An accuracy estimation for a non integer order

An accuracy estimation for a non integer order, discrete, state space model of heat transfer process Krzysztof Oprzedkiewicz, Wojciech Mitkowski, and Edyta Gawin AGH University, A. Mickiewicza 30,30-59 Krakow, Poland State Higher Vocational School in Tarnow, A. Mickiewicza 8, 33-100 Tarnow, Poland {kop,wojciech.mitkowski}@agh.edu.pl {e_gawin}@pwsztar.edu.pl

Abstract. In the paper an accuracy analysis for non integer order, discrete, state space model of heat transfer process in one dimensional plant is presented. The proposed model is a discrete version of time - continuous, non integer order, state space model proposed previously by Authors. The discretization of integro/differential operator was done with the use of backward difference method. The accuracy and convergence of the discussed model was considered as a function of model order and memory length necessary to proper estimation of non integer order order operator. Tests were done with the use of PLC and SCADA based experimental system. Results of experiments show that the proposed, discrete model assures the good performance in the sense of MSE cost function, but its size is relatively high. Keywords: non integer order systems, discrete time systems, heat transfer process, PSE approximation

1

An Introduction

Main areas of application the fractional order calculus in automation are: fractional order control and modeling of processes with dynamics hard to describe with the use of another approaches. Fractional order control covers mainly particularly Fractional Order PID controllers (FO PID). FO PID controllers have been presented by many Authors and their usefulness has been proven ( see for example: [4], [7], [28], [30], [26]). A PLC implementation of FO controller was presented for example in [29]. However, the practical implementation of FO controllers and models causes a number of problems, generated mainly by the fact, that the fractional order differentiation/integration operator is impossible to exact implementation ant it requires to use approximations, possible to digital implementation. It can be done with the use of PSE (Power Series Expansion), CFE (Continuous Fraction

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Krzysztof Oprzedkiewicz, Wojciech Mitkowski, Edyta Gawin

Expansion) approximation or discrete version of ORA (Ostaloup Recursive Approximation) approximation . This paper is intented to present an accuracy analysis for the discrete time, state space, non integer order model of heat transfer process. The considered discrete-time model is directly derived from continuous-time model proposed by authors in papers: [24], [23]. The non integer order operator with respect to time was described with the use of PSE approximation. The paper is organized as follows: at the beginning any elementary ideas from non integer order calculus are remembered, next the proposed discrete fractional order transfer functions with delay are given. Next the experimental PLC based system is shown and experimental results and main conclusions are discussed.

2 2.1

Preliminaries Elementary ideas

The presentation of elementary ideas will be started with define a non integer order, integro-differential operator. It is expressed as follows (see for example [13]): Definition 1. The non integer order integro - differential operator  α d f (t)  α>0 α    dt 1 α = 0 α . 0 Dt f (t) = ∫t   −α  f (τ )(dτ ) α < 0 

(1)

a

where a and t denote time limits to operator calculating, α ∈ R denotes the non integer order of the operation. Next an idea of Gamma Euler function (see for example [14]) can be given: Definition 2. The Gamma function ∫∞ Γ (x) =

tx−1 e−t dt.

(2)

0

The fractional-order, integro-differential operator (1) can be described by different definitions, given by Gr¨ unvald and Letnikov (GL definition), Riemann and Liouville (RL definition) and Caputo (C definition). The digital modeling of FO operator can be most naturally done with the use of GL definition and it will be presented here:

An accuracy estimation for a non integer order, discrete, state space model...

3

Definition 3. The Gr¨ unvald-Letnikov definition of the FO operator ([4],[25]) t

GL α 0 Dt f (t)

−α

= lim h h→0

[h] ∑

cj f (t − lh).

(3)

l=0

where: α

In (4)

(α) l

cj = (−1)( l )

(4)

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

(5)

Definition 4. The Riemann - Liouville definition of the FO operator RL α 0 Dt f (t)

1 dN = Γ (N − α) dtN

∫∞ (t − τ )N −α−1 f (τ )dτ.

(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 definition is described as underneath: Definition 5. The Caputo definition of the FO operator C α 0 Dt f (t)

1 = Γ (N − α)

∫∞ 0

f (N ) (τ ) dτ. (t − τ )α+1−N

(7)

If the RL or C definition is considered, the Laplace transform can be also given (see for example [13]) as a generalization of Laplace transform for integer order case: Definition 6. 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), .

k=0

α > 0, n − 1 < α ≤ n ∈ N

(8)

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Krzysztof Oprzedkiewicz, Wojciech Mitkowski, Edyta Gawin

Definition 7. 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), .

(9)

k=0

α > 0, n − 1 < α ≤ n ∈ N

Consequently an inverse Laplace transform can be given as underneath (see for example [14] 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 The GL definition is limit case of fractional order backward difference: Definition 8. The fractional order backward difference ( ) L 1 ∑ l α (∆ x)(t) = α (−1) x(t − lh) l h α

(11)

l=0

In (11) L denotes a memory length necessary to correct approximation of non integer order operator. The discrete, fractional order state equation using definition (11) is written as follows (see for example [19]): { (∆α L x)(t + h) = Ax(t) + Bu(t) (12) y(t) = Cx(t) where x(t) ∈ RN is a state vector, u(t) ∈ Rr is a control, y(t) ∈ RM is an output. A, B and C are state, control and output matrices respectively. If we shortly denote k-th time moment: h ∗ k by k, then equation (12) turns to: { (∆α L x)(k + 1) = Ax(k) + Bu(k) (13) y(k) = Cx(k) The solution of state equation (13) takes the form: x(k + 1) = (hα A − c1 I)x(0) +

L ∑ l=1

cl+1 x(k − l) + hα Bu(k)

(14)

An accuracy estimation for a non integer order, discrete, state space model...

3 3.1

5

A considered plant and its non integer order, state space model The experimental system

Let us consider an experimental heat plant shown in figure 1. It has the form of a thin copper rod 260[mm] long. For further considerations it will be assumed, that the length of rod is equal 1.0. This implies that during further considerations localization and length of heater and RTD sensors will be expressed with respect to 1.0. The rod is heated with the use of an electric heater of the length ∆x0 = 0.14 localized at one end of rod. An output temperature is measured with the use of Pt100 sensors long ∆x located in points: 0.29, 0.50 and 0.73 of rod length. The input signal of the system is the standard current signal from range 0 − 20[mA]. It is amplified to the range 0 − 1.5[A] and next it is the input signal for the heater. Signals from the Pt100 sensors are read directly by analog input module in the PLC. Data from PLC are read with the use of SCADA. The whole system is connected via PROFINET. The temeprature distribution with respect to time and length is shown in the figure 2.

Fig. 1. The experimental system

The fundamental mathematical model describing the heat conduction in the plant is the partial differential equation of the parabolic type with the homogeneous Neumann boundary conditions at the ends, the homogeneous initial condition, the heat exchange along the length of rod and distributed control and observation. This equation with integer orders of both differentiations was presented by many papers, for example in [21].

3.2

The non integer order, time continuous model of the heat plant

The presented in this section the non integer order model with respect to both time and space coordinates has been discussed in:[23], the non integer order

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Krzysztof Oprzedkiewicz, Wojciech Mitkowski, Edyta Gawin

140

Temperature [oC]

120

100

80

60

40

20 300 200 200 150 100 time [s]

100 0

50 length [mm]

Fig. 2. The spatial-time temperature distribution in the plant

model with respect to time was given in [24]. The proposed form of it is motivated by the fact that the non integer order differentiation is expected to better describe the processes running during the heat exchange and diffussion in the plant. Assume that non integer order difference with respect to time is described by Caputo definition and non integer order difference with respect to length is described by the Riesz definition. Then the non integer order heat transfer equation has the following form:  ∂ β Q(x, t)  C α  − Ra Q(x, t) + b(x)u(t) Dt Q(x, t) = a   ∂xβ    ∂Q(0, t)   = 0, t ≥ 0  dx ∂Q(1, t)  = 0, t ≥ 0   dx    Q(x, 0) = 0, 0 ≤ x ≤ 1   ∫1  y(t) = y0 0 Q(x, t)c(x)dx

(15)

Where α, β > 0 denote non integer orders of the system, a, Ra denote coefficients of heat conduction and heat exchange. Now we need to express (15) as an inifinte dimensional state equation in the Hilbert space, analogically, as it was presented in [21], but with the assuumption, that the both differences are non integer order. The proposed state equation is written as follows:  C α   Dt Q(t) = AQ(t) + Bu(t) Q(0) = 0   y(t) = y0 CQ(t)

(16)

An accuracy estimation for a non integer order, discrete, state space model...

where:

 ∂ β Q(x)   AQ = a − Ra Q,    {∂xβ 2 }   D(A) = Q ∈ H (0, 1) : Q′ (0) = 0, Q′ (1) = 0 , a, Ra > 0,   { }    H 2 (0, 1) = u ∈ L2 (0, 1) : u′ , u′′ ∈ L2 (0, 1) ,    CQ(t) = ⟨c, Q(t)⟩, Bu(t) = bu(t)

7

(17)

The following set of the eigenvectors for the state operator A creates the orthonormal basis of the state space: { 0, i = 0 hi = √ (18) 2cos(iπx), i = 1, 2, ... Eigenvalues of the state operator are expressed as underneath: λβi = −aπ β iβ − Ra , i = 0, 1, 2, ....

(19)

and consequently the state operator has the form: A = diag{λβ1 , λβ2 , λβ3 , ...}

(20)

Next, the spectrum σ of the state operator A is expressed as underneath: σ(A) = {λβ1 , λβ2 , λβ3 , ...}

(21)

The input operator B has the following form: B = [b0 , b1 , b2 , ...]T where bi = ⟨b, hi ⟩, b(x) denotes the control function: { 1, x ∈ [0, x0 ] b(x) = 0, x ̸∈ [0, x0 ] The output operator C is written as underneath:   Cs1 C =  Cs2  Cs3

(22)

(23)

(24)

Rows of output operator C are expressed as follows: Csj = [csj,0 , csj,1 , csj,2 , ...] j = 1, 2, 3 where csj,i = ⟨c, hi ⟩, c(x) denotes the output sensor function: { 1, x ∈ [x1 , x2 ] c(x) = 0, x ̸∈ [x1 , x2 ]

(25)

(26)

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Krzysztof Oprzedkiewicz, Wojciech Mitkowski, Edyta Gawin

Coordinates x1 and x2 depend on sensor location on the rod and they are equal:   x = 0.29 : x1 = 0.26, x2 = 0.32  x = 0.50 : x1 = 0.47, x2 = 0.53   x = 0.73 : x1 = 0.70, x2 = 0.76 From (23) and (26) it turns out, that the control function b(x) and output function c(x) are the interval constant functions. The solution of state equation (16) can be calculated with the use of Laplace transform for Caputo operator with assumptions that initial condition is equal zero: Q(x, 0) = 0, 0 ≤ x ≤ 1 and state and control operators are described by (20) - (23). If we assume, that the control signal has the form of the Heaviside function u(t) = 1(t) then we obtain the solution as underneath:

yj (t) = y0j

∞ ∑ (Eα (λβi tα ) − 1(t)) ⟨b, hi ⟩⟨c, hi ⟩, λβi i=1

(27)

j = 1, 2, 3 and consequently the whole output of the system is expressed as follows: T

y(t) = [y1 (t), y2 (t), y3 (t)]

(28)

Notice that the proposed non integer order model described by (15) - (27) for orders: α = 1 and β = 2 turns to integer order model presented for example in [21]. The setting: α = 1 or β = 2 allows us to obtain suitable ”partial non integer order models” (PNIO) with non integer order differences with respect to length or time respectively. The tests of accuracy and convergence for these models si given in [22]. Next, the non integer order model described by (16) - (27) is an infinite dimensional model. Its practical application requires us to apply its finite dimensional approximation. This can be obtained by ”cutting” further nodes in state equation (16) and consequently calculating solution (27) and (28)as a finite sum expressed by (29). Consequently operators: A, B and C are interpreted as matrices. yj (t) = y0j

N ∑ (Eα (λβ tα ) − 1(t)) i=1

λβ

⟨b, hi ⟩⟨c, hi ⟩,

(29)

j = 1, 2, 3 In (29) N denotes the order of finite approximation. Its correct estimation is a crucial problem during use of presented model. If the time-continuous model is considered, the sensible order is equal N = 22 [23]. The size of model for discrete case can be estimated analogically and it will be shown in the next section.

An accuracy estimation for a non integer order, discrete, state space model...

3.3

9

The fractional order, discrete time state space model of the system

If we employ (16) to (13), the discrete time, state space, fractional order equation describing the heat distribution in the rod turn to the following form:  L Q+ (k + 1) = A+ Q+ (k) − ∑ c Q+ (k − l) + B + u(k) l+1 (30) l=1  + y (k) = C + Q+ (k) where:

 + α α  A = diag{h λβ1 − c1 , h λβ2 − c1 , ....} B + = hα B   + C =C

(31)

The above equations describe an infinite - dimensional model, directly derived from (16). Its finite dimensional approximant of size N + is described analogically, but operators (31) are interpreted as matrices. The solution of discrete state equation can be calculated directly with the use of formula given by (30). Digital implementation of model (30) requires us to set the memory length L assuring the resonable accuracy. Its sensible estimation is a crucial problem during digital implementation of FO models. Typically this parameter causes problems during implementation, because it needs to be relatively big. Unfortunately, the order of the considered model is high also (N = 22 for time continuous case). This implies, that estimation of sensible orders N + and L before digital implementation the presented model is a crucial problem. It can be solved numerically with the use of approach presented for example in paper [22]. This will be presented in the next section.

4

The experimental results

Experiments were done with the use of system presented above, the discrete time model was examined with the use of typical MSE cost function: (see for example [11]): 3 Ks 1 ∑∑ M SE = e+ (32) j (k) 3Ks j=1 k=1

In (32) e+ (k) is the difference between experimental step response ye+ (k) measured in time moments k = 1, ..., Ks and step response of model y + , calculated with the use of (30) along the same time grid. The goal of experiments was to estimate values of finite model size N and memory legth L assuring the acceptable accuracy of the presented, discrete time, non integer order model. To do it the values of cost function (32) were calculated for different values of N and L and next 3D plotted. The results are shown in table 1 and figure

10

Krzysztof Oprzedkiewicz, Wojciech Mitkowski, Edyta Gawin Table 1. Cost function MSE (32) for different orders N and all tested models L N 8 10 12 14 16 18 20 22

20

40

60

80

100

120

140

160

0.1134 4.0430 1.3441 0.1137 2.2272 0.1680 0.2011 0.2711

0.1027 3.8739 1.2896 0.1026 1.7243 0.1365 0.2059 0.2157

0.0901 3.7606 1.2591 0.0868 2.0186 0.1048 0.1716 0.1443

0.0794 3.6906 1.2401 0.0733 1.9760 0.0823 0.1446 0.0994

0.0714 3.6906 1.2282. 0.0632 1.5203 0.0669 0.1252 0.0742

0.0655 3.6193 1.2207 0.0560 1.5110 0.0565 0.1115 0.0582

0.0612 3.6011 1.2160 0.0508 1.4624 0.0493 0.1017 0.0484

0.0582 3.5890 1.2130 0.0472 1.4975 0.0444 0.0948 0.0426

The dependence cost function MSE for different orders N and all tested models

5 4

MSE

3 2 1 0 5 10 200

15

150 100

20 50 25 N

0

L

Fig. 3. The dependence cost function MSE for different orders N and all tested models

From figure 3 it can be concluded, that the order N and memory length L assuring the good performance are equal respectively: N = 18 and L = 100, what gives the summarized size of memory required to implementation the discussed algorithm equal 1800. This can be problematic for older, smaller devices with bounded resources, but recent digital control platforms are more powerful.

5

Final conclusions

Final conclusions from the paper can be formulated as underneath: – The proposed, discrete, state space, non integer order model is able to properly describe the experimental heat plant and its performance in the sense of MSE cost function is satisfying.

An accuracy estimation for a non integer order, discrete, state space model...

11

α=0.9319, β=2.0306, N=18, L=100, a=0.000561, Ra=0.0548 130 response object response model

120 110

temperature [°C]

100 90 80 70 60 50 40 30

0

50

100

150 time [s]

200

250

300

Fig. 4. Step response object and model for 1-3 outputs with non-integer order (α = 0.9319, β = 2.0306) and optimal order N = 18, L = 100, a = 0.000561, Ra = 0.0548

– The sumamrized size of memory required to implementation is relatively huge, what can be a problem during implementation the proposed model at smaller device with limited memory capacity. Fortunately, possibilities of recent devices are significantly better. – The presented discrete model is planned to implement at PLC platform to use it in Model Based Control or Model Based Fault Detection System. Acknowledgments. The paper was sponsored by AGH University grant no 11.11.120.817 .

References 1. 1. Al Aloui M. A. Dicretization Methods of Fractional Parallel PID Controllers Proc of 6th IEEE International Conference on Electronics, Circuits and Systems, ICECS 2009, pp 327-330. 2. Berger H. (2012) Automating with STEP 7 in STL and SCL: SIMATIC S7-300/400 Programmable Controllers, SIEMENS 2012. 3. Berger H. (2013) Automating with SIMATIC: Controllers, Software, Programming, Data, SIEMENS 2013. 4. Caponetto R., Dongola G., Fortuna l., Petras I. (2010), Fractional Order Systems. Modeling and Control Applications, World Scientific Series on Nonlinear Science, Series A, vol. 72, World Scientific Publishing. 5. Chen Y.Q., Moore K.L. (2002) Discretization Schemes for Fractional-Order Differentiators and Integrators, IEEE Transactions on Circuits and Systems - I: Fundamental Theory and Applications, vol. 49, No 3, March 2002

12

Krzysztof Oprzedkiewicz, Wojciech Mitkowski, Edyta Gawin

6. Das S. (2008) Functional fractional calculus for system identification and controls Springer, Berlin 2008, 7. Das S., Pan I. (2013) Intelligent Fractional Order Systems and Control. An Introduction. Springer 2013, 8. Douambi, Charef and Besancon (2007) Optimal approximation, simulation and analog realization of the fundamental fractional order transfer function Int. J. Appl. Math. Comp. Sci. 2007, vol.17 No.4. pp455-462. 9. Dzieliski, A.; Sierociuk, D.; Sarwas, G. (2010) Some applications of fractional order calculus. Bulletin of the Polish Academy of Sciences: Technical Sciences, 2010, 58.4: 583-592. 10. Frey G., Litz L. (2000) Formal methods in PLC programming Proc. of the IEEE Conf. on Systems Man and Cybrenetics SMC 2000, Nashville, Oct 8-11, 2000 pp. 2431-2436 11. Isermann R., Muenchhof M. (2011) Identification of Dynamic Systems. An Introduction with Applications Springer 2011. 12. John K. H., Tiegelkamp M. (2010) IEC 61131-3: Programming Industrial Automation Systems. Concepts and Programming Languages, Requirements for Programming Systems, Decision Making Aids. Springer 2010. 13. Kaczorek T.(2011), Selected Problems in Fractional Systems Theory, SpringerVerlag. 14. Kaczorek T., ROgowski K. (2014), Fractional Linear Systems and Electrical Circuits, Bialystok University of Technology, Bialystok. 15. Lewis R.W. (1998) Programming industrial control systems using IEC 1131-3 IEE 1998 16. Luo Y., Chen Y.Q., Wang Ch.Y., Pi. Y.G. (2010) Tuning fractional order proportional integral controllers for fractional order systems. Journal of Process Control 20 (2010) pp. 823831 17. Mitkowski W., Oprzedkiewicz K. (2007) Optimal sample time estimation for the finite-dimensional discrete dynamic compensator implemented at the soft PLC platform in: 23rd IFIP TC 7 conference on System modelling and optimization : Cracow, Poland, July 2327, 2007 : book of abstracts, eds. Adam Korytowski, Wojciech Mitkowski, Maciej Szymkat ; AGH University of Science and Technology. Faculty of Electrical Engineering, Automatics, Computer Science and Electronics. Krakow ISBN 978-83-88309-0. pp. 7778. 18. Mitkowski W., Oprzedkiewicz K. (2013) Fractional-order P 2Dβ controller for uncertain parameter DC motor In: Advances in the theory and applications of noninteger order systems : 5th conference on Non-integer order calculus and its applications : [45 July 2013], Cracow, Poland , eds. Wojciech Mitkowski, Janusz Kacprzyk, Jerzy Baranowski. Springer , Lecture Notes in Electrical Engineering ; ISSN 1876-1100 ; vol. 257). ISBN: 978-3-319-00932-2 ; e-ISBN: 978-3-319-00933-9. pp. 249259. 19. Mozyrska D., Pawluszewicz E. (2011) Fractional discrete-time linear control systems with initialisation, International Journal of Control 2011, pp 17. 20. Oprzedkiewicz K., Chochol M., Bauer W., Meresinski T. (2015) Modeling of elementary fractional order plants at PLC SIEMENS platform in: Advances in modelling and control of non-integer order systems : 6th conference on Non-integer order calculus and its applications, 2014 Opole, Poland, eds. Krzysztof J. Latawiec, Marian ukaniszyn, Rafa Stanisawski. Switzerland : Springer International Publishing, cop. 2015, Lecture Notes in Electrical Engineering ; ISSN 1876-1100 ; vol. 320, pp 265-273.

An accuracy estimation for a non integer order, discrete, state space model...

13

21. Oprzedkiewicz K.,Mitkowski W., Gawin E. (2015) Application of fractional order transfer functions to modeling of high order systems MMAR 2015 : 20th international conference on Methods and Models in Automation and Robotics : 2427 August 2015, Midzyzdroje, Poland : program, abstracts, proceedings (CD). Szczecin : ZAPOL Sobczyk Sp.j., [2015] + CD. ISBN: 978-1-4799-8701-6, 978-1-4799-8700-9. ISBN: 978-83-7518-756-4. S. 127. full text CD:pp. 11691174. 22. Oprzedkiewicz K.,Mitkowski W., Gawin E. (2016)An estimation of accuracy of Oustaloup approximation. Challenges in automation, robotics and measurement techniques : proceedings of AUTOMATION-2016, March 24, 2016, Warsaw, Poland, eds. Roman Szewczyk, Cezary Zieliski, Magorzata Kaliczyska. Switzerland : Springer International Publishing, cop. 2016. (Advances in Intelligent Systems and Computing ; ISSN 2194-5357 ; vol. 440). ISBN: 978-3-319-29356-1 ; e-ISBN: 978-3-319-29357-8. pp. 299307. 23. Oprzedkiewicz K.,Mitkowski W., Gawin E. (2016), Parameter identification for non integer order, state space models of heat plant In: MMAR 2016 : 21th international conference on Methods and Models in Automation and Robotics : 29 August01 September 2016, Midzyzdroje, Poland ISBN: 978-1-5090-1866-6. ISBN: 978-837518-791-5. pp. 184188. 24. : Oprzedkiewicz K., Gawin E. (2016) Non integer order, state space model for one dimensional heat transfer process, Archives of Control Sciences ; ISSN 1230-2384,2016 vol. 26 no. 2, pp. 261275. full text avalilable at: https://www1degruyter-1com-1atoz.wbg2.bg.agh.edu.pl/downloadpdf/j/acsc.2016.26.issue2/acsc-2016-0015/acsc-2016-0015.xml. 25. Ostalczyk P. (2012) Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains, International Journal of Applied Mathematics and Computer Science 22(3): 533538. 26. Ostalczyk P. (2016) Discrete Fractional Calculus. Applications in control and image pro-cessing, Series in Computer Vision, vol. 4, World Scientific Publishing 2016. 27. Padula F., Visioli A. (2011) Tuning rules for optimal PID and fractional-order PID controllers. Journal of Process Control 21 (2011) 6981 28. Petras I. (2009) Fractional order feedback control of a DC motor. Journal of Electrical Engineering, vol 60, no 3 2009, pp 117-128. 29. Petras I. (2009)Realization of fractional order controller based on PLC and its utilization to temperature control, Transfer inovci 14/2009 30. Petras, I. (2012) Tuning and implementation methods for fractional-order controllers. Fractional Calculus and Applied Analysis, vol. 15, no. 2, 2012. 31. Petras I: http://people.tuke.sk/igor.podlubny/USU/matlab/petras/dfod1.m 32. Petras I: http://people.tuke.sk/igor.podlubny/USU/matlab/petras/dfod2.m 33. Podlubny I.(1999), Fractional Differential Equations, Academic Press, San Diego. 34. Sierociuk D., Macias M. (2016) New recursive approximation of fractional order derivative and its application to control Proc of 17th International Carpathian Control Conference (ICCC) pp. 673-678, 35. Stanislawski R., Latawiec K.J.,Lukaniszyn M. (2015) A Comparative Analysis of Laguerre-Based Approximators to the Grnwald-Letnikov Fractional-Order Difference, Mathematical Problems in Engineering Vol 2015, Article ID 512104, 10 pages http://dx.doi.org/10.1155/2015/512104 36. Valerio D., da Costa J.S. (2006) Tuning of fractional PID controllers with ZieglerNichols-type rules. Signal Processing 86 (2006) pp. 27712784. 37. Vinagre B.M. Podlubny I., Hernandez A., Feliu V.(2000) Some Approximations of fractional order operators used in Control Theory and applications in: Fractional Calculus and Applied Analysis, vol. 3, no.3, 2000, pp 231-248,

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Krzysztof Oprzedkiewicz, Wojciech Mitkowski, Edyta Gawin

38. Vinagre B.M., Chen Y.Q., Petras I. (2003) Two direct Tustin discretization methods for fractional-order differentiator/integrator, Journal of the Franklin Institute 340 (2003) 349362.