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Analysis of the compensation algorithm stability of disturbance for the phase control systems
Analysis of the compensation algorithm stability of disturbance for the phase control systems Darja Gabriska*, German Michalconok**, Pavol Tanuska***, Tomas Skulavik**** Faculty of Material Sciences Technology in Trnava, Slovak University of Technology, 917 24 Trnava, Slovak Republic. * (Tel: +421 918-60-64-21; e-mail: [email protected]), ** (Tel: +421 918-60-64-21; e-mail: [email protected]), *** (Tel: +421 918-60-64-61; e-mail: [email protected]), **** (Tel: +421 918-60-64-61; e-mail: [email protected]). Abstract. The main component of error of phase control systems of drivers is a periodic component caused by external disturbances. This error can be reduced using compensation principle. In fact, the compensation is a positive feedback and thus the setting of control parameters is very parameters sensitive. The paper deals with setting conditions, which guarantee the convergence of compensation process of control error. The result of the presented analysis is a setting technique of phase control systems. Keywords: disturbance control, control precision, system sensitivity, drives, stability analysis 1.
INTRODUCTION
A characteristic feature of the stabilization systems of motion is an existence of the program cyclic operation modes and a dominance of phenomena caused by the abrasion in the bearing. The periodic component of these phenomena is repeated periodically during a continuous rotation of a gear shaft [1]. A disturbance in such objects consists of permanent component and random one as well. The permanent component is significant and it is periodic [3] Classic control methods (Kanjil, P. 1995, Franklin et al., 1990) do not allow to achieve the error equal zero during dynamic processes. Therefore using of positive feedback is recommended (Beling, T. & Bric, L., 1991). The stability of the positive feedback can be achieved by phase control. The stability conditions have been derived for the phase control loop as they are presented in this paper. 2. FILTERING OF THE PERIODIC DISTURBANCE The control error during periodic changes of permanent component can be described as follows (Michalconok, 2005)
E(s) (s)
Q' (s) F ' (s) - (s)Wf (s) sTc 1- e 1 - e sTc
where Е'(s) is a periodic component of the error. Тc - time period of the error and action recurrence. (s) W(s) /[1 W(s)] the transfer function of the error control system. W(s) – the transfer function of the open loop control.
(1)
Q'(s) – a periodic component of the control action. Wf(s) – the transfer function of the disturbance. F'(s) – a periodic component of the disturbance. The presence of the periodic reference input and the disturbance is a source of periodic dynamic control error. This fact allows the use of an error to generate an additional signal to compensate the control error. The source of this error can be any. In this case, the feedback control system is of the synchronous filter character. Thus the control action is in the form
Ek (s) E(s) K(s) (s) sTC sTC 1 e 1 e 1 esTC
where E'k(s) is a periodic component of compensation K'(s) applied is the function (s) W (s) /[1 W (s)] A total compensation of periodic disturbance is possible using WK(s) 1/(s) as follows from (2).
(2)
the error after the closed-loop transfer (3) components of the correction block The structure of the
control algorithms (Michalconok et al., 1986) is shown in Fig.1.
which is physically easy feasible. Thus the system of recurrent equations has the form (Michaľčonok, G. & Bezák, T. 2009)
Enk (s) En (s) Wi (s) (s) esTc Kn1(esT ) K n* ( z) Dk ( z) Z{W f (s) En (s)
(7)
[Dk' ( z) - Dk ( z) Z{Wi (s) W f (s) (s)}]z -N K n*-1
where Wi (s)We (s)W f (s) Then, steady state value of the compensation action at a harmonic error signal E() is described by the expression * (e jT ) K
z - N Dk ( z) Z{W f (s) E(s)} 1 - z - N [Dk' ( z) - Dk ( z)Z{W f (s) E(s)}]
(8)
where z = ejT, and the steady state value of error when the compensation is active is (9) k jT
E E( j) Wi ( j) ( j) K (e
Fig.1. Control scheme The symbols used in this scheme are as follows. Т1 – a sampling period of the master loop. Dp(z1) – transfer function of the master controller. ZOH(s) – transfer function of the zero order hold of master control loop. Wo(s) – transfer function of the controlled system. Df(z1) – transfer function of the antialiasing filter. ZOH1(s) – transfer function of the zero order hold when the signal of the master loop sampled with period Т1 is sampled in compensation loop with period Т2 (generally, Т2 Т1). Dk(z2 ) – transfer function of the correction filter. N = Tc/T2 - number of buffer size. D'k(z2) – transfer function of the correction filter in the positive feedback loop. ZOH2(s) – transfer function of the zero order hold when the signal of the positive feedback loop sampled with period Т2 is sampled in master control loop with period Т1. D'f (z1) – the moving average filter. We define the ratio of the transform of the compensation signal К*(z) to the transform of the error Е*(z) as the transfer function, H(z) for Euler integration is (4) H (z) (1 z 1) Z{(s) / s}
)
To evaluate system parameters influence of the error caused by quantization in time the additive error can be simplified expressed by (10) ( j) [Wi ( j) ( j) T ] K ( j) E
( j) K ( j 2 T ) . where K
The convergence analysis can also be carried out using a simplified expression which is true up to ω /T. (11) H ( j) D (e j ) W ( j) W ( j) ( j) T k
f
i
or, by considering the filter phase compensation
R( j) Dk (e j ) Wf ( j) Wi ( j) / T ( j)
(12)
The results of convergence analysis and the results of steady state computation are given in Fig.2 and 4b. The performed analysis of the discrete compensation algorithm shows that it is not suitable to choose the frequency of quantization 2Т as considerably higher than the cut-off frequency of the master loop, both for the reason of convergence, as well as for the necessity to reduce the volume of memory.
Independent on transfer function Ф(s) the expression Ф(s)/s can be split into two parts s( ) 2 b 2 and s( s ) s[(s )2 2 ] Z transformation of them is in the form (5) -T -T
(1- e
)z/(z -1) (z - e
)
and
z z2 - ze-T (cosT b sin T ) z - 1 z 2 - 2ze-T cos T e- 2T
(6)
At Т→, expressions (5) and (6) tend to 1/(z-1), and expression (4) tends to z-1. This proves that the convergence of the compensation process can be ensured by the forward reading of the discrete compensation signal at one sample,
Fig.2a. Magnitude frequency response compensation of (12).
D'(z)=(z^-1+z^-2+z^-3)/4
20 20
0 E1( )
20
40 40 0.1 0.1
1
10 4
Tc = 20pi, T = pi/4
Fig.2b Magnitude frequency response of compensation signal (12). D'(z)=(z^-1+z^-2+z^-3)/4
Fig.4a Magnitude frequency response of compensation error (9). D'(z) = 1
20 20
20
10 0 E2( )
K1( ) 0
20
10 40
20 0.01
0.1
1
10
Fig.3a. Magnitude response of compensation value Tc = 20pi, T =frequency pi/4 (8).
40 0.1 0.1
1
10 4
Tc = 20pi, T = pi/4
Fig.4b. Magnitude frequency response of compensation error (9). 3. FILTERING OF RANDOM DISTURBANCES
D'(z)= 1
20 20
If controlled processes are of a random nature subjected to the normal density function, then these processes are determined by mathematical expectation and correlation function.
10
K2( )
0
10
20 20 0.01 0.01
0.1
1
10 4
Tc = 20pi, T = pi/2
Fig.3b. Magnitude frequency response of compensation value (8).
Mathematical expectation of random process in the case of phase control is given by a system of recurrent equations (6) obtained from the analysis of a sequence of system changes during a compensation process. From (6) implies, that corresponding impulse transient function of the given system has the form of a complicated periodic damped function W (t ) W1(t Tc ) Wk (t Tc ) W1(t 2Tc ) Wk (t 2Tc ) (t 2Tc ) W1(t 3Tc ) Wk (t 3Tc ) (t 3Tc ) where (t) Wk' (t) W1(t) Wk (t) and symbol “*” means convolution integral
t 0
h (t ) h ( ) d
h(t )
1
2
then in the steady state
0
The complexity of (7) and the uncertain dependence of the corresponding correlation function on system parameters do not allow using it for random processes analysis of phase control system. For this reason it is possible to solve that problem using the analysis of the spectral density of random signals. It is known that the relation between a spectral density of output signal y(t) in a steady state and input stationary random signal x(t) is given by expression 2 (13)
GYY () K ( j) Gxx (),
where K(j) is the frequency characteristic of the system. The random processes analysis of control system with phase control is not justified using (13) because the frequency characteristic represents steady state of the system as response to input harmonic signal with constant amplitude and frequency as well. But real random processes duration is less than the delay of signal Tc. Similar to the analysis of compensation process convergence we consider a sequence of states of the system in Fig.1with respect to the error in the system without compensation, which is a stationary random disturbance with spectral density GEE() . During the first working period of compensator the spectral density of error G1EE () GEE () and a compensation signal with spectral density is generated. 2 (14) G1 () W ( j) G () KK
k
KK
Since the delay of the output signal of the buffer output is greater than the duration of correlation function of the signal E(j), at the second period of the storage device there will be steady value of the spectral density 2 (15) G2 () [1 W ( j) ( j) ] G ( EE
k
EE
And the spectral density of regenerated compensation action will be 2
2 GKK () {Wk ( j) 2
WK ( j) [WK ( j) WK ( j) ( j)] } GEE () At the next cycles of compensator the spectral density will be as follows:
k GEE () {1
[W ( j) W K
K ( j) ( j)
i 2
] } GEE ()
i 0
2
n GKK () {Wk ( j)
n 1
[W ( j) W K
K ( j) ( j)
i 0
Since a workable system always has
i 2
] } GEE ()
WK ( j) ( j)
2
[1 [WK ( j) WK ( j) ( j)]
2
} GEE () (17)
From this expression it is seen that the introduction of compensator inevitably leads to an increase in the variance of deviation of the control system output and its value is given by 2
WK ( j) ( j) { } GEE () d (). (18) 2 ' [1 [W K ( j) WK ( j) ( j)] ]
DE'
The error dispersion reducing of the control system with compensator can be achieved by reducing part of error signal (the input of the buffer) as well as by increasing the rate of convergence of the compensation process. We note that a decreasing of part of error signal decreases the convergence rate of compensation process. Therefore the expression DE'
2
2 WK ( j) ( j) { } GEE () d () (19) 2 [1 [W K' ( j) WK ( j) ( j)] ]
can be minimized by setting The convergence condition requires correction filters with zero phase, which is physically feasible using discrete realization. In general, the random processes in a discrete system are not stationary and consist of periodic changes of their characteristics which are equal to sample period. The analysis of such system is carried out in a discrete form which can be considered as a stationary system. In the steady state, the spectral density of the additional error in the discrete system (see Fig. 1) is given by ' ( z, ) GEE
2 Z {WK ( j) ( j)} Ge ( z) EE
' ( z) D ( z)Z{W (s) W (s) (s)} 2 1 DK f i K
}, (20)
Where e GEE ( z) DK ( z) DK ( z 1)
R (m)z e
m
m
is the equivalent discrete spectral density of the signal at the output of the first quantizing element. This spectral density is given by the equivalent correlation function
2
n GEE () {1 Wk ( j) ( j) n 1
(16)
WK' ( j) Wk ( j) ( j) 1
Re ( )
1 2
W ( j) f
2
GEE () e j d,
(21)
which is computed at discrete points = mT Using the modified Z-transform in (21) reflects the non stationary nature of the spectral density of the additional error caused by the action of the discrete correction device. It should be noted that the correlation function obtained from the spectral density G' EE (z,) is an impulse function defined only for discrete values of = mT.
The change of does not allow to compute intermediate values, but there is a possibility to determine the dependence of the dispersion on time within the sampling period. For a computation of the dispersion of error caused by the action of compensator device
DE' ( )
T 2
/T
G
' jT , ) d EE (e
(22)
/T
Using (20) complex transformations of the spectral density of output signal and transfer functions of the continuous part of a system as well have to be performed. As in the previous section, the presence of an effective low pass filter on the output and input of discrete compensator allows to use the first approximation to compute the spectral density of the additional error G1EE (e jT ) 2
D1K (e jT ) W f ( j) We ( j) ( j) T
2
1 DK ( z) DK ( z) Wi (s) W f (s) (s) T
2
GEE ()
(23)
It should be noted, that the expression (21) is valid for ω -/T ω /T in which the variance is computed according the expression (23). An analysis of the expression (23) shows that the implementation of compensator device increases the variance of the error less than 2/ (2 - ) times. 4. CONCLUSIONS The use of a positive feedback control loop of control error compensation allows to increase an accuracy of controlled processes. The main problem of this solution is compensation process stability. The paper presents an analysis of correction filters impact on process convergence of the control error compensation. This analysis shows that a stable compensation process is achievable by a suitable choice of the type of a filter and its parameter values setting. An implementation of synchronous filter does not cause an instability of the master control loop because the signal delay in a dead time block is relatively sufficient large. But on the other side an analysis of compensation process convergence has to be carried out. Based on of steady state analysis can be stated that the convergence of compensation process is guaranteed if the filter phase compensation
R( j) Wk ( j) Wk ( j) F( j)
is completely located inside of the unit circle of Z-plane. An analysis of steady state error values and compensation action with discrete compensator shows, that the compensation process convergence can be achieved by the forward reading of signal from the buffer and the sample period decrease as well. The implementation of compensator causes an increase of error dispersion since the output signal of storage device appears in time exceeding the duration of correlation function of error signal. The analysis of steady-state value of the spectral density of signal errors as the sum of an infinite series when
convergence conditions are fulfilled shows that an implementation of compensator increases the error dispersion less than 2 / (2 - ) time. In practice, there exists a wide class of control systems with periodic disturbances. The presented method of a control accuracy increase allows to improve the quality of the control systems with cyclic disturbances. The future research in the given area will be focused on a searching of applicability limits of the presented methods. 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 0201-51884-8, New York. Kanjil, P. (1995). Adaptive prediction and predictive control. Peter Peregrinus Ltd, ISBN 0-86341-1932, London. Michalconok, G. et al. (1986). A control system of the electric drive, the copyright certificate USSR №1262676, Б.И. № .37, 1986. Michalconok, G. (2005). Synchronic filters at control of elastic plants, CO-MAT-TECH 2005. Pp 80-84, ISBN 80-227-2286-3, Trnava, 20-21 October 2005. Michalconok, G.; Bezak, T. (2009): Convergence analysis of the error compensation process in the phase control systems. In: Annals of DAAAM and Proceedings of DAAAM Symposium. - ISSN 1726-9679. - Vol. 20, No. 1 Annals of DAAAM for 2009 & Proceedings of the 20th international DAAAM symposium "Intelligent manufacturing & automation: Focus on theory, practice and education" 25 - 28th November 2009, Vienna, Austria. - Vienna: DAAAM International Vienna, 2009. ISBN 978-3-901509-70-4, s.0921-0922.
INTRODUCTION
A characteristic feature of the stabilization systems of motion is an existence of the program cyclic operation modes and a dominance of phenomena caused by the abrasion in the bearing. The periodic component of these phenomena is repeated periodically during a continuous rotation of a gear shaft [1]. A disturbance in such objects consists of permanent component and random one as well. The permanent component is significant and it is periodic [3] Classic control methods (Kanjil, P. 1995, Franklin et al., 1990) do not allow to achieve the error equal zero during dynamic processes. Therefore using of positive feedback is recommended (Beling, T. & Bric, L., 1991). The stability of the positive feedback can be achieved by phase control. The stability conditions have been derived for the phase control loop as they are presented in this paper. 2. FILTERING OF THE PERIODIC DISTURBANCE The control error during periodic changes of permanent component can be described as follows (Michalconok, 2005)
E(s) (s)
Q' (s) F ' (s) - (s)Wf (s) sTc 1- e 1 - e sTc
where Е'(s) is a periodic component of the error. Тc - time period of the error and action recurrence. (s) W(s) /[1 W(s)] the transfer function of the error control system. W(s) – the transfer function of the open loop control.
(1)
Q'(s) – a periodic component of the control action. Wf(s) – the transfer function of the disturbance. F'(s) – a periodic component of the disturbance. The presence of the periodic reference input and the disturbance is a source of periodic dynamic control error. This fact allows the use of an error to generate an additional signal to compensate the control error. The source of this error can be any. In this case, the feedback control system is of the synchronous filter character. Thus the control action is in the form
Ek (s) E(s) K(s) (s) sTC sTC 1 e 1 e 1 esTC
where E'k(s) is a periodic component of compensation K'(s) applied is the function (s) W (s) /[1 W (s)] A total compensation of periodic disturbance is possible using WK(s) 1/(s) as follows from (2).
(2)
the error after the closed-loop transfer (3) components of the correction block The structure of the
control algorithms (Michalconok et al., 1986) is shown in Fig.1.
which is physically easy feasible. Thus the system of recurrent equations has the form (Michaľčonok, G. & Bezák, T. 2009)
Enk (s) En (s) Wi (s) (s) esTc Kn1(esT ) K n* ( z) Dk ( z) Z{W f (s) En (s)
(7)
[Dk' ( z) - Dk ( z) Z{Wi (s) W f (s) (s)}]z -N K n*-1
where Wi (s)We (s)W f (s) Then, steady state value of the compensation action at a harmonic error signal E() is described by the expression * (e jT ) K
z - N Dk ( z) Z{W f (s) E(s)} 1 - z - N [Dk' ( z) - Dk ( z)Z{W f (s) E(s)}]
(8)
where z = ejT, and the steady state value of error when the compensation is active is (9) k jT
E E( j) Wi ( j) ( j) K (e
Fig.1. Control scheme The symbols used in this scheme are as follows. Т1 – a sampling period of the master loop. Dp(z1) – transfer function of the master controller. ZOH(s) – transfer function of the zero order hold of master control loop. Wo(s) – transfer function of the controlled system. Df(z1) – transfer function of the antialiasing filter. ZOH1(s) – transfer function of the zero order hold when the signal of the master loop sampled with period Т1 is sampled in compensation loop with period Т2 (generally, Т2 Т1). Dk(z2 ) – transfer function of the correction filter. N = Tc/T2 - number of buffer size. D'k(z2) – transfer function of the correction filter in the positive feedback loop. ZOH2(s) – transfer function of the zero order hold when the signal of the positive feedback loop sampled with period Т2 is sampled in master control loop with period Т1. D'f (z1) – the moving average filter. We define the ratio of the transform of the compensation signal К*(z) to the transform of the error Е*(z) as the transfer function, H(z) for Euler integration is (4) H (z) (1 z 1) Z{(s) / s}
)
To evaluate system parameters influence of the error caused by quantization in time the additive error can be simplified expressed by (10) ( j) [Wi ( j) ( j) T ] K ( j) E
( j) K ( j 2 T ) . where K
The convergence analysis can also be carried out using a simplified expression which is true up to ω /T. (11) H ( j) D (e j ) W ( j) W ( j) ( j) T k
f
i
or, by considering the filter phase compensation
R( j) Dk (e j ) Wf ( j) Wi ( j) / T ( j)
(12)
The results of convergence analysis and the results of steady state computation are given in Fig.2 and 4b. The performed analysis of the discrete compensation algorithm shows that it is not suitable to choose the frequency of quantization 2Т as considerably higher than the cut-off frequency of the master loop, both for the reason of convergence, as well as for the necessity to reduce the volume of memory.
Independent on transfer function Ф(s) the expression Ф(s)/s can be split into two parts s( ) 2 b 2 and s( s ) s[(s )2 2 ] Z transformation of them is in the form (5) -T -T
(1- e
)z/(z -1) (z - e
)
and
z z2 - ze-T (cosT b sin T ) z - 1 z 2 - 2ze-T cos T e- 2T
(6)
At Т→, expressions (5) and (6) tend to 1/(z-1), and expression (4) tends to z-1. This proves that the convergence of the compensation process can be ensured by the forward reading of the discrete compensation signal at one sample,
Fig.2a. Magnitude frequency response compensation of (12).
D'(z)=(z^-1+z^-2+z^-3)/4
20 20
0 E1( )
20
40 40 0.1 0.1
1
10 4
Tc = 20pi, T = pi/4
Fig.2b Magnitude frequency response of compensation signal (12). D'(z)=(z^-1+z^-2+z^-3)/4
Fig.4a Magnitude frequency response of compensation error (9). D'(z) = 1
20 20
20
10 0 E2( )
K1( ) 0
20
10 40
20 0.01
0.1
1
10
Fig.3a. Magnitude response of compensation value Tc = 20pi, T =frequency pi/4 (8).
40 0.1 0.1
1
10 4
Tc = 20pi, T = pi/4
Fig.4b. Magnitude frequency response of compensation error (9). 3. FILTERING OF RANDOM DISTURBANCES
D'(z)= 1
20 20
If controlled processes are of a random nature subjected to the normal density function, then these processes are determined by mathematical expectation and correlation function.
10
K2( )
0
10
20 20 0.01 0.01
0.1
1
10 4
Tc = 20pi, T = pi/2
Fig.3b. Magnitude frequency response of compensation value (8).
Mathematical expectation of random process in the case of phase control is given by a system of recurrent equations (6) obtained from the analysis of a sequence of system changes during a compensation process. From (6) implies, that corresponding impulse transient function of the given system has the form of a complicated periodic damped function W (t ) W1(t Tc ) Wk (t Tc ) W1(t 2Tc ) Wk (t 2Tc ) (t 2Tc ) W1(t 3Tc ) Wk (t 3Tc ) (t 3Tc ) where (t) Wk' (t) W1(t) Wk (t) and symbol “*” means convolution integral
t 0
h (t ) h ( ) d
h(t )
1
2
then in the steady state
0
The complexity of (7) and the uncertain dependence of the corresponding correlation function on system parameters do not allow using it for random processes analysis of phase control system. For this reason it is possible to solve that problem using the analysis of the spectral density of random signals. It is known that the relation between a spectral density of output signal y(t) in a steady state and input stationary random signal x(t) is given by expression 2 (13)
GYY () K ( j) Gxx (),
where K(j) is the frequency characteristic of the system. The random processes analysis of control system with phase control is not justified using (13) because the frequency characteristic represents steady state of the system as response to input harmonic signal with constant amplitude and frequency as well. But real random processes duration is less than the delay of signal Tc. Similar to the analysis of compensation process convergence we consider a sequence of states of the system in Fig.1with respect to the error in the system without compensation, which is a stationary random disturbance with spectral density GEE() . During the first working period of compensator the spectral density of error G1EE () GEE () and a compensation signal with spectral density is generated. 2 (14) G1 () W ( j) G () KK
k
KK
Since the delay of the output signal of the buffer output is greater than the duration of correlation function of the signal E(j), at the second period of the storage device there will be steady value of the spectral density 2 (15) G2 () [1 W ( j) ( j) ] G ( EE
k
EE
And the spectral density of regenerated compensation action will be 2
2 GKK () {Wk ( j) 2
WK ( j) [WK ( j) WK ( j) ( j)] } GEE () At the next cycles of compensator the spectral density will be as follows:
k GEE () {1
[W ( j) W K
K ( j) ( j)
i 2
] } GEE ()
i 0
2
n GKK () {Wk ( j)
n 1
[W ( j) W K
K ( j) ( j)
i 0
Since a workable system always has
i 2
] } GEE ()
WK ( j) ( j)
2
[1 [WK ( j) WK ( j) ( j)]
2
} GEE () (17)
From this expression it is seen that the introduction of compensator inevitably leads to an increase in the variance of deviation of the control system output and its value is given by 2
WK ( j) ( j) { } GEE () d (). (18) 2 ' [1 [W K ( j) WK ( j) ( j)] ]
DE'
The error dispersion reducing of the control system with compensator can be achieved by reducing part of error signal (the input of the buffer) as well as by increasing the rate of convergence of the compensation process. We note that a decreasing of part of error signal decreases the convergence rate of compensation process. Therefore the expression DE'
2
2 WK ( j) ( j) { } GEE () d () (19) 2 [1 [W K' ( j) WK ( j) ( j)] ]
can be minimized by setting The convergence condition requires correction filters with zero phase, which is physically feasible using discrete realization. In general, the random processes in a discrete system are not stationary and consist of periodic changes of their characteristics which are equal to sample period. The analysis of such system is carried out in a discrete form which can be considered as a stationary system. In the steady state, the spectral density of the additional error in the discrete system (see Fig. 1) is given by ' ( z, ) GEE
2 Z {WK ( j) ( j)} Ge ( z) EE
' ( z) D ( z)Z{W (s) W (s) (s)} 2 1 DK f i K
}, (20)
Where e GEE ( z) DK ( z) DK ( z 1)
R (m)z e
m
m
is the equivalent discrete spectral density of the signal at the output of the first quantizing element. This spectral density is given by the equivalent correlation function
2
n GEE () {1 Wk ( j) ( j) n 1
(16)
WK' ( j) Wk ( j) ( j) 1
Re ( )
1 2
W ( j) f
2
GEE () e j d,
(21)
which is computed at discrete points = mT Using the modified Z-transform in (21) reflects the non stationary nature of the spectral density of the additional error caused by the action of the discrete correction device. It should be noted that the correlation function obtained from the spectral density G' EE (z,) is an impulse function defined only for discrete values of = mT.
The change of does not allow to compute intermediate values, but there is a possibility to determine the dependence of the dispersion on time within the sampling period. For a computation of the dispersion of error caused by the action of compensator device
DE' ( )
T 2
/T
G
' jT , ) d EE (e
(22)
/T
Using (20) complex transformations of the spectral density of output signal and transfer functions of the continuous part of a system as well have to be performed. As in the previous section, the presence of an effective low pass filter on the output and input of discrete compensator allows to use the first approximation to compute the spectral density of the additional error G1EE (e jT ) 2
D1K (e jT ) W f ( j) We ( j) ( j) T
2
1 DK ( z) DK ( z) Wi (s) W f (s) (s) T
2
GEE ()
(23)
It should be noted, that the expression (21) is valid for ω -/T ω /T in which the variance is computed according the expression (23). An analysis of the expression (23) shows that the implementation of compensator device increases the variance of the error less than 2/ (2 - ) times. 4. CONCLUSIONS The use of a positive feedback control loop of control error compensation allows to increase an accuracy of controlled processes. The main problem of this solution is compensation process stability. The paper presents an analysis of correction filters impact on process convergence of the control error compensation. This analysis shows that a stable compensation process is achievable by a suitable choice of the type of a filter and its parameter values setting. An implementation of synchronous filter does not cause an instability of the master control loop because the signal delay in a dead time block is relatively sufficient large. But on the other side an analysis of compensation process convergence has to be carried out. Based on of steady state analysis can be stated that the convergence of compensation process is guaranteed if the filter phase compensation
R( j) Wk ( j) Wk ( j) F( j)
is completely located inside of the unit circle of Z-plane. An analysis of steady state error values and compensation action with discrete compensator shows, that the compensation process convergence can be achieved by the forward reading of signal from the buffer and the sample period decrease as well. The implementation of compensator causes an increase of error dispersion since the output signal of storage device appears in time exceeding the duration of correlation function of error signal. The analysis of steady-state value of the spectral density of signal errors as the sum of an infinite series when
convergence conditions are fulfilled shows that an implementation of compensator increases the error dispersion less than 2 / (2 - ) time. In practice, there exists a wide class of control systems with periodic disturbances. The presented method of a control accuracy increase allows to improve the quality of the control systems with cyclic disturbances. The future research in the given area will be focused on a searching of applicability limits of the presented methods. 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 0201-51884-8, New York. Kanjil, P. (1995). Adaptive prediction and predictive control. Peter Peregrinus Ltd, ISBN 0-86341-1932, London. Michalconok, G. et al. (1986). A control system of the electric drive, the copyright certificate USSR №1262676, Б.И. № .37, 1986. Michalconok, G. (2005). Synchronic filters at control of elastic plants, CO-MAT-TECH 2005. Pp 80-84, ISBN 80-227-2286-3, Trnava, 20-21 October 2005. Michalconok, G.; Bezak, T. (2009): Convergence analysis of the error compensation process in the phase control systems. In: Annals of DAAAM and Proceedings of DAAAM Symposium. - ISSN 1726-9679. - Vol. 20, No. 1 Annals of DAAAM for 2009 & Proceedings of the 20th international DAAAM symposium "Intelligent manufacturing & automation: Focus on theory, practice and education" 25 - 28th November 2009, Vienna, Austria. - Vienna: DAAAM International Vienna, 2009. ISBN 978-3-901509-70-4, s.0921-0922.