Frequency method for determination self oscillations in control systems with a piezo actuator for astrophysical research

Abstract

For the control system with a piezo actuator in astrophysical research the condition for the existence of self-oscillations is determined. Frequency method for determination self-oscillations in control systems is applied. By using the harmonious linearization of hysteresis and Nyquist stability criterion the condition of the existence of self-oscillations is obtained.

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Introduction
A piezo actuator is used in astrophysics for image stabilization and
scan system.1–19 Frequency method for determination self-oscillations
in scan system is applied.20–46 for Nyquist stability criterion of self-
oscillations at harmonious linearization of hysteresis characteristic of
a piezo actuator.
Condition of self-oscillations
The scan system with a piezo actuator is used for astrophysical
research in system adaptive optics. Nyquist stability criterion of self-
oscillations at harmonious linearization of hysteresis characteristic2, 20–40
of a piezo actuator has the form
( )
( )
max
1
l gm
W WE
α
Ω=−
where
α
is the imaginary unit,
Ω
- the frequency of self-
oscillations,
( )
Ωα
l
W
- the frequency transfer function of the linear
part,
( )
maxmg
EW
- the transfer function of the hysteresis part,
maxm
E
- amplitude of the electric eld strength for
m
axis.
For the scan system with a piezo actuator for astrophysical research
the condition of self-oscillations is written
( )
( )
max
10
l gm
W WE
α
+Ω =
.
The condition of self-oscillations is determined in the form
( )
( )
max
1
mg
lEW
W
−=Ωα
here the left side of this equation has the form of the amplitude-
phase characteristic of the linear part of the system, and the right
side of the equation has the form of the inverse amplitude-phase
characteristic of the hysteresis link of the piezo actuator with the
inverse sign minus.
Preisach hysteresis function a piezo actuator has the form22-40
( )
[ ]
mi
t
mi
E,S,t,EFS 
sign0
0
=
here
t
,
i
S
,






0
i
S
, m
E
and
m
E

sign
- the time, the deformation,
the initial deformation,, the strength of electric eld and the sign
velocity.
The symmetric hysteresis the deformation22-40 a piezo actuator has
the form
2
max 2
max
1 sign
n
m
i mi m mi m m
m
E
S dE E E
E
γ

=−−




02 0
,maxmi mi mi m mi i m
d d aE S E
γ
=+=
here ,
mi mi
d
γ
,
0
i
S
,
n
- the piezo module, the hysteresis
coecient, the relative deformation for
0=
m
E
, and the power 1,
2, 3, ….
The transfer function of the linear part of the scan system with a
piezo actuator for elastic-inertia load22, 37-46 has the form
( )
22 21
l
l
t tt
k
Wp Tp T p
ξ
=++
After transformations we have this condition for the scan system
with the PZT actuator at the power
1=n
in the form
( )
22 02
max
11
8
12
3
mi
t tt mi mi m
ll
TTd aE
kk
γ
ξα
απ
=
−Ω Ω−+ +
+
here
4
3
mi l
tt
k
T
γ
πξ
Ω=
For the scan system with the PZT actuator
l
k
= 3.2⋅108 V/m,
0
33
d
=
4⋅10-10 m/V,
33
γ
= 0.8⋅10-10 m/V,
33
a
= 3.1⋅10-22 m3/V3,
t
T
= 10-3 s,
t
ξ
= 10-2 the frequency is determined
Ω
=1.1⋅103 s-1 with error of 10 %.
The frequency transfer function of the symmetric hysteresis the
deformation of a piezo actuator is received in the form
( ) ( ) ( )
maxmaxmax mmmimg EEESEW =
then
( ) ( ) ( )
max max maxg m mi m mi m
WE qE qE
α
′
= +
Aeron Aero Open Access J. 2024;8(2):115‒117. 115
©2024 Afonin. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and build upon your work non-commercially.
Frequency method for determination self-
oscillations in control systems with a piezo actuator
for astrophysical research
Volume 8 Issue 2 - 2024
Afonin SM
National Research University of Electronic Technology, Russia
Correspondence: Afonin SM, National Research University of
Electronic Technology MIET, Moscow, Russia,
Email
Received: June 20, 2024 | Published: July 02, 2024
Abstract
For the control system with a piezo actuator in astrophysical research the condition for
the existence of self-oscillations is determined. Frequency method for determination self-
oscillations in control systems is applied. By using the harmonious linearization of hysteresis
and Nyquist stability criterion the condition of the existence of self-oscillations is obtained.
Keywords: frequency method, control system, piezoactuator, hysteresis, self-oscillations,
astrophysical research
Aeronautics and Aerospace Open Access Journal
Research Article Open Access

Frequency method for determination self-oscillations in control systems with a piezo actuator for
astrophysical research 116
Copyright:
©2024 Afonin
Citation: Afonin SM. Frequency method for determination self-oscillations in control systems with a piezo actuator for astrophysical research. Aeron Aero Open
Access J. 2024;8(2):115‒117. DOI: 10.15406/aaoaj.2024.08.00198
For
1=n
( ) ( )
,
max max
42 8
33
mi mi
mi m mi mi m
qE dqE
γγ
ππ
⋅⋅
′
= =−=−
⋅
For
2=n
( ) ( )
,
max max
424 32
3 5 15
mi mi
mi m mi mi m
qE dqE
γγ
ππ
⋅⋅⋅
′
= =−=−
⋅⋅
For
3=n
( ) ( )
,
max max
4 2 4 6 192
3 5 7 105
mi mi
mi m mi mi m
qE dqE
γγ
ππ
⋅⋅⋅⋅
′
= =−=−
⋅⋅⋅
For
n
to
1+n
( ) ( ) ( )
,
max ( ) max ( 1) max
2
21
mi m mi mi n m mi n m
n
q E dq E q E
n−
′′
= = +
For
1+n
( ) ( )
( )
,
max max
4 2 4 6 ... 2
3 5 7 ... 2 1
mi
mi m mi mi m
n
qE dqE n
γ
π
⋅⋅⋅⋅⋅ ⋅
′
= = − ⋅⋅⋅⋅ ⋅ +
The stability criterion and frequency method are used.
Discussion
By using of frequency method the paramete rs of self-oscillations
are obtained in the scan system. Nyquist stability criterion is used
for calculation the self-oscillations in the control system with a piezo
actuator at harmonious linearization of hysteresis characteristic of a
piezo actuator.
Conclusion
For the scan system its condition of self-oscillations is determined.
For calculation the self-oscillations frequency method is applied at
harmonious linearization of hysteresis characteristic of a piezo actuator.
Acknowledgments
None.
Conicts of interest
The authors declare that there is no conict of interest.
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