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Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications Porto, Portugal, July 19-21, 2006

FRACTIONAL-ORDER SYSTEM IDENTIFICATION USING COMPLEX ORDER-DISTRIBUTIONS Jay L. Adamsa, Tom T. Hartleya, Carl F. Lorenzob a

Department of Electrical and Computer Engineering, The University of Akron, Akron OH 44325, USA b NASA Glenn Research Center, Cleveland, OH, 44135 USA

Abstract: This paper discusses the identification of fractional systems using the concepts of complex order-distribution. Based on the ability to define systems using complex orderdistributions, it is shown that system identification in the frequency domain using a leastsquares approach can be performed. Four examples are presented to show the utility of the identification method. Copyright © 2006 IFAC Keywords: Fractional calculus, fractional-order systems, complex order-distributions, system identification. 1. INTRODUCTION Fractional-order systems have been attracting increasing attention in the engineering area. Discussions of the subject are presented by Podlubny (1999); Oldham and Spanier (1974); and Miller and Ross (1993). A larger number of physical systems have behavior that can be best described by fractional system theory, including long lines, electrochemical processes, chaos, dielectric polarization, and viscoelastic materials (Podlubny, 1999). The fractional calculus has allowed the expansion of integer order operators to the concepts of variableorder and distributed-order operators (Lorenzo and Hartley 2002); (Kobelev and Kobelev 2004). In a real-order fractional systems, identification is hard because, unlike the integer-order case where the maximum order dictates the only possible orders present, the maximum fractional order of the system tells nothing about the other fractional orders present. Hartley and Lorenzo (2003) used real orderdistributions to address the problem of identification in fractional-order systems for real derivatives. Oustaloup et al., (1999) used complex-order operators as a basis to achieve robust CRONE control. The potential for improved modeling of physical systems using complex-order operators is a major motivator for the present paper. A complex derivative, when taken by itself, results in a complex time-response, whose meaning is not fully understood. It has been shown in Hartley, et al. (2005a) that complexconjugate symmetric operators result in real time responses. That is, when a complex-order operator is combined with its conjugate, a real function results, rather than a complex function. This result will be utilized in this paper with the objective of improved dynamic modeling. Hartley, et al., (2005b) introduced the concept of complex order-distributions and explored several

distributions and their dynamic responses. The purpose of the present paper is to apply conjugateorder operators and complex-order distributions to the identification of dynamic systems having complexorder. We now introduce the problem of identification of complex-order dynamic systems. For real integerorder system identification, the maximum order dictates the finite number of possible orders present. For the identification of real fractional-order systems, three choices are present; the number of operators, the order of the operators, and the coefficients of the operators. The extension to complex-order operators increases the dimension of the problem, and further requires the use of complex numbers for the order of the operators. The present paper will first review identification using real order-distributions. The mathematical structure from the real case will then be generalized to the case of complex order-distributions. This paper concludes with the identification of several real-order and complex-order systems. For reference, the uninitialized Riemann-Liouville definition of the fractional integral is given by t (t − τ )q −1 f (τ )dτ , q ≥ 0 , −q (1) 0 dt f (t ) = ∫ Γ(q ) 0 and the fractional derivative is given as the integer derivative of a fractional integral, −p t d (t − τ ) p f (τ )dτ , 0 < p < 1 , (2) 0 d t f (t ) = dt ∫0 Γ(1 − p ) For derivatives of order higher than one, multiple integer derivatives are taken of the proper fractional integral. Of importance is the Laplace transform of uninitialized fractional integrals and derivatives, which is given as (3) L 0 d tq f (t ) = s q F (s ) , for all q.

{

}

In Hartley and Lorenzo (2003), a pseudo-inverse was used to find the vector of k m .

2. ORDER-DISTRIBUTIONS An order distribution is an extension of collections of order to a continuum of order over a band. A differential equation with a real order-distribution is given by u  (4)  k (u )s u du  X (s ) = P(s )X (s ) = F (s ) , ∫ u    where k (u ) is the real order-distribution and is considered a weighting on the order of the derivative in the differential equation (Hartley and Lorenzo 2003). A differential equation with a complex orderdistribution is given by max

min

∞∞   ∫ ∫ k (u , v )s u + iv du dv  X (s ) = P (s )X (s ) = F (s ) ,    − ∞− ∞ 

(5)

where k (u , v ) is the complex order-distribution and again is considered a weighting on the order of the derivative in the differential equation (Hartley, et al., 2005b). Table 1 shows some order-distributions, both real and complex, and gives their equivalent Laplace transforms. 3. SYSTEM IDENTIFICATION USING REAL ORDER-DISTRIBUTIONS The differential equation containing the order distribution is given by ∞   ∫ k (u )s u du  X (s ) = F (s ) ,    −∞ 

(6)

where the inverse of the transfer function, P (s ) , is given by ∞ 1 F (s ) (7) P (s ) = = = ∫ k (u )s u du , G (s ) X (s ) −∞ and the frequency response is given by ∞ 1 u (8) P(iω ) = = ∫ k (u )(iω ) du . G(iω ) −∞ Assuming that the integral in Equation 8 is convergent, it can be replaced with an Euler approximation, as M

P(iω ) ≈ ∑ k m (iω )

u 0 + m⋅∆u

∆u ,

(9)

m =0

where ∆u is the sample width and

(iω1 )u (iω 2 )u

1

1

M

(iω1 )u (iω 2 )u

2

L

2

L O

M

(iω R )u (iω R )u 1

2

L

(iω1 )u (iω 2 )u

M

M

M

(iω R )u

M

The method of system identification presented in the previous section can be extended to a complex order distribution. Because only complex orderdistributions which yield real responses are of interest, the distribution must be complex-conjugate symmetric, a fact that will used to simplify the problem. The differential equation containing the order distribution is given by ∞∞  (11)  ∫ ∫ k (u + iv ) ⋅ s u +iv dudv  X (s ) = F (s ) .    −∞−∞  The transfer function of the system is . (12) 1 X (s ) H (s ) =

F (s )

=

∞∞   ∫ ∫ k (u + iv ) ⋅ s u +iv dudv     −∞−∞ 

In order to make the calculations easier, the inverse of each side of Equation 12 is taken, resulting in ∞ ∞ 1 u + iv ∫−∞−∫∞k (u + iv ) ⋅ s dudv = H (s ) = P(s ) . (13) Each integral in Equation 13 can be written as a sum, using an Euler approximation. Doing so results in N

M

P(s ) ≈ ∑∑ k m,n s u0 + m⋅∆u +iv0 +in⋅∆v ∆u ⋅ ∆v ,

(14)

n = 0 m =0

where ∆u and ∆v are the width of the division of area in the q-plane, u 0 is the minimum real point in the distribution, and k m, n is given by

k m.n = k (u 0 + m ⋅ ∆u + iv 0 + in ⋅ ∆v ) . (15)

Using the frequency response necessitates finding the frequency characteristic of Equation 14, which is given by N

M

P (iω ) ≈ ∆u ⋅ ∆v ∑ ∑ k m ,n (iω ) 0

u + m⋅∆u + iv0 + in⋅∆v

.

(16)

n =0 m =0

Note that, (17) k m,n = k m, N −n , because the distribution must be conjugate symmetric. Equation 16 can be written as

k m is the height

of the sampled order-distribution. Equation 9 must be satisfied at every data point in a sample frequency response, so the system can be written in matrix form as  (iω1 )u0   (iω )u0 ∆u  2  M  (iω )u0  R

4. SYSTEM IDENTIFICATION USING COMPLEX ORDER-DISTRIBUTIONS

 k 0   k1  M   k  M

  P(iω1 )       P(iω 2 )  =   M       P(iω ) R   

(10)

N

(

P (iω ) ≈ Q ∑ (iω ) 0 n =0

u + iv n

(iω )u +iv 1

n

(iω )u +iv 2

n

L

(iω )u

M

+ ivn

 k 0 ,n     k1,n  k   2,n   M  k   M ,n 

)

(18) or N1

(

P(iω ) ≈ Q ∑ (iω ) 0 n=0

u + ivn

+ (iω ) 0

u −ivn

L

(iω )u

M

+ ivn

+ (iω )

u M −ivn

 k 0,n   M  k   M ,n 

)

(19) where Q is given by

Q = ∆u ⋅ ∆v ,

(20)

u m = u 0 + m ⋅ ∆u ,

(21)

v n = v0 + n ⋅ ∆v ,

(22)

u m is given by

v n is given by

5. EXAMPLES

and N 1 is given by N N even .  , N1 =  2 N −1  , N odd  2

(23)

This Equation can be written as P(iω ) ≈ Q ⋅ Ω(iω ) ⋅ K , where K is given by

(

K = k 0, 0 L k M , 0

(24)

k 0,1 L k M ,1 L k 0, N1

L k M , N1

and Ω(iω ) is given by Ω(iω ) = (Ω1 (iω ) Ω 2 (iω ) L Ω N (iω )) , where

(

Ω n (iω ) = (iω ) 0

u + ivn

+ (iω ) 0

u −ivn

L

(iω )u

M

+ ivn

+ (iω ) M

u −ivn

)

T

The first example is the most simple; the transfer function 1 (30) G1 (s ) = , s is considered. Because the distribution is, in general, complex, the magnitude of the complex orderdistribution is given. Figure 1 shows the identified system. The identified system matches the actual system perfectly for the chosen order grid.

,

(25) (26)

). (27)

K is an ((M + 1) ⋅ ( N 1 + 1)) × 1 column vector, and Ω(iω ) is a 1 × ((M + 1) ⋅ (N1 + 1)) row vector. Equation 26 must be satisfied at every frequency data point, ω k , So

writing Equation 26 for each frequency point yields 1 Q ⋅ Ω(iω1 ) ⋅ K = = P(iω1 ) G(iω1 ) 1 Q ⋅ Ω(iω1 ) ⋅ K = = P (iω 2 ) G (iω 2 ) Q ⋅ Ω(iω1 ) ⋅ K =

M

1 = P (iω K ) , G (iω K )

which can be written in matrix form as  Ω(iω1 )   P(iω1 )      ( ) Ω i ω   P(iω 2 )  . 2  Q K = M  M       Ω(iω )  P(iω )   K  K 

(28)

(29)

If the number of frequency points is larger than ((M + 1) ⋅ (N1 + 1)) , then a least-squares, or matrix pseudoinverse, solution can be utilized for solving for  Ω(iω1 )    . The matrix, K  Ω(iω 2 )  , however, does tend to  M     Ω(iω ) K   become singular as Q gets small and as the number of

frequency samples gets large. The Matlab pseudoinverse command pinv was used to solve for K . This command is a single-value decomposition based command and is more numerically stable. Premultiplication by a scaling matrix would improve the conditioning of the matrix. It is also expected that using orthogonal matrices, as described in Rolain, et al.(1995) would also improve the conditioning of the problem, but was not yet explored.

Fig 1. Identified Distribution for Example One. The second example is a more complicated real-order system considered in Hartley and Lorenzo (2003). The transfer function is 1 . (31) G2 (s ) = 2 s − 1.4s1.5 + s − 1.4s 0.5 + 1 The magnitude of the identified system is given in Figure 2. Again the identified distribution matches perfectly the system whose frequency data was generated. Consider the effects of increasing the number of real values that may be used. Rather than allowing five values, nine are allowed. Figure 3 shows the results of this increase in potential derivative orders. The results for u ≥ 1 are exactly as expected, but for u < 1 , the distribution differs from the known distribution. To compare the identified plant with the original system, the Bode plots of both the approximation and the original system are given in Figure 4. The Bode plots over this range are identical, so the approximation is acceptable. The identified systems are different because the algorithm is more numerically reliable with a larger ∆u . As ∆u decreases, the condition number of Ω(iω ) increases, leading to less numerical reliability. The third example considered is given by the transfer function 1 . (32) G (s ) = 3

(

)

s + 0 . 19 ⋅ s 0 .5 + i + s 0 .5 − i + 1

Figure 5 shows the distribution that was identified for the system. The resulting order-distribution looks much like the given system. However, because the identification gives non-zero weighting to orders that are not present in the transfer function of the system, the frequency data, as shown in Figure 6, must be compared. The identified system matches the given system well over the range 10 −2 ≤ ω ≤ 10 6 . Thus, over the frequency range, the identification can be utilized. The identification is not exact due to numerical issues with calculating a pseudoinverse.

G 4 (s ) =

1

. (33) s + 0.63 ⋅ s + s 0.7828−i⋅0.5546 + 1 Figure 7 shows the distribution that was identified for the system. In order to better see the distribution, a contour plot of the order-distribution is given in Figure 8. The resulting order-distribution has peaks close to the expected peak locations. Because the given system is not identical to the identified system, once again the frequency responses of the systems must be compared. Figure 9 shows the frequency responses of the original system and the identified system. The approximation is acceptable over the range 10 −2 ≤ ω ≤ 10 6 , because the Bode plots of both systems are identical over this range.

(

0.7828+ i ⋅0.5546

)

Fig. 2. Identified Distribution for Example Two.

Fig. 5. Identified Distribution for Example Three.

Fig 3. Identified Distribution for Example Two.

Fig. 6. Bode Plots for Example Three.

Fig. 4. Bode Plots for Example Two The fourth example considered is given by the transfer function

Figure 7. Identified Distribution for Example Four.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the NASA Glenn Research Center. REFERENCES Hartley, Tom T., Carl F. Lorenzo, and Jay L. Adams (2005a). Conjugated Order Differintegrals. Proceedings of ASME DETC05 Conference.

Figure 8. Contour for Example Four

Hartley, Tom T., Jay L. Adams and Carl F. Lorenzo (2005b). Complex Order-Distributions. Proceedings of ASME DETC05 Conference. Hartley, Tom T. and Carl F Lorenzo (2003). Fractional-Order System Identification Based on Continuous Order-Distributions. Signal Processing 83, 2288-2300. Kobelev L.Y. and Y. L. Kobelev (2004) The Fractional Derivatives with Orders as Functions (Variable Order) and Their Applications to Problems of Natural and Social Sciences and Proceedings of IFAC FDA04 Engineering. Conference.

Figure 9. Bode Plots for Example Four 5. CONCLUSIONS A frequency domain identification method for distributed complex-order systems has been created and demonstrated. The method is shown to provide complex-order distributions that reasonably match the input order-distributions, and whose frequency responses exactly match those of the input orderdistributions. Future required work will include identification in the presence of noise and improvement of the numerical robustness of the algorithm to allow greater resolution of the order. The technique for identification was shown as viable for several systems over a given bandwidth for fractional-order and complex-order systems. Because the identified system uses conjugated derivatives, the resulting system is real in the time-domain. This technique will be improved on by increasing the numerical robustness of the algorithm by improving the conditioning of the Ω-matrix. Pre-multiplication by a scaling matrix is expected to increase the conditioning of the problem, as is the use of orthogonal polynomials.

Lorenzo, Carl F. and Tom T. Hartley (2002). Variable Order and Distributed Order Fractional Operators. Nonlinear Dynamics 29 57-98. Miller, K.S. and B. Ross (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York. Oldham, K. B. and J. Spanier (1974). The Fractional Calculus: Integrations and Differentiations of Arbitrary Order. Academic Press, New York. Oustaloup, Alain, Jocelyn Sabatier, and Patrick Lanusse (1999). From Fractal Robustness to the CRONE Control. Fractional Calculus and Applied Analysis 2 (1), 1-30. Podlubny, Igor (1999). Fractional Differential Equations: An Introduction to Fractional Derivative, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, New York. Rolain, Y., R. Pintelon, K.Q. Xu, and H. Vold (1995). Best conditioned parametric identification of transfer function models in the frequency domain. IEEE Transaction on Automatic Control AC-40 (11), 1954-1960.

Table 1: Sample Order-Distributions Order Distribution

Equivalent Laplace Transform

 k , u − δu ≤ u ≤ u + δu k (u ) =  0, otherwise

 sinh (δu ⋅ ln(s ))   P(s ) = 2ks u  ln(s )  

k (u ) = ke

−

(u − u )2

1

P(s ) = kσ u π s u e 4

σ u2

σ u2 ln 2 ( s )

k , u + i ⋅ v = u ± i ⋅ v k (u ) =  0, otherwise

P (s ) = 2ks u cos (v ⋅ ln (s ))

 u − δu ≤ u ≤ u + δu  k, k (u, v ) =  − δu ≤ v ≤ δu 0, otherwise

 sinh (δu ⋅ ln (s ))sin (δv ⋅ ln (s ))   P (s ) = 4ks u  ln 2 (s )  

 u − δu ≤ u ≤ u + δu  k, k (u , v ) =  v − δu ≤ v ≤ v + δu 0, otherwise

 sinh (δu ⋅ ln (s ))sin (δv ⋅ ln (s ))  cos(v ⋅ ln (s )) P (s ) = 8ks u  ln 2 (s )  

k (u , v ) = ke

−

(u −u )2 − v 2 σu

σv

 1 (σ u −σ v ) ln 2 ( s )   P(s ) = s u σ uσ v π  e 4   