Control Of Fractional-order Systems Research Articles

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%.

A Novel Fractionalized PID controller Using The Sub-optimal Approximation of FOTF

In the last two decades, fractional calculus has been rediscovered by scientists and engineers and applied in an increasing number of fields, namely in the area of control theory. Recently, many research works have focused on fractional order control (FOC) and fractional systems. It has proven to be a good mean for improving the plant dynamics with respect to response time and disturbance rejection. In This work we use the Sub-optimal Approximation of fractional order transfer function to design the parameters of PID controller and we study the performance analysis of fractionalized PID controller over integer order PID controller.

Robust H∞ Control for Fractional Order Systems with Order α (0 < α < 1)

In the paper, H∞ and robust H∞ control for fractional order systems (FOS) with order 0<α<1 are studied. Firstly, necessary and sufficient conditions of H∞ control and state feedback controller design are proposed. Then, robust H∞ control for FOS with uncertainty is studied, and state feedback controller is designed. These conditions are based on linear matrix inequalities (LMI) and can be easily solved by the LMI toolbox. Finally, the effectiveness of these conditions is verified by two numerical examples.

An experimental analog circuit realization of Matsuda’s approximate fractional-order integral operators for industrial electronics

Analog circuit realization of fractional order (FO) elements is a significant step for the industrialization of FO control systems because of enabling a low-cost, electric circuit realization by means of standard industrial electronics components. This study demonstrates an effective operational amplifier-based analog circuit realization of approximate FO integral elements for industrial electronics. To this end, approximate transfer function models of FO integral elements, which are calculated by using Matsuda’s approximation method, are decomposed into the sum of low-pass filter forms according to the partial fraction expansion. Each partial fraction term is implemented by using low-pass filters and amplifier circuits, and these circuits are combined with a summing amplifier to compose the approximate FO integral circuits. Widely used low-cost industrial electronics components, which are LF347N opamps, resistor and capacitor components, are used to achieve a discrete, easy-to-build analog realization of the approximate FO integral elements. The performance of designed circuit is compared with performance of Krishna’s FO circuit design and performance improvements are shown. The study presents design, performance validation and experimental verification of this straightforward approximate FO integral realization method.

Current PIλ Control of the Single-Phase Grid Inverter

In a grid-connected power generation system, the grid-connected current of the inverter is sensitive to nonlinear factors such as periodic disturbance of grid voltage, which results in grid-connected current waveform distortion. By establishing a single-phase photovoltaic grid-connected inverter control system model, designing an inverse current fractional-order PI (PIλ or FO-PI) controller and the dynamic and steady-state performance, antidisturbance and grid connection inversion characteristics of the system are simulated and compared under the action of the integer-order PI controller and fractional-order PI controller. The quality of the inverter grid-connected current is analyzed by using the fast Fourier transform (FFT). The simulation results show that the fractional-order control system can reduce the total harmonic distortion (THD) of the grid-connected current and dynamic performance and antidisturbance ability of the improving system while satisfying the steady-state performance indexes.

Optimal State Control of Fractional Order Differential Systems: The Infinite State Approach

Optimal control of fractional order systems is a long established domain of fractional calculus. Nevertheless, it relies on equations expressed in terms of pseudo-state variables which raise fundamental questions. So in order remedy these problems, the authors propose in this paper a new and original approach to fractional optimal control based on a frequency distributed representation of fractional differential equations called the infinite state approach, associated with an original formulation of fractional energy, which is intended to really control the internal system state. In the first step, the fractional calculus of variations is revisited to express appropriate Euler Lagrange equations. Then, the quadratic optimal control of fractional linear systems is formulated. Thanks to a frequency discretization technique, the previous theoretical equations are converted into an equivalent large dimension integer order system which permits the implementation of a feasible optimal solution. A numerical example illustrates the validity of this new approach.

Controllability for Retarded Semilinear Neutral Control Systems of Fractional Order in Hilbert Spaces

In this paper, we discuss the approximate controllability for a class of retarded semilinear neutral control systems of fractional order by investigating the relations between the reachable set of the semilinear retarded neutral system of fractional order and that of its corresponding linear system. The research direction used here is to find the conditions for nonlinear terms so that controllability is maintained even in perturbations. Finally, we will show a simple example to which the main result can be applied.

External stability of Caputo fractional-order nonlinear control systems: advances in the Lyapunov function method

In this work, we present an important advance in the Lyapunov function method for the external stability of Caputo fractional-order nonlinear control systems. Benefiting from the very recent development of fractional calculus, the continuous differentiability imposed on inputs and vector field functions as required by the existing Lyapunov function method is successfully removed. This reduction in conditions improves the existing results to a large extent.

An active control for hydrostatic journal bearing using optimization algorithms

Purpose The purpose of this paper is to improve static/dynamic characteristics of active-controlled hydrostatic journal bearing by using fractional order control techniques and optimizing algorithms. Design/methodology/approach Active lubrication has ability to overcome the unpredictable harsh environmental conditions which often lead to failure of capillary controlled traditional hydrostatic journal bearing. The research develops a mathematical model for a servo feedback-controlled hydrostatic journal bearing and dynamics of model is analyzed with different control techniques. The fractional-order PID control system is tuned by using particle swarm optimization and Nelder mead optimization techniques with the help of using multi-objective performance criteria. Findings The results of the current research are compared with previously published theoretical and experimental results. The proposed servo-controlled active bearing system is studied under a number of different dynamic situations and constraints of variable spindle speed, external load, temperature changes (viscosity) and variable bearing clearance (oil film thickness). The simulation results show that the proposed system has better performance in terms of controllability, faster response, stability, high stiffness and strong resistance. Originality/value This paper develops an accurate mathematical model for servo-controlled hydrostatic bearing with fractional order controller. The results are in excellent agreement with previously published literature. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-07-2020-0272

Social spider optimisation based identification and optimal control of fractional order system

Fractional order derivatives and integrals are infinite-dimensional operators and non-local in time. Currently, the researchers are working on the solution of the fractional optimal control problem...

Optimal fractional order control for nonlinear systems represented by the Euler-Lagrange formulation

In this paper a novel control strategy is shown for the control of fractional order systems established in the Euler-Lagrange formulation. The objective is to find an optimal fractional order law w...

Social spider optimisation based identification and optimal control of fractional order system

Social spider optimisation based identification and optimal control of fractional order system

The Microcontroller Implementation of the Basic Fractional-Order Element

The paper presents the implementation of the basic fractional order element sγ on the STM32 microcontroller platform. The implementation employs the typical CFE and FOBD approximations, the accuracy of approximation as well as duration of calculations are experimentally tested. Microcontroller implementation of fractional order elements is known; however, real-time tests of such implementations have been not presented yet. Results of experiments show that both methods can be implemented at the considered platform. The FOBD approximation is more accurate, but the CFE one is faster. The presented experimental results prove that the STM32F7 family processor could be used to develop the embedded fractional-order control systems for a broad class of linear and nonlinear dynamic systems. This is crucial during the implementation of the fractional-order control in the hard real-time or embedded systems.

On the approximate inverse Laplace transform of the transfer function with a single fractional order

The history of fractional calculus dates back to 1600s and it is almost as old as classical mathematics. Although many studies have been published on fractional-order control systems in recent years, there is still a lack of analytical solutions. The focus of this study is to obtain analytical solutions for fractional order transfer functions with a single fractional element and unity coefficient. Approximate inverse Laplace transformation, that is, time response of the basic transfer function, is obtained analytically for the fractional order transfer functions with single-fractional-element by curve fitting method. Obtained analytical equations are tabulated for some fractional orders of [Formula: see text]. Moreover, a single function depending on fractional order alpha has been introduced for the first time using a table for [Formula: see text]. By using this table, approximate inverse Laplace transform function is obtained in terms of any fractional order of [Formula: see text] for [Formula: see text]. Obtained analytic equations offer accurate results in computing inverse Laplace transforms. The accuracy of the method is supported by numerical examples in this study. Also, the study sets the basis for the higher fractional-order systems that can be decomposed into a single (simpler) fractional order systems.

Robust D-Stabilization Analysis of Fractional-Order Control Systems With Complex and Linearly Dependent Coefficients

This article focuses on the robust D-stabilization analysis of fractional-order control systems where each of the system and the controller may be of fractional order. The coefficients of the system are considered as complex linear functions of interval uncertain parameters, so this article deals with fractional-order polytopic systems. First, a necessary and sufficient condition is introduced for the robust D-stabilization of the closed-loop control system based on the zero exclusion condition and the value set concept. Then, the geometric pattern of the value set of the characteristic polynomial is obtained analytically using the exposed vertices. Second, a function is presented to check the introduced condition. Third, the transition points, at which the exposed vertices of the value set may change, are derived to reduce the computational burthen. Fourth, a new function is provided to determine the D-stability robustness radius (margin) of these control systems, that is, determining bounds on the uncertain parameters such that the control system remains D-stable. The achieved results are applicable to systems of incommensurate order and consequently of commensurate order. Finally, numerical simulations and practical experiments based on a three-degrees-of-freedom flight simulator turn table are conducted to illustrate the achieved results.

Multidimensional scaling analysis of generalized mean discrete-time fractional order controllers

The dynamics of discrete-time fractional order control systems depends on the method used for implementing the fractional derivatives and integrals. A reliable numerical approach adopts the generalized mean of the continuous to discrete conversion. The extra freedom provided by the proposed method must be carefully optimized by the user. This paper investigates the use of multidimensional scaling for evaluating and visualizing the performance of generalized mean discrete-time fractional controllers. Two alternative performance indices are adopted for comparing the time and frequency responses of the controlled system when adopting different combinations of the parameters. Numerical experiments with a fractional PID and two linear plants demonstrate the feasibility of the method for comparing and visualizing multiple test cases.