Backstepping Control Design Research Articles

Fixed-time prescribed performance control for nonlinear multi-agent systems with novel uncertainties

This paper addresses an approach to fixed-time prescribed performance control design for multi-agent systems with novel uncertainties, i.e., the uncertainties from system input powers, and a new neural backstepping control design is proposed such that the prescribed performance can be guaranteed within a fixed-time. Unlike existing adaptive results, the upper boundary of stability time is decided only by the design parameters and also is independent of the initial states of the system. By leveraging Lyapunov stability theory and graph theory, it is demonstrated that every follower agent can synchronize to the leader agent, and tracking errors converge to the origin’s neighborhood. Finally, the simulation results verify the effectiveness of the control strategy.

Fuzzy adaptive asynchronous sampled-data consensus control for nonlinear multi-agent systems under DoS attacks

This article investigates the problem of fuzzy adaptive resilient asynchronous sampled-data consensus control (SDCC) for a class of uncertain nonlinear strict-feedback multi-agent systems (MASs) under denial-of-service (DoS) attacks and with a directed graph. Firstly, the upper bounds on the sampling intervals and a distributed sampled-data observer are proposed to estimate the leader's output and its high-order derivatives under DoS attacks. Then, the sufficient conditions to ensure the asymptotic convergence of the designed distributed sampled-data observer are established. Based on the distributed sampled-data observer and backstepping control design technique, a distributed fuzzy adaptive resilient control algorithm is developed. It is proved that the proposed control scheme can guarantee that the controlled MASs are stable, and the consensus errors converge to a small neighbourhood of zero. Finally, a simulation example on heterogeneous vehicular platoons verifies the feasibility of the proposed control scheme.

Adaptive Predefined Time Control for Strict-Feedback Systems with Actuator Quantization

An adaptive predefined-time quantized control issue is considered for strict-feedback systems with actuator quantization. To handle the unknown nonlinearities of a system, the neural networks are first applied to model them. To analyze the predefined-time stability under approximation error, a stability lemma is first introduced. Then, a refreshing predefined-time quantized control strategy is presented. Compared with the existing control studies for actuator quantization, the stability time is not influenced by the initial state and can be set in advance. Furthermore, unlike the available predefined-time control studies, a new parameter adaptive law and virtual controllers are designed. This design not only ensures the predefined-time stability, but overcomes the singularities of system in coventional backstepping control design because of repeating differentiation for virtual controllers.

Deep neural networks-prescribed performance optimal control for stochastic nonlinear strict-feedback systems

This article explores the application of deep neural networks (DNNs) for optimized backstepping control design in a category of stochastic nonlinear strict-feedback systems with unknown dynamics, focusing on prescribed performance. DNNs, distinguished by their ability to handle high-complexity functions and enhance function approximation, are employed to estimate unknown nonlinear functions. The weight updating strategy for each layer of DNNs is determined through a Lyapunov function analysis. Tomitigate potential issues such as the tracking error exploding at uncertain moments, prescribed performance control (PPC) is introduced to constrain the tracking error within a predetermined range. Subsequently, a novel optimal tracking control scheme, integrating the backstepping method and Hamilton–Jacobi–Bellman (HJB) equation, is presented. The results indicate that all signals in the closed-loop system are bounded in probability, and the tracking error is maintained within a set of predefined arbitrarily small residuals. Simulation outcomes affirm the efficacy of the proposed control scheme.

Adaptive finite time command filtered backstepping control scheme of uncertain strict-feedback nonlinear systems

This paper proposes an adaptive finite time command filtered backstepping control scheme for single‐input and single‐output (SISO) uncertain strict-feedback nonlinear systems. The problem of the explosion of complexity existing in the adaptive backstepping control design and the singularity problem are solved by using the proposed adaptive finite time control method. In addition, the compensating signals are introduced to deal with the effect of the known filtering errors caused by the command filters. By using the finite time Lyapunov stability theory, the proposed adaptive finite time control strategy guarantees that all the signals in the closed-loop system are practical finite time stable, and the tracking errors converge to an arbitrarily small neighbourhood of the origin in finite time. Simulation and experimental results of the DC-DC buck converter are given to illustrate the effectiveness of the proposed adaptive finite time control scheme.

Design of adaptive barrier function-based backstepping finite time guidance control for interceptor-target systems

This study introduces an optimal finite-time backstepping sliding mode guidance law, leveraging an adaptive continuous barrier function to examine the interceptor-target system’s motion in cylindrical coordinates. The integration of sliding mode and backstepping controllers enhances the robustness and performance of the proposed guidance method amidst external disturbances and model uncertainties. This combination also improves transient response, reduces chattering, and lowers control efforts. The adaptive continuous barrier function employed in this method eliminates the need for prior knowledge of the upper bounds of uncertainties and disturbances. Additionally, it ensures that adaptation gains are not overestimated, leading to more accurate performance and complete elimination of chattering. The Lyapunov stability theory is used to prove the finite-time convergence of the state trajectories of the interceptor-target system to a predefined neighborhood around the origin, despite external disturbances and uncertainties. A genetic optimization algorithm is utilized to optimally select the parameters of the guidance method. The efficacy of the proposed technique is validated through simulation results and real-time experiments on the Baseline Speedgoat Real-Time Target Machine platform. The method’s effectiveness and performance are further demonstrated through analysis and simulation in the MATLAB/Simulink environment.

Dynamics and adaptive backstepping control of rotor-hybrid foil-magnetic bearings system

The hybrid foil-magnetic bearing (HFMB) is a contactless-type bearing composed of a foil bearing (FB) in the inner side and an active magnetic bearing (AMB) in the outer side. It has the advantages of low drag torque, high DN value (bearing inner diameter times rotating speed), and high supporting efficiency. However, significant subsynchronous vibration will degrade the stability of the rotor-HFMBs system, which is an obstacle to the industrial implementation of HFMBs. To control the rotor vibration, this paper presents a dynamic model of the rotor-HFMBs system with unknown parameters and the design of an adaptive backstepping controller (ABC) for it. The stability of the closed-loop system is established. Simulation results of the rotor-HFMBs system controlled by ABC are compared with the proportional-integral-derivative controller (PID), illustrating that the ABC is capable of eliminating rotor vibration, and it is more effective than the PID.

Adaptive neural network dynamic surface optimal saturation control for single‐phase grid‐connected photovoltaic systems

AbstractAn adaptive neural network (NN) based optimal saturation control scheme is investigated for single‐phase grid‐connected photovoltaic (PV) systems by incorporating dynamic surface control (DSC) and adaptive dynamic programming (ADP) based on the backstepping control design framework. For each backstepping step, a critic‐actor architecture is constructed via reinforcement learning (RL), and the PV system is optimized according to the cost function in the architecture. Due to the nonlinearity, it is difficult to solve the Hamilton–Jacobi–Bellman (HJB) equation. The neural networks (NNs) are employed to approximate the solution of the HJB equation such that the optimal virtual control and the actual controller are obtained. By considering control input symmetric saturation nonlinearity link, constraints on pulse width modulation (PWM) are ensured. On this basis, the combination of backstepping control design and dynamic surface technique is used to overcome the shortcomings of “differential explosion” and simplify calculations. Based on the Lyapunov method, the stability analysis proves that all signals of the closed‐loop PV systems are semiglobally uniformly ultimately bounded (SGUUB). Simulation experiments and comparative results are given to verify the efficacy of the studied control strategy.

Adaptive neural optimal tracking control for uncertain unmanned surface vehicle

This article proposes a novel adaptive neural optimal control scheme for the uncertain unmanned surface vehicle (USV) system. The proposed adaptive neural optimal control scheme is composed of a neural feedforward controller and a neural optimal feedback controller. The neural feedforward controller is designed by using neural networks (NNs) and the backstepping control design technique to make the unknown nonlinear dynamics of USV system to be stabilized. The neural optimal feedback controller is designed based on adaptive dynamic programming (ADP) theory. It is proved that the formulated adaptive neural optimal control scheme can achieve all signals in USV system are uniformly ultimately bounded (UUB) and the cost function is minimized. The simulation and comparison results demonstrate the effectiveness of the proposed optimal control scheme.

A Gradient Dynamics-Based Singularity Avoidance Method for Backstepping Control of Underactuated TORA Systems.

In this paper, a gradient dynamics-based control method is proposed to directly tackle the singularity problem in the backstepping control design of the TORA system. This method is founded upon the construction of an energy-like positive function, which includes an auxiliary variable in terms of the intermediate virtual control law. On this basis, a gradient dynamics is created to obtain a new virtual control command, which is capable of making the auxiliary variable gradually approach zero, thereby mitigating the issue of division by zero. The core innovation is the integration of the gradient dynamics into the recursive backstepping design to overcome the singularity problem and stabilize the system at the equilibrium quickly. In addition, it rigorously proves that all the signals in the closed-loop control system are uniformly ultimately bounded, and the tracking errors converge to a small neighborhood around zero through a Lyapunov-based stability analysis. Comparative simulations demonstrate that the proposed approach not only avoids the singularity issue, but also achieves a better transient performance over other methods.

Barrier Lyapunov Function-based Backstepping Controller Design for Path Tracking of Autonomous Vehicles

This research proposes a novel BLF-based backstepping controller for path tracking of Autonomous Vehicles (AVs) with unknown dynamics and unmeasurable states. The proposed framework includes: (1) forming geometric-dynamic model of the vehicle by combining the dynamics of the vehicle with the kinematics of the visual measurement system, (2) designing a fixed-time Extended-State Observer (ESO) to estimate the unknown dynamics and unmeasurable states, and (3) introducing a BLF-based controller for faster response and more accurate path tracking compared to previous BLF-based controllers. Besides the novelty of the BLF-based controller, by transforming the closed-loop error dynamics into a unified proportional-derivative (PD)-type structure, an intuitive criterion is proposed to provide a systematic procedure for comparing BLF-based controllers. A combined BLF is further proposed based on this performance criterion to eliminate the sensitivity of BLF-based controllers to the magnitude of the constraint. The stability analysis is performed for the fixed-time ESO and the closed-loop control system. MATLAB/CarSim co-simulation is conducted to evaluate the performance of the proposed control system. The outcomes of the work show that the closed-loop control system is exponentially stable. In addition, it can provide a faster response and result in more accurate path tracking compared to previous BLF-based control systems.

Comparative Study of Methods for Robot Control with Flexible Joints

Robots with flexible joints are gaining importance in areas such as collaborative robots (cobots), exoskeletons, and prostheses. They are meant to directly interact with humans, and the emphasis in their construction is not on precision but rather on weight reduction and soft interaction with humans. Well-known rigid robot control strategies are not valid in this area, so new control methods have been proposed to deal with the complexity introduced by elasticity. Some of these methods are seldom used and are unknown to most of the academic community. After selecting the methods, we carried out a comprehensive comparative study of algorithms: simple gravity compensation (Sgc), the singular perturbation method (Spm), the passivity-based approach (Pba), backstepping control design (Bcd), and exact gravity cancellation (Egc). We modeled these algorithms using MATLAB and simulated them for different stiffness levels. Furthermore, their practical implementation was analyzed from the perspective of the magnitudes to be measured and the computational costs of their implementation. In conclusion, the Sgc method is a fast and affordable solution if joint stiffness is relatively high. If good performance is necessary, the Pba is the best option.

Design of Adaptive Finite-Time Backstepping Control for Shield Tunneling Systems with Constraints

This paper focuses on the finite-time tracking control problem of shield tunneling systems in the presence of constraints on the states and control input. By modeling the system based on the LuGre friction model, an effective method of tracking control in finite time is designed to overcome these actual constraints at the same time. First, the constraint on the system state is transformed into a symmetric constraint on the tracking error, and the constraint on control input is handled by designing an auxiliary differential equation. Then, radial basis function (RBF) neural networks are introduced to approximate the uncertainties. Next, using an adaptive finite-time backstepping method and choosing a logarithmic barrier Lyapunov function (BLF), a finite-time controller is designed to realize the finite-time stability of the closed-loop system. Finally, a simulation example is given to verify the correctness and validity of the theoretical results.

A complete control structure based backstepping controller design for stacked multi-cell multi-level SPWM VSC STATCOM

Currently, the installation of FACTS (Flexible Alternative Current Transmission System) devices such as synchronous static compensators (STATCOM) represent a very useful solution to improve both network stability and power quality. The effectiveness of these devices is principally determined by the used static converter (topology and the switching strategy) and the adopted dynamic law characterizing the controller structure. This paper illustrates the stacked five-level, multi-cell STATCOM compensator performances governed by a Backstepping controller technique compared to the use of the conventional PI controller (SPWM) for controlling voltage variation (sags or overshoots) affecting a given network point. The interest in using stacked multi-cell converters topology is to increase the number of voltage levels in order to ensure a good quality voltage waveform at the specified connection point. Therefore, the proposed backstepping controller based on the use of Lyapunov control functions presents a simplified structure and a reduced computational load compared to many others advanced control approaches. The proposed system is verified by running a simulation analysis with Matlab Simulink environment. The simulation results obtained show that the proposed STATCOM configuration ensures good voltage regulation at the common connection point (PCC), correction of the unity power factor on the grid side and good response time via reactive power compensation under variable load conditions.

Output-Feedback Consensus Maneuvering of Uncertain MIMO Strict-Feedback Multiagent Systems Based on a High-Order Neural Observer.

In this article, a distributed output-feedback consensus maneuvering problem is investigated for a class of uncertain multiagent systems with multi-input and multi-output (MIMO) strict-feedback dynamics. The followers are subject to immeasurable states and external disturbances. A distributed neural observer-based adaptive control method is designed for consensus maneuvering of uncertain MIMO multiagent systems. The method is based on a modular structure, resulting in the separation of three modules: 1) a variable update law for the parameterized path; 2) a high-order neural observer; and 3) an output-feedback consensus maneuvering control law. The proposed distributed neural observer-based adaptive control method ensures that all followers agree on a common motion guided by a desired parameterized path, and the proposed method evades adopting the adaptive backstepping or dynamic surface control design by reformulating the dynamics of agents, thereby reducing the complexity of the control structure. Combined with the cascade system analysis and interconnection system analysis, the input-to-state stability of the consensus maneuvering closed loop is established in the Lyapunov sense. A simulation example is presented to demonstrate the performance of the proposed distributed neural observer-based adaptive control method for output-feedback consensus maneuvering.

Backstepping Control Design in Conjunction with an EKF-based Sensorless Field-Oriented Control of an IPMSM

The collector and brushless electronic commutation machines have gained widespread utilization in industrial applications. Among these machines, internal permanent magnet synchronous motors (IPMSMs) have become increasingly popular due to their numerous advantages, including high torque/current and torque/inertia ratios, robust construction, high efficiency, and reliability. However, the incorporation of position sensors in IPMSMs has introduced challenges concerning their application, performance, mass production, and cost. Consequently, the implementation of sensorless control techniques has become essential in drive systems and various applications. This paper proposes a backstepping control approach for achieving speed sensorless control of IPMSMs, employing an extended Kalman filter (EKF). The first step involves developing a comprehensive nonlinear dynamical model of the IPMSM in the direct and quadrature (d-q) rotor frame and obtaining its corresponding state-space representation. Subsequently, backstepping controllers for rotor speed and current tracking are designed to ensure precise tracking and anti-disturbance performance. These controllers are integrated into the field-oriented control (FOC) scheme. The Lyapunov stability theorem is employed to guarantee the asymptotic stability condition of the backstepping controller. To address the challenge of estimating immeasurable mechanical parameters of the IPMSM and accurately tracking the system states, an EKF is designed. This filter enables the estimation of mechanical parameters and achieves high steady-state precision within a finite time. The effectiveness of the proposed methodology is demonstrated through simulations conducted under various dynamic operating conditions, including sudden torque load changes, command speed changes, and parameter variations.

Practical fixed-time neural control for MIMO non-strict feedback nonlinear systems: an adaptive neural network approach

ABSTRACT This paper studies the non-singular practical fixed-time neural adaptive control issues for multi-input and multi-output (MIMO) nonlinear systems with non-strict feedback form. Neural networks (NN) are used to estimate the unknown nonlinearities and deal with the problem of an algebraic loop. Under the framework of the backstepping control design, a practical fixed-time adaptive NN control method is developed by using the adding power integration technology. According to the Lyapunov function theory, it is proved that the closed-loop system is practical fixed-time stable, and the system can track the desired reference signal within a fixed time. Finally, the proposed practical fixed-time control method is applied to a multi-motor control platform, which proves the effectiveness of the control method.

Distributed finite‐time backstepping adaptive containment control for multiple unmanned aerial vehicles with input saturation

AbstractThis paper investigates a distributed adaptive finite‐time containment control scheme for multiple unmanned aerial vehicles subject to external disturbances and input saturation. Combined with a novel indicator of the saturation degree designed by the hyperbolic tangent function, an adaptive method is proposed to deal with the input saturation issue, which considers both symmetry and asymmetry saturation. Moreover, to address the problem of the “explosion of terms” inherent in the traditional backstepping controller design, a fixed‐time sliding mode differentiator is utilized to approximate the derivative of the virtual signal in the command‐filtered backstepping method. Finally, the convergence of the errors and the practicability of the control law are verified by Lyapunov stability analysis and numerical simulations.

Finite-time event-triggered output-feedback adaptive decentralized echo-state network fault-tolerant control for interconnected pure-feedback nonlinear systems with input saturation and external disturbances: A fuzzy control-error approach

In this paper, we introduce a decentralized Event-Triggered fault-tolerant echo-state network (ESN) direct adaptive controller for uncertain interconnected nonlinear systems in pure-feedback form. The proposed controller addresses input saturation, actuator faults, external disturbances, and unavailable states for measurement. Unlike the existing works in the literature that adapts the ESN weights using the tracking error, our method employs the control error that is estimated using a fuzzy inference system, to derive the adaptation laws. The complexity explosion typically seen with recursive back-stepping and Dynamic Surface Control (DSC) designs is completely avoided. Our control scheme addresses four types of state-dependent actuator faults: bias, drift, loss of accuracy, as well as loss of effectiveness. The ESNs approximate unknown ideal control laws, while robust terms are added to enhance the stability of the closed-loop system. Stability analysis ensures that tracking errors converge, in finite time, to a small compact set near the origin due to the strictly positive real (SPR) property. Simulation results of a quadrotor UAV and a mathematical system are presented. A comparative analysis is performed with another approach to emphasize the effectiveness of the proposed method.

Adaptive neural network fault‐tolerant control for uncertain non‐strict feedback nonlinear system with actuator faults and state constraints

SummaryThis article studies a neural network (NN)‐based adaptive fault‐tolerant control (FTC) scheme for uncertain non‐strict feedback systems with time‐varying state constraints and actuator faults. The introduction of asymmetric Barrier‐Lyapunov function (BLF) makes controller design more difficult due to the emergence of actuator faults and state constraints. Therefore, this article designs a fault‐tolerant controller with constraint compensation information under the backstepping control design framework to solve the state constraint asymmetry problem caused by actuator failure. By designing an improved asymmetric time‐varying BLF, the design of the state‐constrained controller will become more realistic and the constraints will be weakened. In the design process, the characteristics of the radial basis function neural network are used to avoid the algebraic ring problem. Actuator failure in this article considers deviation failure and loss of effectiveness. Based on the properties of the exponential function, the improved BLF can make the bounds of the state constraints smaller and smaller, and the bounds of the constraints can change with the desired trajectory. Simulation verified the feasibility of this control method.