FILTERS RESEARCH FOR FREE MOTION EXTRACTION IN SELF-TUNING AUTOMATIC CONTROL SYSTEMS

Technological type control objects specific feature, which distinguish them among many mobile or electro technical types, is more low-frequency parametric disturbances spectral composition than spectral composition of coordinate disturbances. Most often parametric disturbances reveal themselves in changing control object transmission coefficient in the channel “controller control action – control variable”. On a number of occasions transmission coefficient can change in a wide range – in 2…10 times more than initial value. Coordinate disturbances change control variables but don’t change control object features. Besides spectral density of control variable change, under the impact of coordinate disturbances, has its peak in low-frequency range and characterize forced component of automatic control system motion. Parametric disturbances have influence on control variables and on control object properties, in particular on its transition coefficient. In this case, character of transient processes change, i.e. free motion component of the system. Spectral density of control variable changes, under the influence of transition coefficient change, has its peak, in general, in mid-frequency range. In the case, when using filters, even approximately, it is possible to separate out mid-frequency component from the overall process, which characterizes generally free motion of automatic control system, then it is possible to estimate current value of control object transmission coefficient using parameters change of this component. Such control variable components motions separation is used for designing self-tuning ACS. In this case the informative parameter is dispersions difference of control object control variable and control variable of its model in the outputs of linear band pass filters. This dispersion difference is in proportion to current value of transition coefficient. In this article the results of simulation modeling computer experiments of self-tuning automatic control system with opened self-tuning loop are represented. The analysis of filtering efficiency, for three filter model options, are represented as well. The analysis was carried out with three different coordinate disturbances spectral compositions, with the same dynamics of control object and its model and with the different dynamics. Recommendations for filters structures alternatives selection are given. The possibility of optimal parametric tuning of filters is examined.


Introduction
A lot of factors influence technological process regulations variables when the process is considered as control object. These factors are represented as non-controlled coordinate and parametric disturbances. Coordinate disturbances change controlled variables without changing the properties of the control object. Parametric disturbances have influence on controlled variables and on the properties (parameters) of the control object as well. It is important to note that for control object of technological type the parametric disturbances spectral composition is much more low-frequency than spectral composition of coordinate disturbances [1,2]. This specific feature distinguishes control object of technological type from many mobile or electro-mechanical types.
Most often parametric disturbances reveal themselves in changing control object (CO) transmission coefficient in channel "controller control action -control variable". On a number of occasions transmission coefficient can change in a wide rangein 2…10 times more than initial value. Transmission coefficient pattern change can be gradual, e.g. owing to deterioration of -Volume 8, Issue 3 /2016 www.journal-atbp.com heat exchanger heat transfer due to accumulation of scale deposits. Or it can be rapid, like in the mills or presses when properties changing of feed stock occur. In automatic control system (ACS) control variable y(t) change in time relative to set point value y sp often is represented as sum of forced component y for (t), which is determined by specifics of external coordinate disturbances f c (t), measurement noises f n (t), which have influence on CO, and free component y fr (t), which depends on parameters of CO and controller [3]: y(t) = y for (t) + y fr (t) (1) Parametric disturbances f p (t), in particular, CO transmission coefficient k o (t) changes, usually, only scale forced component y for (t). But free component y fr (t) change pattern fundamentally changes: from protracted aperiodic transient processes during decrease of k o (t) and to oscillatory, including divergent, during increase of k o (t).
On the other hand control variable y(t) time change can be represented as additive model of components with different spectral distribution [2, p. 33]: ( ) ( ) ( ) ( ) ( ) l n y t y t y t y t y t , where ( ) y t -constant or gradually changing component governed by y sp and f c (t). In particular case, when y sp = const and f c (t) = const, then in ACS with offset absence ( ) sp y t y ; ( ) l y t -low frequency component governed generally by influence of coordinate disturbances f c (t); ( ) y t -centered mid-range frequency component governed by free motion y fr (t) of ACS, as a matter of fact ( ) ( ) fr y t y t ; ( ) n y t -high frequency component governed by, as a rule, measurement noises ( ) In the case, when using filters, even approximately, it is possible to separate out mid-frequency component ( ) y t from the overall process ( ) y t , which characterizes generally free motion of ACS, then it is possible to estimate current value of CO transmission coefficient ( ) o k t using parameters change of ( ) y t . Such control variable components motions separation is used in [4] for designing self-tuning ACS. In that case ( ) y t component was separated out by high frequency filters. But in that work there are no recommendations on choosing filters structures and parameters and also no optimization criteria is represented.

Problem statement
Let's define specifically the possibility of using of ACS free motion information for designing self-tuning ACS with CO varying transmission coefficient. For that, let's analyze block diagram extract of self-tuning ACS [4], which is shown in fig. 1 where controller control action u(t) in the closed-loop ACS is defined as: The purpose of the filters is to eliminate (to filter out) all components from the signals y(t) and y m (t) except centered midfrequency ones ( )  Let's examine presentation of the dependences (5) - (7) in the frequency domain. In this way, control action u(t) changes in time (5) is being changed, in accordance with [6], with its spectral density S u ( ): where ( ) sp y S spectral density of reference signal action y sp (t); S fc ( ) -spectral density of the coordinate disturbances f c (t) which is applied to the input of the CO; S fn ( ) -spectral density of the noise f n (t), which is applied to the output of the CO; ( ) A j -frequency characteristics of the closed-loop ACS of according transfer channels. Dependences (6) and (7) assume the form: -Volume 8, Issue 3 /2016 www.journal-atbp.com  and its model k m , which, subject to (9)-(10), follows from these expressions: Transition coefficients k o , k m are irrespective of frequency . If it is possible, for the most part, to eliminate consequences of external disturbances f c (t) f n (t) actions on the control variable y(t) using filtration, to make them insignificant, then in (11) Consequently, the expressions (11) and (12) assume the form: From (13) and (14) ( 1 6 ) In this paper the evaluators Calk in the block diagram ( fig. 1) calculate the estimations using dependences (16), because they are easier to implement in a program.
It is important to note that estimations difference From the abovementioned follows that accuracy of the self-tuning procedure in self-tuning ACS depends on the filters operation quality, their capability of division forced component of the ACS motion from the free component. Objective quality indicator in this case can be statistical correlation coefficient, which provide information about level of closeness between variables ( ) a y t and ( ) ma y t in the filters outputs. The goal of this article -to define the approach of filters structure and parameters selection for self-tuning ACS, simplified for the present, i.e. with disconnected self-tuning circuit, and also to compare filtration efficiency, using statistical correlation coefficient, of the variables in the filter outputs in the time of different mismatches between CO and its model and during different spectral densities of coordinate disturbances.
Conditions for carrying out comparative research Computer modeling was chosen as an instrument for the research, because analytical approach towards solving analysis and synthesis problems of non-stationary ACS, with varying transition coefficients objects, have approximate character and were obtained for separate particular cases [5, p. 397].
-Options of CO dynamics descriptions -transfer functions of the "virtual" CO: where: k o , o , T o , T1 o , T2 o -values of the transition coefficient, time delay and time constants, which has been evaluated using recommendations [8].
Step responses of the "virtual" control object are represented in the fig. 2a.
where: k r =2,3; iz =2,7 -values of the transition coefficient and integral action time, which corresponds to optimal value of the integral squared criterion of the control error for the CO option (18).
Step responses of the ACS with the controller (21) and different options of the "virtual" CO are represented in the fig. 2b.   For checking of the fact that the requirement was met, which was pointed out in the introduction to the article, (see relation (2)), about low-frequency character of the coordinate disturbances ( ) c f t in comparison to free motion of the ACS, which can be judged by its amplitude-frequency characteristic (AFC), it is enough to examine fig. 4. Here it is necessary to note, that disturbances modes (22), (23) can be taken as low-frequency in comparison to AFC of the ACS, and the model (24) was ad hoc was chosen with the violation of this condition. where H , L -cut-off frequencies of the filters in the low-frequency and high-frequency ranges. These frequencies will be taken as input parameters of the filter tuning optimization procedure.

Planning of the computer experiments
The goal of carrying out experiments is to research the effectiveness of ACS free motion extraction using the band pass linear filtering. The evaluation of the effectiveness was carried out using (30). Computer experiments intend purposive simulation modeling of the block diagram in fig. 1 in the Simulink environment of the MatLab programs package. ACS options with the "virtual" CO models (18)   The initial values of these frequencies can be selected using following recommendations.
The values of the statistical correlation coefficient It allows to make following conclusions: 1. As a linear filters for self-tuning ACS can be recommended band pass Butterworth filters not less than the fourth order (not less than the second order of the high-pass filter component). 2. AFC of the band pass filter must "embrace" AFC of the closed-loop ACS through the channel of external disturbances, excluding: a) low-frequency areas where spectral density of the control variable changes under the influence of external coordinate disturbances is located; b) high-frequency area, where spectral density of the ACS free motion is getting lower due to inertia specifics of the CO. Instantiation of these recommendations can be done using optimization of the filter cut-off frequencies parameters in the closed-loop ACS with self-tuning. 3. The value of the statistical correlation coefficient during changes of the CO order changes not significantly, which allows to use the simple first order model with delay as a mathematical model of the real CO. 4. Increase in the dispersions estimations values difference in the outputs of the filters during impact on the system from coordinate disturbances with the most wide spectral density indicates about lowering of the filtration efficiency from the consequences of their impact on the control variables of the CO and its model. The reason behind this, in this case, is violation of the abovementioned hypothesis about the more low-frequency character of the coordinate disturbances in comparison to the free motion of the ACS frequency character. If this situation takes place in real life, than according high-frequency part of the CO control variable changes spectrum must be ascribed to measurement noises and must be filtered. 5. The values of the statistical correlation coefficient depend on not precise reflections of the dynamic properties of the CO in its model, in particular, on not precise time delay setting in the model of CO in comparison to the time delay in the "virtual" CO. However, arising phase shifts between CO variables and its model variables don't have much influence on dispersions estimations difference, because these estimations are being calculated by using averaging procedure in the time interval, which is greater than the time delay. Filtration efficiency promotion and, as a consequence, lowering of dispersions estimations difference of the filtered variables can be expected using the decisions, which leads to revelation in these variables of described above negative factors and eliminate their consequences. These decisions needs special design and research.