An innovative fixed-pole numerical approximation for fractional order systems

A novel numerical approximation scheme is proposed for fractional order systems by the concept of identification. An identical equation is derived firstly, from which one can obtain the exact state space model of fractional order systems. It reveals the nature of the approximation problem, and then provides an effective scheme to obtain the desired model. This research project also focuses on solving a knotty but crucial issue, i.e., the initial value problem of fractional order systems. The results generated by the study prove that it can reduce to the Caputo case by selecting some specific initial values. A careful simulation study is reported to illustrate the effectiveness of the proposed scheme. To exhibit the superiority clearly, the results are compared with that of the published fixed-pole finite model method.     

Robust stability analysis of fractional order interval polynomials

The paper deals with the robust stability analysis of a Fractional Order Interval Polynomial (FOIP) family. Some new results are presented for testing the Bounded Input Bounded Output (BIBO) stability of dynamical control systems whose characteristic polynomials are fractional order polynomials with interval uncertainty structure. It is shown that the Kharitonov theorem is not applicable for this type of polynomial. A procedure is given for computation of the value set of FOIP. Based on the value set, an algorithm is presented for testing the stability of FOIP. The results presented in the paper are useful for the analysis and design of Fractional Order Interval Control Systems (FOICS). Examples are given to show how the proposed method can be used to assess the effects of parametric variations on the stability in feedback loops with fractional order interval transfer functions.     

An efficient algorithm for low-order direct discrete-time implementation of fractional order transfer functions

Fractional order systems become increasingly popular due to their versatility in modelling and control applications across various disciplines. However, the bottleneck in deploying these tools in practice is related to their implementation on real-life systems. Numerical approximations are employed but their complexity no longer match the attractive simplicity of the original fractional order systems. This paper proposes a low-order, computationally stable and efficient method for direct approximation of general order (fractional order) systems in the form of discrete-time rational transfer functions, e.g. processes, controllers. A fair comparison to other direct discretization methods is presented, demonstrating its added value with respect to the state of art.     

A novel auto-tuning method for fractional order PI/PD controllers

Fractional order PID controllers benefit from an increasing amount of interest from the research community due to their proven advantages. The classical tuning approach for these controllers is based on specifying a certain gain crossover frequency, a phase margin and a robustness to gain variations. To tune the fractional order controllers, the modulus, phase and phase slope of the process at the imposed gain crossover frequency are required. Usually these values are obtained from a mathematical model of the process, e.g. a transfer function. In the absence of such model, an auto-tuning method that is able to estimate these values is a valuable alternative. Auto-tuning methods are among the least discussed design methods for fractional order PID controllers. This paper proposes a novel approach for the auto-tuning of fractional order controllers. The method is based on a simple experiment that is able to determine the modulus, phase and phase slope of the process required in the computation of the controller parameters. The proposed design technique is simple and efficient in ensuring the robustness of the closed loop system. Several simulation examples are presented, including the control of processes exhibiting integer and fractional order dynamics.     

Efficient method for time-domain simulation of the linear feedback systems containing fractional order controllers

One main approach for time-domain simulation of the linear output-feedback systems containing fractional-order controllers is to approximate the transfer function of the controller with an integer-order transfer function and then perform the simulation. In general, this approach suffers from two main disadvantages: first, the internal stability of the resulting feedback system is not guaranteed, and second, the amount of error caused by this approximation is not exactly known. The aim of this paper is to propose an efficient method for time-domain simulation of such systems without facing the above mentioned drawbacks. For this purpose, the fractional-order controller is approximated with an integer-order transfer function (possibly in combination with the delay term) such that the internal stability of the closed-loop system is guaranteed, and then the simulation is performed. It is also shown that the resulting approximate controller can effectively be realized by using the proposed method. Some formulas for estimating and correcting the simulation error, when the feedback system under consideration is subjected to the unit step command or the unit step disturbance, are also presented. Finally, three numerical examples are studied and the results are compared with the Oustaloup continuous approximation method.     

Robust Takagi-Sugeno fuzzy control for fractional order hydro-turbine governing system

A robust fuzzy control method for fractional order hydro-turbine governing system (FOHGS) in the presence of random disturbances is investigated in this paper. Firstly, the mathematical model of FOHGS is introduced, and based on Takagi-Sugeno (T-S) fuzzy rules, the generalized T-S fuzzy model of FOHGS is presented. Secondly, based on fractional order Lyapunov stability theory, a novel T-S fuzzy control method is designed for the stability control of FOHGS. Thirdly, the relatively loose sufficient stability condition is acquired, which could be transformed into a group of linear matrix inequalities (LMIs) via Schur complement as well as the strict mathematical derivation is given. Furthermore, the control method could resist random disturbances, which shows the good robustness. Simulation results indicate the designed fractional order T-S fuzzy control scheme works well compared with the existing method.     

Adaptive two-degree-of-freedom PI for speed control of permanent magnet synchronous motor based on fractional order GPC

In this paper, an adaptive two-degree-of-freedom (2Dof) proportional-integral (PI) controller is proposed for the speed control of permanent magnet synchronous motor (PMSM). Firstly, an enhanced just-in-time learning technique consisting of two novel searching engines is presented to identify the model of the speed control system in a real-time manner. Secondly, a general formula is given to predict the future speed reference which is unavailable at the interval of two bus-communication cycles. Thirdly, the fractional order generalized predictive control (FOGPC) is introduced to improve the control performance of the servo drive system. Based on the identified model parameters and predicted speed reference, the optimal control law of FOGPC is derived. Finally, the designed 2Dof PI controller is auto-tuned by matching with the optimal control law. Simulations and real-time experimental results on the servo drive system of PMSM are provided to illustrate the effectiveness of the proposed strategy.     

Author’s reply: A comment on the “Robust stability analysis of fractional order interval polynomials”, by Nusret Tan et al.

This author’s reply addresses the comment given in the note mentioned in the title. Theorem 3 given in Tan et al. (2009) [1] uses zero exclusion principle for the stability analysis of Fractional Order Interval Polynomial (FOIP). We show that the constant degree assumption is exist in the definition of zero exclusion principle. Although it has not been clearly stated in Tan et al. (2009) [1] that FOIP of Eq. (1) is a constant degree polynomial, this condition is implicit in the zero exclusion principle. Therefore, Theorem 3 is true under the constant degree assumption which is a requirement for zero exclusion principle.     

Estimation and disturbance rejection performance for fractional order fuzzy systems

This paper gives attention to the issues of output tracking and disturbance rejection performance for a class of fractional order Takagi–Sugeno fuzzy systems in the presence of time-varying delay and unknown external disturbances. More specifically, a new configuration of a fractional order modified repetitive controller that incorporates an improved equivalent-input-disturbance estimator and gain fluctuations in its design is proposed to perform disturbance rejection for the addressed system. By introducing a continuous frequency distributed equivalent model and using the Lyapunov–Krasovskii stability theory, a new set of sufficient conditions ensuring robust asymptotic stability of the resulting closed-loop system is obtained in the framework of linear matrix inequalities. Finally, a numerical example is presented to validate the developed theoretical results, where it is shown that the obtained conditions could force the considered system output to exactly track the given any kind of reference signal by compensating the unknown external disturbance.     

Study of the fractional order proportional integral controller for the permanent magnet synchronous motor based on the differential evolution algorithm

A tuning method of the fractional order proportional integral speed controller for a permanent magnet synchronous motor is proposed in this paper. Taking the combination of the integral of time and absolute error and the phase margin as the optimization index, the robustness specification as the constraint condition, the differential evolution algorithm is applied to search the optimal controller parameters. The dynamic response performance and robustness of the obtained optimal controller are verified by motor speed-tracking experiments on the motor speed control platform. Experimental results show that the proposed tuning method can enable the obtained control system to achieve both the optimal dynamic response performance and the robustness to gain variations.     

Dual mode adaptive fractional order PI controller with feedforward controller based on variable parameter model for quadruple tank process

The Quadruple Tank Process (QTP) is a well-known benchmark of a nonlinear coupled complex MIMO process having both minimum and nonminimum phase characteristics. This paper presents a novel self tuning type Dual Mode Adaptive Fractional Order PI controller along with an Adaptive Feedforward controller for the QTP. The controllers are designed based on a novel Variable Parameter Transfer Function model. The effectiveness of the proposed model and controllers is tested through numerical simulation and experimentation. Results reveal that the proposed controllers work successfully to track the reference signals in all ranges of output. A brief comparison with some of the earlier reported similar works is presented to show that the proposed control scheme has some advantages and better performances than several other similar works.     

An approach to design controllers for MIMO fractional-order plants based on parameter optimization algorithm

The parameter optimization method for multivariable systems is extended to the controller design problems for multiple input multiple output (MIMO) square fractional-order plants. The algorithm can be applied to search for the optimal parameters of integer-order controllers for fractional-order plants with or without time delays. Two examples are given to present the controller design procedures for MIMO fractional-order systems. Simulation studies show that the integer-order controllers designed are robust to plant gain variations.