Trans. of the Society of Instrument and Control Engineers Vol.E-6, No.1, 1/8 (2007) A Numerical Algorithm of Discrete Fractional Calculus by using Inhomogeneous Sampling Data
Fujio Ikeda∗
This paper presents an efficient numerical method to realize discrete models of fractional derivatives and inte- grals which imply derivatives and integrals of arbitrary real order. This approach is based on a class of Stieltjes integrals transferred from the Riemann-Liouville definition. It is to calculate on inhomogeneous sampling periods which are getting longer as the operation points go back toward the initial time. It leads to the effective quality which has low computational costs and enough accuracy. The calculation times and precision of the proposed procedure are compared with those of a conventional procedure for a practical numerical simulation and the effectiveness of this procedure is verified.
Key Words: fractional derivative, fractional integral, Stieltjes integral, discrete model, numerical algorithm
It has been common to employ the G1-Algorithm 1) 1. Introduction or any of the many other similar approaches based on Ordinarily, derivatives of functions are described in the Gr¨unwald-Letnikov differintegral, which is well suited terms of integers, for example, the first derivative of posi- for constructing algorithms suitable for numerical analy- 5)∼11) 12) tion yielding speed, the second derivative yielding accel- sis . Recently, Podlubny described both a geo- eration. The concept has been extended to non-integer metrical interpretation and physical meaning of FD. Ma 13) derivatives, which are known as fractional derivatives and Hori employed these viewpoints in their proposal (FD). Fractional integrals (FI) have also been defined, and for a numerical solution method that has a clear physical these together make up fractional calculus (FC) 1). The analogue. However, these direct discrete procedures all concept of FC was known at the time of Leibniz during suffer from high calculation costs imposed by the require- the 17th century, but it has received little theoretical con- ment of previous history. Practical implementations of sideration since then, because its physical meaning was these algorithms will need to get negotiate this problem questionable and its mathematics was complicated. How- by reducing the computational burden. 4) ever, in recent years much research has been performed The Short Memory Principle , which fixes the range of in physics, rheology, fractals, control systems and other calculations at some limited history and neglects whatever fields where fractional differential equations made up of occurred earlier, is one possible approach for reducing the FD and FI have been reported to show several advan- computations. However, the shorter the remaining calcu- tages 2)∼4). lation domain, the greater the error this principle intro- In order to handle FC operators numerically, the sys- duces, so the researcher is forced to trade-off calculation tems must be transformed into discrete-time system mod- time against precision. els. It is easy to obtain very accurate and concise models This paper describes an inhomogeneous sampling algo- of integer-order differential operators using Tustin trans- rithm (IS-Algorithm) as an essential solution for the issue 14), 15) forms or other well-known trapezoidal integration meth- of high computational costs . It is based on the ods. In contrast, FC operators are global with respect Riemann-Liouville integral, a class of Stieltjes integrals, to time, so there is a strict requirement for the calcula- which were used by Podlubny in the above publication to tions to include the entire time history of the operators. obtain some of his geometrical interpretations. In con- If the size of the time step (i.e., sampling period) is small, ventional methods, numerical integration is carried out in however, the satisfaction of this requirement necessitates the so-called Riemann sum over the variable t (“actual” the use of large amounts of memory, which significantly time) at step ∆τ, i.e., in a discretized model with a con- increases the computational expense of the procedure. stant sampling period. The present method, in contrast, employs the Stieltjes integral, performed over variable Tq ∗ Department of Mechanical Engineering, Kochi National T College of Technology, Monobe 200-1, Nankoku, Kochi 783- (“transformed” time) at a constant step ∆ . The rela- 8508, JAPAN. tion between “actual” time and “transformed” time is as
TR 0001/07/E-601–0001 c 2007 SICE 2 T. SICE Vol.E-6 No.1 January 2007
∞ follows: When the “actual” time steps are constant, the −t z−1 Γ(z) = e t dt , (z>0). (2) “transformed” time is interpreted to pass at a gradually 0 decreasing rate. Conversely, in this method, when the Eq.(1) above is used to define the fractional derivative “transformed” time steps are set constant, the “actual” (FD): The q-th (q ≥ 0,q ∈) derivative of function f(t) time steps do not remain constant in length, but become is given by longer as one moves backward in time. As a result, fewer points in the history are sampled than in the conventional dm Dpf t D−(m−p)f t ( ) = m ( ) methods, greatly reducing the computational costs. Also, dt m t since the influence of the damping characteristics in the 1 d m−p−1 = (t − τ) f(τ)dτ, (3) Γ(m − p) dtm general case of physical dynamic phenomena diminishes t0 as one goes backward in time, this method is less prone to m m − ≤ q also obtained if a power series expansion (PSE) is car- ried out for the fractional differential operator Dq (z−1)= D -0.5f (t) q (1 − z−1)/∆τ . An expression for the discretized model f (Ǽ -1 D f (t) obtained by z-transform in the above equation is given by n−1 q −1 −q j q −j D (z ) =∆τ (−1) z , (10) G1 j j=0 where z−1f(t)=f(t − ∆τ). The sampling period ∆τ in “actual” time τ is constant in conventional discretization procedures such as the G1- Algorithm as shown in Fig.1. In other words, this means that the sampling period ∆T on the “transformed” time Ǽ T q axis Tq is not constant, but increases in length as time L approaches the present, and, correspondingly, decreases Fig. 1 The ”fence” of f(τ)=τ +0.5 sin(2πτ)and its projec- in length as time moves further into the past. As a re- D−1f t D−0.5f t tion ( ) and ( ), for a constant ”actual” sult, the greater the time span in the past for which this time step ∆τ. calculation is performed, the greater the computation is devoted to that past history, and so increasing the time for the example in Fig.1 is f(τ)=τ +0.5 sin(2πτ), span results in a extremely large increase in computa- q =0.5, 0 ≤ τ ≤ 2. A three-dimensional Cartesian space 4) tional costs. The Short Memory Principle and other is drawn with axes (τ,Tq ,f(τ)) with the function in Eq.(7) approaches have been employed to overcome this prob- graphed on the (τ,Tq) plane. Values of f(τ) are drawn lem. When the Short Memory Principle is employed, the vertically above the plane for this domain of τ, to pro- function is not evaluated for the entire domain [t0,t], duce the curved “fence” in the figure. The area of this but rather, over some abbreviated “window” [t − L, t] “fence” corresponds to the line integral along the integra- (see Fig.1) in order to prevent the demand for computa- tion path Tq of function f(τ). The area projected from tional resources. However compressing the domain also the “fence” onto the (τ,f) plane is the so-called “ordinary increases computational error. This problem has ham- first integral” over τ: pered attempts to make the conventional procedures more t −1 efficient. D f(t)= f(τ)dτ. (8) 0 3. Inhomogeneous Sampling Algorithm In contrast, the area of the projection on the (Tq,f) plane corresponds to the value of the FI in Eq.(6) over integra- 3. 1 Procedure for discretizing the fractional tion variable Tq (or in defining Eq.(1)). integral by inhomogeneous sampling As shown in the figure, if “actual” time τ is divided This paper proposes an inhomogeneous sampling algo- into constant time steps ∆τ, the corresponding steps ∆T rithm (IS-Algorithm) which promises to solve the prob- of “transformed” time Tq are not of constant length; the lems faced by conventional procedures. The sampling pe- more distant in the past they are, the shorter they are riod ∆T of “transformed” time Tq(τ) is set to a constant (the finer the divisions), while the further they are in the value, resulting in a discrete model with inhomogeneous future, the longer (coarser) they are. “actual” time sampling period ∆τk. 2. 3 Discretization procedure (G1-Algorithm) Since the calculations of this procedure are performed for conventional fractional calculus in “actual” time, it is necessary to obtain the sampling τ One of the most simple and direct procedures for dis- points k, as shown below. Firstly, when the “trans- T cretizing FC is given in Eq.(4) by simply omitting the formed” time sampling period is set at ∆ , we obtain limit notation. It is called the G1-Algorithm 1): the following relationship from Eq.(7): n−1 q −q j q (D f(t)) =∆τ (−1) f(t − j · ∆τ),(9) G1 j ∆T = Tq(τk−1) − Tq(τk) j=0 1 q q = {(t − τk) − (t − τk−1) } . (11) where the symbol (·)G1 is used to designate the G1- Γ(q +1) Algorithm. An equivalent of the above expression is If the initial time point τ0 is set equal to the present time, 4 T. SICE Vol.E-6 No.1 January 2007 T0.5 (Ǽ) D -0.5f (t) _ǍT f (Ǽ D -1f (t) ǍǼ ǍǼ _ _ Ǽ Ǽ Ǽ Ǽ Ǽ Ǽ Ǽ Ǽ Ǽ Ǽ T q T τ L Fig. 3 If the “transformed” time 0.5( ) ticks steady by the constant ∆T , the “actual” time points ∆τk become Fig. 2 The function f(τ) is the same as Fig.1, but for the coarser in the past. changing ”actual” time steps ∆τ and the constant ”transformed” time step ∆T . the Short Memory Principle 4), limiting the calculation window to some fixed time span L (Fig.2). The upper τ0 = t, then, from the above expression, the “actual” time τ k , , ··· limit of summation then takes the value sampling points k ( =1 2 ) are consecutively cal- culated thus: Lq m = . (17) ∆T · Γ(q +1) 1 τ1 = t −{∆T · Γ(q +1)} q If the discretization procedure in Eq.(15) is re-written as 1 a model discretized using the z-transform, we obtain τ2 = t −{2∆T · Γ(q +1)} q . (12) . . D−q z−1 ( ) IS They are given in general by this expression: ⎧ ⎫ m K 1 K 1 ⎨ − I (k−1) q − I ·k q ⎬ 1 ∆T ∆τ ∆τ τ t − K · k q , = z + z (18) k = I (13) 2 ⎩ ⎭ k=1 ⎧ ⎫ in which K 1 ⎨ K − I ·2 q ⎬ ∆T − I ∆τ 1 = 1+2z ∆τ +2z + ··· . KI = {∆T · Γ(q +1)} q . (14) 2 ⎩ ⎭ Theorem 1. (Discretization of FI by IS-Algorithm) By numerical integration methods such as the trape- Let us now consider specific design parameters for this zoidal rule, the discrete model of IS-Algorithm is given procedure: order of integral q, sampling period ∆T in by this expression: “transformed” time , and calculation window length L if m necessary. When the sampling period in “actual” time −q f(τk−1)+f(τk) D f(t) = ∆T, (15) IS 2 ∆τ =0.1, for example, then, setting the parameters k=1 ∼ −3 q =0.5, ∆T =0.1 determines KI = 7.9 × 10 . The where the symbol (·)IS is used to designate the IS- full model expression would then be written as: Algorithm. Proof. It is obtained by using the trapezoidal rule as −0.5 −1 ∆T −[0.08] −[0.31] an numerical integration method for Eq.(6). D (z ) = 1+2z +2z IS 2 The upper limit of summation m can be calculated us- −[0.71] −[1.26] −[1.96] +2z +2z +2z (19) ing Eq.(7): −[2.83] −[3.85] −[5.03] +2z +2z +2z + ··· q Tq(t) (t − t0) −1 −2 −3 −5 m = = , (16) =0.05 7+4z +2z +2z +2z + ··· . ∆T ∆T · Γ(q +1) where [x] means truncation to the nearest integers. Simi- lar to the conventional procedure, this procedure can use Figure 3 represents the relationship between the two time T. SICE Vol.E-6 No.1 January 2007 5 Tq (Ǽ) f(νk) − f(νk − ∆τ) f˙(νk) = , (22) ∆τ q=1.0 1 q=0.9 νk = t − KD · k 1−q , (23) 1 KD = {∆T · Γ(2 − q)} 1−q . (24) q=0.1 Proof. Eq.(21) is obtained by using the trapezoidal rule as an numerical integration method for Eq.(20). The upper limit of summation l is 1−q l = (t − t0) / (∆T · Γ(2 − q)) Ǽ Fig. 4 The slopes of the transformed time Tq(τ) vary corre- when the calculation is carried out over the entire domain, sponding to the values of q varying from 0 to 1. and 1−q l = L / (∆T · Γ(2 − q)) axes τ and Tq for q =0.5 of Eq.(7) which result the val- ues of τk from a constant ∆T . Thus, the periods between when it is carried out only over window L. consecutive sampling points τk become shorter as time ap- The above is then re-written as a z-transform model: proaches the present, while the points are getting coarser in the past. This reduces the rate of growth in number of p −1 D (z ) sampling points even though the overall time span is ex- IS l K 1 K 1 −1 − D (k−1) 1−p − D ·k 1−p tended, preventing the extreme increase in computational ∆T 1 − z ∆τ ∆τ = z + z costs. 2 ∆τ k=1 Figure 4 also represents the slopes of the transformed K T − D ∆ −1 −1 ∆τ time Tq(τ) corresponding to the values of q varying from = 1 − z + 2(1 − z )z (25) 2∆τ 0 to 1. It describes that the smaller the order, the higher K 1 K 1 − D ·2 1−p − D ·3 1−p the performance of reducing the computational costs. It −1 ∆τ −1 ∆τ +2(1 − z )z + 2(1 − z )z + ··· . is known that several real systems such as the viscoelas- tic materials have the value of their order of derivative 18) around 0.5 . If the sampling period ∆τ is set at 0.1, for example, and 3. 2 Discretization procedure for FD by inho- the design parameters are set to q =0.5 and ∆T =0.1, mogeneous sampling the model is then expressed thus: The R-L definition of the FD (Eq.(3)) involves taking an integer-order derivative of the FI, so the proposed FD 0.5 −1 −1 D (z ) = 0.5 − 0.5z discretization procedure consists of the simple backward IS −[0.08] −1−[0.08] −[0.31] −1−[0.31] + z − z + z − z difference of the FI described in the previous chapter. −[0.71] −1−[0.71] −[1.26] −1−[1.26] We shall therefore skip over the portion of the procedure + z − z + z − z −[1.96] −1−[1.96] −[2.83] −1−[2.83] which is identical to that followed in the preceding chap- + z − z + z − z ter and proceed to the main results. If we limit the order −[3.85] −1−[3.85] −[5.03] −1−[5.03] + z − z + z − z ≤ q< of the differential to 0 1, then, without loss of −[6.36] −1−[6.36] −[7.85] −1−[7.85] + z − z + z − z generality, the FD in Eq.(3) becomes −1 −2 −4 −5 −8 =3.5 − 1.5z − z − z + z − z + ···. (26) d t Dq f t f τ dT τ . ( )= dt ( ) 1−q( ) (20) t0 Theorem 2. (Discretization of FD by IS-Algorithm) 4. Assessment of the Proposed Procedure The trapezoidal calculation for the FD corresponding to Eq.(15), becomes 4. 1 Comparison of calculation steps l Let us compare the number of calculations needed by q f˙(νk−1)+f˙(νk) (D f(t)) = ∆T, (21) the conventional procedures with those needed by the pro- IS 2 k=1 cedure proposed here. For simplicity, the following discus- where sion counts a single time step for any one of the numerical 6 T. SICE Vol.E-6 No.1 January 2007 7 x 10 integration is set at q =0.5, ∆τ =∆T =0.01, and the G1 5 IS calculation window t−t0 = 100[s]. Thus, Nc is observed to increase with the square of the number of sampling points 4 2 in the conventional method, i.e. Nc ∼ O(n ), while the 3 proposed method suppresses the rate of increase; it is a c N more efficient calculation method. 2 4. 2 Assessing error in the proposed method The accuracy of the calculations performed by the pro- 1 posed method was also investigated. The operation ex- 0 amined here is FI, but the results are equally applicable to 0 20 40 60 80 100 t−t [s] 0 FD. Since the proposed method is numerical, uses trape- Fig. 5 Comparison between the proposed method’s calcula- zoids and is based in the Stieltjes integral in Eq.(6), the tion steps and the conventional one. Euler-Maclaurin formula for the error of the trapezoidal rule 19) can be applied. integrations as a single iteration. The single iterations ac- Lemma 1. (Assessment of the error of the trape- tually would consume different amounts of time, however, zoidal rule) Let f(x) ∈ C2[a, b],n=(b − a)/h. Then, since they consist of various operations - calculation of the error given by the trapezoidal rule will be given by binomial coefficients for the conventional procedure, and n algebraic computations for the trapezoidal or other meth- b h ε = f(x)dx − {f (a +(k − 1)h)+f(a + kh)} ods of the proposed technique. 2 a k=1 For the case of conventional computations (G1), if the b − a 2 (2) t − t τ ≤ h max f (θ) . (30) calculation time is 0 and the sampling period is ∆ , 12 a≤θ≤b then the total number of samples is n =(t − t0)/∆τ. The number of calculations necessary for the integration of the Proof. See 19). k k N -th point is , so the total number of calculations ( c) The total error of IS-Algorithm is given by a theorem of G1 is below. Theorem 3. (Assessment of the error of the IS- n t − t0 t − t0 Algorithm) (Nc)G1 = k = +1 2∆τ ∆τ k=1 t − t 2 τ −q −q ( 0) ∆ . ε = D f(t) − D f(t) = 2 1+ (27) IS 2∆τ t − t0 t − t q+1 · T 2 ≤ ( 0) ∆ · In the proposed procedure, however, the number of cal- · q · τ (31) 12 Γ( +1) ∆ culations required for integration at the k-th point is 1 ∆τ (2) + max f (θ) T (k · ∆τ)/∆T . Substituting this relationship in Eq.(7), q +1 2(t − t0) t0≤θ≤t N we find the total number of calculations ( c)ofISis Proof. The error of Eq.(15) at “actual” time k · ∆τ is evaluated by t−t0 n T k · τ ∆τ k · τ q N ( ∆ ) ( ∆ ) ( c)IS = = (28) −q −q ∆T Γ(q +1)· ∆T ε = D f(k · ∆τ) − D f(k · ∆τ) k=1 k=1 IS q+1 T k · τ − T t t − t0 τ q ( ∆ ) q( 0) 2 (2) ∼ ( ) 1 ∆ . ≤ ∆T · max f (θ) = + t ≤θ≤k·∆τ Γ(q +1)· ∆T · ∆τ q +1 2(t − t0) 12 0 q 1 (k · ∆τ) 2 (2) = ∆T · max f (θ) . (32) 12 Γ(q +1) t0≤θ≤k·∆τ The final approximation of Eq.(28) is obtained by substi- tuting in the approximation for the summation Then the error over the entire calculation domain can be n q ∼ q+1 1 1 found by summation k =1∼ n (n =(t − t0)/∆τ): k = n + . (29) q +1 2n k=1 n q Figure 5 shows the numbers of calculations specifically re- 1 (k · ∆τ) 2 (2) ε ≤ ∆T · max f (θ) 12 Γ(q +1) t0≤θ≤k·∆τ quired by each method when, for example, the degree of k=1 T. SICE Vol.E-6 No.1 January 2007 7 Table 1 Calculation times of the two numerical results of Table 2 Calculation errors of the theoretical results for IS D±0.5t2 for ∆τ =∆T =0.01. and the two numerical ones for ∆τ =∆T = t [s] 5 10 20 50 0.1, 0.01. G1 0.66 4.92 37.9 576.6 D−0.5t2 ε of eq.(31) IS G1 IS(FI) 0.06 0.13 0.36 1.31 ∆T =0.1 0.40 0.09 0.60 IS(FD) 0.25 0.67 1.90 7.19 ∆T =0.01 0.04 0.02 0.36 D−0.5 sin 3t ε of eq.(31) IS G1 ∆T =0.1 1.79 0.25 0.31 T . . . . n q 2 ∆ =001 0 18 0 04 0 10 1 q ∆τ · ∆T (2) ≤ k · max f (θ) , (33) 12 Γ(q +1) t0≤θ≤t k=1 D0.5t2 50 which ends the proof of the theorem. The approximate error is estimated by Eq.(31), and the 40 theoretical L = 3 [s] accuracy can be predicted before the calculation is per- L = 2 [s] L = 1 [s] formed. 30 5. Examples of Numerical Calculation 20 Numerical simulations were performed with the IS- 10 Algorithm (IS) proposed in this paper and also with the 0 conventional G1-Algorithm (G1) in order to compare the 0246810 performance of the two processes. The test calculations t [s] a 0.5 2 were performed for the power function f1(t)=t , and the Fig. 6 Comparison between the theoretical solution of D t and the proposed ones for L =1, 2, 3[s]. sine function f2(t) = sin ωt, whose analytical solutions are given by error according to Eq.(31) for the FI of order q =0.5 un- a f t f t t s Dq f t Dq ta Γ( +1) ta−q, der both procedures for 1( ) and 2( ). Here, = 10[ ], 1( )= = a − q (34) Γ( +1 ) the calculation window was the entire domain (L = t), ∞ n 2n+1 q q −q (−1) (ωt) and the sampling periods were ∆τ(= ∆T )=0.1, 0.01[s]. D f2(t)=D sin ωt = t , (35) Γ(2n +2− q) n=0 The error of the IS computation was within the range predicted by Eq.(31) for both f1(t) and f2(t), indicating where the parameters in the above equation were set at that the equation is, indeed, suitable for estimating error q = ±0.5,a = 2 and ω = 3 for the computation. of IS. Furthermore, accuracy was somewhat higher for IS 5. 1 Comparison of computation time for iden- than for G1 for these examples. Similar results were ob- tical cases tained when the calculations were performed with other The calculation window was set for the entire domain parameters. L t ( = ) for both the IS and G1 procedures. Table 1 5. 3 Influence of calculation window : Ä shows the time required to perform a computation for Next, the calculation window L was shortened in order q . f t the =05of 1( ). The sampling period was set at to reduce calculation time, and the effect on calculation τ T . s ∆ (= ∆ )=001 [ ] and the calculation domains were accuracy was examined. Figures 6 and 7 show the results t =5, 10, 20, 50 [s]. The expressions for the FI and FD of IS calculations with FD of order q =0.5 for f1(t) and are exactly the same in G1, but are different in the pro- f2(t). Parameters are set for the calculation time t = 10[s] posed method, so are marked accordingly on the table as and ∆τ(= ∆T )=0.01[s], for f1(t), L =1, 2, 3[s] and IS(FI) and IS(FD). The table indicates that the time re- for f2(t), L =0.1, 0.2, 0.3[s]. Figure 6 shows that, for a quired for the G1 calculation increased dramatically with monotonically increasing function like f1(t), shortening L increase in domain length. This increase was greatly alle- causes a rapid degradation in accuracy, while a periodic viated by the new procedure. It was verified that the new function such as f2(t), however, even a window of L =0.1, procedure offers higher calculation efficiency. which was much shorter than the period of the function, 5. 2 Assessment of calculation precision produced a fairly close approximation of the solution in Table 2 presents the absolute error and the estimated Fig. 7. 8 T. 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He received the B.S. and M.S. degrees in me- Potential applications of the proposed model include prac- chanical engineering from Niigata University, tical control systems employing FDs. Niigata, and the Ph.D. degree in mechanical engineering from Tokyo Metropolitan Univer- References sity, Tokyo, Japan, in 1994, 1996, and 2002, respectively. He is currently an Assistant Pro- 1) K. B. Oldham and J. Spanier: The fractional Calculus, fessor in the Department of Mechanical Engi- Academic Press, New York and London (1974). neering at Kochi National College of Technol- 2) K. S. Miller and B. Ross: An Introduction to the Fractional ogy. His research interests include modeling, Calculus and Fractional Differential Equations, John Wi- analysis and control of fractional differential ley & Sons, New York (1993). systems, and its application to robotics and 3) S. G. Samko, A. A. Kilbas and O. I. Marichev: Fractional process control. Integrals and Derivatives, Gordon & Breach (1993). 4) I. 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