BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. XX, No. Y, 2016 DOI: 10.1515/bpasts-2016-00ZZ Fractional order model of measured quantity errors

Abstract. The paper presents an interpretation of fractional calculus for positive and negative orders of functions based on sampled measured quantities and their errors connected with digital signal processing. The derivative as a function limit and the Grünwald-Letnikov differintegral are shown in chapter 1 due to the similarity of the presented definition. Notation of fractional calculus based on the gradient vector of measured quantities and its geometrical and physical interpretation of positive and negative orders are shown in chapters 2 and 3. Key words: fractional calculus, data processing, measured quantities errors, jitter 1. Introduction The problem of determining the derivative should be ap- The Leibniz notation d f of derivative is defined [1]: proached differently than mathematically by changing the in- dt teger order to non-integer order using different function (for d f (t + dt) f (t) example Gamma function) because this solution can lead to f (t)= lim − = f (1)(t) (1) dt dt 0 dt breaking the laws of physics [9–11]. This radical view is not → where: dt =(t + dt) t is an increment of the independent true because many studies confirm the correctness of fractional − variable t; f (t + dt) f (t) is an increment of function depen- models in phenomena of the real world [2, 12–21]. The prob- − dent on t; f (1)(t) is a first order derivative. lem is not breaking the laws of physics by fractional calculus Derivative of n order is defined: but not clear physical interpretation of how and why chang- − ing integer order into fractional order influences the physical BULLETIN OF THE POLISH ACADEMY OF SCIENCES CONTROL AND ROBOTICS meaning of modelled quantities such as mass, length, time, TECHNICAL SCIENCES, Vol. 67, No. 6, 2019 d d d ... f (t) = f (n)(t)= force, electric current, temperature and others. DOI: 10.24425/bpasts.2019.130887 dt dt dt  dn f (n 1)(t + dt) f (n 1)(t) f (t)= lim − − − (2) 2. Differintegrals of measured function BULLETIN OF THE POLISH ACADEMYdt OFn SCIENCESdt 0 dt → 2.1. Interval error Let f (t) : t R between tn 1 and tn be a TECHNICAL SCIENCES, Vol.where XX, No.n Y, 2016is an order of derivative (multiplicity of the first ∈ − FractionalDOI: order 10.1515/bpasts-2016-00ZZ model of measuredN quantity errors measured function for: BULLETIN OF THE POLISH ACADEMY OF SCIENCES ∈ BULLETINBULLETINBULLETIN OF THE OF OF THE THEPOLISH POLISH POLISH ACADEMY ACADEMY ACADEMY OF SCIENCES OF OF SCIENCES SCIENCES order derivative). TECHNICALBULLETIN OF SCIENCES, THE POLISH Vol. ACADEMY XX, No. Y, OF2016 SCIENCES TECHNICALTECHNICALTECHNICAL SCIENCES, SCIENCES, SCIENCES, Vol. Vol. XX, Vol. XX, No. XX, No.Y, No. 2016 Y, Y, 2016 2016 Equation (2) causes to [2–4]: t f =(t0,t1,...,tn) (6) DOI:TECHNICALBULLETIN 10.1515/bpasts-2016-00ZZ OF SCIENCES, THE POLISH Vol. XX,ACADEMY No. Y, 2016 OF SCIENCES DOI:DOI: 10.1515/bpasts-2016-00ZZ 10.1515/bpasts-2016-00ZZ R. CIOĆ* and M. CHRZAN DOI: 10.1515/bpasts-2016-00ZZDOI: 10.1515/bpasts-2016-00ZZ Fractional order model of measured quantity errors TECHNICAL SCIENCES, Vol. XX, No. Y, 2016 n n m n where tn is produced on the basis of known interval: University of Technology and Humanities in Radom, ul. Malczewskiegod 29, 26-600∑ Radom,( 1) Polandf (t mdt) DOI: 10.1515/bpasts-2016-00ZZ f (t)= lim m=0 − m − (3) Fractional order model of measureddtn dt quantity0 (dt) errorsn dt = t1 t0 = t2 t1 = ... = tn tn 1 (7) FractionalFractionalFractionalFractional order order order order model model model model of of of measured measured measured of measured quantity quantity quantity→ quantity errors errors errors errors − − − − The paper presents an interpretation of fractional calculus forn positiven and! negative orders of functions based on sampled measured FractionalAbstract. order modelwhere: of measureddt = tm tm 1; tm quantity= t mdt; errors= for n > m. Let variable v: − − − m m!(n m)! Abstract. The paper presents an interpretationquantities of and fractional their errors calculusconnected forGrünwald-Letnikov positive with and digital negative signal orders differintegrals processing. of functions The based of derivative any on sampled order− as a is measured function obtained limit and the Grünwald-Letnikov differintegral quantities and their errors connected with digital signal processing. The derivative as a function limit and the Grünwald-Letnikov differintegral v =(v ,v ,...,v )=(f (t ), f (t ),..., f (t )) (8) Abstract. The paper presents an interpretationare shown of in fractional chapter 1 calculus due to the for similaritypositive and ofn the negative presented orders definition. of functions Notation based of on fractional sampled calculus measured based on the gradient0 1 vectorn of measured0 1 n Abstract.Abstract.Abstract.TheThe paperTheare paper papershown presents presents presentsin chapter an interpretation an an 1 interpretationdue interpretation to the similarity of fractional of of fractional fractional of the calculus presented calculus calculus for definition.positive for forpositivepositiveby Notation and replacing and negative and negativeof negative fractional orderswith orders orders calculus of gammafunctions of of functions based functions function basedon the based based ongradient and sampled on on sampledn sampledvectorwith measured ofη measuredmeasured measuredR: quantitiesAbstract. andThe their paper errors presentsconnected an interpretation withquantities digital of signal and fractional its p geometrirocessing. calculuscal The and for derivativepositive physical and interpretation asm a negative function orders limit of positiveand of functions the and Grünwald-Letn negative based orderson sampledikov∈ are differintegral shown measured in chapters 2 and 3. quantitiesquantitiesquantities and and their andquantities their their errors errors errors connectedand connecteditsconnected geometrical with with with digital and digital digital physical signal signal signal p interpretationrocessing. p processing.rocessing. The of Thepositive derivative The derivative derivative and asnegative a as function as a a functionorders function limit are limit limitandshownand theand in Grünwald-Letnthe chapters the Grünwald-Letn Grünwald-Letn 2 and 3.ikovikovikov differintegral differintegral differintegral Let for every interval dt be added an interval Δt named as arequantitiesAbstract. shown and inThe chapter their paper errors 1 presentsdueconnected to the an similarity interpretation with digital of theof signal presented fractional processing. definition. calculus The forNotation derivativepositive of asand fractional a negativefunction calculus limitordersand based of thefunctions on Grünwald-Letn the gradient based on vectorikov sampled differintegral of measured measured are shownareare shown shown inquantities chapter in in chapter chapter 1 dueand 1 1 due totheir due the to to errors similaritythe the similarity similarityconnected of the of of presentedwiththeKey the presented presented digital words: definition. signalfractional definition. definition. processing. Notation calculus, Notation Notation of The fractional data of ofderivative fractional fractional processing,n! calculus as calculus calculus a function measured basedΓ based based(n on limit+ quantities the on1 on)and gradientthe the the gradient gradient errors, Grünwald-Letn vectorΓ vector jitter( vectorη of+ measured1 of of) measuredikov measured differintegralthe interval error and 0 Δt dt. quantitiesare shownKey words: and in chapter its fractional geometri 1 due calcalculus, to the and similarityphysical data processing, interpretationof the presentedmeasured of quantitiesdefinition. positive anderrors, Notation negative jitter. of orders fractional= are shown calculus in based chapters= on 2the and gradient 3. vector of measured(4) ≤| |≤ quantitiesquantitiesquantities andquantitiesare and its and shown geometri its its geometri andgeometri in chapter itscal geometri andcalcal and1 physical and due physicalcal physical to and the interpretation physicalsimilarity interpretation interpretation interpretation of of the positive o of presentedf positive positive and of positive and definition.negative and negative negative and orders Notationm negative orders! orders(n are shownarem of areorders) fractional! shown shown in arem chapters! inΓ shownin( calculuschaptersn chaptersm 2 in and+ chapters based 2 21 and 3.) and on3. 3.m 2 the! andΓ( gradientη 3. m + vector1) of measuredIf t is time then v is measured quantity in time t by an interval Key words: fractional calculus, data processing,1. Introduction measured quantities errors, jitter− − The− problem of determiningdt. In thisthe derivative case Δt can should be interpreted be ap- as an absolute error of KeyKeyKey words: words: words:quantitiesfractionalfractionalfractional andcalculus, calculus, its calculus, geometri data data dataprocessing,cal processing, processing, and physical measured measured measured interpretation quantities quantities quantities errors,of positive errors, errors, jitter jitter and jitter negative orders are shown in chapters 2 and 3. Key words: fractional calculus, data processing, measured quantitiesd errors,By substitutingBy jitter substituting (4) (4) to to (3) (3) thethe direct proached Grünwald-Letnikov Grünwald-Letnikov differently than mathematically by changing the in- 1. Introduction The Leibniz notation dt f of derivative is defined [1]: determining of an interval length. 1.Key Introduction words: fractional calculus, data processing, measured quantitiesderivative errors,derivativeThe jitter of problem non-integerof non-integer of determining order order (G-L) (G-L) isis the producedteger derivative order [3, [3, 4]: to 4]:should non-integer be ap- order using different function (for 1.1. Introduction1. Introduction Introduction d TheTheThe problem problem problem of determining of of determining determining the the derivativethe derivative derivative should should should be ap-be be ap- ap- Let dtη be sum of an interval and its error [7]: 1. IntroductionThe Leibniz notationd f of derivative is definedd [1]: f (t +proacheddtThe) f problem(t) differently of determining than mathematically theexample derivative by Gamma changing should function) thebe ap- in- because this solution can lead to The Leibnizd notationdd fdtof derivative is defined [1]: proachedproachedproached differently differently differently than than than( math1) math mathematicallyematicallyematically by changingby by changing changing the the in- the in- in- TheTheThe Leibniz Leibniz Leibniz1. notation Introduction notation notationf offf derivativeofof derivativedtd derivative is defined is is defined defined [1]: [1]: [1]: f (t)= lim proachedThe− problem differently= f p of(t) determining than math(1)Γematically(η+ the1) derivative by changing should the be in- ap- dt = dt + Δt (9) The Leibnizdt notationdtdt f of derivative is defineddt [1]: dt 0 tegerηdt order to non-integerm order usingbreaking different the laws function of physics (for [9–11]. This radical viewη is not dtd teger→tegerteger order order orderdproached to non-integer to to non-integer non-integer differently∑ orderm=0 order order( than using1) using usingmathm! differentΓ(ηematically different differentm+1) functionf (t function function bymdt changing (for) (for (for the in- The Leibniz notation f of( + derivative) ( is) defined [1]: exampletegerf (t order)= Gamma to non-integer function)− order becau− usingsetrue this because different solution− many function can(5) studies lead (for to confirm the correctness of fractional d dt f t dt where:f t dt =((1) t + dt) exampletexampleis an incrementη Gamma Gammalim function) function) of the independent becau becausese this thisη solution solution can can lead lead to to(5) where: d d d d f (ft()=t +ff((tdtlimt++)dtdtf)(f)t(t+)ffdt((tt)))−(1)f ((t1()1))= f (t) example(1)(1) Gammadtexampleteger order function) Gammadt to0 non-integer becau function)se this orderbecau(dt solution)se using this can different solution lead to function can lead (for to f (t)=ff((tt)=)=limdt limlim dt 0− −− =dtf==f(ft) ((tt)) (1)(1) (1)(1) − breaking the→ laws of physics [9–11].models This in radical phenomena view is of not the real world [2, 12–21]. The prob- dt dtdt dt d0dtfdt(t)=00 limdt→ dtdtf (t + dt)variable− f (t)=t;ff (t(+t)dt) breakingfbreaking((1)breakingt) is theanexample increment the laws the laws laws of Gamma physics of of of physics physics function [9–11].function) [9–11]. [9–11]. depen- This becau This This radicalse radical radical this view solution view view is not is is can not not lead to dt →→ dt 0 dt (1) − breakingt0 thetn laws of physics [9–11].lem This is not radical breaking view the is lawsnot of physicsdtη = bytη fractionaltη = tη calculustη = ... = tη tη (10) → f (t)= lim→ − = f(1) (t) (1)where:truep because= − many. studies confirm the correctness of fractional 1 0 2 1 n n 1 where: dt =(t + dt) t is an incrementdent on t of; f the(t independent) is atrue firsttruetrue because order because becausebreaking derivative. many many many studiesdtt the0 studies studies tn laws confirm confirm of confirm physicsthethe correctnessthe [9–11]. correctness correctness This of fractional of of radical fractional fractional view is not − − − − where:where:where:dt =(dtdt=(t=(+tdtt++)dtdt)t) istt anisis increment andt− an increment0 increment ofdt the of of the independent the independent independent where:modelstrue becausep =� in phenomena¡� many. studiesη of the confirm real worldthebut correctness not [2, clear 12–21]. physical of The fractional prob- interpretation of how and why chang- variablewhere:where:dtt; =(fdt−( t=+t− −(+tdt +dt) dt)) f→(t tt) isisis an an an increment increment incrementDerivative of the of of independent the function of independentn ordermodels depen- varimodelsmodels is- defined: inIn phenomena (5)in in phenomena phenomena the notationdt of the of ofdt the real the real realis world treated world world [2, 12–21]. [2, as[2, 12–21]. 12–21].the functionThe The The prob- prob- prob- power and tt ff((tt++dtdt)) ff((tt)) − ηmodelstrue because in phenomena many studies of the confirm real worldthe correctness [2, 12–21]. of The fractional prob- variablevariablevariablet; variablefwhere:(;t;+ dt)tdt; f(=(1(f)t(t+)t dtis+) andt−isis) increment anf¡ an(tt) increment incrementisis an incrementof function of of function function of of depen- function the depen- depen- independent− depen-(dt) lem. This is not approach breaking leads theη to laws problems of physicsing with integer clearby fractional mathemati- order into calculus fractional order influences the physical dent( onable1(1)) t ;t−;f f (−t−( +t) dtis) a first f (t) orderis an increment derivative. of function dependentlemlemlem is not is is not breaking notIn breaking breaking(5) the the notation the laws the laws laws of dt physics of ofis physics physicstreated by fractional byas by the fractional fractional function calculus calculus calculus power t =(t ,t ,...,t ) (11) (1) ( ) ¡− − lemmodelsη is not in breaking phenomena the of laws the of real physics world by [2, fractional 12–21]. The calculus prob- η η0 η1 ηn dentdentdent on t on; onfdenttvariablet;;f(ft on) is( (tt;t); a)tfis;f firstis1(f1 a()( at(t) firstt first order+is) isdta orderfirst ordera) derivative. first orderf derivative.( derivative.t order) isderivative. an derivative. increment of function depen-cal( anddtbut) . physical notThis clear approach interpretation physical leads interpretation to of problems an ordermeaning with of of how fractional clear of and modelledmathe why calcu-- chang- quantities such as mass, length, time, Derivative( of) n order− is defined: d dbutbut notbutd not clearnotlem clear clear physical is physical physical not breaking interpretation interpretation interpretation the laws of how of of physicshow how and and and why by why why fractionalchang- chang- chang- calculus DerivativeDerivativeDerivativedentDerivative of n on of ofDerivativeordertnn; forderorder1 of is(tn) defined:− isofis isorder defined: an defined:–order first is order defined: is defined: derivative. ... lusf [2,matical(ingbutt) 5, 6] integernot and= especially clearf physical(n order)(t physical)= in intointerpretation relation fractional interpretation to interpretation of order anforce, order of influences how electric of of fractional and an current, thewhy integer physical chang- temperaturewhere: and others. − −− inging integering integer integerbut order order not order into clear into into fractional physical fractional fractional order interpretation order order influences influences influences of the how the physical the andphysical physical why chang- Derivative of n− order is defined: dt dt dt calculusmeaninging integer [2, of5, order6] modelled especially into fractional quantities in relation order suchto interpretation influences as mass, length,of the an physical time, t = t + nΔt (12) meaningmeaning meaningorder of derivative modelled of of modelled modelled for quantities examplequantities quantities such as such such a as tangent mass, as as mass, mass, length,to a length, length, function time, time, time, or a ηn n d −d d (n) n (n 1)integermeaninging integerorder of derivative(n modelledorder1) into for quantities fractionalexample as such ordera tangent as influences mass,to a function length, the physical time, d ddd dd d dd... f (t) (n(n))= f d(t)= f measurement− (force,t + dt) electric off a− path current,(t) (derivative temperature of velocities and others. change). The ...d... d (())d =(=n) (()=)=(n) ( )= force,force,force, electric electric electric current, current, current, temperature temperature temperature and and others.and others. others.2. Differintegrals of measured function ( ) ... dt fdt(t)...ff tt dt= ff(t)f(ft)==tt f (tn)=f t lim orforce, meaninga measurement electric− of modelled current, of a path temperature quantities(derivative(2) suchof and velocities others. as mass, change). length, time,Let g be a function where its values are equal values of f t dt dtdtdt dtdt dtd dtdtd d  (dtn) dt 0 interpretationsdt of G-L differintegral are shown in [7] but they n dt dt ...(ndt1) f (t) = f(n (1t))= , → (2) Theforce, interpretations electric current, of G-L differintegral temperature2.1. andare shown Interval others. in [7] error but Let fin(tt)η::t R between tn 1 and tn be a n nn d (n 1()n(n 11)) f − (t(n+1dt()n(n)11)) f − (t) have borders due to the possibility of substitution of any num- ∈ − d dd n f dtff (−tdt−+((tdtt+(+n)dtdt1))f) ffwhere(−t−) ((tt)(n)n 1)N is an order of derivativethey2. have Differintegrals (multiplicityborders due to of the of the possibility measured first of substitution function of any d f (t)=− lim f − (t +−dt)− f −∈ (t) 2.2. Differintegrals(2)2. Differintegrals Differintegrals of measured of of measured measured function function functionmeasured function for: g(tη )=g(tn + nΔt)=v (13) f (t)=ff((tt)=dt)=limn limlim dt 0 − −− dt (2) (2)(2) ber for2.η Differintegralsand properties of gamma of measured function. function This problem is n nn ndtfdt(t)=00 lim (n 1) order− derivative).(n 1) (2) number for η and properties of gamma function. This problem dt dtdt dtdn 0 dt→0 dtf −dtdt(t + dt) f − (t) 2.1. Interval error Let f (t) : t R between tn 1 and tn be a dt→ f→(→t)= lim dt − 2.1.2.1.2.1. Interval(2)shown Interval Interval2. in error theDifferintegrals error error workLet [8].LetLetf (t)ff:((ttt))::t ofRt between measuredRRbetweenbetween∈ tn 1tnt functionandn 11andandtn be−tntn abebe a a t f =(Functionst0,t1,...,tn)f (t) and g(t) are shown(6) in fig. 1. where n Nn is an order→ of derivativeEquation (multiplicity (2) causes of the to first [2–4]:is2.1. shown Interval in the work error [8].Let ∈f∈(t) : t R between− −− tn 1 and tn be a wherewherewheren nNn isNN anisis order∈ andt an order order of derivative of ofdt derivative derivative0 (multiplicity (multiplicity (multiplicitydt of the of of the first the first first measuredmeasuredmeasured function function function for: for: for:∈ ∈ − ∈orderwhere∈∈ where derivative).n Nn is N an is orderan →order of of derivative derivative (multiplicity (multiplicity of of the themeasured first first functionmeasured2.1.The Intervalproblem for: function errorof determining for:Let f (t) the: t derivativeR between shouldtn 1 andbe tn be a orderorderorder derivative). derivative). derivative).where n∈ 2 is an order of derivative (multiplicityn of the firstn m n ∈where tn is produced− on the basis of known interval: orderorder derivative). derivative).N d ∑m=approached0(measured1) differentlyf function(t mdt for:than) t f mathematically=(t0,t1,...,tn) by changing the 1(6) Equation∈ (2) causes to [2–4]: m t =(ttf ft=(=(,tt0t,...,0,,t1t1,...,t,...,) tntn)) (6) (6)(6) EquationEquationEquationorder (2) (2)causes (2) derivative). causes causes to [2–4]: to to [2–4]: [2–4]: f (t)= lim − f − 0 1 t f =(n t(3),t ,...,tn) (6) EquationEquation (2) causes (2) causes to [2–4]: to [2‒4]: dtn dt 0 integer order(dt)n to non-integer order 0using1 different functiondt = (fort1 t0 = t2 t1 = ... = tn tn 1 (7) n m n → where tn is produced ont thef =( basist0,t1 of,..., knowntn) interval: − (6) − − − Equationn (2) causesnn to [2–4]:n (nn ) f (t mdt) wherewherewheret isexampletnt producednisis produced produced Gamma on the on onfunction) the basis the basis basis of because known of of known known this interval: solution interval: interval: can lead to n nn d n ∑mmn =mm0 1 mn n where tn is producedn! on the basis of known interval: d dd n f (∑t)=m=∑∑0m(mlim==010()( 11)) fm−(mt fwhere:f(m(tmdtt mdt)mdtdt)−)= tm t ; tm(3)= t breakingmdt; the =laws of physicsfor n [9‒11].> m. This Letradical variable view isv: not (()=d)=n −∑−mm=0( 1) −−mn f (t mdt) m 1 m m!(n m)! f ( ntn)=ff ttdtlimnflim(limt)=dt− 0 n −nn−(dtm )n − (3)−(3) (3)− (3) − where tn is produceddt =−t1 t on0 = thet2 basist1 = of... known= tn interval:tn 1 (7) dtn dtdt dtdn 0dtdt 00 lim→ (∑dt)(n(dtdt()) 1)Grünwald-Letnikov n f (t mdt) differintegrals(3) true dtbecause=dtdt oft1== anymanyt1t10 = order tstudies0tt02==t− is2t21 confirm obtained=t1t...1===...− ...thetn== correctnesstntn 1tntn 11− of fractional− (7) (7)(7) dt→ →→ dt 0 m=0  (dt)m dt−−= t1 −t−0 = t2 t1 = ...−−−=−−tn tn=(1 , ,..., (7))=( ( ), ( ),..., ( )) f (t)= lim→ − n  −n! n (3) models in phenomena− of−− the real world− − [2, 12‒21]. −Thev −problemv0 v1 vn f t0 f t1 f tn (8) where: dt =dttnm tm 1dt; tm0=nt nmdtn by;n!( replacingdtnn!)=!n forwithn gamma> m. functionLet variable and ndtwithv:= t1η t0R=: t2 t1 = ... = tn tn 1 (7) where:where:where:dt =dtdttm==tmttmm t1mt;mtm11;=;tmtmt =−=mdttt →mdt;mdt;; = == mn form!forn(fornn>! nmmn)>!>. mm.. LetLet variableLet variable variablev: vv:: − − − − where: dt =−t−m −tm 1; tm =mt −mmdtmm!(;nmm!m(!nn()n!=mm)!)! n−! for n > m. is notLet breaking variable the vlaws: of ∈physics by fractional calculus but not Grünwald-Letnikov−where:−− dt = tm −− tm −− differintegrals 1; tm = t mdt;− (m ofmn)− any−= m! order(n n!m)! is for obtained n > m. Let for every interval dt be added an interval Δt named as Grünwald-LetnikovGrünwald-Letnikov− differintegrals differintegrals¡ − of of any any order orderm is is!(n obtained −+ obtained m)! v =(v ,v ,...,v )=(f (t ), f (t ),..., f (t )) (8) Grünwald-Letnikovwhere:Grünwald-Letnikovdt = differintegralstmn ¡tm 1; differintegralstm = oft any¡mdt order; ofm isany= obtainedm order!(n m) is! for obtainedn > m. clear Letphysical variable interpretation0v: 1 ofn how and0 why changing1 integern by replacingn nn −with− gamma − function  and n with η : ( + v)=(vv=(v=(0,vvv10,...,0,,vv1(1,...,v,...,n)=(+vvnn)=())=(f (t0f)f(,(tf0t0())t,1,f)f(,...,(t1t1)the),...,,...,f (t intervalnf))f((tntn)))) error(8) (8) and(8) 0 Δt dt. byby replacing replacing Grünwald-Letnikovwithwithmn gamma gamma function function differintegrals andn andnnwithwith of anyηnη! order−:: is obtainedR Γ n 1order into fractionalv Γ=(ηv0 ,orderv11,..., influencesvn)=(f (thet0) physical, f (t1),..., meaningf (tn)) of ≤|(8) |≤ by replacingby replacingGrünwald-Letnikovm withmm gammawith function gamma differintegrals and functionwith of andη anynRwith order: RRη= is∈ obtainedR: = (4) mn n   (∈ ∈∈ ) η ( + Let) for everyv(=(v interval0,v+1,...,) dtvn)=(be addedf (t0),If anft(tis interval1) time,..., f then(Δtnt))namedv is measured as(8) quantity in time t by an interval by replacing (m) with gamma functionm and! n n withm ! ∈m R!:ΓLetn Let forLetmmodelled for every for1 every every interval quantitiesm interval! intervalΓ ηdt suchmbedtdt addedbe be1as addedmass, added an length, interval an an interval interval time,Δt namedforce,ΔΔttnamednamed electric as as as by replacing   m with gamma function and −with η2 R: − theLet interval for every error− interval and 0 dtΔbet addeddt.dt. an In interval this caseΔtΔnamedt can be as interpreted as an absolute error of   n!   Γ(n + 1) Γ(η + 1) ∈thethe intervalthe interval intervalcurrent, error error errortemperature and and and0 0 0Δ andt ΔΔ tothers.tdt. dtdt.. n! nn!! n Γ(nΓ+=Γ((n1n+)+11))(n + )Γ(ηΓ=Γ+((Byηη1+)+ substituting11)() + ) (4) to(4) (3)the theLet interval direct for every Grünwald-Letnikov error and interval 0 ≤|dtΔt|≤be addeddt. an interval Δt named as =( ==! )  Γ( = =1=+ ) Γ( η (4)1+(4)(4)) If t is time≤| then≤|≤|v|≤is measured|≤|≤≤| |≤ quantitydetermining in time t ofby an an interval interval length. (mm!!((nn)mmm!))!n! m(m!!ΓΓ(=(nnm+!mΓm+)+n(11))+m )m(m1!!ΓderivativeΓ(=(ηηm+!mΓm+)η+( of11))+ non-integerm )1 If(4)t orderisIfIft timetisis (G-L) timethe time thenIf t interval thenis thenisv timeis producedv measuredvis errorthenis measured measuredv and [3,is quantity measured 4]: 0 quantity quantityΔt in quantity time indt in time. timet by intt anbyby time interval an ant interval intervalby an interval m! n m !m!(nm−n!!Γmn)! mm!Γ1(Γn −n mm!1+Γ 1η) mm!Γ1(Γη η− m1+ 1) dt. In this case Δt can be≤| interpreted|≤ Let asdt anη absolutebe sum of error an interval of and its error [7]: − By−− substituting− −=− (4)− to− (3) the− direct=−− Grünwald-Letnikov− dt. .(4)dtdt In(4).. this In In this this caseIf case casetΔist timecanΔΔttcancan be then interpreted be bev is interpreted interpreted measured as an as quantity as absolute an an absolute absolute in time error error errort ofby an of of interval ByBy substitutingBy substituting substitutingm!( (4)n (4)m to(4))! (3)to to (3)m the(3)!Γ the( thedirectn direct directm + Grünwald-Letnikov1) Grünwald-Letnikov Grünwald-Letnikovm!Γ(η m + 1) 2.determiningdt Differintegrals. In this case of anΔt intervalofcan measured be length. interpreted function as an absolute error of By substituting− (4) to (3)− the direct Grünwald-Letnikov− determiningp determiningdetermining ofΓ an( ofη of+ interval an1 an) interval interval length. length. length. dt = dt + Δt (9) derivative of non-integer order (G-L) is producedη [3, 4]: determiningmdt. In this caseof anΔ intervalt can be length. interpreted as an absolute error ofη derivativederivativederivative of non-integer of of non-integer non-integer order order order (G-L) (G-L) (G-L) is produced is is produced produced [3,d 4]: [3, [3, 4]: 4]: ∑m=0( 1) Let dtη be sumf (t ofmdt an) interval and its error [7]: derivativeBy substituting of non-integer (4) to order (3) (G-L) the direct is produced Grünwald-Letnikov [3, 4]: LetLetdt−Letdtbedtηmη sumbe!Γbe(η sum sum ofm+ an of1 of) interval an an− interval interval and and andits error its its error error [7]: [7]: [7]: f (t)= lim η2.1.determining LetIntervaldt−η be error. sum of an Let of interval an f (t interval) : t length. (5) R andbetween its error tn 1 [7]: and tn be derivative of non-integer order (G-L) isdt producedη dt [3, 4]:0 (dt)η 2 where: ¡ *e-mail: [email protected] m Γ(η+1) → a measuredLet dt functionη be sum for: of andt intervalη = dt and+ Δt its error [7]: (9) η p pp ∑m mΓm(η+Γ1Γ1(η()η++11)) f (t mdt) dtη dt=dtηηdt==+dtdtΔ+t+ΔΔtt (9) (9)(9) η ηη d ∑∑( 1()( 1mp1)=)0 m mf!Γ(Γt((ηηff+((tmmdt1t+) 1mdt))mdt))t0 tn dtη = dt + Δt (9) d dd Manuscriptη f (∑t)=m submitted=0mmlim==00 2018-11-06,∑m!Γ(m(ηm−!Γ! revisedΓm(1η(+)η1m)m 2018-12-30++where:11)) − andp f=2019-02-21,(t −−mdt .) (5) dtη = tη1 tη0 = tη2 tη1 = ... = tηn tηn 1 (10) ( )=ff((tt)=d)=η limlim − −−mp=0 − −−mm!Γ(Γη−(ηη+m−+1−)1) dt (5)(5) where: dtη = dt + Δt − (9) − − − η f ηtη dtinitiallylimη f accepted(t)=dt limfor publication0 2019-04-05,−η ηη (publisheddt−) in December�−(5) 2019� where:where:(5)where:η tf = (t0, t1, …, tn) (6) dt dtdt dtdη 0dtdt 00 → ∑m(=dt0()((dtdt1))) m!Γ(Inη ηm (5)+1) thef (t notationmdt) dt iswhere: treated as the function power dt→ →→ dt 0 − (dt)− − and η f (t)=t0 tnlim→ ηη (5) where: dt = t t = t t = ... = t t (10) where:t0 tnt0t0pt=ntn − dt . 0 (dt( )) . This approach leads to problems withη clearη1 mathemati-η0 η2 η1 ηn ηn 1 dt −− t dttn dt dtη dt=dtηηtη==tηtη1t1η =tηtη0t0η==tηtη2t2η =tηtη1...1===...... tη==n tηtηntnη tηtηnn 11 (10)−(10)(10)tη =(tη ,tη ,...,tη ) (11) where:where:where:p =pp==− dtdt. .�. 0 �→ 1 dtη0= tη 2− tη 1= tη − tη = ...n=1 tηn − tη (10) 0 1 n where:dt�� p =�� dt− . η cal and physical interpretation of an order− − of− fractional1 − −0− calcu-2 1 − −−− −− n 1 �InBull. (5)� Pol. the� Ac.: notationt0η Tech.tn�ηη 67(6)dt 2019is treated as the function power − − − 1023− In (5)InIn (5) the(5)where: the thenotation notation notationp = dt −dtdtis. treatedisis treated treatedη as the as as the functionthe function function power power power and dtη = tη1 tη0 = tη2 tη1 = ... = tηn tηn 1 (10) Inη (5) the notationdt dt is treatedlus [2, as 5, the 6] especiallyfunctionand power inandand relation to interpretation of− an integer− where: − − (η( ))ηη (dt) . This approach� � leadsη to problems with clear mathemati- and t =(t ,t ,...,t ) (11) (dt)dtdt. This.. This( Thisdt approach)Inη approach. approach (5) This the leads approach leads notation leads to pr to tooblems leads pr prdtoblemsoblems tois with pr treatedoblems with with clear clear clear aswith mathemati- the mathemati- mathemati- clear function mathemati- power tt =(=(ttη,,tt ,...,,...,η0 ttη1)) ηn (11)(11) cal and physical interpretation of anorder order derivative of fractional for example calcu- asand a tangenttη to=(ηη atη function0 ,tηηtη010,...,=(ηη11t orηn ), atηηnn,...,t ) (11) (11)tηn = tn + nΔt (12) calcal andcal and andphysical physical physicalη interpretation interpretation interpretation of an of of order an an order order of fractional of of fractional fractional calcu- calcu- calcu- η η0 η1 ηn cal(dt and) . physicalThis approach interpretation leads to ofproblems an order with of fractional clear mathemati- calcu- where: t =(t ,t ,...,t ) (11) lus [2, 5, 6] especially in relation tomeasurement interpretation of of a an pathwhere: integerwhere: (derivativewhere: of velocities change).η Theη0 η1 ηn luslus [2,lus 5, [2, [2, 6] 5,lus 5, especiallycal 6] 6] [2, especiallyand especially 5, 6]physical especiallyin relation in in relation interpretation relation in to relation interpretation to to interpretation interpretation of to an interpretation order of an of of of integer an an fractional integer integer of an integer calcu- where: Let g be a function where its values are equal values of f (t) order derivative for example as ainterpretations tangent to a function of G-L differintegralor a are shown in [7] buttηn they= tn + nΔt (12) orderorderorder derivative derivative derivative for for example for example example as a as as tangent a a tangent tangent to a to to function a a function function or a or or a a where: tηn =tηtηntnn==+tntnn+Δ+tnnΔΔtt in tη : (12)(12)(12) orderlus [2, derivative 5, 6] especially for example in relation as a tohave tangent interpretation borders to a due function of to an the integer or possibility a of substitution of anytηn num-= tn + nΔt (12) measurementmeasurementmeasurement of of a a path path of (derivativea (derivative path (derivative of of velocities velocities of velocities change). change). change). The The The g(t )=g(t + n t)=v (13) measurementmeasurementorder of a derivative path (derivative of a for path example (derivative of velocities as a of tangent change).velocities to aThe change). function The or a Let g be a function wheretηn = itstn values+ nΔt are equal values ofηf(12)(t) n Δ interpretations of G-L differintegralber are for shownη and in properties [7] butLet they ofLetgLet gammabegg abebe functionLet a a function. functiong functionbe awhere function where This where its problem values its where its values values are its is arevalues equal are equal equal values are values values equal of f ofvalues( oft)ff((tt)) of f (t) interpretationsinterpretationsinterpretationsmeasurement of G-L of of G-L G-L differintegral of differintegral differintegral a path (derivative are are shown are shown shown of in velocities [7] in in [7] but [7] but they but change). they they The in tη : interpretations of G-L differintegralshown are shown in the in work [7] [8].butin t theyηinin: tηtη:: Functions f (t) and g(t)(are) shown in fig. 1. havehave borders bordershave borders due due to to the due the possibility topossibility the possibility of of substitution substitution of substitution of of any any num-of num- any num- in tηLet: g be a function where its values are equal values of f t have bordershaveinterpretations due borders to the possibilitydue of to G-L the differintegralpossibility of substitution of substitutionare of shown any num- in of [7] any but num- they (( )=)=g(t(η()=++g(tn)=)=+ nΔt)=v (13) ber for η and properties of gamma function. This problem is in tη : g(tηgg)=tηtηg(tgng(g+t tnt)=nnΔtn)=gnΔ(Δttv+ nvΔv t)=v (13)(13)(13) (13) berber forber forη forandberηhaveηandand propertiesfor borders propertiesη propertiesand dueproperties of gamma of to of gamma the gamma possibility of function. gamma function. function. This offunction. substitution This This problem problem problem This is of problem is any is num- is η n shown in the work [8]. Functions f (t) andg(gtη(t)=) areg( showntn1+ nΔ int)= fig.v 1. (13) shownshownshown in the inshown inber the work the for work workin [8].η theand [8]. [8]. work properties [8]. of gamma function. This problemFunctionsFunctions isFunctionsFunctionsf (t)ffand((tt))andgand(ft()gtgare()(tandt))are shownareg shown( shownt) are in fig. inshown in fig. 1.fig. 1. in 1. fig. 1. shown in the work [8]. Functions f (t) and g(t) are shown in fig. 1. 1 1 11 1 1 R. Cioć and M. Chrzan

where tn is produced on the basis of known interval:

dt = t1 t0 = t2 t1 = … = tn tn 1. (7) ¡ ¡ ¡ ¡ Let variable v:

v = (v0, v1, …, vn) = ( f (t0), f (t1), …, f (tn)). (8)

Let for every interval dt be added an interval ∆t named as the interval error and 0 ∆t dt. ∙ j j ∙ If t is time then v is measured quantity in time t by an inter- val dt. In this case ∆t can be interpreted as an absolute error of determining of an interval length. Let dtη be sum of an interval and its error [7]:

dtη = dt + ∆t (9)

where:

dtη = tη1 tη0 = tη2 tη1 = … = tηn tηn 1 (10) ¡ ¡ ¡ ¡ and

tη = (tη0, tη1, …, tηn) (11) Fig. 1. Functions f and g where:

∆ ∆ tηn = tn + n t. (12) Function g is taking into consideration an interval error t (12). Gradient ∇sη(tη, v) for dt 0 is a linear approximation ! Let g be a function where its values are equal values of of g between next values of tη (Fig. 2). f (t) in tη: Fig. 1. Functions f and g. Fig. 2. Gradient vectors of g function.

g(tη) = g(tη + n∆t) = v. (13) Assuming f (t f ) is a function obtained by measured quanti- By substituting (12) to (16): Functions f (t) and g(t) are shown in Fig. 1. ties v every dt, g(tη ) function takes into consideration an inter- d vn+1 vn d Assuming f (tf ) is a function obtained by measured quanti- val error. The difference between both functions will be bigger g(tηn )= lim − = g(tn+1) (17) dt dtη 0 dt + Δt dt ties v every dt, g(tη) function takes into consideration an inter- → if anval interval error. errorThe differenceΔt is bigger. between both functions will be bigger Second derivative of g function in t is produced: Inif dynamic an interval models error ∆ oft is real bigger. systems with fast changes of an n+1 2 d d input signalIn dynamic an interval models error of isreal taken systems into with consideration fast changes as theof d g(t + ) g(t ) dt n 1 − dt n mostan important input signal factor an interval in precise error modelling is taken into based consideration on the em- as 2 g(tn)= lim = dt dtη 0 tη tη piricalthe datum. most important It is used factor to: modelsin precise of modelling vibration based and velocity on the → n+1 − n empirical datum. It is used to: models of vibration and velocity vn+2 2vn+1 + vn transducers, models of gas and liquid flows, models taking into lim − (18) transducers, models of gas and liquid flows, models taking into 2 consideration the deviation from true periodicity of a presum- dtη 0 (dt + Δt) consideration the deviation from true periodicity of a presum- → ably periodic signal (jitter). Generally, the n order derivative of g in t is formulated as: ably periodic signal (jitter). − 0 n 1 n 1 n d − d − d n 1 g(t1) n 1 g(t0) 2.2.2.2. Derivative Derivative of functionof function with with an an interval interval errorerror. Let ssηη bebe dt − − dt − n g(t0)= lim = a function of variables tη (11) and v (8) from Section 2.1. dt dtη 0 dt + Δt a function of variables tη (11) and v (8) from chapter 2.1. → The gradientThe gradient of s of( tsη,(vtη), functionv) function can can be be formulated formulated as as fol fol-- n m n η η ∑m=0( 1) vn m lows [1]: − m − lows [1]: lim n (19) dtη 0 (dt + Δt) →  ∂ ∂ ∇s (t ,v)= s (t ,v)i + s (t ,v) j (14)(14) η η ∂t η η ∂v η η 2.3. Derivative of non-integer positive order and its inter- pretation Let dtγ stands for change of dt: wherewhere∇sη ∇iss the is the gradient gradient defined defined on on points points ttη situatedsituated on on the the η η γ curvecurveg, i andg, i andj are j are the the standard standard unit unit vectors. vectors. Fig. 2. Gradientdt + vectorsΔt = ofdt ηg function= dt (20) Function g is taking into consideration an interval error Δt By substituting (20) to (19): (12). Gradient ∇sη (tη ,v) for dt 0 is a linear approximation → dn 1 Bull. Pol. Ac.:dn 1 Tech. 67(6) 2019 of g between1024 next values of tη (fig. 2). n − ( ) − ( ) d dtn 1 g t1 dtn 1 g t0 g(t0)= lim − − − = Let vηn be values situated on the gradient vector ∇sη in tηn dtn dt 0 dt η → η (fig. 2): n m n ∑m=0( 1) vn m − m − lim n (21) vη =(vη1 ,vη2 ,...,vηn )=(v0(t0),vη1 (t1),...,vηn (tn)) (15) dtγ 0 (dtγ ) →  n Derivative of g in tη is produced, where g between tηn and Because dt on the left of (21) is the notation of multiplic- γ n tηn+1 is approximated by the gradient vector: ity of dt other than dtη = dt on the right, instead of dt the notation dtη is inserted where: d vn+1 vn g(tηn )= lim − = dt dt 0 t t η = γn (22) η → ηn+1 − ηn d vn+1 vηn+1 and: γ is a factor of changing derivative by Δt (20); n is a g(tn+1)= lim − (16) γ + dt (tη tn+1) 0 tη tn+1 multiplicity of dt ; η : R is a fractional order of derivative. n+1 − → n+1 −

2 Bull. Pol. Ac.: Tech. XX(Y) 2016 Fig. 1. Functions f and g. Fig. 2. Gradient vectors of g function.

Fig. 1. Functions f and g. Fig. 2. GradientFig. 2.Fig. Gradient vectors 2. Gradient vectorsof g function.function. vectors of g function. of g function. Fig. 1. FunctionsFig. 1.Fig. Functionsf 1.and FunctionsgAssuming.ff and gf.. andfg(t. f ) is a function obtained by measured quanti- By substituting (12) to (16): ties v every dt, g(tη ) function takes into consideration an inter- d vn+1 vn d val error. The difference betweenBy substituting bothBy functions substituting (12) will to (12) (16): be bigger to (16): g(tηn )= lim − = g(tn+1) (17) AssumingAssumingf(t f ) isf( at functionf ) is( a function) obtained obtained by measured by measured quanti- quanti-By substitutingBy substitutingBy (12) substituting to (12) (16): to (12) (16): to (16): dt dtη 0 dt + Δt dt AssumingAssumingf Assumingt f isf at functionf isf at f functionis obtained a function obtained by measured obtained by measured by quanti- measured quanti- quanti- → ties v everyvdt, g(t dt) functiong(t ) takes intoif consideration an interval error an inter-Δt is bigger. ties v everyties vdteveryties, g(vtηdtevery) ,function, g(tdtη ),functiong( takestη ) function into takes consideration into takes consideration into consideration an inter- an inter- an inter- d d d vn+1 vvnn+1 vdvn+n 1 vdn d g(t )=g(tlim)= lim− =−Secondg(=t + derivative)g(t )(17) of g function(17) in tn+1 is produced: val error.val The error.val difference Theerror. difference The between difference between both between functions bothIn dynamic functions both will functions be models will bigger be will of bigger real be bigger systems withg(tη fastnn)=g( changestηlimn )=g(tηlimn of)=− an lim=− g(=t−n+1)g=(tn+1g)(17)(tn+1) (17) (17) dt dt dtηdt 0 dtdtη +0Δdtttdtη +0 Δdtdtt + Δdtt dt input signal an interval error is taken into consideration→ as→ the → 2 d d ifif an an interval intervalif an intervalif error error an intervalΔt errort isis bigger. bigger.Δ errort is bigger.Δt is bigger. d dt g(tn+1) dt g(tn) most important factor in preciseSecond modellingSecond derivativeSecond derivative based of derivativeg onfunction of theg em-function of ingttnfunction+1 inisist produced: produced:n+1 inist produced:g(tisn)= produced:lim − = In dynamicIn dynamic modelsIn dynamic models of real models of systems real of systems realwith systems fast with changes fast with changes fastof an changes of an of an n+1 n+1 dt2n+1 dt 0 t t η → ηn+1 ηn pirical datum. It is used to: models2 of vibration2 andd velocityd d d − inputinput signal signalinput an an signalinput interval interval signal an interval error error an interval is is error taken taken is error into into taken considerationis consideration intotaken consideration into consideration as as the the as the as thed2 2 2 d g(t +d)g(t ddg)(t ) d g(t )d d d d dt g(tn+dt1)g(tndt+dtg1g)(t(ntn)+dt1g)(tn)dt g(tn) vn+2 2vn+1 + vn most important factor in precise modellingtransducers, based on models the em- of gas and liquid flows,g(ttn)= modelsg(limlimtn)= takingg(tlimn)= intolim− − = − = = lim − (18) most importantmost importantmost factor important in factor precise factor in precise modelling in precise modelling based modelling on based the based on em- the on em- the em- 2 2 dt 20 2 dt dt dtηdt 0 dtη ttη0ndt+1η t0ttηηnn+1 tηtηn tη dtη 0 (dt + Δt) piricalpirical datum.pirical datum. It is useddatum. It is to: used It models is to: used models of to:consideration vibration models of vibration and of vibration the velocity and deviation velocity and velocityfrom true periodicity→ of a presum-→ n+1 −→ nn+1 − n+n1 − n → vn+2 v2nv+n2+1v+2vvn+1 +2vvn + v transducers,transducers, models models of gas and of gas liquid and flows, liquidably models flows,periodic models taking signal into taking (jitter). into n+2 − n+n2+Generally,1 n+n2+1 then+n 1n ordern derivative of g in t0 is formulated as: transducers,transducers, modelstransducers, models of gas models and of gas liquid of and gas flows, liquid and models liquid flows, flows, models taking models into taking taking into into limlim lim− lim −2 −2 (18) (18) (18) dtη 0 dtη (dt0 dt+FractionalΔ(t0dt)2 + Δt)2 order2 model− of measured quantity errors considerationconsiderationconsideration the deviation the deviation the from deviation true from periodicity true from periodicity true of periodicity a presum- of a presum- of a presum- η → η →(dt +ηΔ(tdt) + Δ(dtt) + Δt) n 1 n 1 → → → n d − g(t ) d − g(t ) Let s be d dtn 1 1 dtn 1 0 ably periodicably periodicably signal periodic (jitter). signal signal (jitter). (jitter).2.2. Derivative of function withGenerally, anGenerally, interval theGenerally,n errororder the n derivative theordern η derivativeorder of g derivativeinin t oft0 isisg in formulated formulated ofgt0(gtisin)= formulatedt islim as: as: formulated as:− as:− − = − Fractional0 n order0 0 model0 of measured quantity errors a function of variables tη (11) and v (8) from− chapter− 2.1. − dt dtη 0 dt + Δt dnn 11 dn 1 dnnn 111 Fractionaldn 1 n order1 model→ of measured quantity errors n n n d − g(td )− gd(dt−−) g(td )− g(dt −) n m n The gradientLet ofssηbeLet(tη ,sv)befunctiond cand be formulatedd dtnn 11 asg(tdt fol-1)n 1 gdt(tnn1n)111gg((tdtt01n)) 1 g(t0n)1 g(t0) ( ) v 2.2. Derivative2.2. Derivative2.2. of Derivative function of function with of function an with interval an with interval error an interval errorLet sη errorbeLet sηLetbe sη be g(t )= dt − dt −− dtdt −− − dt −= dt − ∑m=0 1 m n m n g(t0)=n g(limlimt0)=g(t0lim)= lim− − =− = = lim − − (19) lows [1]: dtn dtn dtη n0 dtη 0 dt + Δtdt + Δt n a functiona function ofa variables function of variablesttη of(11) variablest andη (11)vt(8)η and(11) fromv (8) and chapter fromv (8) chapter from 2.1. chapter 2.1. 2.1. dt dt dtηdt 0 dtη 0 dtdtη +0Δtdt + Δdtt + Δt dtη 0 (dt + Δt) → → → →  n n m n m n n The gradientThe gradientThe of sη gradient(ttη of,,vs)ηfunction(t ofη ,svη)(functiontη can,v) function be can formulated be can formulated be as formulated fol- as fol- as fol- n ( 1n)m ( n1v)m vm ∂ ∂ ∑m=0( ∑1m)=0(∑mm1v=)n0(mm1v)n m vn m ∇sη (tη ,v)= sη (tη ,v)i + sη (tη ,v) jlim lim(14)− − − − − (19)− (19) lowslows [1]: [1]:lows [1]:lows [1]: lim lim lim2.3.n Derivativen ofn (19) non-integer(19) (19) positive order and its inter- ∂t ∂v dtη 0 dtη (0dtdtη+ Δ0(ttdt) + Δ(dtt) + Δt) → → →  Let dtγ stands for change of dt: ∂ ∂ ∂ ∂ ∂ ∂ pretation ∇s (t ,vs)=(t ,v)=s (t ,vs)i(+t ,wherev)si +(t∇,svsη) jis(t the,v) gradientj(14) defined on points tη situated on the ∇sη (tη∇,vs)=η (tη∇,svηs)=η(t(ηtη,v,)=vs)ηi(+tη ,svη)si(ηt+η(t,ηv,)vis)+ηj(tη ,svη)(j(14)tη ,v) j (14)2.3. Derivative(14)2.3. Derivative of non-integer of non-integer positive positive order andorder its and inter- its inter- γ ∂tt ∂t ∂t∂curvev g∂,vi and∂vj are the standard2.3. Derivative unit2.3. vectors. Derivative2.3. of Derivative non-integer of non-integer of non-integerpositive positive order positive and order its order and inter- itsdt and+ inter-Δ itst = inter-dtη = dt (20) Let dtγγ standsγ for changeγ of dt: Function g is taking intopretation considerationpretationLetpretationdt anLetstands intervaldtLet forstands errordt changestands forΔt change of fordt: change of dt: of dt: where ∇wheresη isiswhere∇ the thesη gradient gradientis∇ thesη is gradient defined defined the gradient defined on on points points defined ontt pointsη situated on pointstη situated ontη thesituated on the on the By substituting (20) to (19): γγ γ γ curve gcurve, ii andg,jjiareandg theji are standard thej standard unit vectors. unit(12). vectors. Gradient ∇sη (tη ,v) for dt 0 is a lineardt approximation+ Δttdt=+dtΔηdtt==+dtdtΔηt ==dtdt = dt (20) (20) (20) curve curve, andcurve, areand the, areand standard theare standard unit the standard vectors. unit vectors. unit vectors. → η η η n 1 n 1 of g betweenFractional next values order model of t of(fig. measured 2). quantity errors n d − d − FunctionFunctiong isisFunction taking takingg is takinginto intog is consideration consideration taking into consideration into consideration an an interval interval an interval error error an intervalΔ errortt Δ errorηt Δt d n 1 g(t1) n 1 g(t0) By substitutingBy substitutingBy (20) substituting to (20) (19): to (20) (19): to (19): g(t )= dt − − dt − = Let vη be values situated on the gradient vector ∇sη in tη n 0 lim (12). Gradient(12). Gradient(12).∇sη Gradient(ttη∇,,vs)ηfor(tη∇,dtsvη)(fortη ,0vdt is) for a linear0dt is a approximation linearn0 is a linear approximation approximation n dt dtη 0 dtη → → nn 11 n 1 nnn 111 n 1 n 1 → → →(fig. 2):→ n n d − d − dd −− d − d − of g betweenof g betweenof nextg between values next ofvalues nextttη (fig. values of t 2).η (fig. of tη 2).(fig. 2). dn n n n 1 g(t1) g(tn1)1 g((t0)) g(t0) ( ) n m n Let vη be values situated on the gradient vector ∇sη in tη ηd d d dtn 1 g =tdt1 nγ n1 gdttn1n 11gg tdtt01n 1 g t0n 1 g t0(22) ( 1) v n n g(t )= dt − dt −− dtdt −− − dt −= dt − ∑m=0 m n m Let vη Letbev valuesbe valuessituated situated on the ongradient the gradient vector ∇ vectorsη in∇tηs in t n g(t0)=n g(limlimt0)=g(t0lim)= lim− − =− = = lim − − (21) Let vηnnLetbe(Fig.v valuesηLetn be2):v valuessituatedηn be values situated on the situated gradienton the on gradient vector the gradient∇ vectorsη in vector∇tηsnnη in∇tsηηn in tdtηnn n dtη n0 dt 0 dt n vη =(vη1 ,vη2 ,...,vηn )=(v0(t0dt),vη1 (tdt1),...,dtηdtvηn0(tndt))η 0 dt(15)ηdtη0 dtη dtη dtγ 0 (dtγ ) (fig. 2):(fig. 2): → → → →  (fig. 2):(fig. 2):(fig. 2): and: γ is Fig.a factor 3. Characteristics of changingn derivative of factorn m byγn(nΔ ∆t,mt dt(20);)n. mn isn a mul- γ ∑ ( 1) ( 1v)n m vn = = Derivative of g in t is produced, where g betweenη : + t∑mand=0( ∑1m)=0(Because∑mm1v=)n0(mm1dtv)n onm v then m left of (21) is the notation of multiplic- vη (vη1, vη2, …, vηn) (v0(t0), vη1(t1), …, vηn(tnη)). (15) tiplicity of dt ; Rlim is aη fractionalnlim − order− of −derivative.− − (21)− (21) vη =(vvη ,=(vη v,...,,vvη ,...,)=(vv0)=((t0),vη((tt1)),,...,v (vtη),...,(tn))v (t(15)n)) (15) limγ limγ limγγ n γ n γ n (21) (21)γ (21) n vη =(vvηη11,=(vη2v2v,...,ηη1=(,vvηηv2nnη,...,)=(1 ,vηv2ηv,...,n0)=((t0)v,ηvvnη)=(01(1(t0t1)),v,...,v0η(1t0(vt)1η,)nvn,...,(ηt1n())tv1η),...,n (t(15)n))vηn (tn(15))) (15)The Fig.fractional 3. Characteristics dtorderγ 0 derivativedtγ of0 dt( factordtγ is )0definedγ(Δdttdt,dt) by)(.dt substituting) dt = dt dt tηn+1 is approximated by the gradient vector: → → →ity of other than η on the right, instead of the The fractional order derivative is defined byη substituting Derivative of g in tη is produced, where g between tη n and equationn (22)Fig. ton 3.(21): Characteristicsn ofnotation factor γ(Δdtt,dt)is. inserted where: DerivativeDerivative of g in oftη gisin produced,tη is produced, where g wherebetweeng betweentη andtη andBecauseBecausedt onBecausedt theon leftdt the ofon (21)left the of is left (21) the of notation is (21) the isnotation of the multiplic- notation of multiplic- of multiplic-Fig. 4. Geometric interpretation of a non-integer positive order deriva- DerivativeDerivative ofDerivativeg in oftη gisin produced, oftηgisin produced,tη is where produced,g wherebetween whereg betweentηnngdandbetweentηn andequationtηn andvn (22)+1 tovn (21): tη n + 1 is approximated by the gradient vector: γγ γ γ n n n tη ist approximatedis approximated by the gradient by the gradient vector: vector: g(tηn )=ity ofThelimdtity fractionalother of dt− thanother order=dt thanη = derivativedtdtη on= dt the is=on right, defined the instead right, by insteadsubstituting of dt the of dt thetive. tηnn+11 ist approximatedηn+1 istη approximatedn+1 is approximated by the gradient by the by gradient vector: the gradient vector: vector: ity of dtityother of itydt thanother of dtdt thanηotherdtdt thanη ondtdt theη on right,dt theon insteadn right, the right,instead of dt insteadthe of dt oftheηdt= γthen (22) dt dtη 0 tηnη+1 ntηnη n m Fig. 4. Geometric interpretation of a non-integer positive order deriva- notationequation→Thenotationdt (22)η fractionalisnotationis−d toinserted inserteddt (21):η isdt insertedorder where: where:η is inserted derivative where:∑m=0( where:1 is) definedm vn m by substituting d v v g(t0)= lim − − (23)(23) d d d vn+1 vnn+1 vdvn+n 1 vn vn+1 ηvηn+1 γ and:γ n γ is a factor of changingtive.Fig. derivative 4. Geometric by interpretationΔt (20); n is of a a non-integer positive order deriva- g(t )=g(tlim)= lim− =− = equation (22)dtn to (21):dt 0 n (dtm) n g(tηnn)=g(tηlimn )=g(tηlimn )=− lim=− g(=−tn+1)== lim − → =(16)n (= 1) v γ (22) + same (characteristics will overlap). Other interpretation can dt dt dtη 0 tηdtη 0 tdttη 0 t ( ) d η =∑γmn=η0 = γFractionalnη =mγnn m order model(22) of(22) measuredtive.(22) quantity errors dt dt ηdt→ tηnnη+11 ttηηηnnn+1 tηdttnη+n1 tηn tηn+1 tn+1 0 tηnn+1 tn+1 − multiplicityn − of dt ; η : R is a fractional order of derivative. → →− → − − → d η gn(t0)= lim n γ n m (23) found in [5–7, 22]. − − − (16)where: η dtis− ad notationdt ofγ 0 fractional∑m=(0dt( derivative;)1) vn ism a natural d vn+1 vη v dt n m same (characteristics will overlap). Other interpretation can d d d vn+1 vnη+nn+111 vvn+ηn1+1 vηn+1 and: γand:isis a a factorγ factorand:d is aγ of offactorgis(t changing changing0 a)= factor of→lim changing ofderivative derivative changing derivative− by by derivativeΔtt (20); by− Δtn by(20);isisΔ a at(23)n(20);is an is a g(tn+1)=g(t )=lim lim − − (16) (16) η γ γ n g(tn+1)=g(tn+1g)=(limtn+1)=lim lim− − −. (16) (16)order(16)where: ofd derivativen η dtisγγ a notation (multiplicityγ+ dtofγ fractional+0 of an+ intervalderivative;(dt ) dt );n ηis ais natural a frac- dt dt ((ttη tt(ntt+η1)) 0(tttnt+η1)2 0t ttηn)+1 tn+1 multiplicitymultiplicity ofdt dt ; ofη :dtR; ηis: a→ fractionalis a fractional order of order derivative. of derivative.foundsame inBull. [5–7,(characteristics Pol. 22]. Ac.: Tech. XX(Y) will overlap). 2016 Other interpretation can dt dt dtηnn+11 − n+η1n+→1 0 ntη+ηn1+nn+111 0n+ttη1nn++110 tηtn++11 tn+1 multiplicitywhere:multiplicityη ofismultiplicitydt a notation; ofη :dtR ; of ofηisdt: fractional aR fractional; ηis: R a fractional derivative;is order a fractional of order derivative.n is of order a natural derivative. of derivative. − →− →−− → − − tional orderdt n of derivative (for Δt = 0) connected withη a natural 2.4. Derivative of non-integer minus order and its inter- Fig. 1. Functions f and g. Fig. 2. Gradient vectors of g function. orderwhere:order of derivative ofd derivativeis a notation(multiplicity (multiplicity of fractional of� of an an interval interval derivative;dt dt);); ηn isisis a a frac frac- natural- found in [5–7, 22]. ordertional by a orderdt factorη ofγ derivative(22). (for ∆t = 0) connected with a natural pretation Let n = 1. On the basis of (22)-(23) and η = γ the 2 2 tional order of derivative (for t =Bull.60) connected Pol.Bull. Ac.: Pol. Tech. with Ac.: XX(Y) Tech. a natural 2016 XX(Y) 20162.4. Derivative of non-integer minus order and its inter- 2 2 2 By substitutingFig. 2.Fig. Gradient 2. (12) Gradient to vectors (16): vectors of g function. of g function. Inorderorder the of case by derivative a where factor Δγ (multiplicityt(22).= 0 thenΔ ηBull.= ofn Pol.anand:Bull. interval Ac.: Pol. Tech.Bull. Ac.:dt XX(Y) Pol.); Tech.η Ac.:is 2016 XX(Y) a Tech. frac- XX(Y)2016derivative 2016 is: Fig. 1.Fig. Functions 1. Functionsf andfgand. g. � pretation Let n = 1. On the basis of (22)-(23) and η = γ the By substituting (12) to (16): ordertional by a order factor ofγ derivative(22). ∆ = (for Δt =η 0)= connected with a natural 2.4. Derivative of non-integer minus order and its inter- Assuming f (t f ) is a function obtained by measured quanti- In the case wheren t 0 then � n n and: d v(tη ) v(t0) Fig. 1.Fig. Functions 1. Functionsf andfgand. g. Fig. 2.Fig. Gradient 2. Gradient vectors vectors of g function. of g function. Inorder the caseby a factorwheredγΔt(22).= 0 then η =dn and: derivativepretation is: Let ng=(t1.)= Onlim the basis1 of− (22)-(23) and η =(28)γ the ( ) g(t ) = f (t ) (24) γ 0 γ γ ties v every dt, g tη function takes into consideration anFig. inter- 2. GradientBy vectors substitutingBy substituting ofdg function. (12) (12) to (16): to (16):vn+1 vn d η 0 n 0 dt dt 0 dt Fig. 1. FunctionsAssumingAssumingf andf (t fg)f. (ist f a) functionis a function obtained obtained by measured by measured quanti- quanti- g(t )= lim − = g(t ) (17)(17) In the case wheredtn Δt = 0= then ηdtn= n and: derivative is: → val error. The difference between both functions will be bigger ηn n+1 . d Δt 0 d d v(tη1 ) v(t0) Fig. 1.Fig. Functions 1. Functionsf andfgand. g. dtFig. 2.Fig. Gradient 2.dt Gradientη vectors0 dt vectors+ ofΔtg function. of gdtfunction. g(t0)= lim − (28) ties vtieseveryv everydt, gdt(tη, )g(functiontη ) function takes takes into considerationinto consideration an inter- an inter- d d →v v v v d d gn(t0)  = fn(t0). (24)(24) Equation (28) forγ minus orderγ γ has theγ formula: Fig. 1. FunctionsifAssuming an intervalAssumingf andfg( error.t f )fis(tΔf at)is functionis bigger. a function obtained obtained by measured by measuredFig. quanti- 2. Gradient quanti- vectorsBy substitutingBy of substitutingg function. (12) (12)to (16): ton (16):+1 n+1n n Factor γ dependsη ond an interval dtnandd an interval error Δt dt d dt 0 v(dttη1 ) v(t0) g(tηng)=(tηn )=lim lim − −= =g(tn+g1()tn+1) (17)(17) dt Δt=0 dt g(t0)=→lim − (28) val error.val error. The Thedifference difference between between both both functions functionsBy substituting will willbe bigger be (12) bigger to (16):Second derivative of g function in t is produced: g(t0) = f (t0) (24) γ γ γ Assuming f (t f ) is a functionties vIntiesevery dynamic obtainedv everydt, g models(dtt byη,)g measuredfunction(tη ) offunction real takes quanti- systems takes into consideration intowith consideration fast changes an inter- ofan aninter- dt dt dtη dt0 η dt0+dtΔt+ Δn+t 1dt dt (20). Because Δt [0,dtη ] then on the branchn borders: dt dt 0 dt Secondd d derivative→ ofv gn→ +function1 vn+v1n inv ntd isd produced: dt  Δt=0 dt Equation (28)d for minus orderv(t→ηγ1has) v the(t0) formula: if anif interval an interval error errorΔt isΔ bigger.t is bigger. Fig. 2. Gradient vectorsBy substitutingBy of substitutingg function.g(tη (12)g)=(tη to(12))=lim (16): tolim (16):− −= η n= +g 1(tn+g1()tn+1) (17) (17)Factor γ depends∈ on an interval  dt and an interval error Δt g(t0)= lim − = ties v every dt, gFig.(tη ) 1.function FunctionsvalAssuming error.val takesAssuming error.f Theand intofg difference The(.t considerationf )f differenceis(t f a) functionis between a function between an obtained inter- both obtained both functions by functions measured by will measured be willd quanti- bigger be quanti- bigger v v2 nd n d d γ γ γ input signal an interval error is taken intoBy consideration substituting (12) as the to (16):Secondn+Second1 derivativeddtn derivativedt ofdtgη offunction0dtgηdtfunctiong(0t+n+dt inΔ1t)+t + inΔttdtisg+( produced:tn)isdt produced: Factor γ depends on an interval dt and an interval error ∆t Equationdt (28) for minusdt 0 orderdtγ has the formula: Assuming f (t f ) is a functionIn obtaineddynamicIn dynamic bymodels measured models of real of quanti- realsystems systems with with fast changesfast changesg( oftη an)= of anlim − = g(tn+→1) dt → (17)n 1dt n 1 (20).Factor Becauseγ Δdependst 1[0,dt on] then an interval on for the branchΔdtt =and0 borders: an interval error Δt − → − val error. The differenceties betweenvtieseveryv botheverydt, functionsg(dttη,)gfunction(tη ) willfunction be takes bigger takes into consideration into consideration an inter- ann inter- d g(tdn)= lim v v v − vd d= d v(tη1 ) t v(t0) ifmost anif interval an important interval error factor errorΔt isΔ bigger. int is precise bigger. modelling based ondt the em- dtη 0 dt + Δt2 dt n+1 n+n1 n (20). Becauseγ = ∈ dt 0, dt then on the branch borders: (25) η1 v dt g(t ) dt g(t g)=(t dt)=ηlim0d limdt− −=dt d=g(t g()t ) (17) (17) g(t0)= lim γ − =γ ties every , η functionvalinput takes error.inputval signal error. into The signal an consideration difference The interval an difference interval error between an error betweenis inter- taken both is taken into bothfunctions intoconsideration functions consideration willd be will bigger as be the bigger as the →vSecondn+1 Secondvn derivative2 d2 derivativeηn ofηn g→function of g(functionηn( in)+1tn+ in)1ηntis(n+ produced:)1 (isn+ produced:)1 n+1 (20). Because Δt 2 [0,dt] then on the branch borders: γ d (v vγ )dt v(t )γ v(vt()tdt) if an interval error Δt is bigger.piricalIn dynamicIn datum. dynamic models It is models used of real to: of systemsreal models systems withof vibration with fast changes fast and changes velocity of an of an d dt d dt dtη dt0dtgηdttdtn0++g1dtttn+−1dt gtdttdtn gdttn  logdt 2dt for Δt = dt dt− lim 1 dt 0 0 dtη−1 0 (29) Assuming f (t ) is a function obtained by measured quanti- By substitutingg(tηn (12))= tolim (16): − = g(tn+1) (17)Δ Δ 1∈ £ ¤ for Δt = 0 dtγ 0 → val error. The differencef betweenmostmost important both important functions factor factor will in precisebe in bigger precise modelling modellingSecond based based on derivative the on em- the ofem-g function ingt(tn)=gis(tn produced:)=lim →lim → v − 2−v +=v = γ g(t0−)= lim ≡ t0t −γ = In dynamic models ofif real anif interval systems an interval error with errorΔ fastt isΔ bigger. changest is bigger. of an dt dtη 0 dt + Δt 22 n+dt221 dt dt0 d 0 d n+2 d n+d1 n γ = (25) dt → dtγ 0  ηdt1 ties v every dt, g(t ) functioninputtransducers, takesinput signal into signal an consideration models interval an interval of error gas an anderror is inter- taken liquid is taken into flows, considerationinto models consideration taking as the into as the→ dtd dtd η η limg(ttnη+ng+1(1)tηn+n−+t1η1)ng(tηnn)g(tn) (18) 1 for t = 0 − →γ − γ if an interval errorηΔt is bigger. d vSecondn+1 Secondvn derivatived derivative of →g function ofdt→g functiondt in−tn+dt in1−istn+dt produced:12is produced: (18) Characteristicslog of γdt(Δ2tdt,dt) arefor shownΔt =Δ indt fig. 3. lim (v1 v0)dt vt (t)dt (29) piricalInpirical dynamicIn datum. dynamic datum. models It is models It used is of used real to: of models systemsto: real models systems of with vibration of with fast vibration changes fast and2 changes velocityand of velocity an of an d g(tdn)=g(tn)=lim limdtη 0 −(dt +−Δt) = = γ= . (25)(25) The minus orderγ derivative (29) is equivalentη1 to the integral input signal an interval errormostconsiderationmost is important taken important into the factor consideration deviation factor in precise in from precise as modellingtrue the modelling periodicity based based ofong( a thetη presum-onn )= em- thelim em- g(−t )2= 2gg((ttn)+1) →v(17)v v v+ v + v dt 0 − ≡γ t0 γ val error. The difference between both functions will be bigger Second derivatived of g functiondt n in+dt1tn+1dtisdt produced:ndtη 0dtη 0 t n+2 tn+2t2 n+21t n+1n n Because for every n : → ( )  ( ) In dynamic models of realtransducers, systemstransducers, with models fast models of changes gas of and gas of liquid and an liquid flows, flows, models modelsdt takingg( takingt )= intodtη intolim0 dt + Δt 2 − dt2 =→ limd→ limdηn+1 −ηn+dη1n− ηdn (18)(18)  logdtN2dt for Δt = dt lim v1 v0 dt v t dt γ (29) most important factor ininputpiricalably preciseinputpirical signal periodic datum. modelling signal datum. an intervalsignalIt an is Itusedinterval based (jitter). is error used to: on error ismodels to: the taken modelsis em- taken of into vibration ofconsideration into vibration consideration and2 velocity and asn thevelocity as the→ d d g(tn+g1()t−n+1)g−(tn2)g(tn2). Characteristics of γ(Δ∈t,dt) are shown in fig. 3. definition wheredt stepγ 0 and− branch≡ oft integration equal dt = if an interval error Δt is bigger. 2 dt dtη d0 Generally,tη d tη the n orderdtdtη derivativedt0 ηdt 0(dt of+dt(dtΔgtin+)dtΔt0t)is formulated as: →  0 input signal an interval errorconsideration isconsideration taken into the consideration deviation the deviation from as the from true trueperiodicity periodicityd of a ofpresum- a presum-→ g(tn+1)n+1 g(gt(ntn)=ng)(tn−)=lim lim→ vn→+2 v−n+22vn−+21 v+n+=v1n +=vn Fig. 3. Characteristicsn of factor ( t,dt). The minus order derivative (29) is equivalent to the integral pirical datum. It is usedmosttransducers, to:mosttransducers, models important important models of factorvibration models of factor in gas of precise and andin gas velocityprecise liquid and modelling liquid flows, modellingSecond flows, models based models derivativebased on taking the on taking em- into the of gem- intofunctiondt in tn2+dt−1 is2 produced: n n n n γmΔ m tη0 t0. Geometrical interpretation of (29) as a surface area is In dynamic models of real systems with fast changes of an g(t )= lim dt− dt =dtη 0dtlimη n 01tlimη t−η tηn−n 1tηn (18) (18)BecauseCharacteristics for every of γ(NΔ:t,dt) are shown in fig. 3. γ most important factor in preciseablyably periodic modelling periodic signal based signal (jitter). on (jitter). the em- 2 n vn+2n 2vn+1 + v→n dt d→−0dt n+01 n+1d − 2 2 Characteristics(dt + Δ oft∈) γ=(∆t, dt) are showndt − inΔ tFig. 3. (26) definition−The minus where order step andderivative branch (29) of isintegration equivalent equal to thedt integral= transducers, models of gaspirical andpirical datum.liquid datum. flows, It is usedmodelsIt is used to: taking models to: models into of vibration of vibrationdt and velocity and velocitydtη 0 Generally,t Generally,t the n theordern order derivativeη derivativeηg(t1()dt− of+g(dtΔ− ofint)+tg0(Δinistt0))t formulated0 is formulated as: as: ∑ shown in fig. 5. consideration2.2.consideration Derivative the deviationof the function deviation from with from true an periodicitytrue interval periodicity error2 of a presum-ofLet as presum-η be→ d limηn+1 d −ηdn dt→n 1 →(18) dtn 1 Because for every n Nmn=: 0 m γ inputpirical signal datum. an Itinterval is used error to: modelsis taken of into vibration consideration and velocity as the d dtg(tn+0 1)−Generally,gg(t(t)=n)− 2thelim −n–order− vderivativen+2 v−n+22v nof+− 1g2 +vinn+ vt0n1 is+= formulatedvn as: n  tη definitiont0. Geometrical where step interpretation and branch of of (29) integration as a surface equal areadt is = transducers,transducers, models models of gas of and gas liquid and liquid flows, flows, models models taking taking into intodt η (ndt +0 Δt) n 1 n 1 n 1 n 1 n ∈ n m m 0 = consideration the deviationably fromably periodic trueperiodic signal periodicity signal (jitter). (jitter). of a presum- g(t )= lim →vn+2 dt−2vn+1 +n =vdtn ηn 0dlim− dlim− dt−+dΔg−t−dtg− t (18) (18)The fractional(dt + orderΔt) derivative= n isdt defined− Δt by substituting(26) Let− n 2. On the basis of (18) and (23) the minus order most important factor in precisea function modelling of variables based ontη (11) the em- and v (8) from chapter2 n 2.1. Generally,Generally,n then theorder→dtorder derivative0dtg derivative(t )0g(t of) in ofg2(0t inis)g( formulated20t is) formulated as: as: ∑ transducers, models of gas and liquid flows, models taking into dt Let sLetdtbeη s 0belim t d− t d − −dtηn 1dtη(18)n 11 (dt1dt+n(Δdt1tdt)+n Δ01 t) 0 then the derivative (23) cann have them n formula:n m m showntη0 int0 fig.. Geometrical 5. interpretation of (29) as a surface area is ably periodic signal (jitter).consideration2.2.2.2.consideration Derivative Derivative the of deviation the function of deviation function from with from truewith an interval periodicity true an interval periodicityGenerally, error of error a presum- of the aη presum-n →orderη derivativeηn+1 gη( ofnt )=g(2int )=tlimis formulatedlim→− −→n − as: −− m −n = = (dt + t) m==0 dt t (26) Fig.derivative 4.− Geometric of function interpretationg in t0 ofwith a non-integer multiplicity positiven = order2 is deriva- pro- pirical datum. It is used to: modelsThe gradient of vibration of sη and(tη ,v velocity) function can be formulated as fol-dtη 0 (dt−n + Δ0nt) 0 0 dn 1 dn 1 ( dn1)1 dn 1 v equation (22) to (21): Δ ∑  − Δ consideration the deviation from true periodicity of a presum- − → dtn dtn dtη dt0 η −0 ∑−dtm=+0 dtΔt+− Δt m− n m n n m m Letshownn = in2. fig. On 5. the basis of (18) and (23) the minus order ablya functionaably periodic function periodic of signalvariables of variables signal (jitter).tη (jitter).(11)tη (11) and v and(8)v from(8) from chapter chapter 2.1. 2.1. Generally,vn+2Generally,d 2vd then+1 n+ thevordern n order derivativen 1 g derivative(nt11)g( oft1−)gn in1 ofgt(gnt0isin1)g formulatedt(−t0is) formulated as: as: d m∑=0 ( 1) vn m tive.duced: 2.2.lows2.2. Derivative [1]: Derivative of function of function with with an interval an interval error errorLet sLetη besη be n 1 n 1 → dt→lim− dt − dt − dt0− 0 (19)then the derivative (23) can haven them=0 formula:m n transducers,ably periodic models signal of(jitter). gas and liquid flows, models taking into n limd − − g(t0d)=g−(t−0)=lim−limdt 0 (18)n −n(dt−+ nt)n n= = n g(t0)= lim ( −) − (27) derivativeLet n of= function2. On theg in basist with of (18) multiplicity and (23)n the= minus2 is pro- order TheThe gradient gradient of sη of(tηs,ηv()tηfunction,v) function can beGenerally, can formulated be formulatedd the n as fol-order as fol- derivativen 1 g(t1 ofn) g innn t210gis(t formulated0) η as: m Δ m d η γ ∑mn=0 n1 nmmvn mm 0 2.2. Derivative of functiona function witha function an of interval variables of variables errort (11)t Let(11) andsηv andbe(8)v from(8) from chapter chapter 2.1. 2.1. dtη dt0 − dt(dt +dtΔdtt)− dtη 0dtη n →01 ∑nmdt=1 ∑0+(m=dtΔ10nt)(+1 Δ1 tn) v1n mvn m dtn dt 0 ∑ n dtm Δt− 2 d d consideration the deviation from true periodicity ofη a presum-η g(t )=− lim → n − n → =d→− d − d − dm− m− − then the derivativeg(t0)= (23)lim can→ havem=0 the−m formula:− (23) d γ g(tη ) γ g(t0) ∂ ∂ n 0 n 1 d nd 1 limn 1limg(nt11)g(−t1)n −1 g(nt01)g(t0) (19)(19)(19) dη dtγ 0 ∑m=0( 1γ)n vn m duced:derivative of function g indt−t0 with1 multiplicitydt− n = 2 is pro- 2.2.lowslows2.2. Derivative [1]: Derivative [1]: of function of function with with an interval an interval errordt errorLet sLetbedtsη dbe0 dt d+ Δt dt dtn n dt m dtnnm nn dt (dt ) g(t )= lim − ablya functionperiodic signalof variables (jitter).tη (11)The and gradientThev gradient∇(8)sη fromof(tηs,ηv of( chapter)=tηs,ηv()tηfunctions, 2.1.ηv()tηfunction,v)i can+ be cans formulatedη be(tηn , formulatedv) j asη fol-(14) asη fol-− g(t ) g(−t )=gg((tt ))=lim dtlimη −dt0 η −0 −(dt +(−dtΔt+) −Δt)= = γ gn(t0)= lim→ n n n− m− (27) same (characteristics2γ 0 will overlap). Otherγ interpretation can Generally,d the n order→dt derivativen 12.3.1 Derivativen ofdtgn 0in1n t000 ofis formulated non-integer→ ∑→m as:=0∑ positive(m=10)( 1m order)v nmmv andn m its inter-For dt 0η d and n =dt1γ the0 geometric∑ ( n interpretation1)m vnm m of the duced:dt dtη 0 d dt d 2.2. Derivative of function with an interval error ∂Lett sη be ∂v − dtn− dt− dtnη 0dtη 0 dt + Δdtt + Δt − − n dt ∑m=0 m=0dt − Δt −2 → − The gradient of sη (tηalows,v function) functionalows [1]: function [1]: of canvariables of be variables formulatedtη (11)tη and(11) asv fol- and(8)v from(8) from chapterg chapter(t0 2.1.)=− 2.1.lim m γ = lim lim − − . (19) (19) d → g(t0)=→lim m − − (27) found ind [5–7, 22]. dt γ g(tη1 ) dt γ g(t0) ∂ ∂ ∂ ∂ n n ∑m=0(n 1) →vn m→ n n where: η is a notationη of fractionalγ n derivative;n n m n mis a natural t − t − a function of variables tη (11) and v (8)s from(t s, chapterv()=t ,v)= 2.1.s (t s,v()ti +,v)i +s (tdts,v()t j ,v) j dt(14)η 0(14)d 1pretationdt + Δdt Let1 dtm stands−dtη for0dtη n change0 (dtn + of(dtΔdttn)+: Δtn) fractionaldt ordersdt derivativedt (23)0 is an angledt ofΔ inclinationt of g2 (t0)= lim η2 d γ − η1 d γ The gradientThe∇ gradientη ∇ ofη ηs of(ηt s,v()tfunctionη,v)η ηfunctionη can be canη formulated beη ηn formulatedη as fol-→ as fol-−lim −− → → (19) m m  γ → ∑m=0 m − 2γ v(t)dtγ g(tη )γ v(t)γdtg(t ) lows [1]: where ∇sη is the gradientη η η η defined on points dtη situated on the n2.3.1 g(2.3.t Derivative1) Derivativen 1 g( oft0) non-integern of non-integer∑ = positive∑( = positive1)( order1)vn orderm andvn and itsm inter- its inter-For dt 0 and n = 1 the geometric  interpretation of the dt− d dtη tt0η dt dt−1t0 dt 0 2.2. Derivative of function with an interval error∂tLet∂stη be ∂v ∂v dtdt−η 0n dt(dt−m+nΔt) m 0 m 0 m m ordertangent of to derivativeg in tn, the (multiplicity same as the of first an interval order derivative.dt); η is a In frac- the g(t )=→ 1 − − − − The gradient of sη (tη ,v) function can be formulated∂ as∂ fol- ∂ ∂ g(t0)= lim →∑ −( 1) γvn=mγ lim lim − −γ − − (19) (19) →γ 2γ= 0lim lim γ (30) lowscurvelows [1]:g, [1]:i∇ands (∇tj sare,v()=t the,v)= standards (t s,v unit()ti +,v vectors.)i +s (t s,nv()tj ,v) j (14) (14)pretationmpretation=0 Let dtmLetstandsdt dtstands+ forΔt change= fordt changeη = ofdtdt of: ndt: n (20)fractionalFor dt orders0 derivative and n = 1 (23) the is geometric an angle interpretation of inclination of of the 2.4. Derivativedt of non-integerdttη 0 minus2γdttη order and its inter- a function of variables tη (11)∂ and v (8) fromη ∂η η chapterη 2.1.η η η η η dtη η η dtη 0 lim2.3.2.3. Derivativedt +− DerivativeΔt of non-integer− ofdt non-integerη 0dtη (19)0 positive(dt positive+(Δdt ordert)+ Δ ordert) and andits inter- its inter-tionalcase where orderthe of derivative approximation (for Δ oft = a0) tangent connected  is a secant with a (fig.natural 4) − dtη 0  2→ ( ) dtγ −  1− ( ) γ lows [1]: wherewhere∇sη ∇issη theis gradient the gradient∂t defined∂t defined on points∂ onv points∂tvη situatedtη situated on the→ on the n → →   → � → tt v t dt t0 v t dt ∇sη (tη ,v)= sηFunction(tη ,v)i +g iss takingη (tη ,v into) j consideration(14) an interval error Δdttη 0 (dt + Δtn) γ γ γ γ tangentfractional to g in orderst , the derivative same as the (23) first is anorder angle derivative. of inclination In the of pretation Let n = 1. Onη the1 tη basis− of (22)-(23)tη and η = γ the The gradient of sη (tη ,v) function can be formulated∂ as∂ fol- ∂ 2.3.∂ Derivative of non-integer→pretationnpretation2.3. positive DerivativeLetm dtLet orderstandsdt of stands andnon-integer for its change for inter- change positive of dt of: orderdt: and its interorderthen- the by geometrica factorn γ (22). interpretations of the derivative are: an an- = lim 2 γ 1 γ (30) curve∂t curveg, i andg, i andj∂arev j theare standard the standard unit unitvectors. vectors. ∑mBy=0 substituting( 1) vn (20)dtmγ + todtΔ (19):t+=Δdtt =η =dtηdt= dt (20)(20) t v(t)dt 2γ t v(t)dt ∂ where(12).where∇ Gradientsη∇∇iss∂ηs theη(∇tηis∇s, gradientηsv the()=t(ηt, gradientv,)=v)s definedforη (tηdt defineds,ηv() ontiη+0,v points is) oni + asη pointslinear(ttηs,situatedηv() approximationttηjη ,situatedv) j on(14) theγ on(14) the m − derivativeGenerally, is: the minusdtη 0  ordertη1 derivativedt  with0 multiplicity n is lows [1]: η η pretation Let dt standslim2.3. for2.3. Derivative changepretation.− Derivative of dt ofLet: non-integer dt of stands non-integer for(19) positive change positive of order dt: order and itsand inter- its inter-caseIntangent where the case to theg where approximationin tn,Δ thet = same0 then asofη thea= tangent firstn and: order( is, a secantderivative.) ( (fig., In 4)) the =→lim − − (30) ∇sη (tη ,v)= sη (Functiontη ,vFunction)i + g issηg taking(tisη ,v taking) j into∂t into consideration∂t(14) consideration→ ∂v an∂v interval an interval error errorΔtdt Δt0 (dt + t)n n 1 γn 1 γ gle of inclination of a secant joining points t0 v0 and tη1 v1 2γ where ∇sη is the gradientcurveof definedgcurvebetweeng, i and ong, i pointsjand nextarej the valuesaretη standardsituated the of standardt on(fig. unit the vectors.unit 2).2.3. vectors. Derivative of non-integerη positiven Δ orderγ dt andγ +dtΔ itsdt+−= inter-Δdtt η==dtdtηd=− dt (20) (20) written as: dtη 0  dt  ∂t ∂v η →pretationBypretation substitutingByd substitutingLet dtLet (20)standsdt (20) tostands for(19): ton change1 (19): forg(t1 change) of dtn of:1 gdt(t0:) thencase the where geometric the approximation interpretationsn of of a then tangent derivative is a secant are: an (fig. an- 4) ( )− ( ) s (t s,v()t ,v)dt dt γ γ dt − dt −γ for the fractional order derivative and an angle of inclination Generally, thed minus→ order derivativev tη1 withv t0 multiplicity n is curve g, i and j are the∂ standardwhere(12).(12).where Gradient∇s unitη Gradientis∇∂ vectors.s theη∇is gradientη the∇η η gradientηfor definedfor defined on0 is points on a0 linear is points atη linearsituated approximationtη situated approximationdt on the ondt the+ Δt = dt =gdt(dtt0)= lim ∆ (20)− = d d n 1 n 1 FunctionLetFunctionvη beg is values takingg is taking situated into considerationinto on→ consideration the→ gradientpretation an interval vector anLet interval∇ errorsη standsin errorΔttη Δ fort change η ofn : dt +n 1 t n=1 dtη =n dt1 .n 1 (20)glethen of inclination the geometric of a secant interpretations( ) joining= points of the( ()t derivative0,v0) and ( are:tη , anv1) an- g(t0)=d −lim − d − (28) s ∇sη (tη ,v)= sη (tη ,v)i +n sη (ttη ,v) j (14) n By substitutingBydt substituting (20)dtη (20)to0d (19):− tod (19):− dtd −γ d −γ of a secant joining pointsg t0 (t ,v ) and (ft t,0v ) for the first1 (24) or- n γ dtγ 0g(t ) γ g(t ) where ∇ η is the gradient definedcurveof g ofcurvebetweeng,g onibetweenandg points, i nextjandare nextη valuesj thesituatedare valuesstandard the of standardtη on of(fig. theunittη (fig. 2). vectors. unit2.3. 2). vectors. Derivative of non-integer positiven n orderdt and→+ Δdt itst += inter-gΔ(dttt =)g=(dtt dtη) =gdt(t )g(t ) (20) (20) dtη 0 0 dtn 1 1 writtenGenerally,d as: dt the minus( ordern 1)γ derivativeη1 dt ( withn 1)γ multiplicity0 n is Function g is taking∂t into(12).(fig. consideration(12). Gradient 2): Gradient∂v ∇sη an(∇tηs interval,ηv()tηfor,v)dt errorfor dt0Δ ist a0 linear is a linear approximation approximation d dγ dtn 1dtn 11η 1dtηn 1dtn 01 0 for the fractional order derivativeΔt=0 and an angle of inclination dt− − → − dt− − By substitutingγ (20)+ to (19):= = g(t )=g(t )=lim lim− −n − −− m −n = = gle of inclination of a secant joining points (t0,v0) and (tη1 ,v1) g(t0)= lim curve g, i and j are the standardLet unitvLetη vectors.bevη valuesbe values situated situated on→ the on gradient the→ gradient vector vectorLet∇sdtη ∇instandssηtdtηin tη forΔt changedtη n ofBydt dtsubstituting0n: 0 (20)dn 1 tod n(20)(19):1 ( dn1)1 dn 1 v der derivative. Difference between the angle β of inclination writtennγ as: dt 0 dn 1 γ dn 1 (12). Gradient ∇sη (tη ,vof) forFunctiong betweenofdtFunctiongnbetween0g isn nextis a takingg linear nextvaluesis taking values intoapproximation of t consideration into of(fig.t consideration(fig. 2).pretation 2). an interval an interval error errorΔtn Δnt dtn dtn dtη dt0 η −0 ∑−m=dt0 dt− m− n m Equationdt−n (28) forη minus order− γ hasdt− the formula:− where ∇s is the gradient defined on points t situatedη on theη By substitutingByd substitutingd (20) to(20) (19):n to1 g (19):(nt11)g(ηt1−)n η1 g(nt01)g(−t0) ofFactor afor secant theγ fractionaldepends joining points onorder an( derivativeintervalt0,v0) anddt andand(t1, anv an1) angle intervalfor the of errorinclinationfirst or-Δt d → (n 1)γ g(tη1 ) (n 1)γ g(t0) Functionη g is taking into consideration→ anη interval error Δt n 1 n 1 → dt→lim− dt − dt − dtn − (21)a secant on f and the angle α of inclination a secant on g will dt− − n 1 dt− − n 1 (12).(fig.v(fig.(12). 2): Gradientη =( 2): Gradientvη1 ,∇vηs2η,...,(∇tηs,vηvη()ntη)=(for,v)dtforv0(tdt00), isvη a10( lineart is1),..., a linear approximationvηnn( approximationtn)) (15) d − g(tγ0d)=g−(t0)=lim limdtγ 0 − ( −γ )n n= =  g(t )= lim d − − d − of g between next values ofLettη v(fig.Letη bev 2).η valuesbe values situated situated on the on gradient the gradientBy substituting vectord vector∇sη ∇in (20)sηtη in tot (19):η n 1 g(t1n) nn 1 g(t0) n n dtm m derof derivative. a secant joining Difference points between(t ,v ) theand angle(t ,vβ )offor inclination the first or- nγ n 0 tη g(ttη )γ g(t ) curve g, i and j are the standard unitn vectors.n → → dtn + Δntdt= dtηdt= dtdtdt dtη 0dtη n →01 ∑n(20)1 ∑dt( 1ndt)(1 1 n) v1n mvn m (20). Because Δt [0,dt] then on0 the0 branch1 borders:1 dt d d dtη 0 2 (vn(γt1)γ )dtηv11(t ) (γn 1)γ nγ 0 (12). Gradient ∇sη (tη ,v) for dt 0 is a linear approximation g(t )= lim − n − n − → =d→− d m−=0 mη=d0− η dm− m be the bigger as the bigger is a value of interval error Δt. For − dt− − η1 − −0 dt− − Let v be values situatedof g betweenof ong thebetween gradient next nextvalues vector values of t∇ηs of(fig.tinη 2).(fig.t 2). 0 n 1 n 1 n lim limg(t )g(−t ) −g(t )g−(t )− (21)(21) ∈ =g(limt0g)=(t )=→limlimv(t)dt − v(t)=dt dt (31) ηn (fig.v (fig. 2):→=(v v 2):=(,vv ,...,,v v,...,)=(v )=(v η(t v),(vηtn )(,tv),...,(t )v,...,(tvn ))(t )) dt d0 d d d dtγ n 1 dtγ n 11 1dtn γ1 ndtnγ01n 0 a secantder derivative. on f and the Difference angle α betweenof inclination the angle a secantβ of on inclinationg will nγ γ 0 γ γ γ Function g is taking into considerationηDerivativeη η1 ηη of an21 g intervalηin2 ηtnη isη errorn produced,0 Δ0 t0 η01 where1η1 1g betweennηndt nηn ntη(15)n andη(15)− Becausedtη−dt on thedt left−0dt of (21)n−0 −n is(dt the−m)(dt notationn −m) n of multiplic-Δt = 0 the angle of inclination secants on f and g will be the dt dtη 0 dttηt dt0 0 − t0 dt of g between next values of tη (fig. 2). By substitutingd (20) to→ (19):n 1 g(t1) g(nt01)=gg((tt00))=lim lim ∑ ∑( 1)( 1) vn =mvn =m − dt− →  tηη→1 dt−tη −  (fig. 2): Let vLetηn bevη valuesn be values situated situated on the on gradient the gradient vector vector∇sη in∇stηηnin dttηn− n dt −n → →γm=0 m=0 m m n be the bigger as the1 bigger is a value for Δ oft = interval0 error t. For 2 → γ 1 γ nγ (12). Gradient ∇sη (tη ,v) fortηdt is0 approximated is a linear approximation by the gradient vector: g(t0)= lim ity ofdtn−dt otherdt m thandtnη=dt0dtηlimη =0dtlimondtη− thedt right,−η instead− − of dt(21)the(21)a secant on f Fig.and 3. the Characteristics angle α of of inclination factor γ (∆t, adt) secantΔ on g will tη Let vηn be values situated onvn+1 the=(v gradientv=(,vv ,...,, vectorv v,...,)=(∇vsη)=(vin(ttηv)n,v(t )(,tv),...,(t )v,...,n (tv))(t ))(15) (15) ∑ ( 1n ) n →vn mγ→ γ γ n γ n (21) γ = (25) = lim vt (t)dtγ 1 vt (t)dtγ dt (31) (fig.Derivativeη 2):(fig.→Derivativeη 2):η1 ofηη21g ofinη2gtηinn istη produced,ηnis0 produced,0 0 η where01 1η where1 g1dtbetweenηgn betweennηn tηn dtandηtη 0 andn 1 BecauseBecausedtm=η0 ndtη1 ondt m theon left thedt of left0dt (21) of0 (21) is( thedt is notation)( thedt notation) of multiplic- of multiplic- dt 0 η2 η1 n n →n d −lim d −− − → →n (21)n m n m n  Δt =be0 the the bigger angle as oflog the inclinationdt bigger2dt is secantsfor a valueΔt on= ofdtf intervaland g will error beΔ thet. For ηlim (v1ttη v0)dt −γ t0 v(t)dt γ nγ (29) of g between next values of tη (fig. 2). n 1notationg(t1) dtn 1isgγ( insertedtn0) where:γ γ( 1)( 1)v v n n =→γ lim 1 v(t)dt  v(t)dt dt (31) (fig. 2):vη =(vη1 ,vη2 ,...,vηn )=(v0(t0),vη1 (t1),...,vηn (tn)) (15) d dtdtγ 0 dt (dtn) ∑m=0∑m=0 m n mm n m Bull. Pol. Ac.: Tech. XX(Y) 2016 dt 0 − ≡ t0 3 tη + tηis+ approximatedis approximatedd by the by gradient the gradientv vector:n+1 vector:vn ity− ofnitydt− ofotherdt−mnother thann thandtη =dtηdt=ondt theon right, the right, instead− instead− of dt of thedt the dt→η 0 − Let v be values situatedn onDerivative1 thenDerivative1 gradient of g vector ofin tg inist produced,s isin produced,t where whereg betweengg(betweent0)=t limandt and →∑Becausem= Because( 1dt) ondt v thenon=m left thelim of leftlim (21) of (21)is− the is− notation the notation of. multiplic- of multiplic-(21) (21)Δt = 0 the angle of inclination secants on f and g will be the  ttη t0  ηn vη =(vηvη=(1 ,vvηη21,...,,vη2v,...,gηηn(t)=(η∇vn )=ηηn )=(v0(limt0η)vn,0v(ηt01)(,t1v)η−,...,1 (t1)v,...,=ηnn (tnv))ηn (tηnn))dt(15)ηn0(15) 0 η ηm dtγ 0dtγ 0 γ n γ n →  1  dt 0 dt n η limnotationnotationdt−dtη isdt insertedis inserted− where: where:ηγ (21)= γn(dt )(dt ) n (22)n Characteristics of γ(Δt,dt) are shown in fig. 3. vη Derivative=(vη ,vη of,...,gvinη )=(t isv produced,0(t0),vη (t where1),...,dtvgη between(tn)) (15)t η andtηn+1 tBecauseηn dt on→ theγ left of (21) isγ then notation of→ multiplic-→   Bull. Pol. Ac.:Because Tech. for XX(Y) every 2016 n N: The minus order derivative (29) is equivalent to the integral3 (fig. 2): 1 2 n η tηn+1 tisηn+ approximated1 is1 approximatedd n byd the by gradient theηn→ gradientvn+ vector:vn+ vector:vn vn dt ity0 ofitydt ofother(dtdt )other than thandtη =dtdtη =ondt theon right, the right, instead instead of dt ofthedt the 2 1 −1 → n γ mn n  n n Because for every n N: γ DerivativeDerivative of g in ofgtg(ηtηinisng)=t( produced,ηtηnis)=lim produced,lim wherev − whereitygv− ofbetween=dtg =betweenothertη thanandtη dtand=∑Becausedtm=0onBecause( the1ηdt) right,onηdtv thenon insteadm left the of left of (21)dt of (21)isthe the is notation the notation of multiplic- of multiplic- definition where step and branch of integration equal dt = tηn+1 is approximated by the gradientd vector: n+1 ηn+1 n n n η notationand:notationγ dtis aisdt factor insertedm nis inserted− of where: changingη where:=ηγn= derivativeγn by Δt (20); n(22)is(22) aBull. Pol. Ac.: Tech. XX(Y)∈ 2016 3 dt dt dtη dt0 ηtη +0 tη +−tηn tηn lim − γ (21)γ n n Derivativevη =(vη ,v ofη g,...,invtη )=(is produced,v0(t0),vη where(tg1)(t,...,ng+d1betweenv)=η d(tn)) tlimηn (15)and→ v→nn+1 vnBecause+v11n vn dt η on the(16) leftγ of (21) is theSinceγ notationn dt γon the of= left multiplic-+ of= (21) is the notation of multiplicity of n 1 2 n tηn+1 tisηn+ approximated1 isdt1 approximatedn by(t the by gradient thet ) gradient0 vector:t −notation vector:−t dt is inserteddt ity where:0multiplicity ofitydt ofother(dtdt ) ofother thandt ; thandtη: dtγdtη ison adt fractional theon right, the right,order instead of insteadn derivative. of dt ofthedt the n n n m m tη0 t0. Geometrical interpretation of (29) as a surface area is g(tηngη)=n(+tη1 n )=limn+1 lim ηn−+1 −n=+1 = → γ  η= R n tη is approximatedd by the gradientd vector:vn+d1 vn − →vn+ity1vn of+−v1ηdtn+1votherηn+1 than dtη = dt ondt the otherη right, thanη instead dtη dt of ηondt= thetheηγ nright,= γn instead of tdt thet notationn (22)n (22) (dt + Δt) = dt − Δt (26)(26) − n+1 dt dt dtη 0dttηηn+01 tηn+tη1n tηn notationand:and:notationγ isdtη γ aisis factordt inserteda factoris of inserted changing where: of changing where: derivative derivative by Δ by(20);Δ (20);is a is a ∑ shown in fig. 5. g(tηn )= lim g(tn+−g1()=tn+1=)= lim lim→ → − − η n (16)(16) m Derivative of g in tη is produced, where gdbetweend tηn andvn+1notationv−nBecause+vn1 − vdtn dtis insertedon the left where: of (21)η =dt isγnthe is inserted notationγ where:γ of+ multiplic-+ (22) m=0   dt dtη dt0 tηn+dt1 tηn (tη (tηtn+1) tn+01)tηn0+1tηn+tn1+1 tn+1 multiplicitymultiplicity of dt of; dtη :;Rη :isR a fractionalis a fractional order order of derivative. of derivative. d →vd dv g(tηnng)=+(1t−ηn+)=lim1 −→ limvn→+1−vn+v1η−n=+1vηn=+1 and:γ and:γ isγ ais factor a factor of changing of changingn derivative derivative by Δ byt (20);Δt (20);n isn a is a Let n = 2. On the basis of (18) and (23) the minus order tη is approximated by the2 gradientn+ vector:1 −n ity− of dt−other than dtη = dt on the right, instead ofηdt= γtheηn=Bull.γn Pol. Ac.: Tech. XX(Y)(22) 2016(22) n+1 g(t )= lim g(−tn+g1dt()=t=n+1dt)=limdtη lim0dttηηn+01 tη−nt+η1n −tηn (16) (16) γ γ + + then the derivative (23) can have the formula: d ηn vn+1 vηn+1 → → and: γ ηis a factor of changing derivative by Δt (20); n is a derivative of function g in t with multiplicity n = 2 is pro- dt dtη 0 t dt dtt (tηn+ (tηtnn+1) tn0+1t)ηn+0−1 tηn+tn−1+1 tn+1 multiplicityη =multiplicityγn of dt of; ηdt: ;Rη :isR a(22) fractionalis a fractional order order of derivative. of derivative. 0 g(tn+1)= lim→ ηn+1 η−n 1 − 1 −→(16)→notation− −dt is insertedγ where:+ n n m dt d (t 2 t2 )vnd+0 1t−dvn t vn+1 vnmultiplicity+vη1n+1vηn+1 of dt ; η : and:isand: aγ Bull. fractionalis aγ Pol. factoris Ac.: a orderfactor Tech. of changing of67(6) of derivative. changing 2019 derivativeBull. derivativeBull. Pol. Ac.:by Pol.Δ Tech. Ac.: byt (20);Δ Tech. XX(Y)t (20);n XX(Y)is 2016 an 2016is a d ( 1) v duced: ηn+1 n+1 g(ηtn+1g()=t n+)=1 lim lim − − (16) (16)R ∑m=0 n m 1025 d g(tη )= lim− →vn+1−n+v1η−n=+n1+1 and: γ is a factor of changing derivative byγ Δtγ (20);+ n+ is a g(t0)= lim − − (27) n dt dt (t (t t ) t 0 t) 0 t t t multiplicityη =multiplicityγn of dt of; ηdt: ; η :is a fractional(22)is a fractional order order of derivative. of derivative. η γ n n n m m g(tn+1dt)= limdtη 0 tη −tηn ηn+1 ηn+1(16)n+1 ηn+1 ηn+n1+1 n+1 R R Bull. Pol.Bull. Ac.: Pol. Tech. Ac.: Tech. XX(Y) XX(Y) 2016 2016 dt dt 0 dt Δt 2 d d 2 →2 n+1 − −→ → − − γ + → ∑m=0 m − d γ g(tη ) γ g(t0) dt (tη tn+1) 0 tηn+−1 tn+1 multiplicity of dt ; η : R is a fractional order of derivative. dt 1 dt n+1 − → Bull. Pol. Ac.: Tech. XX(Y) 2016 g(t0)= lim − − − 2 d vn+1 −vηn+1 and: γ is a factor of changing derivative by Δt (20); n is a γ 2γ γ ( )= − For dt 0 and n = 1 the geometric  interpretation of the dt− dtη 0 dt− g tn+1 2lim 2 (16) γ + Bull. Pol.Bull. Ac.: Pol. Tech. Ac.: Tech.XX(Y) XX(Y) 2016 2016 → → dt (tη tn+1) 0 tη tn+1 multiplicity of dt ; η : R is a fractional order of derivative. t t 2 n+1 − → n+1 − Bull. Pol. Ac.: Tech. XX(Y) 2016 fractional orders derivative (23) is an angle of inclination of η2 γ η1 γ tt v(t)dt t0 v(t)dt tangent to g in tn, the same as the first order derivative. In the η1 − = lim 2γ (30) 2 Bull. Pol. Ac.: Tech. XX(Y) 2016 dtη 0  dt  case where the approximation of a tangent is a secant (fig. 4) → − then the geometric interpretations of the derivative are: an an- Generally, the minus order derivative with multiplicity n is gle of inclination of a secant joining points (t ,v ) and (t ,v ) 0 0 η1 1 written as: for the fractional order derivative and an angle of inclination n 1 n 1 n d − d − of a secant joining points (t0,v0) and (t1,v1) for the first or- d (n 1)γ g(tη1 ) (n 1)γ g(t0) dt− − − dt− − der derivative. Difference between the angle β of inclination nγ g(t0)= lim γ dt− dtη 0 dt− a secant on f and the angle α of inclination a secant on g will → tη2 tη1 be the bigger as the bigger is a value of interval error Δt. For = lim v(t)dtγ v(t)dtγ dtnγ (31) Δt = 0 the angle of inclination secants on f and g will be the dtη 0 tt − t →  η1  0 

Bull. Pol. Ac.: Tech. XX(Y) 2016 3 FractionalFractional order model order of measured model of measuredquantity errors quantity errors

Fractional order model of measured quantity errors

Fig. 3. CharacteristicsFig. 3. Characteristics of factor γ( ofΔt, factordtFractional). γ(Δt,dt order). model of measured quantity errors

The fractionalFig.The 3. fractional order Characteristics derivative order of derivative factoris definedγ(Δt, isdt by) defined. substituting by substituting equation (22)equation to (21): (22) to (21): Fig. 4. GeometricFig. 4. interpretation Geometric interpretation of a non-integer of a non-integer positive order positive deriva- order deriva- tive. tive. The fractionaln ordern derivativen is definedmn n bym substitutingn d d ∑m=0( 1)∑m=0(vn 1m) vn m − m −− m − Fig. 4. Geometric interpretation of a non-integer positive order deriva- equation (22) toη g (21):(t0)= ηlimg(t0)= lim γ n γ n (23) (23) dt dtdtγ 0 dtγ (dt0 ) (dt ) sametive. (characteristicssame (characteristics will overlap). will overlap). Other interpretation Other interpretation can can n → n → m n  dn d dn ∑m=0( 1) m vn m found in [5–7,found 22]. in [5–7, 22]. where: ηwhere:is ag notation(t0)=η is alim of notation fractional of− fractionalderivative;FractionalFractional− derivative;n is order a natural order(23) modeln is model a ofnatural measured of measuredFractional quantity quantity order errors model errors of measured quantity errors dt dtη dt dtγ 0 (dtγ )n order of derivativeorder of derivative (multiplicity→ (multiplicity of an intervalFractional FractionalFractional ofFractionalFractionalFractionalFractional anFractionalFractionalFractionaldtFractionalFractional interval); η order orderis order order order aorder orderdt order order orderfrac- order order model); model modelη model model model model model modelis model model model ofsame ofa of frac- ofmeasured ofmeasured of measuredof of of ofmeasured ofmeasured measuredof measured (characteristics measured measured measured measured measured quantity quantity quantityFractional quantity quantity quantity quantity quantity quantity quantity quantity quantity errors errors errorswill errors errors errors errors errors ordererrors errors errors overlap).errors model Other of measured interpretation quantity can errors n tionalwhere: orderd tionalis of a derivative notation order of of derivative (for fractionalΔt = 0) (forconnected derivative;Δt = 0) connected withn is a natural with a natural2.4.found Derivative in [5–7,2.4. 22]. Derivative of non-integer of non-integer minus order minus and order its inter- and its inter- dtη � � order byof derivative aorder factor byγ (22). a(multiplicity factor γ (22). of an interval dt); η is a frac- pretationpretationLet n = 1.Let Onn the= 1. basis On of the (22)-(23) basis of and (22)-(23)η = γ andthe η = γ the tionalIn the order case ofIn where derivative the caseΔt = where (for0 thenΔΔt t=η==0)0n connected thenand:η = n withand: a natural derivative2.4. Derivativederivative is: of non-integer is: minus order and its inter- � n n n n pretation Let n = 1. On the basis of (22)-(23) and η = γ the order by a factor γ d(22). d d d d d v(tη1 ) v(vt0()tη1 ) v(t0) ( ) =( ) ( =) ( ) derivative is: g(t0)= limg(t0)= lim− − (28) (28) In the case where Δη tg=t00 thenη gη t=0 n and:f t0 n f t0 (24) R. Cioć(24) and M. Chrzandtγ dtdtγγ 0 dtdtγ γ 0 dtγ dt Δdtt=0 dt Δt=0 dt → → n  n d v(t ) v(t ) d  d Equation (28)Equation for minus (28) fororder minusγ hasη1 order the formula:γ0has the formula: Factor γ dependsFactor γ ondependsg( ant ) interval on= andt intervalandf (t an) dt intervaland an error interval(24)Δt error Δt γ g(t0)= lim −γ (28) dtη 0  dtn 0 dt dtγ 0 dt (20). Because(20).Δ Becauset [0,dtΔ] thent Δt=[0 on0,dt the] then branch on the borders: branch borders: →( ) ( () ) ( ) Fig. 3. Characteristics of factor ( t,dt). d d v tη1 v vt0tη1 v t0 ∈ ∈ γ Δ Equation (28)g for(t0)= minuslimg order(t0)=γ haslim− the formula:= − = Factor γ depends on an interval dt and an interval error Δt dt γ dt dtγγ 0 dtγ γ0 dt γ  t = t = − − → −→ − 11 for Δ for0 Δ 0 t t (29) (20). Becauseγ =Δt [0γ,dt=] then on the branch borders: (25) (25) d v(t ) η1v(t ) η1 ∈ γ η1 γ0 γ γ The fractionallogdt 2dt orderlogdtfor derivative2dtΔt =fordt is definedΔt = dt by substituting γlimg(t0()=v1 vlim0)dt(v1 v−0)γdtv(t)dt= v(t)dt . (29) (29) dt dtγ 0 −dtγ 0 ≡−dtt ≡ t equation (22)1 to (21): for Δt = 0 Fig. 4.− Geometric→ interpretation→  0− of a non-integer 0 positive order deriva- tη Characteristicsγ =Characteristics of γ(Δt,dt of) areγ(Δ shownt,dt) are in fig. shown 3. in fig. 3.(25) tive. γ 1 γ  logdtn2dt for Δt =n dt m n The minusThe orderlim minus( derivativev1 orderv0)dt derivative (29) is equivalentv (29)(t)dt is equivalent to the integral(29) to the integral Because forBecause every nd for every: n : ∑m=0( 1) m vn m Thedt γminus0 order− derivative≡ t (29) is equivalent to theγ inte- γ N N − − → 0 dt = dt = ∈η g(t0)= ∈lim γ n definition(23) graldefinition where definition step where where and branch step step andand of branchbranch integration of integration equal equal equal dt n dtγ 0 n (dt ) Characteristics of γ(Δt,dt) are shown→ in fig. 3.  sameγ (characteristics will overlap). Other interpretation can n n n m n m n m m tη0Thet0 minus.dt Geometricalt η=0 t orderη t 0. t Geometrical0 derivative. Geometrical interpretation (29) interpretation interpretation is of equivalent (29) as of aof surface to (29)(29) the as integral areaa a surface surface is area is (dtFig.+nFig. 3.Δt Characteristics) 3.(= Characteristicsdt + Δt) = ofdt factor of− factorΔγt(ΔFig.dttγ,(dtΔ−t 3.),.dtΔ Characteristics)t. (26) of(26)− factor γ(Δ−t, 0dt¡). Because for everyd n N: ∑ ∑ shown inareafound fig.shown is 5. shown in in [5–7, fig. in Fig. 22]. 5. 5. γ where: dtη is∈ a notationm=0 ofm fractionalm=0 m derivative; n is a naturaldefinition where step and branch of integration equal dt = Fig.Fig.Fig.Fig.Fig.Fig.Fig.Fig. 3.Fig. 3.Fig. 3.Fig.Fig. 3.Characteristics 3.Characteristics 3.Characteristics 3. 3. 3. 3.Characteristics Characteristics3. Characteristics 3.Characteristics Characteristics Characteristics Characteristics Characteristics Characteristicsn   of of of offactor offactor offactor of of of offactor factorof factor offactor factor factor factor factor factorγγγγ(((Δ(ΔγγΔγtγ(t(tγ(,tΔ,γ(Δγ,Δdt,γ(dtΔ(tdtγ(tΔt(Δ,Δ,t(),Δ)dtt,dt)tΔ.)dtt.,dt,.t.,tdt),dt)dt,)Fig..dt).dt.).)).)..).. 3. Characteristicsn = of factornγ=(Δt,dt). order of derivativen (multiplicityn n ofm anm interval dt); η is atη frac-0Lett0. Geometrical2.Let On the2. interpretation basis On of the (18) basis of and (29)of (23) (18) as thea and surface minus (23) area the order minus is order then the derivativethen( thedt + derivative (23)Δt) can= have (23) the can formula:dt have− theΔt formula: (26) − ThetionalThe fractional fractional order of order derivative order derivative∑ derivativem (forTheΔ ist = defined fractionalis0) defined connected by order bysubstituting substitutingwith derivative a naturalderivativeshown is in defined2.4. fig.derivative of function5. Derivative by substituting ofg functionin oft0 non-integerwithg in multiplicityt0 with minus multiplicityn order= 2 is and pro-n = its2 inter- is pro- n n m=0 n  � mn m Fig.Fig. 4. Geometric 4. Geometric interpretation interpretation of aFig. of non-integer a 4. non-integer Geometric positive positive interpretation order order deriva- deriva- of a non-integer positive order deriva- equationTheTheTheequationTheTheTheorderTheTheTheThe fractionalThe fractional fractionalThe (22)d fractional fractional fractional fractional by fractional fractional fractional (22)fractional fractional to a factor(21): to order order order (21):d order order order order order orderγ order order order(22). derivative derivative derivative derivative derivative derivative∑ derivative derivative derivative derivativem derivative derivative=equation0( is is is1) is∑ isdefined isdefined isdefined ism isv is(22) definedis defined=nThe isdefined defined defined0 defined definedm defined( defined to fractional1 by by by(21):) by by byv bysubstituting substitutingby substitutingbyn by by by substituting substituting substitutingm substituting substituting substituting substituting substituting substituting orderduced: derivativeLet n =pretationduced:2. is Ondefined theLet basis byn = substituting of1. (18)On the and basis (23) of the (22)-(23) minus orderand η = γ the then the derivativeg(t (23)0)= canlimg have(t0)=n thelim formula:n− n − n− − (27) (27)tive.Fig.Fig.Fig.Fig.Fig.Fig.tive.Fig. 4.Fig. 4.Fig. 4.Fig.Fig.Fig. Geometric4. Geometric4. 4.Geometric 4. 4. 4. Geometric4. Geometric4. 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(fig. 4)Δt of. 4) For a tangent is a secantdtη (fig.dt0η t 4)t0ηtηtη1η111ttηtηηt1η1t1tηt1ηηt1η1t1η11 dt−γdt −0000000000000 dtηγ 0  nγ tηdt1 −  0 →  tangenttangenttangenttangenttangenttangenttangenttangenttangenttangenttangenttangentsametangent to to to to tog toinclinationg tog to(characteristics to toin toing togining toggingtingtingntintngninn,in,int,int,tn theinn thetn the the,tn,t,tnn,t then thet,nsecants the,n, the same, samethe the,same samethe the the same same same same same same same same will sameas ason as as as as the asthef asthe the as asand asoverlap). theas the asthe the first firstthe thefirst firstthe the theg first first first firstwill firsttangent first order first order firstorder order first order order orderbe order orderOther order order order order derivative.the derivative. derivative. derivative. to derivative.derivative. derivative.same derivative. derivative. derivative.g derivative. interpretation derivative. derivative.in (characteristicstn, In In theIn In In Inthe Inthe Inthe the In Insame In theIn the Inthe the the the the canthe the as the first order==== derivative.===limlim=lim=lim=→==limlimlimlimlimlimlim→limlim Inv( thet)dt−−−−−−−−−−−− v(t)dt → =dt lim(30)(30)(30)(30)(30)(30)(30)(30)(30)(30)(30)(30)(31)(30) − (30) dt 0 2222γγγγ222γ2γγ2γ22γ2γγ2γγ 2γ n thencasecasecasecasecasecasecasethencasecasecasecase the wherecase whereΔcase where wheret where where where the= wheregeometric where where where where where0 thegeometric the the the the the the the approximation approximationthe theapproximation approximationtheangle the the approximation approximation approximation approximationinterpretations approximation approximation approximation approximation approximation interpretations of inclination ofthen of of of of ofa ofa ofa a of of tangentof tangenta ofthetangent atangent ofa thesecantsa ofatangent atangent atangentcase atangent atangent geometrictangenttangent the derivativetangent tangent iswhere is derivativeis is on isa isa isa ais issecant issecant isa secantfa secantisa isa asecant ainterpretationsandsecantthe asecant are:asecant asecant secant secant secant secant approximation are:g (fig. (fig.(fig. an(fig.will (fig. (fig. (fig. (fig. an-(fig. an(fig. (fig. (fig.4) 4) (fig.4) 4) be an-4) 4) 4) 4) 4) 4)the 4) 4) of4) the of derivative a tangentis are:dtdtdtdt aηηηdtηdt secantdt andtηdtηdtη0dt0η0dt0ηdtηη an-η00η0000t0t0 (fig.η0 4)dtdtdt−dt−dt−dtdtdt−dt−dt−−dt−dt−−−−t−0  dtη 0  dt−  where: d is a notation of fractional derivative; n is a natural foundwill in [5–7,overlap). 22]. Other interpretation can found in [5‒7, 22]. Generally,Generally, the the minus minus→→→→→→ order→→→→→→ order1 derivative derivativeGenerally, with with the multiplicity minus multiplicity ordern→is derivativen is with multiplicity n is dtη glethenthenthenthenthenthenthengle ofthenthenthenthen thethen the thethen theinclination of the the the the geometric geometricthe thegeometricinclination geometricthe the the geometric geometric geometric geometric geometric geometric geometric geometric geometric of interpretations interpretations ainterpretations interpretations ofsecant interpretations interpretations interpretations interpretations ainterpretations interpretations interpretations interpretationssecant interpretations joininggle joining of of of of points of ofthe ofinclinationthe theof the ofthen of ofpointsthe of the theof the derivative derivativethe thederivative derivativethe the( the derivative derivativetthe derivative derivative derivative derivative, derivative derivative(v derivativet geometric) of,vand aare: are:) are: are:secantand are: are:( are: are:t are: are: anare: an are:an an are:(, interpretations ant anv anan- an-an an-joining an-an an an) an an-, an-an an-v an- an- an- an- an-) an- points of(t the,v derivative) and (t , are:v ) an an- order of derivative (multiplicity of an interval dt); η is a frac- 0 00 0 η1 η11 1 writtenGenerally,Generally,Generally,writtenGenerally,Generally,Generally,Generally,Generally,Generally,Generally, as:Generally,Generally,0 as:0 the the the the the the the minus theminusthe minus the the the minus minus minus minusηminus minus minus1 minus minus order order order1 order order order order order order order order order derivative derivative derivativewritten derivative derivative derivative derivative derivative derivative derivative derivative derivative as: with withGenerally, with with with with with with with with withmultiplicity multiplicity withmultiplicity multiplicity multiplicity multiplicity multiplicity multiplicity multiplicity multiplicity multiplicity multiplicity multiplicity the minusnn nnis isnnisisnnnisn orderisnisnisnisisisisis derivative with multiplicity n is forglegleglegleglegleforgle theof ofgle ofgleglegleBull.gle of inclinationof inclinationof inclinationtheof fractional of of of inclinationof inclination inclinationof inclination Pol. inclinationinclination fractional2.4.inclination inclination inclination Ac.: Derivative of ofTech.order of of aof aof aof order ofsecant ofsecant of asecantofsecant aXX(Y) aof a derivativesecant asecant asecant asecant a asecantsecantofsecantsecant derivativesecant joining joiningnon-integer2016 joining joining joiningfor joining joining joining joining joining joining joining joining and the points points points points and pointsgle fractionalpointsan points points points points pointsminus pointspoints angleof an(((t(t0tt0 inclination0(0,( angle,(t,(t,vtv0( 0vt(v0(order,t0(,t0 of,t0()v0),tv order0)vt),0,v0and0,andv0,vand)0andv, inclination))v0 of0)v0and)0and)and 0)and)and()and( inclinationand(t( derivativeandtandηtt ofηη(((t (t,tη,(itsηt(, aη,v(vtη(tvvtη(1η,t1secantη,1 t,1η)v),interηv)v),,v1,1v1,v)1v,))v11)v1)1)1 and)-)) joiningis an written angle points as: of(t inclination0,v0) and (tη ,v1) 3 tional order of derivative (for Δt = 0) connected with a natural 2.4. Derivative of non-integer minus order and1111 its1111111 inter-11 writtenwrittenwrittenwrittenwrittenwrittenwrittenwrittenwrittenwrittenwrittenwritten as: as: as: as: as: as: as: as: as: as: as: as: n 1 1n 1 writtenn 1 as:n 1 n 1 n 1 of aof secant a secantpretation. joining joining points Let points n =(t 1.,(v oftOn),v aandthe) secant andbasis(t ,(v joiningtof), v(22,for) for23) the points theand first firstη(t or- =,v γ or- )theand (t n,v n) for the first or-d − d −g(t g()tn ) d − d −g(t g)(t ) d − g(t ) d − g(t ) � forforforforforforforfor the theforfor the theforforpretationfor the the the the fractional thefractional the fractionalthe fractional the the fractional fractional fractional fractional fractional fractional fractional fractional fractionalLet order order order order order ordern order order order order= order order order derivative derivative derivative derivative1. derivative derivative derivative derivativeOn derivative0 derivative derivative derivative derivative0 the0 and and0 andbasis and and and and andfor and andan anand anand an and1 of an an an anglethe anangle angle anangle an1 an(22)-(23)1 an angle anangle angle angle fractionalangle angle angle1 angle angle of of of of of of inclinationof inclination of inclination inclination of of of andofinclination inclinationof inclination inclination inclination inclination inclination inclinationorder inclinationη 0= γ derivative0 the d1 andd1 an angle ofdt inclination(ndt 1)(γn 1)γη1 dη1 dt (ndt 1)(γn 1)γ0 0dt (n 1)γ η1 dt (n 1)γ 0 order by a factor γ (22). −ddddnnnn−dd1d−1nd1n1nddndnd1n−1dn1n1n11111 − −−ddddnnnn−dd1d−1nd1n1nddndnd1n−1dn1n1n11111 − − dn 1 − − − dn 1 derder derivative. derivative.derivative Difference Difference is: between betweender derivative. the the angle angleβ Differenceofβ inclinationof inclination between the anglennnnn nngnnn(nnβtn0ng)=of(t0)= inclinationlimlim −−− −−−−−−−−− n g(t0−−)=−n−−−−−−−−lim− − − ofofofofof aof aof aofofof secantof secantaderivative asecantof a a a secanta secanta secanta secant a secant secant secant secant secant joining joining joining joining joining joining joining is:joining joining joining joining joining points points points points points points points points points points points points(((t(t0tt00(0,(,(t,(t,vtv0(0vt(v0(t,0(t,0t,0()v0)t,v0)vt),0,v00,andv0,andv)0vand,)and)v00)v0)0and)and0)and)and)ofandand(and((andt(tand1tt11( a1,(,(t,(t,vtv1(1vt( secantv1(,t1(,t1,t1()v1),tv1)vt),1,v11,forv1,forv)1vfor,)for)v11)v1)1for)for1)for)for the) joiningforthefor thefor theforfor the the the the firstthe first the firstthe first the the first first first first firstpoints first firstor- or-first or- or-first or- or- or- or- or- or- or- or- or-(t0,vdt0dd)ddtandddγdddddddγ(t1,v1dt) fordt0 the0 ((n( firstn(nn(1(1n(1n)1()nγ)n(γ)(γn(γ1ng1(gn1 or-)g(n)1γ)(nγ(1)γ1(t(1γg)tg)1ηtγg)tηγ1dtg)γη((1)γgdt(1gtγ1(gt1)tgη)(ηt()gη)(tη1(t1tη1(η)t1η)dtγt)η11)η1)1))1))γ((n(dn(nn(1(1n(1n)1()nγ)n(γ)(γn(γ1ng1(gndt1)g(n)1γ)(nγ(1)γ1(t(1γg)tg)10tγg)t0γ1g)γ0(0)(γg)(gtγ)(g)t0tg0(0t(g0()t0()t)t0(0)t0t)0)0))) (n dt1)γ gγ(tη1 ) (n 1)γ g(t0) In the case where Δt = 0 then η = n and: − − η → η →dtdtdtdt−−−dtdtdtdt−dt−dt−dt−−dt−dt−−−−−−−−−− −−−−−dt−dt−dt−dt−−−dt−dtdt−dt−dt−dt−dt−−dt−dt−−−−−−−−−η−→ dt− − − − dt− − aderderder secantderderderader derivative.der derivative.der derivative.der secantderder derivative. derivative. derivative. derivative. derivative. derivative.on derivative. derivative. derivative. onf andf Difference Difference Differenceand Differencethe Difference Difference Difference Difference Difference Difference Difference Differencethe angle angle between between betweenα between between betweenof between betweenaα between between between between secantinclinationof the theinclination the theder the the the onangle theangle theangle the the the angle anglederivative. anglef angle angle aangle angleand angle angle secantβββ aβofβofβof thesecantββββofofβ inclinationof inclinationβof inclinationofof onangleofof inclination inclinationofDifference inclination inclination inclination inclinationg inclination oninclination inclinationwillαg willof inclination betweennnnnγγ theγγgn agngnγn(γ(γn( secantγtn(gtngγ0tngtγ0 angleγgn0(0γ)=(g)=(gγt)=(gt)=tg0(0t(g0(t)=0(t)=t)=0(0t)=0t)=0)=0)= onlim)=limβ)=limlimlimlimoflimglimlimlimlimwilllimlim inclination γγγγ γγγγγγγγγnγ g(t0)= lim γ dtdtdtdtdtdtdtdtdtdtdtdt dtdtdtdtηηηdtηdtdtdtηdtηdtη0dt0η0dt0ηdtηηη00η0000000 dtdtdtdtdtdtdtdtdtdtdtdtdt dtη 0 dt n n d v(tη ) v(t0) −−−−−−−−−−−− →→→t→η→→→2→→→→t→η2 tη−1−−t−−η−−1−−−−−− tη2 → tη1 − d d beaaaa secant secanta secanta secanttheabeaa secanta secanta secanta secanta secant secantthe secantbigger secant secant on on on on bigger on on on onff onf onf on asonandand onfandfandfffandfand thefand asfandfand theand theand theand theand the biggerthe the the the angle anglethe theangle anglethe the biggerthe angle angle angleg angle angle( angle angle anglet isangleααα)= aαofαof isαofbeα valueαααofof inclinationαaof inclinationαlim inclinationof theofofof value inclinationof inclination inclinationof inclination inclination inclination ofinclination bigger inclinationa inclination secantinterval of1 interval a a− asa secant secant asecant a ona athe asecant errorasecant asecant asecant asecant secantf secant secant bigger secant. errorand on on onΔ on ont ong ong. theg on on onΔwillon Forwillg isonwillggtg.gwill anglegwillgwill agwill Forgwillwillwill value(28)will(28)willα of inclinationinterval error a secantΔt. For on g γwillγ γ γ nγ nγ (31) γ γ nγ g(t0) = f (t0) (24) γ 0 γ γ = =limlim v(t)vdt(t)dt v(=t)vdt(limt)dtdt dt v((31)t)dt(31) v(t)dt dt (31) η n dt dt 0 dt ttηtηtη2η222ttηtηtη2η2t2tηt2ηtη2ηt22η22 ttηtηtη1η111ttηtηtη1η1t1tηt1ηtη1ηt11η11 tη2 tη1 dt dt Δbebebet bebe=be theΔ thebe thebebebetbe0 the thebe the= the bigger biggerthe thebigger the the0 the bigger bigger bigger bigger thebigger anglebigger bigger bigger bigger as as angleas as asthe asthe ofas the as as as theas the asthe inclinationthe biggerof biggerthe thebigger the the the bigger bigger inclinationbigger bigger bigger bigger bigger bigger bigger is is is isa isa isa isΔ secants isvalue isvalue isa valueta isa isa= avalue avalue avaluesecants avalue avalue value value0 value value of of theofbe on of ofinterval ofinterval ofinterval the of ofangle of onfinterval of interval ofinterval intervaland interval interval intervalbigger intervalf intervaland of errorg error errorwill errorinclination error error errorgas error error error errorwillΔ errorΔΔ the betttΔ.t.ΔΔ..Δt ForΔtthe ForΔtbebigger.Δ Fort For.Δ.tΔ.tt For.t For. For.t For.the. Forsecants For For For For is a valueon f and ofdt intervalηgdtwill0η 0t bet error thett Δt.γγ Forγγ−γγγγγγ−tγ0γγ t0 dtηγγγγ 0γγγγγγγγnγnnnγtγtγγnnnγnγγnγnnγnγγnγγ − t0 γ γ nγ Δt=0 → ======limlim=lim=lim=→==limlimlimlimlimlimlim→limlimη1 vvηvv(1((t(tvvt)tv)v()()dtv(dttvdt(tvtv()t()v()tdt()tdttdt()t)dt)tdt)dtdt)dtdt  vvvv(((t(tvvt)tv)v()()dtv(dttvdt(tvtv()t()v()tdt()t→dttdt()t)dt)tdt)dtdt)=dtdtdtdtdtdtdtlimdtdtηdtdt1dtdt. (31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)(31)v(t)dt v(t)dt dt (31)  ΔΔΔtttΔtΔ=Δ=Δ=tΔtΔtΔtΔ=0t=Δ0t=t0=t=t= the= the0Equation=0 the0=00 the0 the0 the0 the0 angle anglethe theangle the the the angle angle angle angle angle angle angle angle angle of (28)of of of ofinclination ofinclination ofinclination of of of forinclinationof inclination ofinclination inclination inclination inclination inclination inclination inclinationminus secants secants secants order secants secants secants secants secants secants secants secants secantsγΔ on ont onhas= on on on onff onf onf0 onthe onandand onfandfandf theffandfandfand formula:fandfandgandgand anglegandandwillwillggwillgggwillgwillgwillgwillgwillwill bewillof bewill bewill be be beinclinationthe thebe the be be be be the bethe the the the the the the the secants ondtdtdtdtηηηdtηdtdtdtfηdtηdtη0dt0η0dtand0ηdtηηη00η000t0t0tt0gtt0t ttwillttttttttttttt be the−−−−−−−tt0−t0−t0−0−−tt0t0t00tt0t0t00t0 dtη 0 tt − t0 Factor γ depends on an interval dt and an interval error Δt Equation (28) for minus order γ has the formula: →→→→→→→→→→→→ηηη1η111ηηη1η11η1ηη1η11η11   →  η1    (20). Because Δt [0,dt] then on the branch borders: Bull.Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. XX(Y) XX(Y) 2016 2016 Bull. Pol. Ac.: Tech. XX(Y) 2016 3 3 3 d v(tη1 ) v(t0) ∈ Bull.Bull.Bull.Bull.Bull.Bull.Bull.Bull.Bull. Pol.Bull. Pol.Bull. Pol.Bull. Pol. Pol. Pol. Pol. Ac.: Ac.:Pol. Pol.Ac.: Pol. Pol. Pol. Ac.: Ac.: Ac.: Ac.: Tech.Ac.: Tech.Ac.: Tech.Ac.: Ac.: Ac.: Tech. Tech. Tech. Tech. Tech. Tech. Tech. XX(Y)Tech. XX(Y) XX(Y)Tech. XX(Y) XX(Y) XX(Y) XX(Y) XX(Y)g XX(Y) XX(Y) XX(Y)( XX(Y) 2016 2016t 20160)= 2016 2016 2016 2016 2016 2016 2016 2016 2016lim Bull.− Pol. Ac.:= Tech. XX(Y) 2016 333 333333333 3 dt γ dtγ 0 dt γ 1 for Δt = 0 1026 − → − Bull. Pol. Ac.: Tech. 67(6) 2019 tη γ = (25) γ 1 γ  logdt 2dt for Δt = dt lim (v1 v0)dt v(t)dt (29) dtγ 0 − ≡ t →  0 Characteristics of γ(Δt,dt) are shown in fig. 3. The minus order derivative (29) is equivalent to the integral Because for every n N: γ ∈ definition where step and branch of integration equal dt = n n t t . Geometrical interpretation of (29) as a surface area is (dt + t)n = dtn m tm (26) η0 − 0 Δ ∑ − Δ shown in fig. 5. m=0 m   Let n = 2. On the basis of (18) and (23) the minus order then the derivative (23) can have the formula: derivative of function g in t0 with multiplicity n = 2 is pro- n n m d ∑m=0( 1) vn m duced: − η g(t0)= lim n n− (27) dt dtγ 0 dtn mΔtm 2 d d → ∑m=0 m − d γ g(tη ) γ g(t0) dt− 1 − dt− γ 2γ g(t0)= lim γ For dt 0 and n = 1 the geometric  interpretation of the dt dtη 0 dt → − → − fractional orders derivative (23) is an angle of inclination of tη2 γ tη1 γ tt v(t)dt t0 v(t)dt tangent to g in tn, the same as the first order derivative. In the = lim η1 − (30) dt 0  dt 2γ case where the approximation of a tangent is a secant (fig. 4) η → − then the geometric interpretations of the derivative are: an an- Generally, the minus order derivative with multiplicity n is gle of inclination of a secant joining points (t ,v ) and (t ,v ) 0 0 η1 1 written as: for the fractional order derivative and an angle of inclination n 1 n 1 n d − d − of a secant joining points (t0,v0) and (t1,v1) for the first or- d (n 1)γ g(tη1 ) (n 1)γ g(t0) dt− − − dt− − der derivative. Difference between the angle β of inclination nγ g(t0)= lim γ dt− dtη 0 dt− a secant on f and the angle α of inclination a secant on g will → tη2 tη1 be the bigger as the bigger is a value of interval error Δt. For = lim v(t)dtγ v(t)dtγ dtnγ (31) Δt = 0 the angle of inclination secants on f and g will be the dtη 0 tt − t →  η1  0 

Bull. Pol. Ac.: Tech. XX(Y) 2016 3 Fractional order model of measuredFractional quantity order errors model of measured quantity errors

FractionalFractional order orderFractional model modelFractional of oforder measured measured order modelFractional model quantity quantityof measured of order measured errors errors model quantity quantity of errorsmeasured errors quantity errors FractionalFractional order order model model of of measured measured quantity quantity errors errors FractionalFractional order order model model of of measured measured quantity quantity errors errors Fractional order model of measured quantity errors

Fig. 8. Block diagram of systemFig. integration 8. Block with diagram jitter: ofdt system= 0.0001 integration with jitter: dt = 0.0001 (sample time), η = 1.001 (fractional(sample order). time), η = 1.001 (fractional order). − − dtdt == .. Fig.Fig. 8. 8. Block Block diagram diagramFig. 8.Fig. Block of of 8. system system Block diagram integration integration diagramFig. of 8.system Block of with with system integration diagramjitter: jitter: integration of with system0000010001 jitter: with integration jitter:dt = 0dt.0001= with0.0001 jitter: dt = 0.0001 (sample(sample time), time), == 11..001001 (fractional (fractional order). order).Fig.Fig. 8. 8. Block Block diagram diagram of of system system integration integration with with jitter: jitter: dtdt ==00...00010001 (sampleηη (sample time), time),η = η(sample1.=0011 (fractional.001 time), (fractionalη = order).1.001 order). (fractional order). −− − − (sample(sample time), time),− ηη == 11...001001 (fractional (fractional order). order). Fig.Fig. 8. 8. Block Block diagram diagram of of system system integration integration with with jitter: jitter: dtdt==00...00010001−− Fig. 8. Block diagram of system integration with jitter: dt = 0.0001 (sample(sample(sample time), time), time),ηη== 11...001001 (fractional (fractional order). order). (sample time),−η−= 1.001 (fractional order). − Fig. 10. Signals with and withoutFig. jitter 10. effect. Signals with and without jitter effect. Fig. 5. GeometricFig. 5. Geometric interpretation interpretation of a non-integer of a non-integer minus order minus deriva- order deriva- tive. tive. Fractional order model of measured quantity errors Fig.Fig. 10. 10. Signals SignalsFig. with with 10.Fig. an an Signalsdd 10. without without Signals with jitter jitterFig. an withd effect. effect. without10. an Signalsd without jitter with effect. jitter and effect. without jitter effect. Fig. 5. Geometric interpretationFig.Fig.Fig. 5. 5. Geometric 5.Fig. Geometric Geometric of 5. aBecause non-integer Geometric interpretation interpretation interpretation integral interpretation minus of of order a of isnon-integera a non-integer non-integer deriva- operation ofFig. a non-integer 5. minus Geometric minus minus opposite order order minus order interpretation deriva- deriva- order deriva- to derivative deriva- of a non-integer minus order deriva- Fig.Fig. 10. 10. Signals Signals with with an andd without without jitter jitter effect. effect. tive.tive.tive.Becausetive.Because integral integral is is operation operation oppositetive. opposite to derivative to derivative(New- By using double integral of (37) we can calculate a volume. tive. Fig. 6.Fig.Fig. Geometric 9. 6. Jitter Geometric intermodellingpretation inter (Signalpretation of multiple SamplingFig. of multiple 9. non-integer Jitter block). modelling non-integer minus (Signal order minus Sampling order block). (Newton-Leibnizton-Leibniz(Newton-Leibniz formula) formula) [1]: formula) [1]: [1]: A triple integral can calculate a mass of a solid with known den- derivativederivative for Δt = for0.Δt = 0. Fig.Fig. 10. 10. Signals Signals with with an andd without without jitter jitter effect. effect. Fig. 5. Geometric interpretation ofb a non-integer minus order deriva- sity. Currently no physical interpretations are known regarding Fig. 10. Signals with and without jitter effect. b b Because integraltive.Because isBecauseBecause operationBecause integral integral integral opposite integral is is is operation operation tooperation isf derivative( operationt)dt opposite opposite= oppositeBecauseF(b opposite) toF to to(integral derivativea) derivative derivative to derivative is operationthe(32) infinity oppositemultiple integral to derivative of oneFrom variable this an(38). interpretation From this an of presented minus order derivative f (tf)(dtt) dt= =F(Fb)( b) F(Fa() aFig.)− 6. GeometricFig.Fig. 9.(32) 9. Jitter Jitter(32) interFig.From modelling modellingFig.pretationFig. 6. 6. this Geometric 6.Fig. Geometric Geometric (Signal ofan(Signal 6. multiple interpretation Geometric inter Sampling Sampling inter interpretation non-integerpretationpretation inter block). block). ofpretation of presented of multipleminus of multiple multiple of orderFig. non-integermultipleminus non-integer 6. non-integer Geometric order non-integer minus derivativeminus minus inter order order minuspretation order order of multiple non-integer minus order (Newton-Leibniz formula)(Newton-Leibniz(Newton-Leibniz(Newton-Leibniz [1]:(Newton-Leibniz formula) formula) formula)a formula) [1]: [1]: [1]:a [1]:¡(Newton-Leibniz− formula) [1]:interpretationFig. 9.Fig. Jitter 9. of modelling Jitter presented modellingFig. (Signal minus 9. Jitter (Signal Samplingorder modelling Samplingderivative block). (Signal block).is limited Sampling to block). a derivative for Δt = 0.derivativeisderivative limitedderivativederivative for to for forΔ thetΔ=tΔ one=t0. for=0.0.Δ iterationt = 0. ofisFig.Fig. limited minus 9. 9. Jitter Jitterderivative order to modelling modelling the derivative one for iterationΔ (Signal (Signalt = (36).0. Sampling Sampling of minus block). block). order derivative (36). Z γ bBecauseandand and integralb b isb operationb opposite to derivativeb theLet one dtiterationγ be an of integration minus orderstep derivativeLet anddt[ t(36).,bet an] be integration an integration step and [t ,t ] be an integration Fig. 6. Geometricγ interpretation of multiple0 non-integerηn minus order 0 ηn f (ft()ftdt()tdt)=dtf=F(=tF()bdtF()b(=)b)FF(F(aFb()a)()aFrom)FromF(a) this thisFig.Fig. an an 9. 9.(32) interpretationJitter interpretation Jitter(32)f(32)(t) modelling modellingdt(32)=LetF( b ofdt of (Signal) (Signal presented presentedbeF an(a Sampling Sampling)integration minus minus block). block). step order order and(32) derivative derivative t0, tη n be an integration (Newton-Leibnizf (t)dt = F(b) formula)F(a) [1]: (32) b b FromFrom thisbranch. an this interpretation an Because interpretationFrom thethis of definite an presented interpretationof presentedbranch. integral minus minuscan Because of order presented be order derivative written the derivative definite minus as a sum order integral derivative can be written as a sum a −a a a a F(b)−−F−(a) − aFig.af (t 9.)dt Jitterderivative modelling for− Δ (Signalt = 0. SamplingFromFrom this this block). an an interpretation interpretation of of presented presented minus minus order order derivative derivative ( )= F (b) F(a) =isis limited limiteda f (t) todt to the the one one iterationbranch. iteration Because of of minus minus the order orderdefinite derivative derivative integral can (36). (36). be written as a sum F�(a)=F� alimb lim − −= lim lim is(33) limited(33)isof limited(33) todefinite the to one the integrals,is iteration limitedone iteration thus to of the minus on of one theof minus order basisiterationdefinite£ order derivative of¤ integrals,(29): of derivative minus (36). thus order (36). on derivative the basis (36). of (29): and and (b a) 0 b a (b a) 0γγ b a ofγ definite integrals, thusisis limitedon limited the basis to to the the of one(29): one iteration iteration of of minus minus order order derivative derivative (36). (36). and andand (b a) 0 − →b a and(b a) 0LetLet− b→dtdtabebe an anLet integration integrationdt be anγ step step integration and and [[tt00,,ttγη stepηnn]] bebe and an an[ integrationt integration,t ] be an integration − →f (t)dt =−F(b) −F(a−)FromFrom→ this this− an an interpretation interpretation(32)− Let of ofdt presented presentedbe anLet integration minus minusdt be order order an step integration derivativeγ derivativeγ and0 ηn[t0,tη stepn ] be and an[ integrationt0,tηn ] be an integration a − branch.branch.b b b b Because Because the the definite definite integral integral can canb be beLetLet written writtendtdt bebe as as an an a a sumintegration sumintegration step step and and [[tt00,,ttηηnn]] bebe an an integration integration where F is a primitiveb function of f . From this anbranch. interpretation Becausen of the presented definite minus integral order cann derivative be written as a sumt t F(whereb) whereFF(a F)is is a a primitive primitiveFF(bF()b()b functionfunction)FfF(F(taFb)()dta)()a )of ofF (ffa..) isisis limited limited limitedf (ft()ftdt()t todt to) todt thef the the(t)F one onedt one(b) iteration iteration iterationbranch.F(a) of ofBecause of minus minus minusbranch. theorder order order definitef Because( derivativet derivativet derivativeη)ndt integral thed (36). (36). (36). definite can be integral writtentη1 can asηn abe sum written as a sum η1 F (a()=()=)=F (lima)= lim−a =−==lim= (aofoflim)=a definitea definitea integrals, integrals,(33) (33) thus thusd on on= the the basis basis of ofa (29): (29):branch.branch. Because Becauseγ the the definite definite integral integralγ can canγ be be written written as as a a sum sumγ F�(a)= lim and F� −F� �aa � =limlimlim −− (33)limlimF�(a)is limitedγγγlim( ) toof(33) the(33) definite oneof− definite iteration integrals,g(t integrals,0)=lim ofof thus minus definitelim on thus order the integrals, onv basis( derivative thet)dt basis(33) of= thus (29):limg of (36).( ont0 (29):)= the basislimv(t of)dt (29):v+(t)dt = lim v(t)dt + Exchanging(bExchanging(b(ab)a)a0) ( 0bgenerally0 ab) b generally0ba a signsab F signsa(b0((ab(a))b) aand)Fa0)�( 0bfaLet0Let(bta()b)and bwith)dtdta(0bafa bebeafractionalb)t an an0witha integration integrationb fractionala η step step(b and anda) [[[tt0tγ00,,,tttηbηnnn]]]bebea an an integration integrationη γ γ γ (b a) 0 bExchanginga (b generallya) 0 b signsa F� a and ft with γ fractional dt dt  0 oftof definite definitedt integrals, integrals,− dt 0 thus thustdt on on0 the thet0 basis basis of of (29): (29):dt 0 t0 − → − −−→→−→→− →−−− − − −−→→→− −→−Let−−dt→−be an− integration− step− → and→ [t0−0,tηn ] be an integration→ 0 → → derivativederivative− derivative (23) (23) and and (23)v v(t( and)t):: −v(t): branch.branch.b Because Because the the definite definite integral integral can can be be written written as as a a sum sum   F(b) F(a) a f (nnt)dt ttηη t ttηη t tη tη wherewherewhereFwhereFisFis ais aprimitive a primitiveF primitiveis a primitive function function function function of of off .f .fwhere. of f . F isdbranch.d a primitive Because functionn thennn definite of f . η2 integraltηn can be11 writtenηn astη a sum2 d n d where F is a primitive functionF�(a)= oflimf . − = lim ofof definite definite integrals, integrals,(33)dt thus thus on on the the basis basisγγ of ofn (29):γ (29):tηn γγtηn γ1 tη1 γ tη1 γ n n m n gg((tt0t0)=)= limlimηn d vγv((tt))dtdt v==(td)limdtlim+ ...γ+nn vv((tγt))dtdtv(+t+)dt v=(γttt )dtγ +g(...γt )++ v(t)γtdttt = g(t0)+ ExchangingExchangingExchangingn Exchanging(db generallya) generally generally0 generallybn signs signsa signs∑FF� signs(mFa(()a((nb)aand1)andFa))andExchanging�(fa0()ft()fvandtb(n)withtof)withmηwithηa definitef ( fractionalt) generally fractionalηwith fractionaln integrals, fractionaltγγ γ signsvg((tt)0dt thus)=Fg((t onalim)=) theandlim basisfγγ(tv)g((t ofwitht)dt)= (29):vdd(t= fractional)dtlimlim= limv(t)vdt(t)ηηdtn=nv(+tlimγ)dt 0+ v(t)dt ηη+11 γ Exchanging generallyd signs F�−(a)→and ∑f (t) with( m1)= fractional0� � −vn→mm dtdt−− t v(tdtdt)dt0 η00 tt00 η � 0tγ γdtdt η 00 0 tt00 tγ γ γ tη dt γ γγ tη dtγγ − g(t0)= mlim−=0 −m − = 0lim dt−→→ dt dtη1 0dtdtt0 0→→t dt ggdtη((ntt001)=)=0tdt limt lim00 1 t vv−(dt(tt))dtdt0 ==t limlimn 1 vv((tt))dtdt ++ g(t0)=η lim γ − γ−n = lim γ γ −n → →− 0 ηη →−→ 0 →γγγ 0 → 0 γγγ − derivative (23) andderivativev(derivativetderivative): ηderivative (23) (23)dt (23) and and and (23)γv(vt()vt: and()dtt:): v0(t): γ n (dtderivative) nn (23)γ anddt v0γ(t)n:(dt ) dtdt−− dtdt 00 ttt00  dtdt 00 ttt00 Fig. 11. Integration of signals withFig. and 11. without Integration jitter of effect. signals with and without jitter effect. wheredtF is a primitivedt 0 function→ (dt of f).   dt 0 ttηη(→dt ) tttηηnnn ttηηnn tttηη111 →→ →→ →   dd →n  22 tη d tηγγγ ttηηnd ddtη γγγ d tη dγ  γ (34)γγ (34)2tηn 2 γ γγ 2 tη1n γ d n d d Exchanging generallyn n n signsnmFtmnm(nan) andm n f (t) withggd(t(ηtttn0t fractional0η)=t)=nηn vvlim(lim(limttηtγn)n)dtγdt(34)γ ++...... γvvm(+(+ttt))ndtdtg(=t=ηv1vγ)+(lim(limlimtt))dtdtγ... + ==tηvnv((gttt)()dttdtgηgγttn(tη(ηtt+0+10γ)+)=)+γ γglim(tη1 )+(v1 ...v+t0ttηηγ)nndt +g(tηn 1 )= lim (v1 v0)dt + n n n n n mn n ηn� γ n ηη dtdtγγγ v(γt)dtv(t+dtdt)γdtγγ... ++ ... +v(γvt()tdtγ)γdtv+2(2t)...dt=+γ γ = g(vt0(t)+)gdt(t )+=γ −g(t )+dtddγ d dd( 1)d v ∑∑m∑=m0=m(=0(0∑1()m1=)10)( vv(1ntv)nvdtmnmdm vdtndt−−m t0 t tvtηgη(v(tt()v0tdt()=∑)tt0dt)mdt=0v00(lim(tttt0)00dt1)dt− vttη(ηvtn)dtm =00limttt0dt00 t−dtdt−v−(tv)dt(−t)dtdt− +dtγγ γ0 −γ 0dt− γγγ 0 0 − d ∑m=0 wherem nv ism a primitivet0 m functionmm− m of g−. η0 0 11 →→dt γ 0tη m t nn 11 →→dtγ ttη0 0 t vv((tt))dtdtdtt−→++...dt...++ (39)vv((tt))dtdtdt == →gg((tt00)+)+ g(t )= derivativewhereg(−gt0(gt)=(0 (23)tv)=0)=isglim a( andtlim0 primitivelim)=−v(t=)lim: − function−− − of g−. −===glim(limt0lim)=dt=−limlim − 1t0 η1 −−− = lim η1n 1 t0ηn 1γ   ηn 1 −  γ γ− γγ γ 0 lim η ηη ηn γ γ γ γ lim γ γnγn n γn n η γ γ γ γγ γ γnγn n γ →n γ n γ → − −γ tttηηn − (tvttηηFig.Fig.v 11. 11.)dt Integration Integration+ ... +dtdtand−− of of signalswithout signals(v with withv jitter and and)dt effect without without are jitter jitter shownand effect. effect. without in the fig. jitter 10. effect Signals are shownafter in the fig. 10. Signals after dtη dtγ 0 dtdtdt (dtdtγ ) Convertingdtdtdt0 0 0dt dt (34):0γ(dt(0dt(dt) )(dt)(dtγ ) ) dt dtdtdt0 0 0dt(ttdttηdtη(dt(dt0)0) )(dt )(dt ) tttηηnnn limdt (v20  v(1dt)dt ) 1+1 ... + lim (limvn vn2n 11 )dt1 (39) Fig. 11.limFig. Integration 11.n Integrationn of1Fig. signals 11.of(39) signals Integration with and with without andof signals without jitter with effect. jitter and effect. without jitter effect. → Converting →→ (34):→ → →     →→→dd 222→→t γγγ d dd   t dtγ γγdγ 0 → −dd γγ dtγ 0dtγ −0 −−− dtγ 0 − − n n m n tηngg((vtvt((ttηt))+2(34)))+dtdtγ(34)(34)...... ++++...... (34)++d gg((tt vv(η(ttnt))=))=dtdt→dlimlim==d((vv ggv(vd(ttt)))+dt(34))+dt ++d → →γ γ γintegration→ (classical andFig.Fig. fractional) 11. 11.integration Integration Integration are (classicalshown of of signals signals in theand with with fig. fractional) and and 11 without without are jitter jitter shown effect. effect. in the fig. 11 (34) γγ v(tηη)11dt γg(t γγ)+ηη...nn +11 γγgγ(t 11dγγdγ)=0000lim (v v )ddtd + d ∑m=0( t1η)n m vn m dtdtnt0 v(t)dtγ dt+dtη...1 +g(tη1−−)+v...(tdt+)dtγdtg(00dttηdtηn=1g)+1−(−tη...n 1+)=gγ (t )+lim1g(t(ηv0n1 1 )=v0)dtlim+(v1 v0)dt + γγ where v is a primitivetηn function of −g. n d −−tttηη111 dt −−γtttηηnnn 111 dtγ γ−−− (dtγ( )+)+0 γ...... −++ (( γ )=)= (( )) ++ and 12. Absolute error of signals (classical and fractional) are wherewherewheregv(twherev0is)=vis ais aprimitive a primitivelimv primitiveis a primitive function function function− function of ofg of.gγ.g.wheren of=dg.γlimvn is a primitivet function− dt−The of g. graphicalt dt−− interpretation→dt→ − dtγγ gg− ttηη1The1dt ofdt− (39) graphical0 as−− aγγ g surfaceg interpretationdtttηηnn 110 area−limlim is ofvv11 (39)andvv00 12. asdtdt Absolutea surface error area of is signals (classical and fractional) are where v is a primitive functionη ofdtgγ . 0 lim γγv(nt)dt γ =(dtdt)γ 0 g(t0 γ)=ηn1 −− γγηn 1  dtdt − → γγ → dtdt andandFig.Fig. without without→ 11.−− 11. Integration Integrationdt jitterdt jitterγγγ 00 effect effect of of−− signals signals are are shown shown with with and and in in withoutthe withoutthe fig. fig. jitter jitter10. 10. Signalseffect. Signals effect. after after dt Convertinglim dt (34):γ v(t()dt )=(dt ) g(t0)=η (dt ) limlim ((vv22 vv11))dtdt ++−...... ++ limlim γ((vvnn −−γvvnn 11))dtdt (39)(39)γ −− γ γ and→→ withoutandγ without jitter jittereffectand effectwithout are shown are jitter shown inshown effect the in fig. the are in 10. fig.the shown Signals fig.10. in Signals 13 the after (in fig. relation after 10. Signals to integration after signal without ConvertingConvertingConvertingConvertingdt (34):γ → (34): (34):0 (34):0 t0   dtConvertingη →ddtd (34): γγ ddshownlim in fig.(vlim2 6(vv and12γ)γdtv 7.1+)dt The...lim++( markedshownγγv...γlim2 +(vvlim1 inn) areadt fig.(vvn+n in...1 6) thevdt and+n γ fig.1γlim)(39) 7.dtFig.( 6Thev 11.(39)n repre- marked Integrationvn 1)dtshown area of(39) signalsin in the theγγ withfig. fig.and 6 13 repre- (inwithout relation jitter to effect. integration signal without Converting (34): t0 → d dt(34)dt 00 −− ddtγ 0 γ − dtdt 00 −−γ −dt−γ 0 γ − − integrationintegrationγ (classical (classical and and fractional) fractional) are are shownand shownand without without in in the the jitter jitter fig. fig. effect effect11 11 are are shown shown in in the the fig. fig. 10. 10. Signals Signals after after → n γγγgg((tttηη111)+)+...... →→++ γγγgg((tttηηnnn 111)=)=dt lim0limlim →(−→(vv11 dtvv00))dtdt0 ++−limlimγdt ((v0v22 −vv11))dtdt−dt ++...0...++−integrationlimlim−((integrationvvnn vvn (classicaln 11))dtdt (classicalintegration(39)(39) and fractional) and (classical fractional) are and shown are fractional) shown in the in fig. arethe 11 shownfig. 11 in the fig. 11 n dtdt n g(tη )+...dtdt+sents Δtg→−−=(tη0→ anddtdt)=γγγ natural00lim−−(v number1→ vsents0)→dtdtdt ofγγγΔ+ηt0→0=order.0−− and The natural marked→ numberdt areadtγγγ 00 of ηjitter−−order.−− - the The ideal marked signal). area Fromjitter fig. - 13 the it ideal arises signal). that fractional From fig. 13 it arises that fractional t where v is a primitivetηntηntnηn functiontηn of g. n n n n nm−− γ tηn 1 −− γ nn 1 γ andand 12. 12. Absolute Absolute error error of of signals signals (classical (classicalintegrationintegration and and fractional) fractional) (classical (classical are are and and fractional) fractional) are are shown shown in in the the fig. fig. 11 11 ηn d γ γ γ γlimγγnγndnddmγ(n 1d) TheThedt graphical graphicalvn m The interpretation interpretationγ graphicaldt(35)γ n d interpretation of− of (39) (39)→→dt as as0 a a of surface surface− (39) as area area→→ a surface is is area is and→→ 12. Absolute error of signals (classical and fractional) are γ γ n lim γ ( 1∑) vn m− (35)inThe the−γ graphicalγγ fig. 7 representsThe interpretation graphical→Δt =0in interpretationwhere of theγγγ (39) fig.η is as 7 not represents aandand of surfacea natural without without (39)Δ asareat number. jitter jitter a=0 issurface where effect effectandintegrationη arearea are 12.is shownnot shown Absolute is a natural whereand in in the the error12. number. fig.jitter fig. Absolute of 10. 10. error signals Signals Signals isintegration error (classicalcoded after after of signals into where and its (classical fractional) order jitter has error and lower are is fractional) coded into are its order has lower v(t)dt =(limdtlimlim) limv(gvt()vtdt()t)=dt)dt=(γv(=(t=()dtdt∑dtdtdt) =() )0 dtg(g)t−0(gt)=(0t)=0)=glim(mt0)=−limlimlimv(t()(vdtv22 =(vv11))dtdtdt )++...... ++g(tlimlimlim0)=((vvnn vvnn 11))dtdt (39)(39) lim Convertingγ γ γ (34):γη 0 dt 0 −mη=η0η mshownshownη γ− in in fig. fig.γγγ 6 6 and and 7. 7. The The marked markedγη γγγ area area in inTheThe the the� graphical graphicalfig. fig. 6 6 repre- repre-γ interpretation interpretationshownshownand without in in of of the� the (39) (39) fig. fig. jitter as as 13 13 aeffect a (in (in surface surface relation relation are shown area area to to isintegration isintegration in theandand fig. 12. 12. 10.signal signal Absolute Absolute Signals without without error errorafter of of signals signals (classical (classical and and fractional) fractional) are are dtγ 0 dtdtdt0 0t0dtt0t0 0 t0 m=0→dtdtdt dt dt 0 t0dtdt shown00lim−−( inv2 fig.v1) 6dtdt and+ ...dt 7.dt + The00lim marked−−(vn−− v arean 1)dt in the.(39) fig. 6 repre- shownshown in the in fig. the 13 fig.shown (in 13 relation in (in the relation fig. to integrationabsolute 13 to (in integration relation error signal values to signal without integration (0.01-0.14%) without signal thanwithout classical integration t0 →→→ → →   → →→ γ shownIn bothFig. in cases 7. fig.Fig. Geometricshown 6 the 7. and→→ negativeGeometric in 7.γ inter fig. 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Tech. Tech. minus minusAc.: XX(Y) XX(Y) XX(Y)Tech. order order2016 2016 XX(Y) 2016 derivative derivative 2016 Bull. (29)-(31) (29)-(31) Pol. Ac.: has has Tech. been been XX(Y) defined defined 2016 on on 5 5 5 4 derivative In mathematics there are known multipleSignals integra- (outputgarding signals thefrom infinity SignalBull. Pol.Bull.Bull. multiple Sampling Ac.: Pol. Pol. Tech. Ac.: Ac.: integral block) XX(Y) Tech. Tech. XX(Y) XX(Y)2016 withof one 2016 2016 variableThe minus (38). order derivative (29)-(31) has been defined on 55 … f (x1, …, xm)dx1 … dxm. (37) ZZZ Z Bull.Bull. Pol. Pol. Ac.: Ac.: Tech. Tech. XX(Y) XX(Y) 2016 2016 55 4 D Bull. Pol. Ac.: Tech. XX(Y) 2016 Bull. Pol. Ac.: Tech. XX(Y) 2016 5 So that the multiple integration of one variable is produced:

… f (x)dx1 … dxn (38)

ZZZD Z Fig. 6. Geometric interpretation of multiple non-integer minus order where n is a number of iteration of x. derivative for ∆t = 0

Bull. Pol. Ac.: Tech. 67(6) 2019 1027 R. Cioć and M. Chrzan

Fig. 7 represents ∆t = 0 where η is not a natural number. In 6 both cases the negative order derivative is the one step upper Darboux integral, where an integral step and an integral branch are equal.

2.6. Jitter effect. Jitter definition and its effect can be found in works [23‒25]. Generally, jitter effect changes real measured input signal by inaccuracy of work of sampled systems and the read signal (behind the sampled system) is different than the measured signal (before the sampled system). In this case next operation has result with jitter error, for example an integration operation of signal. Block diagram of this example is shown in the Fig. 8 where input signal is sampled with and without jitter effect and obtained signals are integrated by classical integrator and fractional integrator built on base of the equation (39). Jitter modelling (Fig. 9) was created on the base of work [26]. Signals (output signals from Signal Sampling block) with and without jitter effect are shown in the Fig. 10. Signals after integration (classical and fractional) are shown in the Fig. 11 and 12. Absolute error of signals (classical and fractional) are shown in the Fig. 13 (in relation to integration signal without jitter – the ideal signal). From Fig. 13 it arises that fractional integration where jitter error is coded into its order has lower Fig. 7. Geometric interpretation of multiple non-integer minus order absolute error values (0.01‒0.14%) than classical integration derivative for ∆t = 0 (0.03‒0.3%). 6

Fig. 8. Block diagram of system integration with jitter: dt = 0.0001 (sample time), η = –1.001 (fractional order)

Fig. 9. Jitter modelling (signal sampling block)

1028 Bull. Pol. Ac.: Tech. 67(6) 2019 Fractional order model of measured quantity errors

3. Conclusions

A definition of fractional order derivative and integral dedicated d n to describe a measured quantity error and its notation η have dt been proposed. In the definition an order of derivative is a product of nat- ural order and γ factor which determines a relation between an interval (an increment of independent variable t) and an interval error (20). In the special case where an interval error is equal 0 (∆t = 0), γ = 1 and the real positive order derivative (23) becomes natural order derivative. From this the natural order derivative is special case of the real order derivative. The minus order derivative (29‒31) has been defined on the basis of the definition of positive order derivative of func- tion (23). If an integral step and an integral branch are equal the negative order derivative of function is the upper Darboux integral. Fig. 10. Signals with and without jitter effect In the case of positive and negative order of derivative of measured function, an interval error can interpreted as an error of determining of an increment dt of independent measured variable t. The definition presented can be used in various ways. Gen- erally the definition takes into consideration an interval error of independent measured value. In real measurements an interval error can be interpreted as an error of determining of sample time branch of measuring signal arising for example from an action of electronic parts of sampling systems (from not ideal sampling of measuring cards and jitter effect). In mathematics it can be an error arising from rounded values of a differential to defined place after comma. In physics an interval error can be influenced by factors connected with drag air and friction of moving objects.

References [1] T.M. Apostol, Calculus, John Wiley & Sons (1968). Fig. 11. Integration of signals with and without jitter effect [2] S. Das, Functional Fractional Calculus for System Identification and Controls, Springer-Verlag Berlin Heidelberg (2008). [3] K. Miller and R. Bertram, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley & Sons, Inc. (1993). [4] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999). [5] I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation”, Fract. Calc. Appl. Anal. 5(4) (2002). [6] R.S. Rutman, “On physical interpretation of fractional integra- tion and differentiation”, Theor. Math. Phys. 105(3) (1995). [7] R. Cioć, “Physical and geometrical interpretation of Grünwald- Let- nikov differintegrals, measurement of path and acceleration”, Fract. Calc. Appl. Anal. 19(1) (2016), DOI: /10.1515/fca-2016‒0009. [8] R. Cioć, “Grünwald-Letnikov derivative, analysis in range of first order”, Frontiers in Fractional Calculus. Book Series, Current Developments in Mathematical Sciences 1, Bentham Science Publishers (2017). [9] R. Sikora, “Fractional derivatives in electrical circuit theory – critical remarks”, Archives of Electrical Engineering 66(1) Fig. 12. Integration of signals with and without jitter effect – zoom (2017).

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