Control Of Fractional-order Systems Research Articles

Adaptive Fuzzy Control for Fractional-Order Networked Control Systems with Input Time Delay and Data Loss

This paper considers the adaptive fuzzy control of fractional-order nonlinear networked control systems subjected to network-induced input delay and data loss. To approximate unknown functions, fuzzy logic systems are employed. Furthermore, the Pade approximation method and an intermediate variable are introduced to eliminate the impact of input delay, and an adaptive fuzzy controller is designed using backstepping technology. Based on fractional-order Lyapunov stability theory, the proposed method can ensure that all signals are uniformly ultimately bounded, and the tracking error can converge to a small region of the origin. Two simulation examples are provided to verify the viability of the control method.

Robust H∞ adaptive model following control for fractional-order systems with polytopic and bounded uncertainties subject to input saturation

This paper investigates a type of robust observer-based tracking control for the well-known inverted pendulum as an uncertain fractional-order system with two-norm bounded uncertainties subject to input saturation along with external disturbance that focuses on the case of a fractional-order α such that [Formula: see text]. The main goal is to design a robust model following tracking control servomechanism by the side of stability analysis. This paper addresses the well-known quadratic Lyapunov function in addition to the Gronwall–Bellman lemma and the sector condition of the saturation function stability synthesis. This paper proposes an outstanding strategy to achieve the best model following robust observer in the cast of output-feedback control subject to input saturation in the framework of an uncertain fractional-order system based on the linear matrix inequality (LMI) solution. The LMI procedure can be used to achieve static output-feedback tracking control and observer gain. Several valuable theorems support this approach, and various simulation scenarios are available to showcase its effectiveness.

Comments on “Global leader-following consensus in finite time for fractional-order multi-agent systems with discontinuous inherent dynamics subject to nonlinear growth”

Despite the proposed results on the non-existing of finite time stable equilibria for fractional order systems (Shen and Lam, 2014), the commented paper (Wang et al., 2020) has shown finite time convergence of fractional order control systems via a couple of theorems. This comment demonstrates that the proofs of the given theorems in Wang et al. (2020) are incorrect.

Discrete Impulsive Signal Observer for Fractional-Order Control Systems and Its Consumer Electronic Circuit Application

In the process of actual electronic equipment signal transmission, interference is inevitable. How to accurately estimate the device status to ensure valid signal travelling is very meaningful. This paper mainly investigates a new transmission signal estimation strategy for fractional order control systems with unknown inputs. Compared with integer order modeling, fractional order modeling can better describe and report the essential process of system signal transmission. By using the fractional order output derivative-based method and convex optimization technique, an effective discrete-time impulsive observer design algorithm is proposed to stabilize the error dynamic systems. Especially, the resulting criterion reflects well the relationship between the fractional order and the pulse signal, and then is applied on a fractional order electronic circuit. It implies that the given conclusions are of significance in signal estimation.

A Study on Fractional Power-Law Applications and Approximations

The frequency response of the fractional-order power-law filter can be approximated by different techniques, which eventually affect the expected performance. Fractional-order control systems introduce many benefits for applications like compensators to achieve robust frequency and additional degrees of freedom in the tuning process. This paper is a comparative study of five of these approximation techniques. The comparison focuses on their magnitude error, phase error, and implementation complexity. The techniques under study are the Carlson, continued fraction expansion (CFE), Padé, Charef, and MATLAB curve-fitting tool approximations. Based on this comparison, the recommended approximation techniques are the curve-fitting MATLAB tool and the continued fraction expansion (CFE). As an application, a low-pass power-law filter is realized on a field-programmable analog array (FPAA) using two techniques, namely the curve-fitting tool and the CFE. The experiment aligns with and validates the numerical results.

Modified fractional order social media addiction modeling and sliding mode control considering a professionally operating population

<abstract><p>With the advancement of technology, social media has become an integral part of people's daily lives. This has resulted in the emergence of a new group of individuals known as "professional operation people". These individuals actively engage with social media platforms, taking on roles as content creators, influencers, or professionals utilizing social media for marketing and networking purposes. Therefore, in this article, we designed a six-dimensional fractional-order social media addiction model (FOSMA) in the sense of Caputo, which took into account the professional operations population. Initially, we established the positivity and boundedness of the FOSMA model. After that, the basic regeneration number and the equilibrium points (no addiction equilibrium point and addiction equilibrium point) were computed. Then, the local asymptotic stability of the equilibrium points were proved. In order to investigate the bifurcation behavior of the model when $ R_0 = 1, $ we extended the Sotomayor theorem from integer-order to fractional-order systems. Next, by the frequency analysis method, we converted the fractional order model into an equivalent partial differential system. The tanh function was introduced into the scheme of sliding mode surface. The elimination of addiction was achieved by the action of the fractional order sliding mode control law. Finally, simulation results showed that fractional order values, nonlinear transmission rates, and specialized operating populations had a significant impact on predicting and controlling addiction. The fractional-order sliding mode control we designed played an important role in eliminating chatter, controlling addiction, and ensuring long-term effectiveness. The results of this paper have far-reaching implications for future work on modeling and control of fractional-order systems in different scenarios, such as epidemic spread, ecosystem stabilization, and game addiction.</p></abstract>

An analytical approach to finite time H∞ event-triggered state feedback control of fractional order systems with delay

This paper investigates finite time H∞ event-triggered state feedback control problem of fractional-order systems with delay. Based on Laplace trasform and “inf-sup” norm, a delay-dependent sufficient condition for designing H∞ event-triggered control is established in terms of the Mittag-Leffler function and Linear matrix inequalities. A numerical example is given to show the effectiveness of the obtained result.

Multivariable Fractional-Order Controller Design for a Nonlinear Dual-Tank Device

Fractional calculus is defined by expanded integer order integration and differentiation. In this paper, multiple mathematical models of a nonlinear dual-tank device are precisely formulated by fractional calculus. Using the accurate model, a multivariable fractional-order controller is designed for nonlinear devices. The merits of the fractional-order design include: (1) control of multivariable nonlinearities, (2) compensation of uncertainties, and (3) elimination of coupling effects. Simulations and experiments are conducted to verify the precision of the fractional order models and the effectiveness of the multivariable fractional-order control system design.

Influence of models approximating the fractional-order differential equations on the calculation accuracy

In recent years, the usage of fractional-order (FO) differential equations to describe objects and physical phenomena has gained immense popularity. Due to the high computational complexity in the exact calculation of these equations, approximation models make the calculations executable. Regardless of their complexity, these models always introduce some inaccuracy which depends on the model type and its order. This article shows how the selected model affects the obtained solution. This study revisits seven fractional approximation models, known as the Continued Fraction Expansion method, the Matsuda method, the Carlson method, a modified version of the Stability Boundary Locus fitting method, and Basic, Refined, and Xue Oustaloup filters. First, this work calculates the steady-state values for each of the models symbolically. In the next step, it calculates the unit step responses. Then, it shows how the model selection affects Nyquist plots and compares the results with the plot determined directly. In the last stage, it estimates the trajectories for an example of an FO control system by each model. We conclude that when employing approximation models for fractional integro-differentiation, choosing the appropriate type of model, its parameters, and order is very important. Selecting the wrong model type or wrong order may lead to incorrect conclusions when describing a real phenomenon or object.

Anti-windup scheme-based control for fractional-order systems subject to actuator and sensor saturation

This paper addresses the problem of dissipative-based stabilization for fractional-order nonlinear control systems with uncertainties, time-delay, external disturbances, actuator faults, actuator and sensor saturations. Specifically, a non-fragile reliable dynamic output feedback controller is designed for the purpose of stabilization of the addressed system. Due to the effect of saturation phenomena occurring in both the actuator and sensor, the anti-windup loops framed with the help of the observer design are incorporated into the controller design. Some sufficient conditions are formulated in terms of linear matrix inequalities by choosing a suitable Lyapunov function thereby assuring the dissipative-based stabilization. The derived results are thus validated by two numerical examples including the practical model of two-stage chemical reactor.

Motion Strategy of Drilling Anchor Manipulator Based on Intelligent Optimization Algorithm

The drilling and anchoring robot is an important equipment to realize the intellectualization of fully mechanized coal mining. The control effect of its manipulator directly affects the safety and efficiency of the support operation of the drilling and anchoring robot. The control of drilling anchor manipulator usually adopts PID controller, but due to the limitation of integer order PID control algorithm and traditional empirical parameter adjustment, it is difficult to find a group of parameters with the best control effect in a short time, resulting in the failure of timely and accurate positioning of the end of the manipulator. In this paper, the combination of numerical modeling and simulation analysis is used to adjust the parameters of fractional order PIλDμ controller by using search algorithm (GPS) and applied to the motion control of manipulator; Based on the independent joint control theory, the single input single output system model of hydraulic cylinder at the joint of manipulator is established by using Matlab-Simulink software and fractional PIλDμ control technology. The fractional order control system is regulated by input shaping feedforward control technology(IS). The step response effect of the hydraulic system is further analyzed by setting input feedforward controllers with different gain ratios. The numerical simulation results show that the control strategy of “Input Shaping(IS) + Intelligent optimization algorithm (GPS) parameter adjustment + Fractional PIλDμ control technology” can effectively improve the spatial accurate positioning of mining manipulator. The research expand the application of intelligent control technology in the field of mining equipment.

Predictive Generalized Active Disturbance Rejection Control for Fractional Order Systems with Time Delay

Abstract In this paper, two different active disturbance rejection control techniques based on notion of Smith Predictor are proposed for control of fractional order stable systems with time delay. In the proposed Smith Predictor based Fractional order linearised active disturbance rejection control (SP-FO-LADRC) technique, fractional order dynamics are treated as a part of generalized disturbance and concept of Smith Predictor is used to obtain delay less output, which is fed to an extended state observer (ESO) and an integer order control strategy is developed. To further ameliorate control performance of this technique, complete plant information of commensurate fractional order system is used to formulate another new technique, namely Smith Predictor based fractional order generalized active disturbance rejection control (SP-FO-GADRC), which incorporates fractional ESO and leads to formulation of fractional order control law. An extensive simulation study and robustness analysis conducted on three different examples is testimony to efficacy of proposed techniques.

Consensus tracking for a class of fractional-order non-linear multi-agent systems via an adaptive dynamic surface controller

In this paper we investigate bottlenecks in adaptive dynamic surface control (DSC) and unveil an innovative consensus tracking controller to track the desired trajectory for a class of fractional-order multi-agent systems with non-linear dynamics. The study derives an algorithm by implementing graph theory and the DSC method. The main approaches in the control of fractional-order systems are the DSC and the adaptive DSC techniques to avoid the computational complexity of fractional-order virtual control law. According to these techniques, a virtual control law is formulated and the proposed controller is passed through a fractional-order dynamic surface. By employing the DSC and adaptive DSC laws, we demonstrate that the desired consensus tracking between agents can be ensured. To verify the performance of the new approach, we simulate the desirable scenarios and evaluate the results against a popular adaptive sliding mode technique.

Dual-Loop Voltage–Current Control of a Fractional-Order Buck-Boost Converter Using a Fractional-Order PIλ Controller

Based on the fact that the inductor and capacitor are of a non-integer order by nature, to provide a more accurate theoretical basis for the optimal control of the converter, the fractional-order model of the Buck-Boost converter in the continuous mode of current is established according to the fractional-order calculus theory. The fractional-order PIλ control system of the fractional-order Buck-Boost converter is designed to compare the performance of the integer-order PI controller with the fractional-order controller. Secondly, the sparrow search algorithm is applied to the optimal design of the fractional-order PIλ control system of the fractional-order Buck-Boost converter to improve the system’s phase margin, stability, and robustness. Finally, the simulation is verified on the Matlab/Simulink simulation platform and compared with the integer-order PI controller.

Delayed reset control design for uncertain fractional-order systems with actuator faults via dynamic output feedback scheme

In this work, the stabilization of uncertain fractional-order dynamical control system with actuator faults and external disturbances will be addressed. Specifically, a fractional-order reset control system consisting of two sub-systems namely flow and jump subsystems will be designed. Based on the triggering conditions of reset law, a particular subsystem of the control system will be invoked. Meanwhile, the uncertainty and time-delay will be incorporated in the reset control system. Then the output of the reset control system, that is the control input of the system is designed to be fault-tolerant due to the presence of actuator faults. Further, the fractional-order dependent linear matrix inequalities (LMIs) will be derived by an appropriate choice of a Lyapunov function and other matrix inequalities. Thus, by solving the obtained LMIs, the gain values can be obtained guaranteeing the stabilization of the closed loop system with prescribed disturbance attenuation level. Later, the theoretical results are authenticated by providing numerical simulation results.

Robust Stability Analysis of Smith Predictor Based Interval Fractional-Order Control Systems: A Case Study in Level Control Process

The robust stability study of the classic Smith predictor-based control system for uncertain fractional-order plants with interval time delays and interval coefficients is the emphasis of this work. Interval uncertainties are a type of parametric uncertainties that cannot be avoided when modeling real-world plants. Also, in the considered Smith predictor control structure it is supposed that the controller is a fractional-order proportional integral derivative (FOPID) controller. To the best of the authors' knowledge, no method has been developed until now to analyze the robust stability of a Smith predictor based fractional-order control system in the presence of the simultaneous uncertainties in gain, time-constants, and time delay. The three primary contributions of this study are as follows: i) a set of necessary and sufficient conditions is constructed using a graphical method to examine the robust stability of a Smith predictor-based fractional-order control system—the proposed method explicitly determines whether or not the FOPID controller can robustly stabilize the Smith predictor-based fractional-order control system; ii) an auxiliary function as a robust stability testing function is presented to reduce the computational complexity of the robust stability analysis; and iii) two auxiliary functions are proposed to achieve the control requirements on the disturbance rejection and the noise reduction. Finally, four numerical examples and an experimental verification are presented in this study to demonstrate the efficacy and significance of the suggested technique.

Command-filtered adaptive containment control of fractional-order multi-agent systems via event-triggered mechanism

In this paper, the containment control problem of a class of fractional-order nonlinear multi-agent systems is studied, in which the multi-agent system contains unmeasured states and system nonlinearity. An adaptive neural network backstepping controller combined with event-triggered mechanism is proposed to ensure all followers can converge to the convex hull spanned by the leaders. The command filter is introduced into the proposed fractional-order control system to obtain fractional derivatives for virtual control functions continuously and avoid “explosion of complexity.” From the Lyapunov stability theory, all the signals can remain semi-global uniformly ultimately bounded in the closed-loop system. Numerical example and simulation results confirm the feasibility of the proposed control method.

Finiteness conditions for performance indices in generalized fractional-order systems defined based on the regularized Prabhakar derivative

In the present paper, we introduce the generalized fractional-order control systems with the regularized Prabhakar fractional derivative and investigate the BIBO stability for this kind of systems. For such kind of systems with step functions at their inputs, we establish some conditions imposed on the parameters of the corresponding fractional-order transfer functions under which the steady state error is zero and the performance indices are finite. Further, in order to examine the analytical obtained results, several generalized fractional-order transfer functions are considered. Then, we show that the parameters of the regularized Prabhakar fractional derivative play significant roles in finiteness of the performance indices in the systems.

Formal Verification of Fractional-Order PID Control Systems Using Higher-Order Logic

Fractional-order PID control is a landmark in the development of fractional-order control theory. It can improve the control precision and accuracy of systems and achieve more robust control results. As a theorem-proving formal verification method, it can be applied to an arbitrary system represented by a mathematical model. It is the ideal verification method because it is not subject to limits on state numbers. This paper presents the higher-order logic (HOL) formal verification and modeling of fractional-order PID controller systems. Firstly, a fractional-order PID controller was designed. The accuracy of fractional-order PID control can be supported by simulation, comparing integral-order PID controls. Secondly, the superior property of fractional-order PID control is validated via higher-order logic theorem proofs. An important basic property, the relationship between fractional-order differential calculus and integral-order differential calculus, was analyzed via a higher-order logic theorem proof. Then, the relations between the fractional-order PID controller and integral-order PID controller were verified based on the fractional-order Grünwald–Letnikov definition for higher-order logic theorem proofs. Formalization models of the fractional-order PID controller and the fractional-order closed-loop control system were established. Finally, the stability of the fractional-order control systems was verified based on established formal models and theorems. The results show that the fractional-order PID controllers can be conducive to the control performance of control systems, and the higher-order logic formal verification method can ensure the reliability and security of fractional-order control systems.

Fractional-order control for nano-positioning of piezoelectric actuators

Piezoelectric actuators are widely used in the field of high-precision positioning control. However, due to the inherent hysteresis nonlinearity of piezoelectric actuators, it can seriously affect their accuracy in tracking and positioning systems, and even cause system instability. In order to suppress the hysteresis characteristics of piezoelectric, based on the analysis of piezoelectric characteristics and the characteristics of classical Bouc–Wen model, a Bouc–Wen model is adopted to describe the hysteresis characteristics of piezoelectric actuators, an artificial bee colony (ABC) algorithm is used to identify Bouc–Wen. The maximum error of the model is 2.49%. According to the established hysteresis model, its inverse model is obtained as the feedforward controller, and the inverse model compensation method is used to realize the open-loop control of the system. Since the feedforward controller based on the inverse model cannot eliminate the modeling errors and external disturbances of the inverse model, a fractional-order controller is designed on the basis of inverse compensation, and a compound control method combining feedforward control and feedback control is adopted. Experiments show that the fractional-order control system has fast response speed, small steady-state error and simple structure compared with other nonlinear controllers, positioning error is about 3%.