Recursive Backstepping Research Articles

Consensus for Heterogeneous Multiagent Systems: Output Rate-Coded Secure Control.

This article investigates an output rate-coded secure control based on backstepping for the consensus of heterogeneous multiagent systems (MASs) with output-triggering condition. Due to the nondifferentiable virtual control inputs caused by discontinuous triggering signals, it presents a technical obstacle in implementing the recursive backstepping, rendering previous results inapplicable. To address this problem, an auxiliary high-order filter is elaborately constructed to guarantee the required-order derivatives of virtual control inputs. As this filter is driven by local triggering consensus error, it allows our MASs to have different system orders and relative degrees. Besides, to reduce communication bit and strengthen communication security, we propose a novel distributed rate-coded algorithm from an encoding-decoding viewpoint. When it is triggered, agent's output is encrypted into an L-length codeword, then transmitted to neighbors without exposing any sensitive system states or real control inputs. It is shown that all the closed-loop system signals are ultimately bounded, and the mean square consensus tracking error can be reduced by appropriately selecting control parameters. Simulations illustrate our theoretical finding.

Enhanced composite controller for PV/PMSG/PEMFC and BESS‐based DC microgrids voltage regulation: Integrating integral terminal sliding mode controller and recursive backstepping controller

AbstractThe variability of renewable energy sources due to weather patterns often leads to a mismatch between power generation and consumption within microgrids (MGs). This challenge is exacerbated when integrating bio‐renewable units, complicating stability maintenance in MGs. This research work introduces a novel solution to address this issue: a composite controller merging an integral terminal sliding mode controller with a recursive backstepping controller for direct current MGs (DCMGs). The proposed DCMG incorporates solar photovoltaic units, wind farms based on permanent magnet synchronous generators, proton exchange membrane fuel cells fuelled by hydrogen gas, an electrolyser, battery energy storage systems, and DC loads. First, a comprehensive mathematical model for the components within the DCMG is developed to design the proposed composite controller. This controller not only overcomes the inherent limitations and convergence issues of conventional SMCs but also ensures stable DC‐bus voltage and maintains power balance across various operational conditions. Moreover, a fuzzy logic‐based energy management system is introduced to regulate power flow, considering factors like battery state of charge and renewable energy sources' total power output. The control Lyapunov function confirms the DCMG system's asymptotic stability. Finally, the proposed controller's effectiveness is validated through simulations on both MATLAB/Simulink and Arduino Mega 2650 processor‐in‐the‐loop platforms under various operational conditions. In both platforms, the proposed controller surpasses an existing controller in terms of settling time, overshoot, and tracking error of the DC‐bus voltage.

Adaptive Free‐Will Arbitrary Time Tracking of a Class of Uncertain SISO Nonlinear Systems

ABSTRACTThis paper investigates the free‐will arbitrary time (FWAT) tracking problem for a class of single‐input single‐output (SISO) strict‐feedback nonlinear systems without/with parametric uncertainties. First, we design an FWAT backstepping tracking controller for strict‐feedback nonlinear systems without uncertainties in a recursive manner. It is shown that the proposed virtual controllers and their derivatives are continuously differentiable for the backstepping design, and the tracking error converges to zero within the pre‐specified settling time independent of design parameters and initial conditions and remains at zero after the pre‐specified settling time. Second, we address the adaptive FWAT tracking problem of SISO strict‐feedback nonlinear systems with parametric uncertainties in the practical stability sense. The continuously differentiable and nonsingular time‐varying adjustment functions for practical FWAT tracking are developed to transform error variables for the recursive backstepping design. Then, the adaptive FWAT backstepping tracking design and stability analysis strategy are established to ensure practical FWAT convergence within the pre‐specified settling time independent of design parameters and initial conditions. The designed adaptive virtual and actual controllers are continuous for all . Furthermore, a practical adaptive prescribed‐time tracking scheme is provided using the proposed practical adaptive FWAT tracking scheme. Finally, the effectiveness of the theoretical approaches is confirmed through a simulation example and an experimental result.

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.

Deep Entropy-Learning based virtual inertia control for VRFB regulation considering Phase-Locked loop dynamics

The virtual inertia control (VIC) technique is often adopted as a prominent mechanism to address the inertia deficiency of modern integrated power systems with high penetration of sustainable energy units. In such control mechanisms, a phase-locked loop (PLL) is required to estimate the grid frequency. However, utilization of such frequency measurement platforms is accompanied by large frequency fluctuations due to its inherent dynamics. In particular, this paper proposes an adaptive integral recursive backstepping (I-RBSC) based on virtual inertia for the regulation of vanadium redox flow battery (VRFB) in stand-alone Micro-Grids (SMGs). Unlike previous works that neglected the effect of frequency measurement devices, this work investigates the effect of PLL dynamics in virtual inertia control. The I-RBSC controller is designed by an entropy-driven actor-critic neural networks (NNs) to adaptively adjust the tunable coefficients during its interaction with the SMG. By training the ability of deep NNs, the entropy is maximized to regulate the system output. The VRFB unit can be discharged up to 90% which can quickly respond to the randomness of loads and sustainable energy units. The transient outcomes of SMG reveal the feasibility of I-RBSC (designed by entropy learning) to address the challenges of low inertia and PLL dynamics. The real-data of wind turbine and solar radiation are utilized to simulate the SMG in a more realistic framework from a systematic point of view. The comprehensive analysis under typical scenarios confirms the supremacy of the suggested VIC controller over prevalent state-of-the-art virtual inertia controllers including conventional VIC, fuzzy logic, backstepping, and model-predictive control (MPC).

Jerk Chaotic System: Analysis, Circuit Simulation, Control and Synchronization with Application to Secure Communication

Dynamical behaviors of 3D Jerk system were examined in terms of equilibrium, stability, dissipative, and phase space attractor in this study. The system’s practical applications were demonstrated through circuit realization and synchronization scheme via active backstepping control with its effectiveness demonstrated in secure communication. The viability of the theoretical model of 3D Jerk system was confirmed using electronic circuit workbench designed in MultiSIM enviroment. A nonlinear feedback controller was designed using the recursive backstepping technique to control and track a desire function. For secure communication application, active backstepping method was adopted to synchronize two identical chaotic systems evolving from different initial conditions. It was demonstrated that when the controller was activated, the systems synchronize successfully. The results of the active backstepping designed controllers were numerically applied in the area of secure communication, with the variable of the drive being encrypted information transmitted through a coupling channel. Using an additive encryption masking scheme, the encrypted signal was a superposition of sinusoidal information specified by period function and chaotic carrier generated from a variable of the Jerk system. The transmitted information signal was successfully retrieved from the chaotic response signal using an inverse function decryption algorithm, thereby confirming the effectiveness and robustness of the designed controller.

Design of a novel neuro‐adaptive excitation control system for power systems

AbstractThis manuscript proposes a robust excitation control strategy for synchronous generators using backstepping theory and an artificial neural network with a radial basis function to improve power system performance during disturbances and parametric uncertainties. The artificial neural network is used to estimate unmeasurable quantities and unknown internal parameters of a recursive backstepping control. Lyapunov theory is used to carry out the stability analysis and to deduce the online adaptation laws of artificial neural network parameters (weights, centres and widths). To validate the performance of this approach, simulations are performed on an IEEE 9 bus multi‐machine power system. Different test results, compared with those of an existing non‐linear adaptive controller, confirm the high robustness of the proposed method against disturbances and uncertainties.

Neuro-Based Consensus Seeking for Nonlinear Uncertainty Multi-Agent Systems Constrained by Dead-Zone Input

The topic about consensus target track seeking for high-order nonlinear multi-agent systems (MASs) with unmodeled dynamics, dynamic disturbances, and dead-zone input is considered in the paper. Using the strong nonlinear map characteristic of radial basis function neural networks (RBFNNs), the complex functions arising from recursive procedure are simplified. Also, inspired by input-to-state practical stability (ISpS), the authors construct a dynamical signal in order to counteract the impact of unmodeled dynamics and dynamic disturbances. The bounded inequality expression has been applied to tackle the unknown input of dead zone. Based on this, consensus control protocol suitable for nonlinear constraints has been constructed by using the recursive backstepping technique and adaptive neural network method. Theoretical analysis indicates not only the uniform boundary of all signals in the closed-loop under the neuro-based consensus controller, but uniform ultimate convergence of consensus tracking errors. The final simulations also confirmed the correctness of the theoretical analysis.

Control of DC–DC Buck Converters Using Robust Composite Backstepping and Integral Terminal Sliding Mode Approaches

This paper proposes a new composite scheme to design a robust voltage controller for DC-DC buck converters in which the composite structure combines the recursive backstepping approach with an integral terminal sliding mode controller (ITSMC). In this work, the proposed ITSMC is designed to overcome the one-degree limitation of existing ITSMCs and difficulties for the finite time convergence in standard sliding mode controllers (SSMCs) as it incorporates a quick reaching law. Moreover, the singularity problem in existing SSMCs is tackled using a modified integral sliding surface. The backstepping strategy is used to ensure the voltage monitoring with a higher degree of accuracy. The robustness of the proposed composite approach is ensured by capturing the variations in parameters, input voltage, and mismatch in the model; as all these factors are included in the dynamical model of the DC-DC buck converter based on which the proposed composite controller is designed. The robustness and stability of the proposed composite controller are simultaneously analyzed using the concept of the Lyapunov stability theory. The theoretical and analytical findings for the proposed composite robust backstepping ITSMC (RBITSMC) are further verified through extensive simulation studies in the MATLAB/Simulink platform. Performance comparisons of the proposed RBITSMC are also presented against an existing adaptive nonsingular fast terminal sliding mode controller (ANFTSMC). Finally, experimental results from the application of the proposed control strategy on a laboratory prototype are presented to justify its application on the practical system.

Robust hybrid nonlinear control approach for stability enhancement of a constant power load DC-DC boost converter

This paper presents a robust hybrid super-twisting reaching law-based sliding mode controller in conjunction with a complementing recursive backstepping controller for DC-DC boost converters with constant power loads (CPLs). The key control goal is to keep the DC-bus voltage steady and constant while continuously powering the CPL. The nonlinear dynamical model of the DC-DC boost converter is translated into Brunovsky’s canonical forms, utilizing the precise feedback linearization technique to attain this control goal. It is worth mentioning that the dynamical model has been modified with bounded external uncertainties to demonstrate the restraint performance for huge loads and input voltage variations. The proposed hybrid controller is then constructed using this model. Furthermore, to ensure the large signal stability of the whole system under the designed control law, the Lyapunov stability theory is used. Numerical simulation analysis are carried out in MATLAB/Simulink to validate the performance of the designed robust controller for a DC–DC boost converter feeding a CPL when the operating point change. The results show that the designed controller outperforms both existing nonlinear controllers and a traditional PI controller in terms of ensuring the system’s stability and fast response in a variety of operating modes. Finally, in order to confirm the theoretical design and simulation results of the designed robust hybrid controller, experimental results are also provided in this paper.

Shimizu–Morioka's chaos synchronization: An efficacy analysis of active control and backstepping methods

This research study inspects the effectiveness of synchronization methods such as active control and backstepping control from systematic design procedures of a synchronized Shimizu–Morioka system for the same parameter. It aimed to achieve synchronization between the state variables of two identical Shimizu–Morioka chaotic systems by defining the proposed varieties of the error dynamics coefficient matrix. Furthermore, this study also aimed to designed an active controller that enables the synchronization of these systems. The use of designed recursive backstepping nonlinear controllers was based on the Lyapunov function. Furthermore, it also demonstrated the stability of the synchronization of the nonlinear identical Shimizu–Morioka system. The new virtual state variable and establishment of Lyapunov functionals are used in the backstepping controller to stabilize and reduce errors between the Master (MS)/Drive (DS) systems. For comparison, the complexity of active controllers is verified to be such that the designed controller's effectiveness based on backstepping is attainable in engineering applications. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed synchronization strategy with the Runge–Kutta (RK-4) algorithm of fourth order through MatLab Simulink.

Adaptive Neural Network Control Using Nonlinear Information Gain for Permanent Magnet Synchronous Motors.

In this study, an adaptive neural network (NN) control using nonlinear information (NI) gain for permanent magnet synchronous motors (PMSMs) is proposed to improve control and estimation performance. The proposed method consists of a nonlinear controller, a three-layer NN approximator, and NI gain. The nonlinear controller is designed via a backstepping procedure for position tracking. The commutation scheme is designed to implement the PMSM control without the direct-quadrature (DQ) transform. The three-layer NN approximator is designed to estimate the unknown complex function generated by the recursive backstepping process. The NI gains are designed to enhance the control and estimation performance according to the increased tracking errors owing to the load torque and the desired position variations. All of signals in the closed-loop system guarantee the semiglobal uniformly ultimately boundness (UUB) using the Lyapunov stability theorem and the input-to-state stability (ISS) property. The performance of the proposed method was validated by experiments performed using a PMSM testbed.

Saturated-command-deviation based finite-time adaptive control for dynamic positioning of USV with prescribed performance

This paper investigates the finite-time adaptive dynamic positioning (DP) control problem of an unmanned surface vehicle (USV) with prescribed performance subject to time-varying disturbances, unknown model parameters and input saturation. First, the error transformation is performed to provide a framework for the design of model-free prescribed performance DP controller, which eliminates the need for accurate prior information of the model and external disturbances. Then, a saturated-command-deviation based compensated law is proposed by incorporating the fuzzy supervisory approach to account for the excess control efforts resulted by input saturation. Subsequently, the fuzzy logic systems (FLS) integrated with minimum learning parameters technique are employed to approximate the model-based and high-order terms, such that the explosion of computational complexity is removed and requirements for accurate modeling are eliminated. Meanwhile, to suppress the negative effects induced by time-varying disturbances and unknown approximation errors, the adaptive technique is adopted to estimate the upper boundary of compounded disturbances online. Finally, by incorporating the softening sign signal and piecewise function into recursive backstepping procedure, a DP controller is developed to avoid potential singularity and achieve finite time convergence. Theoretical analysis and simulation results successfully illustrate the effectiveness of the proposed scheme.

Adaptive Event-Triggered Control of Uncertain Nonlinear Systems Using Intermittent Output Only

Although rich collection of research results on event-triggered control exist, no effort has ever been made in integrating state/output triggering and controller triggering simultaneously with backstepping control design. The primary objective of this article is, by using intermittent output signal only, to build a backstepping adaptive event-triggered feedback control for a class of uncertain nonlinear systems. To do so, we need to tackle three technical obstacles. First, the nature of the event triggering makes the transmitted output signal discontinuous, rendering the regular recursive backstepping design method inapplicable as the repetitive differentiation of virtual control signals is literally undefined. Second, the effects arisen from the event-triggering action must be properly accommodated, but the current compensating method only works for systems in normal form, thus a new method needs to be developed in order to handle nonnormal form systems. Third, as only intermittent output signal is available, and at the same time, the impacts of certain terms containing unknown parameters (arising from event triggering) need to be compensated, it is rather challenging to design a suitable state observer. To circumvent these difficulties, we employ the dynamic filtering technique to avoid the differentiation of virtual control signals in the backstepping design, construct a new compensation scheme to deal with the effects of output triggering, and build a new form of state observer to facilitate the development of output feedback control. It is shown that, with the derived adaptive backstepping output-triggered control, all the closed-loop signals are ensured bounded and the transient system performance in the mean square error sense can be adjusted by appropriately adjusting design parameters. The benefits and effectiveness of the proposed scheme are also validated by numerical simulation.

Terminal sliding mode observer based–asymptotic tracking control of electro-hydraulic systems with lumped uncertainties

Since the existence of lumped uncertainties caused by uncertain parameters and unmeasured external load, the disturbance suppression is always a significant issue in research of electro-hydraulic system (EHS). In this paper, a type of terminal sliding mode observer (TSMO) is introduced for EHS, based on which the disturbances can be converged in a finite time. Then, based on TSMO, a controller is designed via recursive backstepping method to realize the displacement tracking control of EHS. Finally, simulated results are given to verify the feasibility of the proposed theoretical conclusions and a series of contrast experimental results are also given to illustrate the better performance of TSMO and proposed control strategy than others.

Fuzzy‐based adaptive control design for stochastic nonlinear time‐delay systems via event‐triggered mechanism

SummaryThis article elaborates an adaptive fuzzy event‐triggered control design for stochastic nonlinear time‐delay systems with full‐state constraints. The prescribed constraints are ensured by employing barrier Lyapunov function. The fuzzy logic systems are exploited to approximate lumped unknown nonlinear functions. By cleverly choosing Lyapunov–Krasovskii functional, the adverse effect of time‐delay is effectively overcome. Meanwhile, the event‐triggered mechanism is adopted to save communication resource. It is illustrated that all signals of the controlled system are bounded and all states are maintained the prescribed regions at every step of recursive backstepping design. Finally, simulation results elucidate the validity of the devised scheme.

Decentralized adaptive control of uncertain interconnected systems with triggering state signals

Backstepping is a powerful technique of control design for uncertain systems and has been well received. As it involves differentiating virtual control signals recursively, the involved signals must be differentiable. However, when the measured states are only transmitted based on event-triggered conditions, such requirement will not be met, making the recursive backstepping design procedure inapplicable. Thus it is still an open problem to study how an event-triggered scheme is employed to transmit state signals in decentralized control of uncertain interconnected systems with dynamic interactions. In this paper, we present a solution by constructing a new decentralized adaptive backstepping-based control algorithm capable of coping with discontinuity caused by state-triggering. With aid of appropriately chosen Lyapunov functions, all signals in the closed-loop system are ensured globally uniformly bounded. Simulation results are presented to demonstrate the effectiveness of the proposed methodology.

Implementation of adjustable variable step based backstepping control for the PV power plant

The most recent research challenge in sustainable photovoltaic (PV) energy is to maintain a high efficiency of PV arrays. Our contribution in this issue is reducing the power loss caused by environmental condition change and parametric variations. A non linear recursive Backstepping maximum power point tracking (MPPT) controller based on an adjusted variable step Perturb and Observe (P&O) algorithm (VS-Backstepping) is proposed to gather the maximum power with fast converge time from the PV system. The trade off between the high accuracy and the fast tracking speed cannot be realized through single MPPT techniques. Thus this work combines two individual MPPT controllers. However, the accuracy for finding the maximum power is highly related to the part of MPP tracking control using the Backstepping approach as well as the rapidity to reach the MPP is managed with the adjusted variable step P&O algorithm. To further evaluate the effectiveness of the proposed method (VS-Backstepping), simulation and experimental results on a commercial PV panel are presented and compared to an individual fixed step P&O algorithm and a variable step P&O (VS-P&O) controller.

A DSC approach to adaptive dynamic region‐based tracking control for strict‐feedback non‐linear systems

As an extension of the conventional set-point control problem, the dynamic region-based tracking control scheme with obstacle avoidance is proposed for a class of uncertain strict-feedback non-linear systems. A novel adaptive tracking controller is designed by a fusion of artificial potential field, recursive backstepping approach, neural networks, dynamic surface control technique, calculus method, and Lyapunov stability theory. In the proposed control scheme, the objective region cannot be required to have a regular shape or a fixed size for the passibility of the system in constrained space. The region tracking error is transformed into a new virtual error variable for recursively designing a dynamic surface controller, and the dimension of neural network inputs can be greatly reduced, especially for high-order systems. The Lyapunov theorem is used to confirm the stability and uniform boundedness of all closed-loop signals. Simulation results are provided to demonstrate the effectiveness of the proposed controller.

Toward Large-Signal Stabilization of Floating Dual Boost Converter-Powered DC Microgrids Feeding Constant Power Loads

In modern dc microgrids (MGs), floating dual boost converters (FDBCs) take the advantages of both interleaved modulations and higher voltage gains, and the FDBC topology has been shown rather suitable for high-power applications. For a given dc MG, those power electronic loads and motor drives would behave as constant power loads (CPLs) that have negative incremental input impedances causing serious stability issues. To make the FDBC-enabled dc MG more adaptable to the increasing penetration of CPLs, this article proposes a compound stabilizer that is capable to stabilize all states in the converter in the large-signal sense. The stabilizer is synthesized by nonlinear disturbance observers and a recursive backstepping controller. The observers provide disturbance estimations to the controller, which neutralizes the disturbances in a feedforward manner and also guarantees the large-signal stabilization of MG under Lyapunov theorems. Practical parameter identification procedures are provided, and it is shown that the proposed stabilizer contributes the wider stability margin to the FDBC than the commonly used proportional–integral (PI) control. Finally, hardware experimentations substantiate the effectiveness and feasibility of the large-signal stabilizer.