribed performance control (PPC) technique and the command-filtered backstepping approach. The adaptive law is derived to estimate the bounds of unknown environmental disturbances, the PPC technique is introduced to guarantee the prescribed transient and steady-state performance of synchronization control and the command filter is employed to avoid the direct analytic derivation of the virtual stabilizing function in the conventional backstepping design. The theoretical analysis proves that the designed control law can realize the synchronization navigation between the supply and the main ships with prescribed performance, while guaranteeing the uniform ultimate boundedness of all signals in the synchronization closed-loop control system of the supply ship. Simulation results on a supply ship demonstrate the effectiveness of the designed synchronization navigation control law and the robustness to unknown environmental disturbances.

Robust circumnavigation of a heterogeneous multi-agent system

This paper focuses on the design of robust control laws for a heterogeneous multi-agent system composed of omnidirectional and differential-drive mobile robots under the leader–follower scheme and considering the distance and orientation measurements. It is assume that the agent leader is an omnidirectional mobile robot moving freely in the plane while the rest of the agents are the followers. The control laws are designed by means of the Backstepping approach. It is proved that, although the control laws do not need information about the velocity of the leader, the followers will circumnavigate the leader. Numerical simulations and real-time experiments exhibit the performance of the proposed control strategy.

Cascaded H-Bridge Inverters for UPS Applications: Adaptive Backstepping Control and Formal Stability Analysis

In this letter, an advanced adaptive controller is designed for N-cascaded H-bridge multilevel inverters, each fed by a DC voltage source. The studied system provides an added-value to Uninterruptable Power Supply applications, and is able to generate a precise sinusoidal voltage using a low dimensional LC filter. The main objectives of this letter are: i. generating a low THD sinusoidal voltage, ii. developing adaptive control laws able to regulate the system even with the presence of parameters' variations, and iii. proving the stability of the designed system using some tools from Lyapunov stability, Lasalle's invariance principle and persistency of excitation. This latter proves that the closed-loop control system is uniformly exponentially stable. To achieve these objectives, the adaptive backstepping approach is used. The performance of the system is checked with the use of MATLAB/SIMULINK/ SimPowerSystems environment, where the synthesized regulator meets its objectives.

Cooperative output regulation for a network of parabolic systems with varying parameters

This paper is concerned with the cooperative output regulation problem for a network of agents with different dynamics described by parabolic PDEs subject to spatially- and time-varying parameters. Firstly, a networked controller is designed achieving output synchronization for identical finite-dimensional reference models, which deliver the state of the global reference model required for the synchronization to the parabolic agents. The latter can be subject to local disturbances acting in-domain, on all boundaries and on the anti-collocated output to be controlled. The cooperative output regulation problem is solved by designing local output feedback regulators for the parabolic agents. This requires the solution of time-varying regulator equations and the design of disturbance observers for parabolic systems with spatially- and time-varying coefficients. For this, a systematic backstepping approach is provided and it is shown that cooperative output regulation with exponential convergence is ensured for the resulting multi-agent system. The results of the paper are applied to the cooperative output regulation of a heterogeneous network of four parabolic agents in the presence of local disturbances.

Full-State Constrained Adaptive Fuzzy Finite-Time Dynamic Surface Control for PMSM Drive Systems

This paper proposes an adaptive fuzzy finite-time dynamic surface control method for permanent magnet synchronous motors (PMSMs) with full-state constraints. By combining with the finite-time stability theory, the $$\log $$ -type barrier Lyapunov function is constructed to constrain the angular speed, stator current and other state variables of PMSMs in the predefined compact sets. Then, the first-order filter is employed to solve the shortcoming of “complexity of differentiation” in backstepping approach and the fuzzy logic systems are used to approximate the nonlinear functions. Finally, it is demonstrated that the presented control approach realizes fast tracking effect and the boundedness of all signals. Comparison of simulation results is included to illustrate the superiority of the new design scheme.

3D Space Trajectory Tracking of Underactuated AUVs using Back-Stepping Control and Time Delay Estimation

Abstract In this paper, a robust controller using the back-stepping approach and the time delay estimation(TDE) is proposed for the 3D space motion control of torpedo-like autonomous underwater vehicles(AUVs). Due to the lack of control inputs, the vehicle dynamics does not satisfy the matching condition and has nonlinear dynamics including rigid dynamics and hydrodynamics. By utilizing a relevant coordinate transform, it is shown that the tracking error dynamics of the vehicles can be represented as a strict-feedback form. Using the error dynamics, the back-stepping controller is designed to gain robustness against the unmatched dynamics. The nonlinear dynamics of the vehicle is compensated by using the TDE which estimates the dynamics robustly utilizing the system inputs and outputs. By combining the back-stepping scheme and the TDE, the proposed controller has robust tracking performance against model uncertainties and disturbances. Owing simplicity of the TDE, moreover, the proposed controller can be implemented without necessity of whole dynamic model of the vehicles. The simulation results using the REMUS model demonstrates the robustness of the proposed controller against the model uncertainties and disturbances.

Finite-Time Fault-Tolerant Control for a Robotic Manipulator With Output Constraint and Uncertainties

This paper proposed a finite-time backstepping control for a robotic manipulator under the presence of actuator fault, saturation constraints, output constraints, and external disturbance to obtain requirements about the robustness, fast convergence, and high accuracy tracking performance. To manage the above challenges, the proposed control is designed on a transformed model with the backstepping approach and extended state observer. The transformed model is resulted from converting a constrained system based on a transformation technique. So, it provides an ability for the proposed control to obtain the prescribed performance of the output response. Additionally, an extended state observer is conducted to deal with the lumped uncertainties in the system. The essential characteristic of the proposed control is no required knowledge of the actuator faults and external disturbance to be available. Furthermore, fractional-order terms are added in the control laws to enhance the rate of output responses. To demonstrate the advantages of the proposed control in terms of global asymptotic stability, the Lyapunov approach is used to verify the whole controlled system in theory. The proposed control is applied to a 2-degree of freedom (DOF) manipulator and simulated by MATLAB Simulink. Its simulation results are compared to other state-of-the-art methods to exhibit the effectiveness of the proposed control.

Command Filtering and Barrier Lyapunov Function‐Based Adaptive Control for PMSMs with Core Losses and All‐State Restrictions

With the troubles of core losses and all‐state confined to certain limitations which are the innate traits of permanent magnet synchronous motors (PMSMs), this article develops a command filtered adaptive backstepping approach to follow the track of PMSM’s desired rotor position. To begin with, the RBF neural network technique is utilized to get close to the uncharted nonlinear terms which existed in PMSM’s mathematical model. Meanwhile, an advanced adaptive command filter control methodology is constructed to avoid the computing explosion during the process of backstepping design. Furthermore, to make sure that all the state variables are confined into certain ranges, we employed the barrier Lyapunov function (BLF) at every step of the controllers construction. In addition, an error compensating mechanism is proposed to neutralize filtering errors and only one adaptive law is required. At last, simulation results bear out the superiority of the aforementioned control scheme.

A 3DOF Path-Following Controller for a Non-Directionally Stable Vessel with Slow Thruster Dynamics

Abstract In this paper, a 3DOF path-following controller for an electric double-ended passenger ferry prototype is presented. The controller is formulated through a 3-step backstepping approach, taking into consideration several challenging physical properties of the vessel, such as a lack of passive damping in the vessel hull, lack of directional stability, and slow thruster dynamics compared to the vessel dynamics. The controller design also features a new thrust allocation approach that allows the thrust allocation of the over-actuated thruster system to be formulated on closed form, which enables us to include the thruster dynamics in the control law. The performance of the suggested 3DOF controller is demonstrated and compared to two other controllers through simulations with a model of the electric passenger ferry.