Operators for Fractional Discrete Approximation in System of Control

IFAC DECOM-TT 2004 SAMPLE PAGES TO BE FOLLOWED EXACTLY Automatic Systems for Building the Infrastructure in Developing Countries IN PREPARING SCRIPTS October 3 - 5, 2004 Bansko, Bulgaria        

OPERATORS FOR FRACTIONAL DISCRETE APPROXIMATION IN SYSTEM OF CONTROL Nadejda Radomirova Radeva Technical university Sofia, Kliment Ohridski 1, Technical University, blok 9, department ANP e-mail: [email protected]   

Abstract: This paper has purpose to cover and systematizes some main methods for discrete approximation. With discrete domains it is possible to use rational approximations for these operators in order to have suitable realizable forms in control applications, several methods for approximations are discussed For discrete models, approximations using Lubich's formula, discrete approximation using numerical integration and power series expansion (PSE), discrete approximation using numerical integration and continued fraction expression (CFE), the trapezoidal rule, and technique to integro-differential operators formulated in the Z domain. Using operators of Euler-Grunvald-Letnikov, Tustin, Simpson, Al-Alaoui and theirs approximations can be used as a numeric realization in programmable controllers. Copyright © 2004 IFAC Keywords: fractional system, discretization, fractional approximate analysis (fractional calculs, fractional-order system, integer-order approximation, discrete fractional approximation)

1. INTRODUCTION Applications of fractional-order models in control theory are relatively new comparing to mentioned applications. In the last two decades the possibility of using fractional order controller has been considered (R. L. Bagley and R. A. Calico, 1991; P. Bidan, 1998; K. Matsuda and H. Fujii, 1993; G. Montseny, J. Audounet, and D. Matignon, 1997; A. Oustaloup, 1983; A. Oustaloup, 1991; A. Oustaloup, 1994; Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, 2000; I. Petras and L. Dorcak, 1999; I. Podlubny,1999; I. Podlubny,1999; S. B. Skaar, A. N. Michel, and R. K. Miller, 1988). In all the cases, the important point for the purposes of this paper, is that the equivalent circuit or the fractional-order controller in its practically realizable form is a finite dimensional integer-order system resulting from the approximation of a fractional-order or in finite dimensional system. The problem of obtaining a continuous or discrete realizable model for a fractional order controller can be viewed as a problem of obtaining a rational

approximation of the irrational transfer function, modeling the fractional controller. In general, there are two possibilities for realizing a controller: a hardware realization based on the use of a physical device, or a software (or digital) realization based on a program, which will run on a computer or microprocessor. In electronics, hardware realizations imply the use of electronic devices or circuits, implementing the required function as an admittance or impedance function. For digital realizations, a finite difference equation is needed. This equation can be obtained by numerical approximation of by performing the inverse Z transform of a discrete transfer function. The aim of this paper is to present methods for rational discrete approximations and theirs application like algorithms in system of control To find a discrete time model of the fractional dynamic system (1) we use discrete operators or their approximations for fractional differentiation and integration. Its application in (1) lead to common equation building for the discrete time transfer function of the fractional dynamic system

G

 z (2), as a discrete analogue of (1). 1

m

G

p p u p

bm p

y

a

n

p

E

m

D

n

E

 b m1 p a

D

p

n1

m 1

n 1

E

˜˜ b0 p ˜˜ a0 p

¦b

0

k

D

G

z

 uz y z

1

1

  Y  z

  Y  z

bm Y z

1

E

m

 b m1 Y z

1

E

an

1

D

n

 a n1

1

D

Where: z (3) – complex variable, connected with the complex variable p and with the sample period

Y  z

T 0 according with (4). The discrete time integro-

complex variable

differential

z

fractional

z

operators

e

T

0

Re z  j Im z ;

p

in

e

T

0

 Re z z

e

T

0

V

cos Z T  j e

e

T

p

0

V

;

Numerical

Using the generation function corresponding to the backward fractional difference rule,

expansion

1  z , and performing the power series (PSE) of 1  z , the Grunwald1

1 D

Letnikov formula for the fractional derivative of order D is obtained: f k §D · ’DT f  nT T0 D ¦  1 ¨ ¸ f   n  k T (5) k 0 ©k¹ Performing the PSE of the function

1  z

k

  Y  z

˜˜b0 Y z

1

1

E

D

0

(2) 0

is shown as a function of the

z or z

1 D

leads

to the formula given by Lubich for the fractional integral of order D f k §D · ’TD f  nT T0D ¦  1 ¨ ¸ f   n  k T (6) k 0 ©k¹ In any case, the resulting transfer function, approximating the fractional-order operators, can be obtained by applying the relationship:

T

0

V

1

.

sin Z T

c o s Z T , Im z

p

Among other mathematical methods, two of them are particularly interesting for this purpose, from a control theory point of view: Discrete approximations using numerical integration and Power Series Expansion (PSE) and Discrete approximations using numerical integration and Continued Fraction Expansion (CFE).

Z  z 1

p

k

0

˜˜ a0

rD

(1) D

(2)

2. METHODS FOR DISCRETE APPROXIMATION

2.1.Discrete Approximations using Integration and Power Series Expansion

1

k

0

¦a

m1

n1

E

n

0

k

1

p

k

e

T

0

V

sin Z T



, (3)

T 0 1 l n z (4)

^

Y  z T0 mPSE 1  z 1

rD

` F z

(7)

where T0 is the sample period, Y  z is the Z transform of the output sequence y  nT0 , F  z is the Z transform of the input sequence f  nT0 , and PSE{u} denotes the expression, which results from the power series expansion of the function u. Doing so gives: Y  z rD D rD  z T0 ma PSE 1  z 1 (8) F z

^

where D

rD

z

`

denotes the discrete equivalent of the

fractional-order operator, considered as processes. Another possibility for the approximation is the use of the trapezoidal rule, that is, the use of the generating function 1  z 1 Z  z 1 2 (9) 1  z 1 It is known that the forward difference rule is not suitable for applications to causal problems (R.Gorenflo, 1997; Ch.Lubich, 1986). It should be mentioned that, at least for control purposes, it is not very important to have a closed-form formula for the coefficients, because they are usually pre-computed and stored in the memory of the microprocessor. In such a case, the most important is to have a limited number of coefficients because of the limited available memory of the microprocessor system.



The approximations, considered in the previous section, lead to discrete transfer functions in the form of polynomials, and this is not convenient, at least from the control point of view. On the other hand, it can be recalled that the continued fraction expansion (CFE) leads to approximations in rational form, and often converges much more rapidly than PSE and has a wider domain of convergence in the complex plane, and, consequently, a smaller set of coefficients will be necessary for obtaining a good approximation. In view of these reasons, a method for obtaining discrete equivalents of the fractional-order operators, which combines the well known advantages of the trapezoidal rule in the control theory and the advantages of the CFE, is proposed here. The method implies: x the use of the generating function 1  z 1 (10) Z  z 1 2 1  z 1 1 is the where z is the complex variable, and z shifting operator, x and the continued fraction expansion (CFE) of

rD

z

§ a zb · ¸ o ¨ ¨ czd ¸ © ¹

rD

 Y  z

rD



; z

rD

§ 1  z 1 · (11) ¨2 1 ¸ © 1 z ¹ for obtaining the coefficients and the form of the approximation. The resulting discrete transfer function, approximating fractional-order operators, can be expressed as:

2.2.Discrete approximations using numerical integration and continued fraction expansion

Z  z 1

Y  z

DrD  z



rD

rD

­°§ 1  z 1 · T0mD CFE ®¨ 2 ¸ 1  z 1 ¹ ¯°©

F  z

½° ¾ ¿° p,q

Pp  z 1

T0mD

Pq  z 1

(12) where T0 is the sample period, CFE{u} denotes the function resulting from applying the continued fraction expansion to the function u, Y (z) is the Z transform of the output sequence y(nT), F(z) is the Z transform of the input sequence f(nT), p and q are the orders of the approximation, and P and Q are polynomials of degrees p and q, correspondingly, in the variable

z 1 .

3. OPERATORS FOR DISCRETE APPROXSIMATION For the discretization of the fractional systems, several rD

types of z – transformations Y are used. All of them are in the class of the linear fractional transformations, based on the common operator (13).

1



rD

§ a  b z 1 o¨ ¨ c  d z 1 ©

· ¸ ¸ ¹

rD

Y  z 1

rD

(13)

This is a operators of Euler-Grunvald-Letnikov (14), Tustin (15), Simpson (16), Al-Alaoui (17).

p

rD

{D

p

p

p

rD

rD

rD

Y  z

rD

{D

{D

{D

rD

rD

1

rD

 Y  z 1

Y  z 1

rD

Y  z 1

Their approximations in series, as

§ 1 ¨ ¨ T © 0

rD

D

rD

rD

1  z 1

§ 2 ¨ ¨ T © 0 § 3 ¨ ¨ T © 0

rD

· ¸ ¸ ¹

 Euler - Grünwald - Letnikov (14)

 1  z ·¸  1  z ¸¹ 1

rD

 Tustin

1

 1  z  1  z ·¸  1  4 z  z ¸¹

§ 8 ¨ ¨ 7T 0 ©

1

1

1

1

1

rD

2

 1  z ·¸  1   z 7 ¸¹

(15)

 Simpson

(16)

rD

 Al - Alaoui

is the fractional power of the operators, are shown in Table 1

(17)

Table 1 Fractional operator for

Operators approximation in Taylor series

p o z transformation p

p

p

ª 1 « «¬ T 0

D

ª 2 « «¬ T 0

D

ª 3 « «¬ T 0

D

D

º » »¼ Euler Grünwald

1  z 1

1  z 1  z 1

1

º » »¼

p

2

D

p Simpson

Fractional operator for

p

ª 1 « «¬ T 0

§ 3 · ¸ |¨ ¨ T ¸ © 0 ¹

D

º » »¼ Euler Grünwald

1  z 1

1

D D 1



z

2!

Letnikov

D

D

1  2 D z

1  4 D z

1

1

 2D

 2D

2

2

z

4D

p

ª 3 « «¬ T 0

D

 1  z º»  1  z »¼ 1  z 1  z 1  4 z  z

ª 2 « «¬ T 0

D

p

D

§ 1 · ¸ | ¨ ¨ T ¸ 0 © ¹

D

1

Tustin

1

1

2

º » »¼

D

p

D

§ 2  D  1 z ¨ ¨ 2  D  1 z ©

p

D

1  z

D

D

§ 1D z ¨ ¨ 1D z ©

˜˜˜˜



Simp

1

n

· ¸ ¸ Euler ¹ Grünwald

1 1

1 1

§ 2  4 D  3 z ¨ ¨ 2  4 D  3 z ©

m

º » »¼ Euler Grünwald

1

2

Tustin

Letnikov

§ 2 · ¸ | ¨ ¨ T ¸ © 0 ¹

§ 3 · ¸ | ¨ ¨ T ¸ © 0 ¹



· ¸ ¸ ¹

Tustin

1 1

· ¸ ¸ ¹

Simpson

Operators approximation in Padé series with

Fractional operator for ª 1 « «¬ T 0

D

Simpson

p o z transformation D

D

p

1

1

˜˜˜˜

3 z

Letnikov

p

· ˜˜˜˜ ¸ ¸ Euler ¹ Grünwald

2

Operators approximation in Padé series with  m

p o z transformation D

D

§ ¨ 1D z ¨ ©

§ 2 · ¨ ¸ ¨ T ¸ © 0 ¹

D

Tustin

º » »¼

D

§ 1 · ¸ | ¨ ¨ T ¸ © 0 ¹

D

1

1

p

Letnikov

1  z 1  z 1  4 z  z 1

D

p

D

§ 1 · ¸ | ¨ ¨ T ¸ © 0 ¹

D

2

n

§ 12  6  D  2 z ¨ ¨ 12  6  D  2 z ©

  D

1

 D

1

2 2

 3D  2 z

 3D  2 z

· ¸ ¸ Euler ¹ Grünwald

2 2

Letnikov

Letnikov

p

p

ª 2 « «¬ T 0

D

ª 3 « «¬ T 0

D

1  z 1  z 1

1

º » »¼

D

p Tustin

1  z 1  z 1  4 z  z 1

1

1

2

º » »¼

D

p

D

Simpson

After the discretization of (1), using the operators (14)-(17) or its approximations and transformations 1

with reordering basing on z - orders, the discrete time transfer function (2) is transformed to (18)



G z

1

Where: c

 uz y z

1

1

is a constant,

c

 z

Pr z 1 Qq

1

D

§ 2 · ¸ | ¨ ¨ T ¸ © 0 ¹

§ 3 · ¸ | ¨ ¨ T ¸ 0 © ¹

D

D

§ 3  3D z ¨ ¨ 3  3D z ©



P r and Q q are 1

polynomial functions of the z (using the full orders). (18) is a discrete transfer function of the fractional dynamic system. Simultaneously the function (18), as a relation of the

1

  D

 D

2 2

1 z

1 z

2 2

· ¸ ¸ ¹

Tustin



2 § 64 D  144 D  92 D  177 D  28 z  ¨ 2 3 2 ¨  3 16 D  25  6 16 D  24 D  25 D  33 z  1 ¨ ¨ 3 16 D 2  25  6 16 D 3  24 D 2  25 D  33 z  1  ¨ ¨  64 D 4  144 D 3  92 D 2  177 D  28 z  2 ©





4

3









2









in the case of full order system, we can use the z -transformation. The reverse z reverse transformation (19) is a tool, that allow to find the y k T0 if the zoriginal function



transformation of y (18)

1

y k T0





 z is known or Z

1

^ y  z ` 1



y z

1

.

(19)

The calculation of the reverse z -transformation can be done using several methods. The result is finding of an effective and easy equation, which can be used as a numeric realization in programmable controllers. 4. CONCLUSION

1

z power polynomial functions (only for full order powers), do not differs the discrete time transfer function for full order system. Therefore, as

In this paper is shown some method for discrete approximations using fractional operators. This is

Discrete approximations using numerical integration and Power Series Expansion (PSE) and Discrete approximations using numerical integration and Continued Fraction Expansion (CFE). of the generating function defined by the trapezoidal rule. Discrete models is a demand and motivation for further work towards practical implementation of fractional-order controllers. PREFERENCES R. L. B a g l e y and R. A. C a l i c o (1991). Fractional order state equations for the control of viscoelastically damped structures. Journal of Guidance, 14, No. 2 304{311. P. B i d a n (1998). Commande diffusive d'une machine electrique: une introduction. In D. Matignon and G. Montseny, editors, Proceedings of the Colloquium FDS'98: Fractional Differential Systems: Models, Methods and Applications, No. 5, 55{68. G. E. C a r l s o n and C. A. H a l i j a k (1961). Simulation of the fractional derivative operator

s and the fractional integral operator 1/ s . Proc. Of the CSSCM, Kansas State U. B., 45, No. 7, 1{22. G. E. C a r l s o n and C. A. H a l i j a k (1964). Approximation of fractional capacitors

1/ s

1

n

by a regular Newton process. IRE

Transactions on Circuit Theory, CT-11, No. 2, 10{213. A. C h a r e f f, H. H. S u n, Y. Y. T s a o, and B. O n a r a (1992). Fractal system as represented by singularity function. IEEE Transactions on Automatic Control, 37, No. 9, 1465{1470. R. G o r e n f l o (1997). Fractional calculus: Some numerical methods. In: A. Carpinteri and F. Mainardi (eds.). Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Vienna { New York. C. H a b a, M. M a r t o s, G. A b l a r t, and P. B i d a n (1998). Composants electroniques a impedance fractionnaire. In D. Matignon and G. Montseny, editors, Proceedings of the Colloquium FDS'98: Fractional Differential Systems: Models, Methods and Applications, No. 5 99{109. D. H e l e s c h e w i t z and D. M a t i g n o n (1998). Diffusive realizations of fractional integrodifferential operators: structural analysis under approximation. IFAC Conference on System, Structure and Control, Vol. 2, Nantes, France, 243-248. A. H e r n _ a n d e z, B. V i n a g r e, and V. F e l i u (1999). Realizaci on de reguladores fraccionarios. Actas de las XX Jornadas de Automatica, Salamanca, 235{240. H. E. J o n e s and B. A. S h e n o i. (1970). Maximally flat lumped-element approximation to fractional operator inmitance function. IEEE Transactions on Circuit Theory 17, No. 1, 125128. C h. L u b i c h (1986). Discretized fractional calculus. SIAM Journal of Mathematical Analysis, 17, No. 3, 704{719.

K. M a t s u d a and H. F u j i i. (1993). H f optimized wave-absorbing control: Analytical and experimental results. Journal of Guidance, Control, and Dynamics, 16, No. 6 1146{1153. K. S. M i l l e r and B. R o s (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley & Sons. Inc., New York. G. M o n t s e n y, J. A u d o u n e t, and D. M a t i g n o n (December 1997). Fractional integrodifferential boundary control of the Euler{Bernoulli beam. 36th IEEE Conference on Decision and Control, San Diego, 4973{4978. K. B. O l d h a m and J. S p a n i e r (1974). The Fractional Calculus. Academic Press, New York London. A. O u s t a l o u p (1983). Systemes Asservis Lineaires d'Ordre Fractionnaire: Theorie et Pratique. Editions Masson, Paris A. O u s t a l o u p (1991). La commande CRONE. Editions Hermes, Paris, A. O u s t a l o u p (1994). La Robustesse. Editions Hermes, Paris A. O u s t a l o u p, F. L e v r o n, B. M a t h i e u, and F. M. N a n o t (2000). Frequency-band complex noninteger differenciator: Characterization and systhesis. IEEE Trans. on Circuits & Systems I: Fundamental Theory and Applications, 47, No. 1, 25{39. I. P e t r a s and L. D o r c a k (1999). Some possibilities of realization of fractional-order controllers. Envirautom, 4, No. 1, 83{90. I. P o d l u b n y (1999). Fractional-order systems O P and PI D -controllers. IEEE Transactions on Automatic Control, 44, No. 1, 208{214. I. P o d l u b n y (1999). Fractional Differential Equations. Academic Press, San Diego S. B. S k a a r, A. N. M i c h e l, and R. K. M i l l e r (1988). Stability of viscoelastic control systems. IEEE Transactions on Automatic Control, 33, No. 4, 348{357. J. A. T e n r e i r o M a c h a d o (2001). Discrete time fractional order controllers, FCAA – Journal of Fractional Calculus & Applied Analysis, vol.4, n.1, pp.47-66, B. M. V i n a g r e, I. P o d l u b n y, A. H e r n a n d e z, V. F e l I u (2003). Some approxsimation of fractional order operators used in control theory and application. FEDER Research Grant IFD970755-C02-01 and VEGA Reserach Grant 1/7098/20 I v o P e t r a s, L u b o m i r D o r c a k (2001). Fractional-Order Control Systems: Modelling and Simulation. Kluwer Academic Publishers.