Discrete transfer function models for non integer order

Discrete transfer function models for non integer order inertial system Krzysztof OPRZ†DKIEWICZ October 31, 2017

Abstract In the paper new, discrete, transfer function models of non integer order inertial plant are proposed. These models can be employed to digital modeling of high order dynamic systems, for example heat transfer systems. Models under consideration use Charef approximation and generating functions expressed by schemes given by Euler, Tustin and Al-Aloui. The practical stability and accuracy for all presented models is analysed also. Results are by simulations depicted.

keywords:

fractional order systems, fractional order transfer function, Charef approximation, generating function, practical stability.

1

An Introduction

Fractional order (FO) models can properly and accurate describe a number of real physical phenomena.This problem was presented by many Authors, for example in: [5], [8], [10], [21], [22], [26], [25]. The possibility of eective modeling fractional order system at each digital platform (PLC, microcontroller, FPGA) is determined by possibility of its discrete, integer order approximation. The time-continuous approximation of elementary operator sα was proposed by Oustaloup in [28], the approximation 1 of elementary inertial transfer function (T s+1) α was proposed by Charef in [3]. The both approximations were also presented in [6]. Ideas of the both approximations are close and they consist in tting Bode magnitude plots of exact and approximated transfer functions. Discrete approximations of basic element 1 s are known also as PSE and CFE approximations (see for example [2], [27]). Fastly convergent digital approximations of FO element was also considered in [17]. However any of these methods does not oer a tool to direct, digital mod1 eling of fractional order inertial plant (Tα s+1) α . Similar problem was analysed in [11]. The paper is intented to present new, proposed by Author, discrete models of non integer order inertial plant. The models are dedicated to implementation at industrial digital platform with bounded resources (for example PLC), employed 1

for example to Model Based Control (MBC) or Model Based Fault Detection (MBFD). The paper is organized as follows: at the beginning some preliminaries from area of fractional order calculus are remembered. Next the considered inertial, fractional order plant and its approximation proposed by Charef are presented. Furthermore its discrete versions, proposed by author are shown and analysed. Finally a numerical example is given. 2 2.1

Preliminaries Elementary ideas

The presentation of elementary ideas will be started with dene a non integer order, integro-dierential operator. It is expressed as follows (see for example [13], [15]):

Denition 1. The non integer order integro - dierential operator  α d f (t)  α>0 α    dt f (t) α = 0 α 0 Dt f (t) = ∫t    f (τ )(dτ )−α α < 0 

.

(1)

0

where t denotes time limit to operator calculating, α ∈ R denotes the non integer order of the operation. Next ideas of complete and incomplete Gamma Euler functions can be given:

Denition 2. The complete Gamma function ∫∞ Γ(x) =

tx−1 e−t dt.

(2)

0

Denition 3. The incomplete Gamma function ∫Tf Γ(x) =

tx−1 e−t dt.

(3)

0

The fractional-order, integro-dierential operator (1) can be described by dierent denitions, given by Grünvald and Letnikov (GL denition), Riemann and Liouville (RL denition) and Caputo (C denition). The digital modeling of FO operator can be most naturally done with the use of GL denition and it will be presented here:

Denition 4. The Grünvald-Letnikov denition of the FO operator ([2],[27]) ( ) α f (t − jh). (−1) j j=0 t

GL α 0 Dt f (t)

−α

= lim h h→0

[h] ∑

2

j

(4)

where

(α) j

is a generalization of Newton symbol into real numbers: ( ) { 1, j=0 α = α(α−1)...(α−j+1) j , j>0 j!

(5)

Denition 5. The Riemann - Liouville denition of the FO operator RL α 0 Dt f (t)

=

∫∞ (t − τ )N −α−1 f (τ )dτ.

1 dN Γ(N − α) dtN

(6)

0

where N −1 < α < N denotes the non integer order of operation and Γ(..) is the complete Gamma function expressed by (2). The Caputo denition is described as underneath:

Denition 6. The Caputo denition of the FO operator C α 0 Dt f (t)

2.2

1 = Γ(N − α)

∫∞ 0

f (N ) (τ ) dτ. (t − τ )α+1−N

(7)

Laplace transform for fractional order operator

If the RL or C denition is considered, the Laplace transform can be also given (see for example [13]) as a generalization of Laplace transform for integer order case:

Denition 7. Laplace transform for Riemann - Liouville operator α α L(RL 0 Dt f (t)) = s F (s), α < 0 α α L(RL 0 Dt f (t)) = s F (s) −

n−1 ∑

sk 0 Dtα−k−1 f (0), .

(8)

k=0

α > 0, n − 1 < α ≤ n ∈ N

Denition 8. Laplace transform for Caputo operator α α L(C 0 Dt f (t)) = s F (s), α < 0 α α L(C 0 Dt f (t)) = s F (s) −

n−1 ∑

sα−k−1 0 Dtk f (0), .

k=0

α > 0, n − 1 < α ≤ n ∈ N

3

(9)

Consequently an inverse Laplace transform can be given as underneath (see for example [15] p.29): L−1 [sα F (s)] =0 Dtα f (t) +

n−1 ∑ k=0

tk−1 f (k) (0+ ) Γ(k − α + 1) .

(10)

n − 1 < α < n, n ∈ Z 2.3

Generating functions

The crossing from contiuous to discrete FO system can be done with the use of so called generating function allowing us to replace an elementary contiunuous-time dierentiator s by its rational approximation: s ≈ H(z −1 ) (see for example[6], p. 19). This approximation can be done with the use of dierent formulas. They are remembered underneath. In each formula h denotes the sample time. The most simple is the Euler approximation expressed by: s≈

1z−1 . h z

(11)

The next one is Tustin approximation (bilinear transformation) given by: s≈

2z−1 . hz+1

(12)

and the last one considered here is Al-Aloui method given by (see [1]): s≈

8 z−1 . 7h z + 17

(13)

All these methods will be used to build a discrete version of Charef approximation. This will be shown in the further parts of paper. 3

The considered inertial, fractional order plant and its approximation

Let us consider the inertial, fractional order plant, expressed by the following, elementary transfer function: Gα (s) =

k . (T s + 1)α

(14)

where k is the steady-state gain of the plant, T denotes the time constant of the plant, 0 < α < 1 is the fractional order of the plant. For the above transfer function analytical formulas of impulse and step responses can be given ([2], pp.8-9) and they have the following forms: yimp (t) =

−t

k

·

(Tα )α

tα−1 e Tα . Γ(α)

(15)

where Γ(α) denotes the complete Gamma function (2). ystep (t) =

k (Tα )α

4

·

( ) Γ α, Ttα Γ(α)

.

(16)

and Γ(α, ..) is the incomplete Gamma function (3). The possibility of modeling the transfer function (14) at MATLAB platform is determined by possibility of its nite-dimensional, integer order approximation. Approximation methods for fractional-order transfer functions has been conisdered by many Authors. For time-continuus systems fundamental results were given by Oustaloup in [28], they are also proesented in [4], [2] and Charef in [3], [9] they can be found also in [2]. Discrete approximations using PSE (Power Series Expansion) and CFE (Continuous Fraction Expansion) methods has ben considered by many Authors, for example: [31], [27], they can be also found in [2]. The approximation proposed by Charef allows us to approximate the transfer function (14) with the use of the following approximation: N∏ −1

Gch (s) =

n=0 N ∏

(1 +

s zn )

= (1 +

n=0

s pn )

LCh (s) . DCh (s)

(17)

where zi and pi denote zeros and poles of approximation, N denotes order of the approximation. An idea of this approximation is to best t the Bode magnitude plot of approximation to Bode magnitude plot of plant in given frequency band. Zeros and poles are calculated with the use of following recursive dependencies (see [2], [3]): 1 p= . √T p0 = p b. z0 = ap0 . ... pn = p0 (ab)n zn = ap0 (ab)n

n = 1...N. n = 1...N.

where:

(18) (19) (20) (21) (22) (23)

∆

a = 10 10(1−α) . ∆ b = 10 10α . ∆ ab = 10 10α(1−α) .

(24)

In (24) ∆ > 0 denotes maximal permissible error of Charef approximation, dened as the dierence between Bode magnitude plot for model and plant, expressed in [dB]. The direct connection between n-th pole of approximation and order α and error ∆ can be at once calculated from (18) - (22) and it has the form: 1 ∆(2n+1−α) (25) pn = 10 20α(1−α) n = 0, ..., N. T

The order of approximation N can be estimated as follows (see [3]): ( ) max T ) N = Int log(ω +1= log(ab) ( ) . 10α(1−α)log(ωmax T ) Int +1 ∆

5

(26)

where ωmax denotes the frequency band, for which the approximation will be applied. 4

Main results

4.1

The proposed discrete models

The discrete transfer function models are obtained after use (11)-(13) to (17) and any elementary calculations are given below. In each case zn and pn are described by (18) - (24), h is the sample time. The model obtained with Euler transformation has the form: N∏ −1

G+ E (z)

n=0 N ∏

= KsE z

z − 1) ( zEn

(27)

.

( pEn − 1) z

n=0

where:

N∏ −1 n=0 N ∏

KsE = h {

n=0

( z1n )

(28)

.

( hp1n )

1 1+hzn . 1 1+hpn .

zEn = pEn =

(29)

Next, the model built with the use of Tustin transformation has the form (30): N∏ −1

G+ T (z)

= KsT (z + 1)

n=0 N ∏

( zTzn + 1) .

(30)

z

( pT n + 1)

n=0

where:

N∏ −1

KsT = h {

(1 +

hzn −2 zn )

(1 +

hpn −2 pn )

n=0 N ∏

.

n=0

zT n = pT n =

hzn −2 2+hzn . hpn −2 2+hpn .

(31)

(32)

and the model obtained with the Al-Aloui approximation is as follows: N∏ −1

(z − zAn ) 1 n=0 ) G+ (z) = K (z + . sT A N 7 ∏ (z − pAn ) n=0

6

(33)

where:

N∏ −1

KsA =

( 17 −

8 7hzn )

( 17 −

8 7hpn )

n=0 N ∏ n=0

 zAn =

pAn =

8 7hzn

.

− 17

8 1+ 7hz n 8 1 7hpn − 7 8 1+ 7hp n

. .

(34)

(35)

The poles od each discrete approximation pE,T,An are functions of time constant T , approximation error ∆, sample time h and non integer order α. This implies that the stability for all the proposed models is determined by these parameters. The stability analysis will be given in the next subsection. 4.2

Practical stability analysis

An idea of practical stability was proposed by Kaczorek in paper [12]. The fractional order system is called practically stable if its approximation is asymptotically stable. This property for all the proposed discrete models will be investigated using the analytical formulae for poles of each model, given in exact form by (29)-(35). The practical stability is described by the following proposition:

Proposition 1. (The practical stability of the discrete Charef approximation) Let us: •

consider fractional order inertial plant described by transfer function (14) with fractional order 0 < α < 1 and time constant T > 0,

•

construct the discrete transfer function models using N -th order Charef approximation (17) with maximal error ∆ and sample time h.

The discrete approximation of FO inertial transfer function (14), expressed by (27), (30), (33) is practically stable for each T > 0, 0 < α < 1,∆ > 0, N and h. Proof. Note that the each proposed discrete approximation will be asymptotically stable if and only if all modules of its poles are smaller than one: |p(E,T,A)n | < 1.

(36)

The n-th pole for time-continuous approximant is described by (25). Notice that pn > 0 for each α, N and ∆. This implies the asymptotic stability of time-continuous Charef approxmation. The asymptotic stability of its discrete version will be shown step by step for Al-Aloui model (33) only. This seems to be sucient because for all the other models the proof will be analogical. The condition (36) for Al-Aloui model takes the form: 8 1 7hpn − 7 8 < 1. 1 + 7hp n

7

(37)

what yields:

 8 1 7hpn − 7   < 1. 8  1+  7hpn ∧    7hp8 n − 17 > −1.  1+ 8

(38)

16 6 >− . 7hpn 7

(39)

7hpn

and nally:

Inequality (39) is met for each h > 0 and pn > 0. This nishes the proof. 5

An Example

As an example les us consider the elementary FO inertial transfer function given by: 1 G(s) = . (40) α (25s + 1)

Its time-continuous step response is given by (16) and it will be interpreted as the standard to accuracy estimation for all tested models. The accuracy of discrete models was tested using the typical MSE cost function: M SE =

K 1 ∑ (ystep (kh) − yCh,E,T,A (k))2 . Ks

(41)

k=1

where: Ks is a number of all collected samples, ystep (kh) is the analytical step response (16) in k-th time moment, yCh,E,T,A (k) are step responses of all the tested discrete models, evaluated at the same discrete time grid. All the experiments were done for N = 8 and ∆ = 0.5[dB], nal time of each simulation was equal 300[s]. Firstly the MSE as a function of sample time was tested. Results are presented in diagrams 1, 2 and 3. It is importatnt to notice that for α = 0.5 the MSE has a minimum for sample time equal circa h ≈ 0.15[s]. This phenomenon needs to be analytically explained. Next step responses of all the considered discrete approximations were calculated. Their time diagrams are given in Figure 4.

8

10-4

8

Euler Tustin Al Aloui

7 6

MSE

5 4 3 2 1 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sample time [s]

Figure 1: MSE cost function as a function of sample time for all discrete approximations, α = 0.25

5.5

10-4 Euler Tustin Al Aloui

5

4.5

MSE

4

3.5

3

2.5

2 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sample time [s]

Figure 2: MSE cost function as a function of sample time for all discrete approximations, α = 0.5

9

10-3

1.5

Euler Tustin Al Aloui

1.45

1.4

MSE

1.35

1.3

1.25

1.2

1.15 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sample time [s]

Figure 3: MSE cost function as a function of sample time for all discrete approximations, α = 0.75

1 Euler approximation Tustin approximation Al Aloui approximation Analytical response (reference)

0.9 0.8

Step responses

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

50

100

150

200

250

300

time [s]

Figure 4: Step responses of all discrete approximations, α = 0.5, T = 25[s],

h = 1, M = 8.

10

6

Final conclusions

Final conclusions from the above considerations can be formulated as follows: • All the considered discrete versions of Charef approximation are practically stable for all values of order α and sample time h, • The approximation employing Tustin method is most accurate for each

order and sample time,

• Very interesting phenomenon was observed for order α = 0.5, where the

accuracy as a function of sample time has a minimum. Its explaination requires us to make more exact analysis. This is planned to do during further investigations.

7

Acknowledgements

This paper was sponsored partially by AGH UST grant no 11.11.120.815. References

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