This study deals with the influence of the second order tunable digital frequency-dependent components' transfer function coefficients on the phase response during those components tuning. The obtained approximations of phase frequency responce dependence on the ripple level allowed us to find the digital filter transfer function of the denominator coefficients values. It is worth noting that this expression works at ripple level up to 40 with a standard error of 0,06% for Chebyshev filter-1 and 0.28% for Chebyshev filter-2. Also in these work we fix the ripple level. With this value, for each cutoff level and ripple level we determine the phase value and to obtain the three-dimensional graph approximation. Theoretical studies have been verified experimentally
At modern epoque the development of information management systems and robotic
systems takes place in accordance with the Industry 4.0 concept. In the framework of
this direction, there is a need to improve existing and develop new approaches to the
components development of such systems with the criteria of mobility, flexibility, the
ability to quickly restructure and adaptability to their operating conditions [
Exiting of the Internet of things (IoT) technology implies the information exchange not only between people and "things", but also between "things", i.e. computerized machines, devices, components, and sensors.
On the one hand, "things" equipped with sensors can exchange data and process information without human engaging to that process, and on the other hand, a human can actively participate in the process, for example, when it comes to "smart devices" or "smart enterprise " concepts.
A smart enterprise is a subcategory of IoT, embodying as its kind, the industrial Internet of things (IIoT). Therefore, today the technological equipment and facilities components are equipped with multifunctional sensors, actuators and controllers.
Using of Internet technologies and wireless networks to them facilitates the data rapid collection, their primary processing and presentation to the operator in a convenient form for prompt and timely decision-making. However, increasing the automation level in all areas of the enterprise allows you to organize production without the people participation. In this regard, robotic industries are in the first place, as are robotic platforms.
The personnel role in this case is reduced to the control of the technological
process and response to emergency situations [
Then, there arises a problem of building the robotic computer systems capable for a comprehensive rearrangement of their characteristics by soft- and hardware tools, depending on the operating conditions to increase the system efficiency is relevant. It should be noted that the signal processing path of such systems uses frequencydependent components (FDC), representing a combination of hard- and software arranged in a complex architecture and including a variety of linkages [11-14].
Well-known is that the FDC responses do uniquely depend on the transfer function coefficients [7].
Then, while calculating the coefficients, we can take into account the peculiar characteristics of both the amplitude-frequency response (AFR) and phase-frequency (PFR) response.
Let's consider the influence of the second order tunable digital frequencydependent components' phase-frequency response values when a smooth amplitudefrequency response while calculating the transfer function coefficients [15]. 2
The influence of the frequency-dependent components transfer function coefficients on the its characteristics Here we shall consider the widely used typical digital filters as a component of signal processing and filtering path. Well known is that for ease of tuning the high-order components are built on the basis of first and second orders frequency-dependent components (FDC) [7].
Therefore, the analysis of the digital filter transfer function coefficients' influence on the digital filter responses properties is carried out using the second-order transfer function, which has the form where a0 , a1 , a2 , b1 , b2 are the real coefficients of the numerator and denominator. When substituting z −1 = e − j
or using Euler's formula z −1 = cos() − j sin () , where is the normalized angular frequency, = the linear frequency and the sampling frequency, respectively, we get a complex transfer coefficient, and based on this coefficient, the amplitude-frequency response (AFR) and phase-frequency response (PFR): at a0 = −a2 [13] are 2 f
fd , 0, , f , f d 2 2 + a1 cos ( ) + 2 cos (2 )
+ 4 a1 H ( ) = a2 a0 a0 0 1+ b12 + b22 + 2b1 (1+ b2 ) cos ( ) + 2b2 cos (2 ) It should be noted that similar equations are used to describe the digital filters, and typical second order links at automatic control systems. Therefore, we consider the influence of the transfer function coefficients onto the frequency response of second order low-pass (LP) digital filters, as the most used frequency-dependent component used at computer systems and control systems.
For polynomial filters, the relation a1 = 2a0 , which makes it possible to get the amplitude frequency response square from the formula (2) upon certain transformations carried out
H 2 ( ) =
4a02 (1+ cos ( ))2 (1− b2 )2 + (b + 2 cos (2 )) (b1 + 2b2 cos ( )) 1 From equation (3) for the Butterworth filter and with respect to PFR type (Fig.1), it can be specified that the phase value at cutoff frequency ( c ) = − . Then the PFR 2 denominator will appear like We define b1 as (b2 +1) cos ( ) + b1 = 0 b1 = − (b2 +1) cos ( ) (2) (3) (4) (5) For the Chebyshev filter-1, this solution is not possible, because here the PFR has different phase values at frequency the cutoff level c set by the ripple value in the bandwidth . It should be noted that in the most mathematical packages, the ripple level is set in RP decibels further recalculated to c as follows c =
1 100.1RP For the Butterworth digital filter, the RP value is constantly -3 dB ( c = 0.707 ), and for other filters it has different values, but the value c is in the range , refer to Fig. 2 and 3. Studies of PFR have shown that one of its main features is not a dependence on the cutoff frequency, but the dependence on the ripple level. This phenomenon is manifested differently in different filters.
After all transformations based on equations (2) and (3), the coefficients b1 and b2 for the Chebyshev filter -1 will have the form b1 = − (1+ b2 ) cos ( c ) + 1− b2 sin ( c )
b2 = 2 1 + 2 + (1 + cos ( c )) − sin ( c ) 2 1 + 2 + (1 + cos ( c )) + sin ( c ) where = tg ( ) from formula (3).
As evidently seen, to calculate the coefficients b1 and b2 it is necessary to know the cutoff frequency PFR values at different ripple level values in the bandwidth. It follows from Figures 2 and 3 that the phase values at the cutoff frequency depend only on the ripple level in the bandwidth, Fig.4 [12]. To be noted is that if the ripple level is constant and the cutoff frequency varies, this value does never change. The approximation of this dependence has the form It is worth noting that this expression works at RP up to 40 with a standard error of 0,06%. The resulting transformation is shown in Figure 5. If we consider the Chebyshev filter-1 phase value as the limit value equal to -π, then the expansion into series allows using it only up to the bandwidth ripple level value equal to 35.
Applying this approach to finding the non-polynomial Chebyshev filter-2 coefficients has its own peculiarities. For the Chebyshev filter-2, this solution is not possible, because here the PFR has different phase values at the cutoff frequency level cs set by the ripple value in the bandwidth . The ripple level is set in RP decibels further recalculated to cs the same as c in the most mathematical packages.
For the Chebyshev filter-2, the phase also has a different value depending on the cs value, Fig. 6 and 7. After all transformations based on equations (2) and (3), the coefficients b1 and b2 for the Chebyshev filter -2 will have the form 2cs 1 + 2 − (1+ cs ) − (1− cs ) cos ( c ) + (1− cs )sin ( c ) 2cs 1 + 2 − (1+ cs ) − (1− cs ) cos ( c ) − (1− cs )sin ( c ) where = tg ( ) from formula (3).
As evidently seen, to calculate the coefficients b1 and b2 it is necessary to know the cutoff frequency PFR values at different ripple level values in the bandstop. If the ripple level is constant and the cutoff frequency varies, this value does never change. From figures 6 and 7 we can conclude that the phase values at the cutoff frequency depend only on the level of ripples in the bandstop, Fig. 8 [15]. The first approximations gave us the dependence of the kind shown in Fig.9. The mean square error in this case is 0.28%.
Transitions to the non-polynomial Cauer filter (elliptic) after all transformations based on equations (2) and (3), the coefficients b1 and b2 will have the form 2c 1 + 2 + (c + cs ) + (c − cs ) cos ( c ) − (c − cs ) sin ( c ) 2c 1 + 2 + (c + cs ) + (c − cs ) cos ( c ) + (1− cs ) sin ( c ) where = tg ( ) from formula (3). As evidently seen, to calculate the coefficients b1 and b2 it is necessary to know the cutoff frequency PFR values at different ripple level values c and cs . If the ripple level is constant c and cs , this value does never change, Fig. 10. To obtain the three-dimensional graph approximation, we go the following way. We fix the ripple level cs or RS.
With this value, for each c or RP we determine the phase value. Thus, having drawn sections for a three-dimensional graph, Fig. 10, one can find the approximating function description = f ( RP) , for a given RS. Such a function will be a quadratic equation and have the form
= − (aRP2 + bRP + c) The coefficient values a, b and c have found for different RS values, table 1. Based on these data, the dependences of graphs a = f ( RS ) , b = f ( RS ) , c = f ( RS ) are plotted, respectively Fig. 11-13. Fig. 12. Tabulated values (1) and approximating function b = f ( RS ) (2) graphs The square error in each case was 0.016%, 0.038%, and 3.14%, respectively, and the approximating functions have the form
Then, the phase value will be described by an approximating function To verify experimentally the exposed theoretical provisions, we used an ultrasonic rangefinder, installed on the robotic system for assessing the distance to the obstacle. The rangefinder improved accuracy has been achieved by tuning the radiation frequency and, accordingly, rearrangement of channel for reflected signal processing. In the reflected signal processing path, we used a digital refrangible filter. Refer to Fig. 14 and 15 for details of filter operation. The obtained approximations of the PFR dependence on the ripple level and equations for finding the denominator coefficients of the digital filters transfer contributes to a more rapid using them in robotic systems. However, the resulting implementations are not quite simple. Therefore, expedient shall be continuing the search for a more convenient implementation on the microprocessor technology.
It should be borne in mind that the values of RP and RS do not change in large ranges. Usually they vary in narrow ranges and rarely. In this case, you can pre-calculate and use some pre-prepared values stored in memory. In the case of a sharp change in working conditions, these formulas can be useful for operational calculation, in order to maintain the best robotic system functioning. 2. Industry 4.0: an overview. Conference Paper (2018).
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