CC BY
121
YflK 621
PARAMETRIC IDENTIFICATION IN MECHANICAL SYSTEMS
V.S. Lovejkin, Doctor of Technical Sciences, Pr.,
Y.V. Chovnjuk, Candidate of Technical Sciences, Assoc. Prof., M.G. Dikterjuk, Candidate of Technical Sciences, Assoc. Prof., Kyiv National University of Construction and Architecture
Summary. Parametric identification in mechanical systems has been offered in order to improve mechatronic models.
Key words: parametric identification, mechanical systems.
Research objectives
It’s necessary to define the equivalent disturbance in order to consider the robust control of motion actuated by electric motor. The explanation and the interpretation of robustness and stiffness in motion control lead to definition of disturbance. The general definition for singleinput and single-output (SISO) linear system is discussed.
There are various proposals to estimate the disturbance d. This paper introduces a disturbance observer. Gopinath’s method is a systematic way to construct such an observer [1]. Once d is estimated, the input of the mechanical system u (t) will be sum of two parts:
U = uref +udls.
(1)
The first term in the right side is a driving input to excite the system. The second term is a compensation to suppress the equivalent disturbance and the system acquires robustness. To cancel the equivalent disturbance, the compensation input is made by using the estimated equivalent-disturbance d. Since d will be delayed by the lag poles in the disturbance observer, the compensation of the equivalent disturbance will be also delayed by the same amount. It’s possible to design such delay as small as possible not to make robust stability deteriorate. The compensation input udls will change the original system into the nominal system without any disturbance.
It is noted that the design of uref comes from the motion reference generator.
Generally total controller will have cascade of the outer loop to bring the desired output and the inner loop by disturbance observer.
The previous design method is applied to the motion system described by:
rda>
jZr = KtIaref ~T!-dt
(2)
Here J - inertia; Kt - torque coefficient of electric motor; 1) - load torque. This approach is successes to design robust motion controller of mechanical system (such as building machine) as well.
The disturbance is load torque. The parameter variations are the change of inertia and the change of torque constant of motor. The output is position detected by position detector. The equivalent disturbance is:
; T,
= —L + J
KL_K1 J J,
\
J ref
(3)
o y
Suppose the first derivative of d is zero. An augmented state equation is:
d
0 “0 10“ 0
CD = 0 0 1 CD +
d 0 0 0 d
K*
Jo
0
ref
(4)
By Gopinath’s method, the following estimation process is obtained:
= £j9 + Zj.
(5)
Zj should satisfy (6), where kx and k2 are free parameters:
d(j> _ ref _
J. to a
dt
-U + uiSL-uy,
ref
. (11)
The second term of (11) is the disturbance torque Td,s
Tdls=Tl+M^-AKtIa. (12) dt
Comparing (2), (3), and (12), the following equation holds:
d_
dt
1 JX) 1 1 O 1 1 1 Jxi 1
3_ 1 1 1 3_
—k^Q
(kl-k22)B + ^-Iaref
J n
Tdls c
(13)
I’Jis contains: 1) mechanical load ( = 2]); 2) va-(6) ried self-inertia torque = AJ(d<x>/dt) ; 3) torque ripple from motor (= AKtIa).
Equations (4) and (6) lead (7):
s ~\~ k2s + k^
s2e——iref
k
s 4- k2s 4- k|
Jn
■d.
(7)
Two poles of the observer for the 5 - parameter of transfer function between input U(5) and output Y(s) are a and P, which are arbitrarily allocated in the complex plane. They satisfy (8):
[a +13 = -k2 I aP = ^j.
(8)
It is worthwhile reconsidering (2). The parameters in (2) are the inertia and the torque coefficient. The inertia will change according to the mechanical configuration of motion system. The torque coefficient will vary according to the rotor position of electric motor due to irregular distribution of magnetic flux on the surface of rotor:
J = J0 + AJ
Kt ~Kto +AKt.
(9)
(10)
By substituting (9) and (10) into (2), (11) holds:
The robust motion controller is designed to cancel the disturbance torque as quickly as possible.
The estimated disturbance torque is obtained from the position 9 and the current reference. According to (1) and:
dis
u =-■
Kr
(14)
compensation input is as follows:
1
jdis 'A j
Kt.
T
dis ■
(15)
The schematic block diagram of robust motion controller has an integrator with high gain equivalently in the forward path. Therefore, the robust motion controller eliminates steady state error.
Equation (7) shows that the disturbance is estimated through low-pass filter. Generally, there is such a low-pass filter in the observer structure. The poles of the observer determine the delay of the low-pass filter GT (s). GT (s) gives a certain effect to the control performance.
The disturbance observer in motion system (a robust motion controller) may be transformed into an acceleration controller as well. Such transformation is possible due to its ability to
a
clarify the feedback effect of the disturbance. If there is no delay in the estimation process, the disturbance is completely canceled out. In fact, since there is definitely some time-delay in the estimation process, the controlled system is not robust in high frequency range determined by 1 -GT (5) = Gs (5). Gs (5) is called a sensitivity function which shows a performance limit of robust control in high frequency range. In most of low frequency area covered by GT (s), the motion system is robust. The transformation: a robust motion controller —> an acceleration controller - shows another interpretation. It’s possible to select nominal inertia and nominal torque coefficient as unity. This case shows that a current reference is also an acceleration reference.
The paper reaches a result that robust motion controller makes a motion system (for example, building machine) to be an acceleration control system. The result implies a versatility of robust motion controller for both position and force control. If position signal is fed back, a high-gain feedback in the robust controller makes stiffness very high. On the contrary, only pure force error feedback makes total stiffness zero since there is no gain to the position.
In the above discussion, the disturbance estimated by (7) is used for a realization of robust mechanical system. In the actual application, the estimated disturbance is effective for not only the disturbance compensation but also the parameter identification in the mechanical system. As defined in (3), the equivalent disturbance d , which is estimated by the disturbance observer, includes the load torque T and the parameter
variation torque (Kt/J)-(Kto/J0) //e/ . Here
the load torque T consists of friction and external force effects in the mechanical system as follows:
= T/nction + T/nctwn co + Text, (16)
where jfnctwn _ C0L1i0mb friction effect, jJnction(S)_visCOsity friction effect, ^-external force effect.
This equation means that the output of the disturbance observer is only the friction effect under the constant angular velocity motion.
This feature makes it possible to identify the friction effect in the mechanical system. The friction effects are well identified in many cases as Stribeck friction model [2]. The external force effect is also identified by using the estimated disturbance. Here it is assumed that the friction effects are known beforehand by the above identification process. By implementing the angular accelerated motion, the system parameter Kto / J0 is adjusted in the observer design so that it is close to the actual value Kt / J . As a result, the disturbance observer estimates only the external force effect as follows:
d = —:——---------------------d\ =
s + k2s + kx 'Kto lJa ->£< u0
к T
_________1_____ external (17)
Conclusions
The identified external force is applicable to sensorless force feedback control in mechanical system (for example, building machines) [3] and is utilized for a realization of mechanical vibration control as well.
References
1. Gopinath B. On the control of linear multiple
input-output systems//Bell System Tech. J. - 1971. - Vol. 50. - №. 3. - P. 1063-1081.
2. Southward S., Radcliffe C., MacCluer C. Ro-
bust nonlinear stick-slip friction compensation // ASME J. Dynamics Syst., Measurement, Contr. - 1991. - Vol. 113. -P.639-645.
3. Murakami T., Ohnishi K. Torque sensorless
control in multi-degree-of-freedom manipulator // IEEE Trans. Ind. Electron. -1993. - Vol. 40. - № 2.
Рецензент: О.П. Алексеев, профессор, д.т.н., ХНАДУ.
Статья поступила в редакцию 20 июня 2007 г.
