Transfer function multidimensional frequency graphs of the control system dynamic links

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Kaluga branch, Bauman Moscow State Technical University

Transfer function multidimensional frequency graphs of the control system dynamic links

Yu.P. Kornyushin, E.V. Klimanova, A.V. Maksimov

Kaluga, 248000, Russia

Abstract

The article proposes a method to increase information content in studying frequency properties of the automatic control systems typical dynamic links based on representation of the link frequency transfer functions in the form of multidimensional graphs. Examples of separate links are provided.

Keywords: control systems, typical links, amplitude-frequency characteristics

Introduction

Relevance and novelty

Studying dynamics of a separate link in the control system and/or of the entire system in the frequency domain is the basic and key stage in designing new complex technical systems. The quality of assessing dynamic properties of the object under study is largely determined by the quality of construction and by the detailed researcher perception of the object properties' graphical interpretation in the frequency domain.

Currently, classical method for such interpretation lies in constructing in the form of two-dimensional graphs of amplitude-frequency, phase-frequency and amplitude-phase-frequency characteristics (AFC, PFC, APFC) of both individual links of the control system and of the entire system [1 -5]. Practice of simulating the complex technical structure dynamics demonstrated that models in the three-dimensional graphics with the same initial data provides the researcher with more information than the two-dimensional models.

There are currently no technological difficulties in constructing AFC, PFC and APFC models in the three-dimensional graphics. Any experience and methodology for using such graphic models in studying the control system dynamics in the frequency domain is missing. The method proposed in the article, according to the authors, makes it possible to start accumulating experience in the use of three-dimensional frequency characteristics of the control system links and to create a methodology for studying the system elements' dynamics in the frequency domain.

Method

Frequency transfer function is used in the classical theory of control systems [1-4] to study the system links dynamics in the frequency domain. By mathematical transformations from the expression of the link or of the entire system complex-valued transfer function in the following form:

,

separate module function depending on the frequency is expressed:

,

as well as the phase versus frequency function:

.

Functions А(щ) and ц(щ) for the object under study are, respectively, the AFC and the PFC.

Representation of the frequency transfer function of a link or the entire system in the form of two functions of frequency, real and imaginary, makes it possible to apply the three-dimensional graphics operations to them and build surfaces of these functions using the personal computer existing mathematical software packages. Authors in work [5] are using this method in preparation of the physics students for development of their imaginative thinking on the examples of elementary and special complex-valued function surfaces, including sine of a complex argument, hyperbolic tangent of a complex argument, Zhukovskiy and Bessel functions, Euler's gamma function and others.

Construction of the typical dynamic link three-dimensional frequency characteristics using the proposed method in this work was carried out based on the Matlab mathematical programming environment [6].

Results

As an example, three-dimensional amplitude-frequency characteristics of the aperiodic link of the first order (Figure 1) and three oscillatory links at different values of the damping coefficient (Figures 2, 3, 4) could be provided. The aperiodic link AFC surface was built based on the following mathematical model of its transfer function [7]:

Here, the following is accepted:

· link gain ;

· link time constant S;

· circular frequency alteration range rad/S;

· frequency alteration step 0.0625 rad/S.

Expression for the AFC is as follows:

. (1)

Figure 1 shows the surface of the amplitude-frequency characteristic of the first-order aperiodic link built in accordance with expression (1):

Figure 1. Surface of the first-order aperiodic link AFC

Transfer function of the oscillation is presented as a complex-valued function and has the following form [8]:

. (2)

General expression for the oscillation link AFC surface in accordance with (2) takes the following form:

. (3)

To construct the oscillation link AFC surfaces, three different values of the о damping factor were taken in expression (3) reflecting dynamics of the typical link of this type operation (Figures 2, 3, 4). frequency information dynamic

Figure 2:

· gain ;

· damping factor ;

· circular frequency alteration range rad/S;

· frequency alteration step 0.0625 rad/S.

Figure 2. Surface of the oscillation link AFC at о = 0.03125

Figure 3:

· gain ;

· damping factor ;

· circular frequency alteration range rad/S;

· frequency alteration step 0.0625 rad/S.

Figure 3. Surface of the oscillation link AFC at о = 0.25

Figure 4:

· gain ;

· damping factor ;

· circular frequency alteration range rad/S;

· frequency alteration step 0.0625 rad/S.

Figure 4. Surface of the oscillation link AFC at о = 0.75

Discussion

Examples provided show distinctive features of the three-dimensional graphics, i.e. creating volumetric image of the object under study. “Relief” of the system link frequency characteristic clearly demonstrates this link behavior in vicinity of the special points. For example, such a point for an oscillatory link is the value of the damping coefficient equal to .

Any researcher has the opportunity to examine the model from any observation point. It becomes possible to highlight with color or shades of gray any areas in the frequency characteristic that could be of interest to the researcher. The model could be represented both as a solid surface, and as a grid of mutually perpendicular lines parallel to the coordinate axes.

Possible application areas of the proposed method

At present, optimization method for synthesizing controllers with a given structure in deterministic and static formulation became widespread in the theory of automatic control. In accordance with this method, the quadratic objective function using the Parseval's rule is reduced to a standard integral [9]. In this case, the synthesis problem is solved analytically, if the synthesized system has low dimension, and the number of required parameters is small. If the system has high dimension, and the number of required parameters reaches several dozens, then difficulties arise due to the fact that the objective function is multi-extreme, and there is no guarantee that the fminsearch standard Matlab function or the Maple NLPsolve function will find the global extremum. In this case, the global extremum search control could be carried out by observing alterations either in the surface of the frequency transfer function of the system with controller, or in the surface of the Fourier transform of the synthesized system output system, comparing them with the corresponding characteristics of a reference system. And, if necessary, changes could be made in the synthesis procedure.

Similar problems arise in the controller synthesis using the method of moments [10]. Approach to solving these problems and overcoming the arising difficulties could also be found using the method proposed in this work.

It could be assumed that the proposed method of the dynamic links transfer function surfaces will allow to successfully solve the problems arising in parametric synthesis of the automatic control system controllers, proceeding from the minimum root-mean-square error (statistical formulation) condition [11].

When synthesizing systems having uncertainty in describing a control object, the control theory is nowadays widely used [9, 10, 12]. In this case, criterion of optimality is the norm in the Hardy space of the complex-valued functions being analytic in the right half-plane. For SISO (Single Input Single Output) systems, the maximum of the frequency transfer function module (AFC systems) is used as the optimality criterion. Thus, when synthesizing robust controllers, 3d graphs of frequency transfer functions could also be used.

Conclusion

The proposed method makes it possible to significantly increase information content of a link and/or system dynamics analysis in the frequency domain ensuring better quality results. The method of representing frequency characteristics of the dynamic links in form of the three-dimensional surfaces is a step towards the development of imaginative thinking skills in developers of complex modern technical systems.

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