Integer Order Approximation Research Articles

The Integer–Order Approximation of Fractional–Order Systems in The Loewner Framework

Abstract This paper deals with the integer–order (finite–dimensional) approximation of a fractional–order (infinite–dimensional) system using the interpolation approach in the Loewner framework. First given a set of interpolation frequencies and the corresponding values of the original noninteger–order transfer function, a Loewner matrix and a shifted Loewner matrix are created. Then, from these matrices an integer–order model in state–space descriptor form is constructed. Its transfer function matches the original system transfer function at the given frequencies. To avoid singularity problems related to data redundancy, a truncated singular value decomposition may finally be applied. Numerical simulations show that the suggested approach to fractional–order system approximation compares favourably with alternative techniques recently presented in the literature to the same purpose.

New integer-order approximations of discrete-time non-commensurate fractional-order systems using the cross Gramian

This paper presents a new approach to modeling of linear time-invariant discrete-time non-commensurate fractional-order single-input single-output state space systems by means of the Balanced Truncation and Frequency Weighted model order reduction methods based on the cross Gramian. These reduction methods are applied to the specific rational (integer-order) FIR-based approximation to the fractional-order system, which enables to introduce simple, analytical formulae for determination of the cross Gramian of the system. This leads to significant decrease of computational burden in the reduction algorithm. As a result, a rational and relatively low-order state space approximator for the fractional-order system is obtained. A simulation experiment illustrates an efficiency of the introduced methodology in terms of high approximation accuracy and low time complexity of the proposed method.

A New Integer Order Approximation Table for Fractional Order Derivative Operators

There is considerable interest in the study of fractional order calculus in recent years because real world can be modelled better by fractional order differential equation. However, computing analytical time responses such as unit impulse and step responses is still a difficult problem in fractional order systems. Therefore, advanced integer order approximation table which gives satisfying results is very important for simulation and realization. In this paper, an integer order approximation table is presented for fractional order derivative operators (sa) where α ϵ R and 0 < α < 1 using exact time response functions computed with inverse Laplace transform solution technique. Thus, the time responses computed using the given table are almost the same with exact solution. The results are also compared with some well-known integer order approximation methods such as Oustaloup and Matsuda. It has been shown in numerical examples that the proposed method is very successful in comparison to Oustaloup’s and Matsuda’s methods.

Derivation of Analytical Inverse Laplace Transform for Fractional Order Integrator

There is considerable interest in the study of fractional order derivative integrator but obtaining analytical impulse and step responses is a difficult problem. Therefore all methods reported on to date use approximations for the fractional derivative/integrator both for analytical based computations and more relevantly in simulation studies. In this paper, an analytical formula is first derived for the inverse Laplace transform of fractional order integrator, 1/sα where α∈R and 0<α<1 using Stirling’s formula and Gamma function. Then, the analytical step response of fractional integrator has been computed from the derived impulse response of 1/sα. The obtained analytical formulas for impulse and step responses of fractional order integrator are exact results except the very small error due to the neglected terms of Stirling’s series. The results are compared with some well known integer order approximation methods and Grunwald-Letnikov (GL) approximation technique. It has been shown via numerical examples that the presented method is very successful according to other methods.

An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators

This paper introduces an integer order approximation method for numerical implementation of fractional order derivative/integrator operators in control systems. The proposed method is based on fitting the stability boundary locus (SBL) of fractional order derivative/integrator operators and SBL of integer order transfer functions. SBL defines a boundary in the parametric design plane of controller, which separates stable and unstable regions of a feedback control system and SBL analysis is mainly employed to graphically indicate the choice of controller parameters which result in stable operation of the feedback systems. This study reveals that the SBL curves of fractional order operators can be matched with integer order models in a limited frequency range. SBL fitting method provides straightforward solutions to obtain an integer order model approximation of fractional order operators and systems according to matching points from SBL of fractional order systems in desired frequency ranges. Thus, the proposed method can effectively deal with stability preservation problems of approximate models. Illustrative examples are given to show performance of the proposed method and results are compared with the well-known approximation methods developed for fractional order systems. The integer-order approximate modeling of fractional order PID controllers is also illustrated for control applications.

$$\fancyscript{H}_2$$ H 2 -norm computation of a class of implicit fractional transfer functions: application to approximation by integer order models

This paper focuses on the $$\fancyscript{H}_2$$ -norm of a class of implicit fractional order transfer functions well suited to describe the input–output behaviour of diffusive systems. First, the analytical expression of the $$\fancyscript{H}_2$$ -norm of this kind of transfer function is established. This result is then used to evaluate the quality of the integer order approximation of such an implicit fractional transfer function.

Approximation of a fractional order model by an integer order model: a new approach taking into account approximation error as an uncertainty

In order to solve some analysis or control problems for fractional order models, integer order approximations are often used. However, in many works, approximation error is not taken into account, leading to results that cannot be guaranteed for the initial fractional order model. The objective of the paper is thus to provide a new methodology that takes into account approximation error and leads to rewriting the fractional order model as an uncertain integer order model.

A rational approximate method to fractional order systems

This paper presents an approximate method for general fractional order dynamic systems. Firstly, a novel piecewise approximate method is proposed for fractional order integrator based on its frequency distributed mode. Based on the above method, an integer order approximation system is constructed to approximate a fractional order system. Theoretical analysis results show that the proposed method can achieve much better performance than the existing schemes for a given order in an interested frequency range. The advantage of the proposed method lies in that the resulting system are standard integer order system, which facilitates systematic stability analysis and controller synthesis in view of the well-developed linear or nonlinear system theory. Numerical simulations are presented to illustrate the effectiveness of the proposed approach in the end.

Design Of Multivariable Fractional Order Pid Controller Using Covariance Matrix Adaptation Evolution Strategy

Abstract This paper presents an automatic tuning of multivariable Fractional-Order Proportional, Integral and Derivative controller (FO-PID) parameters using Covariance Matrix Adaptation Evolution Strategy (CMAES) algorithm. Decoupled multivariable FO-PI and FO-PID controller structures are considered. Oustaloup integer order approximation is used for the fractional integrals and derivatives. For validation, two Multi-Input Multi- Output (MIMO) distillation columns described byWood and Berry and Ogunnaike and Ray are considered for the design of multivariable FO-PID controller. Optimal FO-PID controller is designed by minimizing Integral Absolute Error (IAE) as objective function. The results of previously reported PI/PID controller are considered for comparison purposes. Simulation results reveal that the performance of FOPI and FO-PID controller is better than integer order PI/PID controller in terms of IAE. Also, CMAES algorithm is suitable for the design of FO-PI / FO-PID controller.

H2-norm of fractional transfer functions of implicit type of the first kind

This paper studies the H2-norm, or impulse response energy, of fractional transfer functions of implicit type. The analytical expression of the H2-norm is first derived for an elementary fractional transfer function of the first kind with a single real pole. Series connection of such a transfer function with a pure fractional integrator and with another implicit transfer function of the first kind are then studied. Results developed in the paper are finally used to derive a criterion to evaluate the quality of an integer order approximation for an implicit type fractional order model of the first kind.

A method for the integer-order approximation of fractional-order systems

A procedure for approximating fractional-order systems by means of integer-order state-space models is presented. It is based on the rational approximation of fractional-order operators suggested by Oustaloup. First, a matrix differential equation is obtained from the original fractional-order representation. Then, this equation is realized in a state-space form that has a sparse block-companion structure. The dimension of the resulting integer-order model can be reduced using an efficient algorithm for rational L2 approximation. Two numerical examples are worked out to show the performance of the suggested technique.

Reduced Order Approximation of MIMO Fractional Order Systems

A new two-stage method for reduced integer order approximation of fractional multiple-input, multiple-output (MIMO) systems is proposed. In the first stage, the transfer function matrix (TFM) Gf(s) of the given fractional order MIMO system is obtained and an integer order approximate TFM R(s) is formed by applying an existing approximation method to each fractional order transfer function (FOTF) of Gf(s). In the second stage, a reduced order state space model is formed. The system matrix of the reduced order system is constructed by selecting the dominant poles from the intermediate high integer order model R(s). The input and output matrices are found by matching approximate time moments and Markov parameters of the final reduced order model and the original system. The proposed method has been illustrated by an example.

Approximate Realization of Fractional-Order 2-D IIR Frequency-Planar Filters

Starting with the established concepts of multi-dimensional network resonance, practical-bounded-input-bounded-output (BIBO) stability of multi-dimensional discrete systems, and the integer order approximation of fractional order filters, a new method for the realization of integer-order approximations of 2-D fractional-order systems having frequency-planar beam shaped passbands is proposed. The method results in 2-D infinite impulse response (IIR) filters having nonseparable denominators in their transfer-functions while having guaranteed practical-BIBO stability for zero initial conditions in the corresponding discrete system. The resulting integer-order filters are shown to approximate 2-D fractional order frequency-planar beam shapes in both magnitude and phase leading to a new theoretical tool for use in the emerging area of multi-dimensional fractional-order circuits and systems. Although examples are provided for frequency-planar filters in two dimensions, the proposed methods are equally applicable to other types of transfer-functions having prototypes based on resistively-terminated passive networks defined over any integer number of dimensions.

Fractional order modeling and control for permanent magnet synchronous motor velocity servo system

This paper presents the application of fractional order system on modeling the permanent magnet synchronous motor (PMSM) velocity servo system. The traditional integer order model of the PMSM velocity system is extended to fractional order one in this work. In order to identify the parameters of the proposed fractional order model, an integer order approximation of the fractional order operator is applied and a state-space structure is presented for using the output-error identification algorithm. In real-time PMSM velocity servo plant, the fractional order model is identified according to some experimental tests using the presented algorithm. Two proportional integral (PI) controllers are designed for velocity servo using a simple scheme according to the identified fractional order model and the traditional integer order one, respectively. The experimental test performance using these two designed PI controllers is compared to demonstrate the advantage of the proposed fractional order model of the PMSM velocity system.

Optimal Tuning for Fractional-Order Controllers: An Integer-Order Approximating Filter Approach

This paper offers a systematic framework for the design of suboptimal integer-order controllers based on fractional-order structures. The proposed approach is built upon the integer-order approximations that are traditionally used to implement fractional-order controllers after they are designed. Accordingly, the fractional-order structures are exploited to derive a suitable parameterization for a fixed-structure integer-order controller. The parameters, which describe the structure of the fixed-order integer controller, are the same as those used to describe the original fractional structure. Optimal tuning is then performed in the parameter space of the fractional structure and the resultant set of optimum parameters will determine the optimal integer-order controller. The proposed approach outperforms traditional implementations of optimal fractional controllers and presents an alternative that attempts to capture merits of both fractional-order and integer-order structures.

The Fractional-Order Control Method of Dynamic Elastoplastic Responses of Benchmark Buildings

The elastoplastic phenomenon of the structures will be advent under the action of the strong earthquakes, the presented research on the vibration control of the ones is chiefly concentrated on fuzzy and neuro-controller with the expense of bigger energy. In the paper, the 3-storey benchmark building is used as research object, the control strategy of vibration system with fractional-order is studied. The control method based on acceleration responses output is emphatically concerned as it is usually adopted in most actual application of active vibration control techniques. The concrete courses include the following three steps: the integer-order approximation of fraction-order, the transfer function reduction in frequency-domain which base on Pade approximation, the controller design and simulation. In the last, an example is used to show the feasibility of proposed method.

Rules for selecting the parameters of Oustaloup recursive approximation for the simulation of linear feedback systems containing PI λD μ controller

Oustaloup recursive approximation (ORA) is widely used to find a rational integer-order approximation for fractional-order integrators and differentiators of the form s v , v ∈ (−1, 1). In this method the lower bound, the upper bound and the order of approximation should be determined beforehand, which is currently performed by trial and error and may be inefficient in some cases. The aim of this paper is to provide efficient rules for determining the suitable value of these parameters when a fractional-order PID controller is used in a stable linear feedback system. Two numerical examples are also presented to confirm the effectiveness of the proposed formulas.

Stabilizing fractional-order PI and PD controllers: An integer-order implemented system approach

This paper proposes a scheme for computing the stable regions from which the parameters of fractional-order proportional–integral (PI) and proportional–derivative (PD) controllers can be selected. In particular, a previously investigated method for plotting the stability regions of PI λ and PD μ controllers is extended to their integer-order approximations. Next, the two stability regions are compared and their overlapping sections, referred to as the implementable stability region, are derived. In order for the final closed-loop system to be stable, the designer must choose the controller parameters from the overlapping section of the two stability regions. Simulation results for two example systems are provided to illustrate the effectiveness of the proposed method. The authors believe that the devised scheme can facilitate the design procedure for fractional-order PI and PD controllers.

Computation of frequency responses for uncertain fractional-order systems

This paper proposes an algorithm for computation of Bode magnitude and phase responses for a large class of linear uncertain fractional-order systems using interval constraint propagation technique. It is first shown that the problem of finding the magnitude and phase of the uncertain fractional-order system over a frequency range can be formulated as a interval constraint satisfaction problem and then solved using branch and prune algorithm. The algorithm guarantees that the magnitude and phase responses are computed to prescribed accuracy. The other advantage of the method is that the magnitude and phase can be computed without any kind of integer-order approximation of the given fractional-order system. The proposed algorithm is demonstrated on three examples including a practical application of a gas turbine plant.