analog modeling of fractional switched order derivative using different switching schemes

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IEEE JOURNAL ON EMERGING AND SELECTED TOPICS IN CIRCUITS AND SYSTEMS 1

Analog Modeling of Fractional Switched OrderDerivative Using Different Switching Schemes

Dominik Sierociuk, Member, IEEE, Michal Macias, and Wiktor Malesza

Abstract—The paper presents comparison of two differentswitching schemes of variable order derivatives. The first one isadditive-switching scheme, when change of order is caused byadding a derivative at the beginning of the system. The secondone, introduced in this paper, is reductive-switching scheme,when change of order is caused by removing a derivative at thebeginning of the system. For both methods numerical schemesare given and analyzed. Based on presented switching schemesresults of analog modeling are presented. Results were obtainedusing analog approximations of integrators of orders 0.5 and 0.25.Finally, results of analog modeling were compared with numericalapproach.

Index Terms—Analog realization, fractional calculus, fractionalorder integrators, variable order differentiation.

I. INTRODUCTION

F RACTIONAL calculus can be viewed as generalizationof traditional integer order integration and differentiation

onto noninteger order. The idea of such generalization has beenmentioned in 1695 by Leibniz and l’Hospital. In the end of 19thcentury, Liouville and Riemann introduced first definition offractional derivative. However, it was not until late 1960s, whenthis idea drew attention of engineers. Theoretical background offractional calculus can be found in, e.g., [1]–[4]. Fractional cal-culus turned out to be a very useful tool for modeling behaviorof manymaterials and systems, especially those involving diffu-sion processes. One of such devices, that can be modeled moreefficiently by fractional calculus, is an ultracapacitor, whose ca-pacity can reach thousands of Farads. Fractional order modelsof these electronic storage devices were presented in [5], [6].Recently, the case with the order changing in time, started

to be intensively researched. The variable fractional order be-havior can be met for example in chemistry (when the proper-ties of a system are changing due to chemical reactions), elec-trochemistry, and other areas. In [7], experimental studies of anelectrochemical example of physical fractional variable ordersystem are presented. In [8], the variable order equations wereused to describe a history of drag expression. Papers [7], [9]present methods for numerical realization of fractional variableorder integrators or differentiators. The fractional variable order

Manuscript received February 15, 2013; revised May 17, 2013; acceptedJune 29, 2013. This work was supported by the Polish National ScienceCentre with the decision number DEC-2011/03/D/ST7/00260. This paper wasrecommended by Associate Editor B. Maundy.The authors are with the Institute of Control and Industrial Electronics,

Warsaw University of Technology, 00-662 Warsaw, Poland (e-mail: [email protected]; [email protected]; [email protected]).Digital Object Identifier 10.1109/JETCAS.2013.2273281

calculus can be also used to obtain variable order fractionalnoise [10], and to obtain new control algorithms [11]. Someproperties of such systems are presented in [12]. In [13], thevariable order interpretation of the analog realization of frac-tional order integrators, realized as domino ladders, was pre-sented. The applications of variable order derivatives and inte-grals can be also found in signal processing [4].In [14], derivation and switching scheme for particular type of

fractional variable order derivative is given, and also an analogmodel for this type of definition is presented. That switchingscheme assumes that changing the order is caused by additiona derivative at the beginning of the system (in this paper calledadditive-switching scheme). In this paper, a reductive-switchingscheme is introduced, and numerical switching scheme for thissituation is given. Both switching-types are compared in numer-ical and analog modeling examples. Analog modeling resultsare obtained based on domino ladder approximations of orders0.5 and 0.25.The rest of the paper is organized as follows. Section II

presents existing generalizations of Grunwald–Letnikov frac-tional order derivative definitions for a variable order case.In Section III, two switching order schemes: additive andreductive are presented and their numerical algorithms aregiven. Section IV presents an analog realization of two types ofswitched order integrators, and comparison of obtained resultsto the numerical solutions.

II. FRACTIONAL VARIABLE ORDER GRUNWALD–LETNIKOVTYPE DERIVATIVES

As a base for generalization onto variable order derivative thefollowing definition is taken.Definition 1: Fractional constant order derivative is defined

as follows:

where .The matrix form of the fractional order derivative is given by

[15], [16]

......

where

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2 IEEE JOURNAL ON EMERGING AND SELECTED TOPICS IN CIRCUITS AND SYSTEMS

Fig. 1. Plots of step function derivatives with respect to the first type derivative(given by Def. 2).

......

......

, and

is a time step, is a number of samples.According to this definition, one obtains: fractional deriva-

tives for , fractional integrals for , and the originalfunction for . For the case of order changing withtime (variable order case), three types of definition can be foundin the literature [17], [18].The first one is obtained by replacing a constant order by

variable order . In that approach, all coefficients for pastsamples are obtained for present value of the order, accordingto the following definition.Definition 2: The first type of fractional variable order deriva-

tive is given by

Plots of step function derivatives for , and

(1)

(according to Def. 2) are presented in Fig. 1.The matrix form of the first type of fractional variable order

derivative is given by

......

Fig. 2. Plots of step function derivatives with respect to the second type deriva-tive (given by Def. 3).

where

......

......

The second type of definition assumes that coefficients forpast samples are obtained for order that was present for thesesamples. In this case, the definition has the following form.Definition 3: The second type of fractional variable order

derivative is defined as follows:

The matrix form of the second type of fractional variable orderderivative is given by

......

where

......

......

In Fig. 2, plots of step function derivatives (according to Def. 3)are presented for , and given by (1).The third definition is less intuitive and assumes that coeffi-

cients for the newest samples are obtained respectively for theoldest orders. The following definition applies to this case.


SIEROCIUK et al.: ANALOG MODELING OF FRACTIONAL SWITCHED ORDER DERIVATIVE USING DIFFERENT SWITCHING SCHEMES 3

Fig. 3. Plot of step function derivatives with respect to the third type derivative(given by Def. 4).

Definition 4: The third type of fractional variable orderderivative is given by

The matrix form of the third type of fractional variable orderderivative is given by

......

where

......

......

In Fig. 3, plots of step function derivatives (according to Def. 4)are presented for , and given by (1).

III. SWITCHING ORDER SCHEMES AND ITSNUMERICAL REALIZATION

In this section, the routines and schemes for switching orderderivative are presented. For simplicity, we start with the sim-plest case of order switching, namely, switching between tworeal arbitrary constant orders, e.g., and . This idea will bethen generalized for a multiple-switching (variable order) case.

A. Simple Additive-Switching Order Case

The idea was already presented in [14], nevertheless, we re-call it below. It is depicted in Fig. 4, where all the switches

, change their positions depending on a current value of. If we want to switch from to , then, before switching

Fig. 4. Structure of simple additive-switching order derivative (switching fromto ; configuration before time ).

time , we have: , and after this time:and . At the instant time , the derivative block of com-plementary order is pre-connected on the front of the currentderivative block of order , where

(2)

If , then corresponds to integration of ; and,if , then corresponds to derivative of , withappropriate order .Now, the numerical scheme corresponding to the above

derivative switching structure is introduced.Lemma 1: For a switching order case, when the switch from

order to order occurs at time is a numberof samples, the numerical scheme is

...

...

...

...

where

and

The order , appearing above, is given by relation (2), andstands for the variable order derivative according to the

additive-switching scheme.Proof: The signal incoming to the block of derivative

can be described as follows:



Fig. 5. Structure of simple reductive-switching order derivative (switchingfrom to ; configuration before time ).

...

...

...

...

Until time , the input of -block obtains the original function, so in matrix we have an identity matrix. From

time , the input signal flows through the block of derivative, and we have the sub-matrix that is responsible

for starting the -derivative action from time on. That signalflows to the -block and has the following matrix form:

...

...

...

...

B. Simple Reductive-Switching Order Case

The reductive-switching order case occurs when the initialchain of derivatives is reduced in accordance with changing ofthe variable order. The idea is depicted in Fig. 5, where theswitches , change their positions depending on thecurrent value of . If we want to switch from to , then,before switching time , we have: , and afterthis time: and . At the instant time , the deriva-tive block of complementary order is disconnected from thefront of the derivative block of order , where

(3)

The numerical scheme for reductive-switching case can be ob-tained in a similar way as the additive-switching scheme.Lemma 2: For a reductive-switching order case, when the

switch from order to order occurs at time , the numericalscheme is

...

...

...

...

where

and

The order , appearing above, is given by relation (3), andstands for the variable order derivative according to the

reductive-switching scheme.Proof: The signal incoming to the block of derivative

can be described as follows:

...

...

...

...

Until time , the input of -block is fed with the derivativeof function , thus in matrix one can dis-tinguish the sub-matrix which corresponds toderivative action before switching. From time on, the inputsignal flows directly to the block of derivative , and this cor-responds to the identity matrix. The signal that flows from the-block is described by the following matrix form:

...

...

...

...



Example 1: Consider integration of the function ,with switching variable order taking value , andafter switching time , order , both foradditive- and reductive-switching case. Numerical calculationsare performed for .For reductive-switching case, where , we have

On the other hand, for additive-switching case, where, we have

It follows from the comparison of the matrices corresponding tothe additive- and reductive-switching schemes that in both thesecases the results are divergent (see Fig. 6).

C. Multiple Additive-Switching (Variable Order) Case

In general case, when there are many switchings betweenarbitrary orders, we have the following structure of additive-

Fig. 6. Plots of step function variable-order derivatives from Ex. 1.

Fig. 7. Structure of multiple additive- and reductive-switching order deriva-tives. (a) Additive-switching order derivatives (configuration before time

). (b) Reductive-switching order derivatives (configuration before time).

switching case, presented in Fig. 7(a). When we switch fromthe order to the order at the switch-time instant ,for , we have to set

and the pre-connected derivative block (on the front of the pre-vious term) is of the following complementary order:

where

The numerical scheme describing the already presented gen-eral case of structure allowing to switch between an arbitrarynumber of orders is given below. What is very important, thenumerical scheme of multiple-switching case (when order isswitched in each sample time) is equivalent to the second typeof variable order derivative.Theorem 1: Switching order scheme presented in Fig. 7(a) is

equivalent to the second type of variable order derivative (givenby Def. 3).The proof of Theorem 1 is given in [14].

D. Multiple Reductive-Switching (Variable Order) Case

The structure of reductive-switching scheme is presented inFig. 7(b). This scheme can be viewed as an extension of the one



Fig. 8. Realization of half-order impedance. (a) An analog model. (b) A circuitboard.

depicted in Fig. 5. When we want to switch between the consec-utive orders , then, before the first switch-time, all the switches , are in the -position,

except of , which is in the -position. At the switch-time in-stant , for , we have to set

The complementary orders of the derivative blocks are

hence

IV. ANALOG REALIZATION OF SWITCHING-ORDER SCHEMES

A. Analog Realization of the Half-Order Impedance

In this paper, the following method of half-order impedanceimplementation, introduced in [19] and meticulously investi-gated in [13], [20], will be used. The scheme of this methodis presented in Fig. 8(a).Based on the algorithm described in details in [19], the ex-

perimental circuit boards of half-order impedances were con-structed. The circuit board that has been used in experimentalsetup is presented in Fig. 8(b). The circuit consists of 200 dis-crete elements with k k nF,and nF.

B. Analog Realization of the 0.25 Order Impedance

The method discussed in Section IV-A can be extended ontoa fractional impedance of order 0.25. This can be done by

Fig. 9. Realization of quarter-order impedance. (a) Analog model. (b) Circuitboard.

Fig. 10. Frequency response of the implemented quarter- and half-order im-pedances.

replacing the capacitors in the scheme in Fig. 8(a) by half-orderimpedances, which can be 0.5 order domino ladders. Thisleads to impedance of order , which corresponds to aquarter-order integrator.In Fig. 9(a), the scheme of the approximation of a quarter-

order impedance is shown; half-order impedance were imple-mented as the impedance of domino ladder. The circuitboard presented in Fig. 9(b) has been used as a quarter-orderimpedance. The board contains about 5000 discrete elementsand was designed according to the scheme shown in Fig. 9(a).The main ladder includes 50 branches with the resistors’ values

k and k . The half-order impedanceshave been used in the quarter ladders denoted as on thescheme.A frequency responses of the circuit boards have been shown

in Fig. 10. The phase shift for ideal quarter-order impedance isconstant and equal to , with gain slope db/decade. Forideal half-order impedance we have constant phase andgain slope db/decade. Moreover, the implemented imped-ances are close to the ideal ones in constrained, however quitewide, range of frequencies.



Fig. 11. Analog realization of additive- and reductive-switching schemes.

C. Experimental Setup

Analog models of switching system, used in experimentalsetup and corresponding to the switching schemes depicted inFig. 7(a) and (b), are presented in Fig. 11. The fractional inte-grators were realized using domino ladder approximation pre-sented in Sections IV-A and IV-B. Due to operational amplifiers,signals with inverted polarization were obtained. This requiredamplifiers with gain for each integrator (amplifiers and). In general cases, the scheme based on amplifiers and

contains resistors and impedances chosen accordingto the value of realized orders. As a realization of switchesand , integrated analog switches DG303 were used. In orderto obtain impedance order equal to or , the dominoladder approximations shown in Fig. 8(a) and Fig. 9(a) wereused. The experimental circuit was connected to the dSPACEDS1104 PPC card with a PC.

D. Experimental Results

1) Additive-Switching Results: Numerical examples ofthe second type of variable order derivative were computedin Matlab/Simulink environment, using dedicated numericalroutines [21], developed by the authors.Example 2: Integrator with additive-switching order from

to .The system was build based on the scheme in Fig. 11 withand being the quarter-order integrators, and

k . In this case, according to the additive-switching strategy,the system before switch is a quarter-order integrator, andafter switch the additional quarter-order integrator is added atthe front of existing integrator. Finally, the additive-switchingsystem order is equal to 0.5. Comparison of analog and nu-merical results is presented in Fig. 12(a) and (b). The analogapproximations of the quarter-order integrators have timeconstants different from 1. Due to this fact, the time constantof the resulting variable order integrator changes its value witheach consecutive switch.Results of the identification process, based on numerical min-

imization in time domain, were obtained for each order sepa-rately. In this process, the square error of step responses wereminimized for obtaining parameters of each order integrator. In

Fig. 12. An analog and numerical implementation of additive-switching fromorder to . (a) Result plots of derivative. (b) Differencebetween analog and numerical implementation.

this example, sampling time was chosen as s, andinput signal .For , the following model was obtained:

and for order , the following model was obtained:

Finally, this gives the following variable order integrator (takinginto consideration results of Theorem 1):

where (the switching time s)

and

Example 3: Integrator with additive-switching orders fromto .

In this configuration, according to the scheme in Fig. 11,and are the half-order integrators, and k . Theinput signal was . The parameters of analogmodels were obtained in similar way as in the previous example,separately for both orders. After identification process, the fol-lowing models, for orders and , respectively, were ob-tained:



Fig. 13. An analog and numerical implementation of additive-switching fromorder to . (a) Result plots of derivative. (b) Differencebetween analog and numerical implementation.

which gives the following variable order integrator:

where (for the switching time s)

and

Comparison results of analog and numerical implementation arepresented in Fig. 13(a) and (b).2) Reductive-Switching Results:Example 4: Integrator with reductive-switched orders from

to .In this case, according to the scheme in Fig. 11, structure has

the following parameters: and are the quarter-order inte-grators, k . The identification results were obtainedby numerical minimization of time responses square error withsampling time s, and input signal . Afteridentification process, the following time domain models of or-ders and , and parameters and , respectively,were obtained

Fig. 14. An analog and numerical implementation of reductive-switching fromorder to . (a) Result plots of derivative. (b) Differencebetween analog and numerical implementation.

The following form of the numerical scheme, based on Lemma2, is obtained:

...

...

...

...

The switching time was equal to 0.701 s. The experimental re-sults are presented in Fig. 14(a) and (b).Example 5: Integrator with reductive-switched orders from

to .In this configuration, according to the scheme in Fig. 11:

and are the half-order integrators, k . The identifi-cation results were obtained by numerical minimization of timeresponses square error with sampling time s, andinput signal . After identification process, thefollowing time domain models of orders and , respec-tively, were obtained:



Fig. 15. Analog and numerical implementation of reductive-switching fromorder to . (a) Result plots of derivative. (b) Differencebetween analog and numerical implementation.

The switching time was equal to 0.701 s. Theresults were obtained using switching matrix

, and input signalscaled by coefficients and similarly as in the previousexample. The experimental results are presented in Fig. 15(a)and (b), and shows high accuracy of proposed method.

V. CONCLUSION

The paper presents additive- and reductive-switchingschemes for variable order derivatives. The numerical schemes,based on matrix approach, for that switching schemes aregiven and investigated. It is worth to notice that these twoswitching schemes give different results for the same switchingsequences and input signal. For additive-switching scheme, itis numerically equal to the second type of fractional variableorder definition. The reductive-switching case differs from allknown in literature variable order definitions, and results can beobtained only by the numerical switching scheme introducedin the paper. Moreover, the presented switching schemes wereimplemented using analog modeling methods. Domino ladderstructures of 0.5 and 0.25 orders were used in experimentalrealization of fractional impedances. The results thus obtainedwere compared to the numerical implementations showing theirhigh accuracy. The presented methods can be used in future foranalog modeling of linear and nonlinear switching (variable)fractional order dynamical systems.

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Dominik Ryszard Sierociuk (M’10) received theM.Sc. degree in automatic control and computerengineering and the Ph.D. degree in automaticcontrol and robotics, both from Warsaw Univer-sity of Technology, Warsaw, Poland, in 2002 and2008, respectively. His Ph.D. degree dissertationconcerned the estimation and control algorithms fordiscrete fractional order state-space systems.In 2006, he became a research and teaching assis-

tant and since 2008 he is an Assistant Professor in In-stitute of Control and Industrial Electronics, Warsaw

University of Technology, Warsaw, Poland. His primary research interests arein fractional calculus with an emphasis on modeling and estimation algorithmsof fractional order systems.



Michal Macias received the M.Sc. degree in auto-matic control and robotics at Warsaw University ofTechnology, Warsaw, Poland, in 2011. His masterthesis concerned the heat system control using frac-tional order PID controller. He is currently workingtoward the Ph.D. degree at the Institute of Controland Industrial Electronics, Warsaw University ofTechnology. His main part of research is devoted tosystems modeling based on fractional calculus.He has few years of experience in power industry.

Wiktor Malesza received the Ph.D. degree in au-tomatic control and robotics from the Warsaw Uni-versity of Technology, Warsaw, Poland, in 2009. HisPh.D. degree dissertation concerned the equivalenceof linear and nonlinear control systems invariant oncorner regions.Currently, he is an Assistant Professor in Institute

of Control and Industrial Electronics, at the Facultyof Electrical Enigeneering, Warsaw University ofTechnology, Warsaw, Poland. His main research in-terests are in fractional calculus, linear and nonlinear

control, and positive systems.

analog modeling of fractional switched order derivative using different switching schemes

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