Convergence analysis of the error compensation processin the phase control systems.

Michalconok, German ; Bezak, Tomas

1. INTRODUCTION

A characteristic feature of electric drives of control systems of motion stabilization is the presence of the program cyclic operation modes and the dominance of actions stipulated by abrasion in legs whose regular components in batches iterate at a continuous rotation of the gear shaft. A disturbing action in such objects can be presented as a sum of regular and random components, where a regular component has an essential weight and is governed by a cyclic law of change in time.

Classical methods of regulation (Kanjil, 1995) (Franklin et al., 1990) do not allow us to achieve the information on dynamic errors equal to zero. Therefore, it is suggested to apply the mode of padding compensation of errors via the actuation of a positive feedback (Beling & Bric, 1991). The phase control mode permits to achieve its stability. Thus, the main problem is to find the conditions for its stability, which is analysed here.

2. INFORMATION

The equation of an automatic control system error at a cyclic change of regular actions looks like this:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where E'(s) is a periodic component of an error, [T.sub.c] is a time cycle of the error and action recurrence, [[PHI].sub.[epsilon]](s) = 1/[1+W(s)] is the transfer function of the error management system, W(s) is the transfer function of the open loop control, Q'(s) is a periodic component of the assignment action (program), [W.sub.f](s) is the transfer function of perturbation, F'(s) is a periodic component of perturbation.

The periodic recurrence of control and disturbance actions allows one to use a cyclic component of a dynamic error to shape the padding program of driving compensatory an operation of perturbations irrespective of their nature and an application point in a system. Thus, the error control looks as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

Where [E'.sub.k](s) is a periodic component of an error after introducing the compensation program K'(s); [PHI](s) = W(s)/[1+W(s)] is the transfer function of the assignment action of the management system.

From (2) follows that a full compensation of periodic components of perturbations requires the introduction of a correction device with the transfer function [W.sub.k](s) = 1/[PHI](s) at an evaluation of the compensation program for a cyclic component of an error E'(s). A full structural diagram of the digital automatic control system with the compensation of cyclic disturbance actions [2] is shown in Fig. 1, where [T.sub.1] indicates a sampling period of the basic head loop, [D.sub.p]([z.sub.1])--transfer function of the digital governor of the basic control loop, [W.sub.E](s) --extra polariser of the basic head loop, [W.sub.o](s)--transfer function of the control object, [D.sub.f]([z.sub.1])--transfer function of the presample filter, [W.sub.E]1(s)--transfer extrapolation function at the passage from digitisation of the basic head loop with phase [T.sub.1] to digitisation of the compensation head loop T2 (generally, [T.sub.2] [not equal to] [T.sub.1]), [D.sub.k]([z.sub.2])--transfer function of the correction filter, N = [T.sub.c]/[T.sub.2] --number of storage cells, [D'.sub.k]([z.sub.2])--transfer function of the correction filter in a circuit of the positive feed-back for the storage, [W.sub.e2](s)--transfer function of the extra polariser at the passage from digitisation with phase [T.sub.2] to digitisation with phase [T.sub.1]; [D'.sub.f] ([z.sub.1])--the rectifying filter.

The transfer function of the control system from the off-on signal of compensation [K.sup.*](z) up to the off-on signal of error [E.sup.*](z) in the absence of filters and extra polariser of the zero order looks like this:

H(z) = (1-[z.sup.-1]) Z{[PHI](s)/s}. (3)

Irrespective of transfer function [PHI](s), the expression in curly brackets is divided into the following items:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

Z--transformation of which, accordingly, has a form

(1-[e.sup.-[alpha]T])z/(z-1)([z-e.sup.-[alpha]T]) (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

At T [right arrow] [infinity], expressions (4) and (5) tend to 1/(z-1), and expression (3) tends to [z.sup.-1].

This proves that the convergence of the compensation process can be ensured by the preceding reading of the discrete compensation signal at one clock tick, which is physically easy feasible. In the case of the discrete compensation device, the system of recurrent equations takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

[FIGURE 1 OMITTED]

Where [W.sub.i](s) = [W.sub.e](s)[W'.sub.f](s). Then, the fixed meaning of the compensation action at a harmonic error signal E([omega]) is described by an expression (Michalconok, 2005)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Where z = [e.sup.j[omega]T], and the fixed error meaning, on the existence of compensation,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

For estimation of the influence of system parameters on the padding error, it is possible to present the padding error caused by the presence of quantisation in time, simplistically by an expression

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [K'.sub.[infinity]](j[omega]j = [K.sub.[infinity]](j[omega] + 2[pi][pi]/T).

The convergence analysis can also be carried out using a simplified expression which is true up to w = p/T.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

or, by considering the filter phase compensation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

The results of convergence analysis and the results of steady state calculation are given in Fig. 2a. and Fig. 2b.

The performed analysis of the discrete compensation device shows that it is illogical to select a frequency of quantisation 2T as considerably higher than the cut-off frequency of the basic head loop, both for the reason of security of convergence, as well as for the necessity to reduce the volume of storage.

[FIGURE 2a OMITTED]

[FIGURE 2b OMITTED]

3. CONCLUSIONS

The introduction of the synchronic filter into the automatic control system does not influence stability of the basic head loop (circuit) due to a relatively large signal lag in the storage, but it demands the analysis of the compensation process convergence.

From the analysis of the fixed meanings also follows that the compensation process convergence is determined by the position of the hodograph of the transfer function R(j[omega]) = [W.sub.k]'(j[omega])--[W.sub.k](j[omega])F(j[omega]) inside a circumference of a single radius. The analysis of the fixed error meanings and the compensation action in management systems with the discrete compensation device imply that it is possible to ensure security of the compensation process convergence by means of the preceding reading of a signal from the storage at a simultaneous reduction of the quantisation phase.

4. REFERENCES

Beling, T. & Bric, L.(1991), Phase locked loop control system, Available from: http://www.freepatentsonline.com/4272712.html, Accessed: 2008-06-30

Franklin, F.; Powell, D. & Workman, M.(1990), Digital Control of Dynamic Systems, Addison-Wesley. ISBN 0-201-518848, New York

Kanjil, P. (1995). Adaptive prediction and predictive control, Peter Peregrinus Ltd, ISBN 0-86341-1932, London

Michalconok, G. et al. (1986). A management system of the electric drive, the copyright certificate USSR N[degrees]1262676, EM. M .37, 1986

Michalconok, G. (2005). Synchronic filters at control of elastic plants, CO-MAT-TECH 2005. Pp 80-84, ISBN 80-2272286-3, Trnava, 20-21 October 2005