Fractional Power Terms Research Articles

Nonsingular predefined‐time dynamic surface control of a flexible‐joint space robot with actuator constraints

AbstractTo achieve predefined‐time trajectory tracking control of a flexible‐joint space robot(FJSR) with actuator constraints, a nonsingular predefined‐time dynamic surface control scheme is developed. The input saturation caused by actuator constraints is addressed via the designed predefined‐time anti‐saturation compensator. On this basis, two different control laws are designed for such high‐order nonlinear systems by utilizing the backstepping technique, and a novel nonlinear filter is constructed to filter the virtual control signals, thus avoiding the “differential expansion” phenomenon. Moreover, a singularity‐free auxiliary function is designed to solve the singularity issue generated by the derivative of fractional power terms in the predefined‐time control algorithm framework. The closed‐loop system is proven to be semi‐globally predefined‐timely uniformly ultimately bounded (SGPTUUB) via constructing the suitable Lyapunov function. The difference and effectiveness of the two designed control laws are illustrated by the conducted simulations. Both of them allow the FJSR system to track the desired trajectory in a reasonably predefined time.

Adaptive fixed‐time control for nonlinear systems subject to mismatched uncertainties and external disturbance: An ESO‐based strategy

AbstractIn this paper, the tracking problem is concerned for a class of nonlinear system. The primary objective is to achieve fixed‐time control performance in the presence of nonvanishing mismatched uncertainties and nonparametric external disturbance. Firstly, in order to estimate disturbance, a fixed‐time extended state observer with fractional power term is developed, the order of which is reduced to alleviate peaking value phenomenon. By reasonably switching among different estimation patterns, both transient and steady‐state performances are balanced. Secondly, by means of fuzzy logic system theory, an ESO‐based adaptive fuzzy control strategy with time‐varying gains is presented, and thus the mismatched nonlinearities are approximated, and the external disturbance is compensated in real time simultaneously. Moreover, by constructing a mixed barrier and quadratic type of Lyapunov function, the fixed‐time stability analysis is put forward for tracking error dynamics based on backstepping methodology. Finally, the advantages and effectiveness are illustrated by a numerical example on wheeled mobile robot. The main characteristic of this paper is to integrate ESO methodology into fuzzy control architecture in fixed‐time sense. As a consequence, not only the output tracking purpose, but also desirable transient performance can be achieved.

Neural network-based finite-time command-filtered adaptive backstepping control of electro-hydraulic servo system with a three-stage valve

This paper aims to improve the tracking control performance of the three-stage valve (TSV) controlled electro-hydraulic servo system (EHSS) with parameter uncertainties and other lumped unknown nonlinearities, including unknown dynamics and disturbances. A more accurate nonlinear model of the TSV-controlled EHSS is established and a neural network-based finite-time command-filtered adaptive backstepping control (NNFCABC) method is proposed for the EHSS. Adaptive control is used to deal with the system parameter uncertainties, and the radial basis function neural network (RBFNN) algorithm is introduced to approximate the lumped unknown nonlinearities. The prediction errors of serial-parallel estimation models (SPEMs) and the tracking errors are utilized together to design adaptive laws to estimate the system parameters and the weights of the RBFNNs. The entire control framework utilizes command-filtered control and backstepping techniques. By applying Levant differentiators as command filters and introducing fractional power terms into the virtual control laws and the SPEMs, the proposed NNFCABC theoretically guarantees the tracking performance of the closed-loop control system with finite-time convergence. Comparative simulations and experiments verify the feasibility and superiority of the proposed control scheme.

Saturated exponential super‐twisting sliding mode control for bottom‐following of biomimetic underwater vehicles with unmeasured velocities

AbstractThis article addresses the bottom‐following problem for biomimetic underwater vehicles without translational velocity measurements in the presence of complex uncertainties and actuator saturation. A novel saturated exponential super‐twisting algorithm based integral terminal sliding mode control (SESTA‐ITSMC) scheme is created to ensure fast convergence and strong robustness as well as attenuate the chattering of actuators. Salient features are as follows. Instead of discontinuous term in the conventional super‐twisting algorithm, a novel SESTA is developed by incorporating a fractional power term, which ensures that the system achieves a faster transient response, effectively eliminating the chattering phenomenon. Then, to deal with the unknown states and further improve the anti‐interference ability of the system, the finite‐time dual estimator is devised to fast and precisely estimate the translational velocities and lumped uncertainties. Finally, a finite‐time anti‐windup auxiliary system is proposed to compensate for the saturation constraints on actuators in real time. Finite‐time convergence of estimation errors and bottom following errors is guaranteed by the Lyapunov stability theory. Comparative simulation results fully demonstrate the excellence of the proposed SESTA‐ITSMC scheme.

Adaptive finite-time fault-tolerant control for the full-state-constrained robotic manipulator with novel given performance

Most robot manipulators usually operate under different nonlinearities, such as state constraints, time delays, and actuator input saturation faults. This paper proposes an adaptive finite-time fault-tolerant controller for the full-state-constrained robotic manipulator with novel prescribed performance and time-varying delays. Firstly, a novel prescribed performance function consisting of hyperbolic tangent and hyperbolic cotangent functions is constructed to the full-state performance constraints of the robotic manipulator. Such a design can circumvent the demands for accurate initial conditions of the state variables and guarantee the convergence of the state errors in a prescribed time. Then, based on the finite-time principle, a finite-time command filter and a compensation mechanism with fractional-power terms are used to bypass the “explosion of complexity” and further compensate for the filter errors within a finite time. Moreover, a proper Lyapunov–Krasovskii functional is devised within the controller design to compensate for the time-varying delays. And the radial basis function neural networks are used to estimate the other nonlinearities including actuator saturation faults, and external perturbations. Finally, simulation results exhibit that the controller designed in this paper has a faster convergence speed and minor state errors.

A Generalized Supertwisting Algorithm.

The work proposes a generalized supertwisting algorithm (GSTA) and its constructive design strategy. In contrast with the conventional STA, the most remarkable characteristic of the proposed method is that the discontinuous term in the conventional STA is replaced with a fractional power term, which can fundamentally improve the performance of the conventional STA. It is shown that if the fractional power in the nonsmooth term becomes -1/2, the GSTA will reduce to the conventional STA. Under the GSTA, it will be rigorously verified by taking advantage of strict Lyapunov analysis that the sliding variables can finite-time converge to an arbitrarily small region in a neighborhood of the origin by tuning the gains and the fractional power. Finally, simulation studies are provided to demonstrate the superiority of the theoretically obtained results.

Finite-time formation control of nonaffine nonlinear systems under directed communication interactions

This paper studies the problem of finite-time formation tracking control for networked nonaffine nonlinear systems with unmeasured dynamics and unknown uncertainties/disturbances under directed topology. A unified distributed control framework is proposed by integrating adaptive backstepping control, dynamic gain control and dynamic surface control based on finite-time theory and consensus theory. Auxiliary dynamics are designed to construct control gains with non-Lipschitz dynamics so as to guarantee finite-time convergence of formation errors. Adaptive control is used to compensate for uncertain control efforts of the transformed systems derived from original nonaffine systems. It is shown that formation tracking is achieved during a finite-time period via the proposed controller, where fractional power terms are only associated with auxiliary dynamics instead of interacted information among the networked nonlinear systems in comparison with most existing finite-time cooperative controllers. Moreover, the continuity of the proposed controller is guaranteed by setting the exponents of fractional powers to an appropriate interval. It is also shown that the improved dynamic surface control method could guarantee finite-time convergence of formation errors, which could not be accomplished by conventional dynamic surface control. Finally, simulation results show the effectiveness of the proposed control scheme.

Fixed-Time Fault-Tolerant Tracking Control of Fixed-Wing UAVs with Actuator Fault and Unmatched Disturbances

This paper presents a dynamic surface fixed-time fault-tolerant control strategy for the longitudinal dynamic model of fixed-wing unmanned aerial vehicles (UAVs). Firstly, a novel disturbance observer is constructed to precisely estimate the lumped disturbance. Secondly, without fractional power terms in the designed fixed-time fault-tolerant controller, the potential singular value problem is tactfully avoided, which often exists in the stability analysis of the traditional fixed-time controller design. Thirdly, a novel fixed-time filter is proposed to overcome the phenomenon of “differential explosion” in the backstepping control scheme. Lyapunov stability analysis guarantees that the tracking errors can converge to the neighborhood of the origin in the fixed time. The simulation results verify the effectiveness of the proposed control scheme.

Fuzzy Adaptive Prescribed Performance Fault-Tolerant Control for HFVs with Fixed-Time Convergence Guarantee

A new fixed-time fuzzy adaptive fault-tolerant control methodology is proposed for the longitudinal dynamics of hypersonic flight vehicles (HFVs) in the presence of actuator faults, uncertain dynamics, and external disturbances. In contrast with the conventional fixed-time control schemes that typically contain the fractional powers of errors in their designs, this work develops a low-complexity control structure in the sense of removing the dependence on the need of abovementioned fractional power terms by means of prescribed performance control (PPC) method. Different from the most existing PPC approaches where the initial conditions of tracking errors are required to be known, the newly proposed prescribed performance function (PPF) can relax such restrictions through choosing properly small initial values of PPF. Fuzzy logic systems (FLSs) are employed to handle unknown dynamics, and minimal learning parameter (MLP) technique is incorporated into the design for the purpose of alleviating computation burden. Closed-loop stability is rigorously proved via Lyapunov stability theory, and simulation results are eventually given to validate the effectiveness of the proposed control strategy.

DEH Scheme DGTD-Based Transient Modeling Approach for the Cole–Cole Dispersive Media Using Tustin’s Method

The Cole–Cole model can simulate the electromagnetic properties of biological tissues more accurately in a wide frequency band. The current simulation of wave propagation in the Cole–Cole dispersive media usually uses the finite-difference time-domain (FDTD) method, but its requirements in small grid size and time step interval for curved objects limit its application in calculating multiscale complex biological tissues. The discontinuous Galerkin time-domain (DGTD) method is more suitable for simulating multiscale complex targets because it can use unstructured grids and high-order basis functions. Therefore, this article proposes a transient modeling approach for the Cole–Cole dispersive media based on DGTD method, in which the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> transform is performed on the Cole–Cole model, and the fractional power terms are approximated to integer power polynomials according to the continued fraction theory. Moreover, in order to avoid the frequency-dependent permittivity in Maxwell’s equations, the DEH scheme is constructed and a weak form equation for the constitutive relationship between D and E is created. Maxwell’s equations and the constitutive relationship are updated by the leap-frog time-stepping method. The accuracy and ability of the new method are verified by simulating the plane wave propagation in various Cole–Cole media.

Observer based sliding mode control of PMSM speed regulation system with a novel reaching law

A novel reaching law based sliding mode control of PMSM speed regulation system with uncertainties and unknown external disturbance is proposed in this paper. The proposed reaching law mainly consists of a time-varying gain and a fractional power term of the sliding variable. And the time-varying gain is auto-tuned by the sliding variable such that it is large when the sliding variable is far away from the sliding surface to speed-up the reaching process, and vice versa. Unlike the existing auto-tuning power rate reaching law (PRRL), the proposed reaching law provides a guaranteed reaching time independently on initial conditions. Moreover, the reaching time can be effectively reduced by tuning the gains of the reaching law. Based on this new reaching law, a disturbance observer is designed to estimate the unknown total disturbance and a sliding mode controller is designed for the robust control of PMSM speed regulation system based on the estimated disturbance. Simulations and experiments are carried out to verify the effectiveness of the proposed control method.

Fixed-time composite neural learning control of state-constrained nonlinear uncertain systems

This paper presents a fixed-time composite neural learning control scheme for nonlinear strict-feedback systems subject to unknown dynamics and state constraints. To address the problem of state constraints, a new unified universal barrier Lyapunov function is proposed to convert the constrained system into an unconstrained one. Taking the unconstrained system, a modified fixed-time convergence state predictor is explored, enabling the prediction error for compensating the neural adaptive law to be obtained and improving the learning ability of online neural networks (NNs). Without employing fractional power terms or a complicated switching strategy to build the control law, a new method of constructing a smooth fixed-time dynamic surface control scheme is proposed. This overcomes the potential singularity problem and the explosion of complexity often encountered in fixed-time back-stepping designs. The representative features of our design are threefold. First, it is free of the fractional power terms, yet offers fixed-time convergence. Second, it addresses the state constraint problem without requiring a feasibility check. Third, it constructs a new state-predictor and enhances the approximation accuracy of NNs. The stability of the proposed control scheme is analyzed using the Lyapunov technique. Simulation results are presented to illustrate the effectiveness of the proposed controller.

Barrier Lypunov functions‐based nonsingular fixed‐time switching control for strict‐feedback nonlinear dynamics with full state constraints

AbstractThis work proposes a nonsingular adaptive fixed‐time switching control method for a class of strict‐feedback nonlinear dynamics subject to full state constraints. The peculiarity of this design lies in overcoming the singularity issue that typically appears in the existing backstepping‐based fixed‐time control methods caused by the iterative differentiation of fractional power terms as tracking errors approach to zero, while guaranteeing the nonviolation of full state constraints. Crucial in solving such singularity issue is to skillfully introduce a smooth switching between fractional power and integer power terms, which guarantees that fractional power term is confined within a positive interval all the time. An asymmetric time‐varying barrier Lyapunov function is delicately incorporated into control design, rendering state variables to be within prescribed time‐varying bounds. Besides, radial basis function neural network is employed to handle system unknown nonlinearities. It is rigorously proved that all the closed‐loop signals eventually converge to small regions around origin within fixed‐time. Comparative simulation results are finally given to validate the effectiveness and superiority of the proposed control strategy.

Forecasting China’s per Capita Living Energy Consumption by Employing a Novel DGM (1, 1, tα) Model with Fractional Order Accumulation

The living energy consumption of residents has become an important technical index to promote the economic and social development strategy. The country’s medium- and short-term living energy consumption is featured with both a certainty of annual increment and an uncertainty of random variation. Thus, it can be seen as a typical grey system and shall be suitable for the grey prediction model. In order to explore the future development trend of China’s per capita living energy consumption, this paper establishes a novel grey model based on the discrete grey model with time power term and the fractional accumulation (FDGM (1, 1, tα) for short) for forecasting China’s per capita living energy consumption, which makes the existing model to adapt to different time series by adjusting fractional order accumulation parameter and power term. In order to verify the feasibility and effectiveness of the novel model, the proposed and eight other existing grey prediction models are applied to the case of China’s per capita living energy consumption. The results show that the proposed model is more suitable for predicting China’s per capita energy consumption than the other eight grey prediction models. Finally, the proposed model based on metabolism mechanism is used to predict China’s per capita living energy consumption from 2018 to 2029, which can provide a reference for energy companies or government decision makers.

Numerical solution for the stress near a hole with corners in an infinite plate under biaxial loading

We consider the elastic stress near a hole with corners in an infinite plate under biaxial stress. The elasticity problem is formulated using complex Goursat functions, resulting in a set of singular integro-differential equations on the boundary. The resulting boundary integral equations are solved numerically using a Chebyshev collocation method which is augmented by a fractional power term, derived by asymptotic analysis of the corner region, to resolve stress singularities at corners of the hole. We apply our numerical method to the test case of the hole formed by two partially-overlapping circles, which can include either a corner pointing into the solid or a corner pointing out of the solid. Our numerical results recover the exact stress on the boundary to within relative error $10^{-3}$ for modest computational effort.

Design of a Non-Singular Adaptive Integral-Type Finite Time Tracking Control for Nonlinear Systems With External Disturbances

This paper proposes an adaptive non-singular fast terminal sliding mode control (FTSMC) with integral surface for the finite time tracking control of nonlinear systems with external disturbances. An appropriate parameter-tuning adaptation law is derived to tackle the disturbances. A new fast terminal sliding scheme with self-tuning algorithm is proposed to synthesize the adaptive non-singular fast integral terminal sliding approach. The proposed approach has the following features: 1) It does not require the derivative of the fractional power terms with respect to time, thereby eschewing the singularity problem typically associated with TSMC; 2) It guarantees the existence of the switching phase under exogenous disturbances with unknown bounds; 3) Because of the integral terms in the sliding surface, the power functions are hidden behind the integrator; 4) It ensures chattering-free dynamics. The effectiveness of the proposed approach is assessed using both a simulation and an experimental study. The obtained results showed that the FTSM control technique guarantees that when the switching surface is reached, tracking errors converge to zero at a fast convergence rate. Additionally, the integral term offers one extra degree-of-freedom and since the time-derivative of fractional power terms is not needed in the controller, the proposed switching surface provides a comprehensive framework for singularity avoidance.

Understanding partonic energy loss from measured light charged particles and jets in PbPb collisions at LHC energies

We perform a comprehensive study of partonic energy loss reflected in the nuclear modification factors of charged particles and jets measured in PbPb collisions at = 2.76 and 5.02 TeV in wide transverse momentum (p T) and centrality range. The p T distributions in pp collisions are fitted with a modified power law and the nuclear modification factor in PbPb collisions can be obtained using effective shift (Δp T) in the spectrum measured at different centralities. Driven by physics consideration, the functional form of energy loss given by Δp T can be assumed as power law with different power indices in three different p T regions. The power indices and the boundaries of three p T regions are obtained by fitting the measured nuclear modification factor as a function of p T in all collision centralities simultaneously. The energy loss in different collisions centralities are described in terms of fractional power of number of participants. It is demanded that the power law functions in three p T regions and their derivatives are continuous at the p T boundaries. The Δp T for light charged particles is found to increase linearly with p T in low p T region below ∼5–6 GeV c−1 and approaches a constant value in high p T region above ∼22–29 GeV c−1 with an intermediate power law connecting the two regions. The method is also used for jets and it is found that for jets, the Δp T increases approximately linearly even at very high p T.

An intelligent ABC-based terminal sliding mode controller for load-frequency control of islanded micro-grids

In today’s electric networks, micro-grids are highly integrated into power system regarding their technical, environmental, and economic advantages. Due to the stochastic behavior of loads and intermittent nature of renewable energy resources, the micro-grids are subjected to frequency oscillations especially in the islanded mode of operation. In this paper, an intelligent Terminal Sliding Mode Control (TSMC) based on Artificial Bee Colony (ABC) optimization algorithm is proposed for load-frequency control in islanded micro-grids composed of several energy resources. In TSMC approach, by designing the nonlinear sliding surface and using fractional power terms in surface, it is ensured that the states of system converge to the origin in the finite time. The finite-time stability not only guarantees the finite-time convergence of the state variables, but also provides a function with high accuracy. The parameters of the proposed terminal sliding mode controller are optimized using the presented evolutionary algorithm. The ABC algorithm introduces an intelligent TSMC showing an excellent performance in micro-grid’s load-frequency control. The simulations are implemented in MATLAB, and the results are compared with PID control approach. The obtained results demonstrate the superiority of the proposed approach in damping of frequency oscillations during severe disturbances and uncertain conditions.

An improved continuous sliding mode controller for flexible air‐breathing hypersonic vehicle

AbstractAn improved continuous sliding mode control algorithm is proposed for a flexible air‐breathing hypersonic vehicle (FAHV), including nonsingular fast fixed‐time sliding surface (NFFS) and dual‐layer adaptive continuous twisting reaching law (DACTL). Firstly, the nonlinear control‐oriented model of FAHV is processed using input/output feedback linearization method with the significant flexible effects modeling as unknown matched disturbances. Secondly, a novel NFFS is improved from conventional fixed‐time sliding surface by adjusting power exponent to accelerate convergence rate. In the meanwhile, in order to avoid singularity aroused by fractional power term, an exponential convergent sliding surface is switched when tracking error approaches zero. Thirdly, a DACTL is proposed to realize finite‐time convergence of sliding mode variable with higher convergence precision and less chattering. Dual‐layer adaptive law is utilized to adjust the gain in DACTL based on equivalent control concept so as to enhance robustness automatically and avoid overestimation of control gain. Meanwhile, disturbances can be compensated without knowledge of Lipschitz constants. Ultimately, simulations on longitudinal control of FAHV demonstrate the control algorithm proposed is superior to conventional quasi‐continuous sliding mode controller in the aspect of convergence accuracy and chattering suppression.

Existence and exponential stability in the pth moment for impulsive neutral stochastic integro-differential equations driven by mixed fractional Brownian motion

This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.