Fractional Order Plants Research Articles

Disturbance Rejection for Fractional-Order Time-Delay Systems

This paper presents an equivalent-input-disturbance (EID-) based disturbance rejection method for fractional-order time-delay systems. First, a modified state observer is applied to reconstruct the state of the fractional-order time-delay plant. Then, a disturbance estimator is designed to actively compensate for the disturbances. Under such a construction of the system, by constructing a novel monochromatic Lyapunov function and using direct Lyapunov approach, the stability analysis and controller design algorithm are derived in terms of linear matrix inequality (LMI) technique. Finally, simulation results demonstrate the validity of the proposed method.

Robust stabilization criterion of fractional-order controllers for interval fractional-order plants

The robust stabilization criterion of fractional-order controllers for interval fractional-order plants is proposed in this study. For the interval uncertainties existing in the numerator and denominator of a transfer function with respect to a fractional-order plant, the vertices of the value set corresponding to the characteristic function are offered by the Minkowski Sum. This algorithm for getting the vertices avoids to calculate the redundant vertices, reducing the computational complexity. An auxiliary function reflecting the position relationship between the origin and the value set is defined to investigate the robust stability conditions. The lower and upper limits of the test frequency interval are offered to check the auxiliary function within a finite frequency. Supposing that the fractional-order controller can stabilize the nominal fractional-order plant, the stabilization criterion of the closed loop system with interval uncertainties is provided based on the auxiliary function and other two conditions. Finally, three illustrative examples with fractional-order controllers are given to validate the effectiveness of this robust stabilization criterion.

Graphical tuning method of FOPID controllers for fractional order uncertain system achieving robust ‐stability

SummaryThis paper focuses on the graphical tuning method of fractional order proportional integral derivative (FOPID) controllers for fractional order uncertain system achieving robust ‐stability. Firstly, general result is presented to check the robust ‐stability of the linear fractional order interval polynomial. Then some alternative algorithms and results are proposed to reduce the computational effort of the general result. Secondly, a general graphical tuning method together with some computational efficient algorithms are proposed to determine the complete set of FOPID controllers that provides ‐stability for interval fractional order plant. These methods will combine the results for fractional order parametric robust control with the method of FOPID ‐stabilization for a fixed plant. At last, two important extensions will be given to the proposed graphical tuning methods: determine the ‐stabilizing region for fractional order systems with two kinds of more general and complex uncertainty structures: multi‐linear interval uncertainty and mixed‐type uncertainties. Numerical examples are followed to illustrate the effectiveness of the method. Copyright © 2015 John Wiley & Sons, Ltd.

Robust Stability of Fractional Order Time-Delay Control Systems: A Graphical Approach

The paper deals with a graphical approach to investigation of robust stability for a feedback control loop with an uncertain fractional order time-delay plant and integer order or fractional order controller. Robust stability analysis is based on plotting the value sets for a suitable range of frequencies and subsequent verification of the zero exclusion condition fulfillment. The computational examples present the typical shapes of the value sets of a family of closed-loop characteristic quasipolynomials for a fractional order plant with uncertain gain, time constant, or time-delay term, respectively, and also for combined cases. Moreover, the practically oriented example focused on robust stability analysis of main irrigation canal pool controlled by either classical integer order PID or fractional order PI controller is included as well.

Model reference adaptive control in fractional order systems using discrete-time approximation methods

In this paper, model reference control of a fractional order system has been discussed. In order to control the fractional order plant, discrete-time approximation methods have been applied. Plant and reference model are discretized by Grünwald–Letnikov definition of the fractional order derivative using “Short Memory Principle”. Unknown parameters of the fractional order system are appeared in the discrete time approximate model as combinations of parameters of the main system. The discrete time MRAC via RLS identification is modified to estimate the parameters and control the fractional order plant. Numerical results show the effectiveness of the proposed method of model reference adaptive control.

A graphical tuning method of fractional order proportional integral derivative controllers for interval fractional order plant

In this paper, a novel graphical tuning method of fractional order proportional integral derivative (FOPID) controllers is proposed for a given interval fractional order plant family. Firstly, an approach is presented to solve the problem of robustly stabilizing the interval fractional order plant using FOPID controller. Moreover, some alternative methods are developed to reduce the computational burden of the presented approach. The results obtained here are general and strict proofs are given on these results. Secondly, a new approach is presented to calculate the complete sets of FOPID controller parameters which guarantee the specified H∞-norm constraint for the interval fractional order plant. The developed approach is convenient and flexible. Finally, a unified design framework is proposed. The aim of the unified design is to compute the biggest region which can simultaneously provide internal stability, maintain the classical gain and phase margin and guarantee the modern H∞-norm constraint for the interval fractional order plant. Examples are followed to illustrate the design procedure.

Limit Cycles in Nonlinear Systems with Fractional Order Plants

In recent years, there has been considerable interest in the study of feedback systems containing processes whose dynamics are best described by fractional order derivatives. Various situations have been cited for describing heat flow and aspects of bioengineering, where such models are believed to be superior. In many situations these feedback systems are not linear and information on their stability and the possibility of the existence of limit cycles is required. This paper presents new results for determining limit cycles using the approximate describing function method and an exact method when the nonlinearity is a relay characteristic.

The Fractional Brownian Motion Approach to Analysis of Fractional Control System with Non-Stationary Plant

In this study a fractional Brownian motion model is used to randomize integer-and fractional-order plant parameters. The analysis of these plants in systems with integer-order PID and fractional-order PIλDδ controllers is shown. The aim of the analysis is to compare robustness of both controllers. The quality of control system is measured by integral quality parameter. The fractional Brownian motion gives much flexibility in modeling non-stationary plants, the volatility and standard deviation of fractional Brownian motion can be easily adjusted.

Pareto optimal robust design of fractional-order PID controllers for systems with probabilistic uncertainties

In this paper, a multi-objective optimization approach is used for Pareto optimum design of robust integer-order and fractional-order PID controllers for both integer-order and fractional-order plants with parametric uncertainties. The aim of this paper is to compare the performances of those optimally-designed controllers dealing with parametric uncertainties of both integer-order and fractional-order plants. In this way, four combinations of plants and controllers have been considered in which five conflicting objective functions in both time and frequency domains are used in the Pareto design of both integer-order and fractional-order PID controllers. The results clearly show that effective trade-off design points can be compromisingly explored among the optimal robust integer-order and fractional-order PID controllers obtained using the methodology of this work. Furthermore, the superior robust stability and robust performance of the fractional-order PID controllers will be demonstrated in comparison with those of the integer-order PID controllers.

Tuning fractional order proportional integral controllers for fractional order systems

In this paper, two fractional order proportional integral controllers are proposed and designed for a class of fractional order systems. For fair comparison, the proposed fractional order proportional integral (FOPI), fractional order [proportional integral] (FO[PI]) and the traditional integer order PID (IOPID) controllers are all designed following the same set of the imposed tuning constraints, which can guarantee the desired control performance and the robustness of the designed controllers to the loop gain variations. This proposed design scheme offers a practical and systematic way of the controllers design for the considered class of fractional order plants. From the simulation and experimental results presented, both of the two designed fractional order controllers work efficiently, with improved performance comparing with the designed stabilizing integer order PID controller by the observation. Moreover, it is interesting to observe that the designed FO[PI] controller outperforms the designed FOPI controller following the proposed design schemes for the class of fractional order systems considered.

Discussion on: “Simple Fractional Order Model Structures and their Applications in Control System Design”

Fractional calculus is about the integration or differentiation of non-integer orders. The use of “fractional” is purely due to historical reasons [1]. Using fractional order differential equations is believed to be able to better characterize the nature around us. Using an integer order model is only for our own convenience. Depending on the scale onwhich we characterize the dynamics of a system, more and more evidences are found that using fractional order model is ubiquitous and unavoidable [1, 2, 3]. For dynamic systems and controls, there are four possibilities 1) Integer-order plant model with integer order controller; 2) Integer-order plant model with fractional order controller; 3) Fractional-order plant with integerorder controller; and 4) Fractional-order plant model with fractional-order controller [2]. Case-1 is our traditional (integer-order) control. An example of Case 2) can be found in [3] and an example of Case-4 can be found in [4]. Examples for Case-3 are abundant such as conventional integer-order controls for flexible structures whose transfer functions are irrational or of fractional order. It is interesting to note that, reaction curve based rough modeling technique, although widely used in process industry to get a first order plus delay time (FOPDT) for initial tuning of a working PID controller with techniques such as Ziegler-Nichols tuning rule, has not been put enough emphasis in control textbooks.Webelieve that, every practicing control engineer and control engineering educator should pay attention to the importance of reaction

Limit Cycle Prediction for Fractional Order Systems with Static Nonlinearities

This paper deals with limit cycle prediction for nonlinear systems which have fractional order transfer functions. Describing function method is used for the analysis of fractional order nonlinear systems. It is observed that the limit cycle frequencies obtained from the simulation of the fractional order nonlinear systems, agree with the theoretical values of the limit cycle frequencies obtained from Nyquist plots of the fractional order plant and complex plot of the negative inverse of describing function. The critical gain of the system is estimated from Nyquist plot. The fractional order nonlinear system becomes stable for the gain smaller than the critical gain value. System starts to oscillate for the gain bigger than the critical gain value.

Finding a fractional model from frequency and time responses

An existing method for identifying an integer model from frequency data, developed to be used when synthesising second-generation Crone controllers, is adapted to identify fractional order plants. The modification only allows models with poles but no zeros or zeros but no poles. Two application examples are given, one of them showing how the method can also be used when a time response, rather than a frequency response, is available.

Design of fractional-order PI λD μ controllers with an improved differential evolution

Differential evolution (DE) has recently emerged as a simple yet very powerful technique for real parameter optimization. This article describes an application of DE to the design of fractional-order proportional–integral–derivative (FOPID) controllers involving fractional-order integrator and fractional-order differentiator. FOPID controllers’ parameters are composed of the proportionality constant, integral constant, derivative constant, derivative order and integral order, and its design is more complex than that of conventional integer-order proportional–integral–derivative (PID) controller. Here the controller synthesis is based on user-specified peak overshoot and rise time and has been formulated as a single objective optimization problem. In order to digitally realize the fractional-order closed-loop transfer function of the designed plant, Tustin operator-based continuous fraction expansion (CFE) scheme was used in this work. Several simulation examples as well as comparisons of DE with two other state-of-the-art optimization techniques (Particle Swarm Optimization and binary Genetic Algorithm) over the same problems demonstrate the superiority of the proposed approach especially for actuating fractional-order plants. The proposed technique may serve as an efficient alternative for the design of next-generation fractional-order controllers.

GPC Control of a Fractional–Order Plant: Improving Stability and Robustness

This work deals with the use of Generalized Predictive Control (GPC) with fractional order plants. Low integer–order discrete approximations will be used as models to design the controllers. The stability and robustness of the closed loop system will be studied with the Nyquist criterion. Three techniques will be proposed to enhance robustness: the improvement of the model response at low frequencies, the use of the prefilter T(z−-1), and a new recommendation to choose two of the parameters (the control horizon Nu and the error weighting sequence λ) of the GPC controller.

On Fractional-Order QFT Controllers

We propose the synthesis of robust fractional-order controllers using the principles of quantitative feedback theory (QFT). The resulting controllers are called as fractional-order QFT controllers. To demonstrate the synthesis method, we synthesize proportional-integral-derivative (PID) and more general types of fractional-order QFT controllers for a fractional-order plant, a DC motor, and a multistage flash desalination process.