Implementation of bi-fractional filtering on the Arduino ...

Implementation of bi-fractional ltering on the Arduino Uno hardware platform Waldemar Bauer1 , Aleksandra Kawala-Janik2 1

AGH University of Science and Technology, Department of Automatics and Biomedical Engineering, Al. Mickiewicza 30, 30-059 Kraków Email: [email protected] 2

Faculty of Electrical Engineering, Automatic Control and Informatics, Opole University of Technology, ul. Proszkowska 76/1, 45-758 Opole, Poland Email:[email protected]

Abstract. In this paper application of method based on bi-fractional

ltering on the Arduino Uno was proposed. The authors showed that the implementation of a non-integer order lter on a micro-controller hardware platform is possible and gives promising results. Another aspect of potential implementation of such systems involves using biomedical data  in particular EEG signals, which were applied during the research carried out for the purpose of this paper.

1 Introduction Nowadays the implementation of non-integer order subsystems on various platform is a broadly researched topic.One of the questions of great importance is to design and develop an algorithm for realisation of non-integer order elements in the discrete implementation (see: [1, 25]). Only a very few years ago  the technology based on using biomedical data for the purpose of control was very rare and even voice recognition or touch panels were rarely used and nowadays application of brain signals (EEG) as a source data (although very novel)has become in the past two decades very popular due to the growing interest of researchers all over the world [17, 16, 19]. This is because (in accordance with multiple studies) not only people, but also animals, are able to communicate with computers using biomedical signals [16]. This particularly important for handicapped users, unable to conduct simple tasks such as using keyboard or mouse [17]. The application of EEG signals is a very dicult task, as it relies on real-time analysis and interpretation a data of (frequently) a very poor quality. Therefore a very important role is to choose appropriate signal processing method (see: [16]), and so the concept of using fractional calculus in technical applications became recently very popular, although regarding it theory was developed already in the 19th century. This is because implementation of such ltering (fractional) allows

great exibility in lter shaping, which is not possible while using traditional ltering methods [28] The theory of non-integer order systems can be found in e.g.:[27, 23, 11, 15, 7] and the Oustaloup method was in more detail described in: [24]. It is important to mention that the Oustaloup approximation can be used in simulations, which were described in: [9, 10, 13, 20, 29], in ltering presented in more detail in: [5, 18, 14] and with appropriate care in experiments  presented in: [12, 22]. Its sensitivity and stability problems during discretisation were deeply discussed in: [26, 3, 8]. Dierent method of approximation is based on Laguerre functions (see: [6, 30, 1]). The implementation of the algorithm requires the discretisation of the control system designed in a continuous time domain. The earlier results (see: [4]) prove that the transfer function cannot be directly implemented. The further part of this paper is organised as follows, where the bi-fractional lter is presented in a transmitation and dierential equation form. Then the time-domain Oustaloup approximation is shown. The results of experiments are then presented and the main dierences between the Matlab-Simulink and the Arduino Uno hardware platform lter realisation are discussed. Finally the conclusions and future works are drawn.

2 Bi-fractional lter In this paper the authors focused on the digital realisation of bi-fractional lters. The lter can be given by the following transfer function (see [4]):

G(s) =

c s2α + 2bsα + c

(1)

Where:

 α is base order  b is damping coecient  c is free coecient This formula (1) can be represented in the form of a system of dierential equation with zero initial conditions: C α 0 Dt x(t)

(2)

= Ax(t) + Bu y(t) = Cx(t)

with matrix:



0 1 A= −c −2b



  0 B= 1

  C= c0

Such system can be realized with non-integer order integrators as illustrated with the Fig. 1.

Fig. 1: Block diagram realization of bi-fractional lter in dierential equation form.

3 The Original Ostaloup Non-Integer Time-Domain Approximation The Oustaloup lter approximation with a fractional-order dierentiator G(s) = sα has a large use of potential, especially in applications [21]. An Oustaloup lter can be designed as [24]:

Gt (s) = K

N Y s + ωi0 i=1

s + ωi

(3)

where:

ωi0 = ωb ωu(2i−1−α)/N

(4)

ωb ωu(2i−1+α)/N ωhα

(5)

ωi = K=

s ωu =

ωh ωb

(6) (7)

Because the poles of this approximation spacing from close to −ωh to those very close to −ωb . This spacing is not-linear and more poles are grouping near −ωb . This is one of the main reasons of problems in discretisation process.

Time domain Oustaloup Approximation For zero-initial condition it is possible to describe any part of (3) as follows:

s + ωk0 ⇐⇒ s + ωk

( x˙ k = Ak xk + Bk uk yk = xk + uk

where

Ak = − ωk ,

Bk = ωk0 − ωk

This can be written in vector matrix notation  as below:

(8)

   KB1 A1 0 0 . . . 0  KB2   B2 A2 0 . . . 0         B 3 B 3 A3 . . . 0  x˙ =   x +  KB3  u  ..   .. .. .. . . ..   .   . . .  . . KBN B N B N . . . B N AN   y = 1 1 . . . 1 1 x + Ku 

(9)

or in brief

x˙ = Ax + Bu y = Cx + Du

(10)

This approach proved to be more robust to dierent discretisation schemes (for more details see: [2]). This form will be use in implementation at the Arduino Uno.

4 Digital realization bi-fractional lter at the Arduino Uno The Arduino Uno is a microcontroller board based on the ATmega328. The Atmega328 has 32 KB of ash memory for storing code (of which 0,5 KB is used for the bootloader). It has also 2 KB of SRAM and 1 KB of EEPROM. The board has 14 digital input/output pins (of which 6 can be used as PWM outputs), 6 analog inputs, a 16 MHz crystal oscillator, a USB connection, a power jack and an ICSP header. It contains everything needed to support the microcontroller and requires simply only USB cable in order to be connected to a computer.

4.1

Algorithm of calculation non-integer integrator

In order to perform the realisation of bi-fractional ltering on the Arduino Uno platform (based on Fig. 1) an algorithm to calculate response of non-integer integrator has to be created. At this stage the time-domain Oustaloup approximation is being used. Because the considered approximation is realised in the time-domain  dierential scheme can be applied for creating the needed algorithm. The proposed algorithm for the value approximation of response noninteger integrator will have the following form:

Data: N  order of approximation

A, B, C, D  discreet matrix described approximation xp  integrator state vector Bu , Ax  local variable ut  signal value in time t x  helper integrator state vector Result: yt  non-integer integrator response in time t ; yt = 0; for i = 0 to N − 1 do Bu = B[i] ∗ ut ; Ax = 0; for j = 0 to N − 1 do Ax = Ax + A[i][j] ∗ xp[j];

end

x[i] = Ax + Bu ; yt = yt + C[i] ∗ x[i];

end

xp = x;

return y+ = D ∗ ut

As it was shown above  the complexity of this algorithm can be clearly seen and it is constant and has a value of O(n2 ). One can easily observe the presented algorithm worked only at basic arithmetic operation. For this reasons, it can be easily adapted for any hardware platform e.g.: micro-controllers, PLC, FPGA.

5 Experiments and Results The tested EEG signal was recorder using inexpensive EEG headset  Emotiv EPOC and was ca. 700 seconds long. The tests were carried out during imagery left- and right-hand movements [16]. An experiment has been conducted in order to compare the performance of Matlab Simulink and Arduino Uno with the bi-fractional lter implementation for EEG signal. The described time domain Oustaloup approximations has been discretaised and implemented on both considered platforms. The scheme of ltering with the use of the Arduino Uno we can seen in Fig. 2.

Fig. 2: Arduino Uno/PC connection schema.

The bi-fractional lter settings and time-domain Oustaloup approximation parameters are:

    

N =7 ω = [10−6 , 106 ] α = {0.1, 0.7} b = 1.5 c = 2.24

Result of the perfomed ltering for α = 0.1 is shown in the Fig. 3, 4 and the results obtained for α = 0.7 illustrate the Fig. 5, 6. The average error between Matlab Simulink and Arduino Uno implementation for the ltered signal in time has value of 9.5. Considering the fact that the Arduino Uno represents the oating-point numbers as oat, the presented result is very good. The second case when α = 0.7 gives the average error value of 12.48. This shows that such implementation work was correct and was also conrmed with the analysis in the frequency-domain presented in Fig. 4 and 6.

Fig. 3: Comparison ltration result with Oustaloup time domain approximation and Matlab Simulink for

α = 0.1.

6 Conclusions and Further Research In this paper the implementation method of bi-fractional lter on the Arduino Uno platform was proposed. The authors showed that the implementation of

40 30 Matlab Arduino Uno

Magnitude [dB]

20 10 0 −10 −20 −30 −40 −50 −60 0

1

10

10 Frequency [Hz]

Fig. 4: FFT signal ltered by Oustaloup time domain approximation and Matlab Simulink for

α = 0.1.

Fig. 5: Comparison ltration result with Oustaloup time domain approximation and Matlab Simulink for

α = 0.7.

Fig. 6: FFT signal ltered by Oustaloup time domain approximation and Matlab Simulink for

α = 0.7.

non-integer order lter in micro-controller hardware platform is possible and gives good scientic results. Further work will include dierent types of non-integer order lter prototypes, methods of discretisation and implementation on real-time systems hardware platforms.

ACKNOWLEDGEMENT Work realised in the scope of project titled "Design and application of noninteger order subsystems in control systems". Project was nanced by National Science Centre on the base of decision no. DEC-2013/09/D/ST7/03960.

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