Optimal Control of a Multilayer Electroelastic Engine with a Longitudinal Piezoeffect for Nanomechatronics Systems

Abstract

A electroelastic engine with a longitudinal piezoeffect is widely used in nanotechnology for nanomanipulators, laser systems, nanopumps, and scanning microscopy. For these nanomechatronics systems, the transition between individual positions of the systems in the shortest possible time is relevant. It is relevant to solve the problem of optimizing the nanopositioning control system with a minimum control time. This work determines the optimal control of a multilayer electroelastic engine with a longitudinal piezoeffect and minimal control time for an optimal nanomechatronics system. The expressions of the control function and switching line are obtained with using the Pontryagin maximum principle for the optimal control system of the multilayer electroelastic engine at a longitudinal piezoeffect with an ordinary second-order differential equation of system. In this optimal nanomechatronics system, the control function takes only two values and changes once.

Full text

Article
Optimal Control of a Multilayer Electroelastic
Engine with a Longitudinal Piezoeffect for
Nanomechatronics Systems
Sergey M. Afonin
National Research University of Electronic Technology (MIET), 124498 Moscow, Russia; learner01@mail.ru
Received: 21 May 2020; Accepted: 25 November 2020; Published: 1 December 2020


Abstract:
A electroelastic engine with a longitudinal piezoeffect is widely used in nanotechnology for
nanomanipulators, laser systems, nanopumps, and scanning microscopy. For these nanomechatronics
systems, the transition between individual positions of the systems in the shortest possible time is
relevant. It is relevant to solve the problem of optimizing the nanopositioning control system with
a minimum control time. This work determines the optimal control of a multilayer electroelastic
engine with a longitudinal piezoeffect and minimal control time for an optimal nanomechatronics
system. The expressions of the control function and switching line are obtained with using the
Pontryagin maximum principle for the optimal control system of the multilayer electroelastic engine
at a longitudinal piezoeffect with an ordinary second-order differential equation of system. In this
optimal nanomechatronics system, the control function takes only two values and changes once.
Keywords:
multilayer electroelastic engine; longitudinal piezoeffect; optimal control; Pontryagin
maximum principle; nanomechatronics system; ordinary second-order differential equation;
control function; switching line
1. Introduction
An electroelastic actuator in the form piezo actuator is used for nanotechnology in
nanomanipulators, laser systems, nanopumps, scanning microscopy [
1
]. The piezo actuator is
used in photolithography for nano- and microdisplacements when aligning templates, in medical
equipment for precise instrument delivery during microsurgical operations, in optical-mechanical
devices, in adaptive optics systems, and in adaptive telescopes [
2
]. It is also used in stabilization
systems for optical-mechanical devices, systems for alignment and tuning of lasers, interferometers,
adaptive optical systems and fiber-optic systems for transmitting and receiving information [3].
The application of a multilayer electroelastic engine with a longitudinal piezoeffect is promising
for nanomanipulators in nanotechnology [
4
]. To increase the range of the displacement, a cellular
actuator and a multilayer electroelastic engine are used [
5
]. Mechatronics control systems with a
multilayer electroelastic engine with a longitudinal piezoeffect are used in precision engineering [6].
It is important to solve the problem of positioning a multilayer electroelastic engine with a
longitudinal piezoeffect from an arbitrary state with minimal control time [
7
] with the transition
between individual positions of the optimal control system in the shortest possible time. The Pontryagin
maximum principle is used to derive minimum time control. The prospects for constructing optimal
control systems for nanomechatronics are shown in work [8].
In this work using the Pontryagin maximum principle for optimal systems, which ensures
under optimal control the maximum of the Hamilton function and the minimum of the control time.
The expression of the control function is obtained, which has only two values and changes once.
Appl. Syst. Innov. 2020,3, 53; doi:10.3390/asi3040053 www.mdpi.com/journal/asi

Appl. Syst. Innov. 2020,3, 53 2 of 7
The switching line is obtained for the optimal control of a multilayer electroelastic engine with a
longitudinal piezoeffect in a nanomechatronics optimal system.
2. Optimal Control of a Multilayer Electroelastic Engine
2.1. Characteristics of a Multilayer Electroelastic Engine with a Longitudinal Piezoeffect
Let us consider a multilayer electroelastic engine with a longitudinal piezoeffect with one fixed
face in Figure 1, where Pis the polarization and 3 is the axis and
S3(t)
is the relative deformation along
axis 3 and tis time. The equation for the inverse longitudinal piezoeffect [9] has the form
S3(t)=d33E3(t)+sE
33T3(t)(1)
where
d33
is the longitudinal piezomodule,
E3(t)
is the electric field tension along axis 3,
sE
33
is the
elastic compliance at
E=const
, and
T3(t)
is the mechanic tension along axis 3. From the inverse
longitudinal piezoeffect Equation (1), the adjusting characteristic of a multilayer electroelastic engine is
found for elastic load in the form
ξ(t)
l=d33E3(t)−sE
33Ceξ(t)
S0
(2)
l=nδ,F(t)=Ceξ(t),E3(t)=U(t)/δ
where
ξ(t)
is the displacement of the face engine,
t
is time,
l
is the length of the engine,
n
is the number
of piezolayers,
δ
is the thickness of the piezolayer,
Ce
the stiffness of the load,
S0
is the area of the
engine, F(t)is the force, and U(t)is the voltage.
Apl. Sys. Innov. 2020, 3, x FOR PEER REVIEW 2 of 8
In this work using the Pontryagin maximum principle for optimal systems, which ensures
under optimal control the maximum of the Hamilton function and the minimum of the control time.
The expression of the control function is obtained, which has only two values and changes once. The
switching line is obtained for the optimal control of a multilayer electroelastic engine with a
longitudinal piezoeffect in a nanomechatronics optimal system.
2. Optimal Control of a Multilayer Electroelastic Engine
2.1. Characteristics of a Multilayer Electroelastic Engine with a Longitudinal Piezoeffect
Let us consider a multilayer electroelastic engine with a longitudinal piezoeffect with one fixed
face in Figure 1, where P is the polarization and 3 is the axis and 




tS3 is the relative deformation
along axis 3 and
t
is time. The equation for the inverse longitudinal piezoeffect [9] has the form

















+= tTstEdtS E
3333333 (1)
where 33
d is the longitudinal piezomodule, 




tE3 is the electric field tension along axis 3, E
s33 is
the elastic compliance at const=E, and 




tT 3 is the mechanic tension along axis 3. From the
inverse longitudinal piezoeffect Equation (1), the adjusting characteristic of a multilayer
electroelastic engine is found for elastic load in the form
0
33
333 S
tCs
tEd
l
te
E
















ξ
−=
ξ (2)
δ= nl , 










ξ= tCtF e, δ= 










tUtE3
where 





ξt is the displacement of the face engine,
t
is time, l is the length of the engine,
n
is the
number of piezolayers, δ is the thickness of the piezolayer, e
C the stiffness of the load, 0
S is the
area of the engine, 




tF is the force, and 




tU is the voltage.
Figure 1. The kinematic scheme of a multilayer electroelastic engine with a longitudinal piezoeffect.
The decisions the characteristics for the piezo engine are obtained in the works [10–17].
Therefore, for the multilayer electroelastic engine at longitudinal piezoeffect the adjusting
characteristic has the following form:

















=
+
=ξ tUb
CC
tnUd
tU
E
e
33
33
33
1 (3)
()
lsSC EE
33033 =,
()
E
e
UCCndbtUt 333333 1+==ξ 











where E
C33 and U
b33 are the stiffness of the multilayer electroelastic engine with a longitudinal
piezoeffect and the transfer coefficient, respectively. The multilayer piezo engine with a longitudinal
Figure 1. The kinematic scheme of a multilayer electroelastic engine with a longitudinal piezoeffect.
The decisions the characteristics for the piezo engine are obtained in the works [
10
–
17
]. Therefore,
for the multilayer electroelastic engine at longitudinal piezoeffect the adjusting characteristic has the
following form:
ξ(t)=d33nU(t)
1+Ce/CE
33
=bU
33U(t)(3)
CE
33 =S0/sE
33l,ξ(t)/U(t)=bU
33 =d33n/1+Ce/CE
33
where
CE
33
and
bU
33
are the stiffness of the multilayer electroelastic engine with a longitudinal piezoeffect
and the transfer coefficient, respectively. The multilayer piezo engine with a longitudinal piezoeffect
from a ceramic PZT at
d33
=4
×
10
−10
m/V,
n
=5,
CE
33
=12
×
10
7
N/m,
Ce
=0.6
×·
10
7
N/m,
U
=100 V
results in bU
33 =1.9 nm/V and ξ=190 nm.
In a control system the fransfer function
Wn(s)
multilayer electroelastic engine with a longitudinal
piezoeffect is determined in the form
Wn(s)=Ξ(s)
U(s)=bU
33
Tns+1(4)

Appl. Syst. Innov. 2020,3, 53 3 of 7
Tn=RCn
where
Ξ(s)
and
U(s)
are Laplace displacement
ξ(t)
and voltage
U(t)
in Figure 1,
s
is the transformation
operator,
Tn
is the electrical constant time of the multilayer electroelastic engine,
Cn
is the capacitance
of the multilayer electroelastic engine, and
R
is the matching circuit resistance. For the multilayer piezo
engine with a longitudinal piezoeffect from a ceramic PZT at
R
=10 kOm and
Cn
=0.5
µ
F, the electrical
constant time is Tn=5 ms.
Let us consider the multilayer electroelastic engine with a longitudinal piezoeffect and the electrical
constant time is much larger than the mechanical constant time [
10
] of the multilayer engine at an
elastic inertial load:
Tm=qM/Ce+CE
33
Tn>> Tm
where
Tm
and
M
are the mechanical constant time of the engine at an elastic inertial load and the mass
of the load, respectively. For the multilayer piezo engine with a longitudinal piezoeffect from a ceramic
PZT at an elastic inertial load
M
=0.3 kg,
CE
33
=12
×
10
7
N/m, and
Ce
=0.6
×·
10
7
N/m, the mechanical
constant time is Tm=0.05 ms.
Let us consider the optimal control system with the integrator in a series with the multilayer
electroelastic engine with a longitudinal piezoeffect. In this optimal control system, the integrator
transfer function Wi(s)has the form
Wi(s)=ki
s(5)
Tn>> Ti=1/ki
where kiis the transfer coefficient of the integrator.
Accordingly, for the control system with the integrator for the multilayer electroelastic engine
with a longitudinal piezoeffect, considering the electrical constant time of the multilayer electroelastic
engine and the transfer coefficient of the integrator, the ordinary second-order differential equation is
written as
Tn
d2ξ(t)
dt2+dξ(t)
dt =kU
33u(t)(6)
kU
33 =kibU
33,|u|≤u0(7)
where
ξ(t)
,
u(t)
, and
kU
33
are the displacement, the voltage, and transfer coefficient of the control system
of the multilayer electroelastic engine with a longitudinal piezoeffect, respectively.
For the ordinary second-order differential Equation (6), therefore, the system of differential
equations in Cauchy form is obtained,







dξ1
dt =ξ2
dξ2
dt =−1
Tnξ2+kU
33
Tnu(8)
where
ξ1=ξ
and
ξ2=.
ξ1
as the state variables are selected the displacement and the velocity of the
multilayer electroelastic engine with a longitudinal piezoeffect.
2.2. Application of Pontryagin Maximum Principle for Decision Optimal Control of a Multilayer Electroelastic
Engine
Using the Pontryagin maximum principle [
18
], the control law ensures the transition of the
multilayer electroelastic engine from any initial state to a given end point at a minimum time.

Appl. Syst. Innov. 2020,3, 53 4 of 7
The end point is the origin of coordinates. The Hamilton function for this maximum principle has the
following form
max
uH=max
u





ψ1ξ2+ψ2





−1
Tn
ξ2+kU
33
Tn
u









(9)
where ψ1and ψ2are the functions.
Therefore, for the Hamilton function [19], the system of equations has the form







dψ1
dt =−∂H
∂ξ1=0
dψ2
dt =−∂H
∂ξ2=−ψ1+1
Tnψ2
(10)
The solution to this system of equations has the form







ψ1=A1=const
ψ2=−A1t+A2et
Tn(11)
Therefore, the function
ψ2
changes once the sign (see Figure 2), since the straight line and the
exponent intersect once. According to the maximum principle, from (9) the control signal has two
values, which equal in absolute value and opposite in the sign. The sign of the control signal coincides
with the sign of this function ψ2on Figure 2. The control function uhas the form
u=u0sign ψ2(12)
where u0is the control signal amplitude.
In this optimal system with an ordinary second-order differential equation, in Figure 2,
u
takes
only two values, u0and −u0, and changes once.
The trajectory of the control system of the multilayer electroelastic engine with a longitudinal
piezoeffect in the phase plane consists of two sections, the initial and the final, with the latter passing
through the origin and acts as the switching line. The switching line of the multilayer electroelastic
engine is defined on the phase plane.
Apl. Sys. Innov. 2020, 3, x FOR PEER REVIEW 5 of 8
Figure 2. The control function in the optimal system.
The solution for the system of Equation (8) for const=u has the form





+=ξ
+−=ξ
−
−
n
T
t
U
n
T
t
n
U
eBuk
BeBTutk
2332
12331 , (13)
when the initial location of the system is at the origin and when the time is
t
−=τ , the constants are
found in the form



−=
−=
ukB
uTkB
U
n
U
332
331 , (14)
then, the system of equations is determined in the form















−=ξ
−+τ−=ξ
τ
τ
.euk
uTkeuTkuk
n
T
U
n
U
n
T
n
UU
1
332
3333331
. (15)
From (15), the equation has the form
uk
eU
n
T
33
2
1ξ
−=
τ
. (16)
Accordingly, the time is found in the form







ξ
−=τ uk
TU
n
33
2
1ln . (17)
At the parameters 1
ξ = 0 and 2
ξ = 0, the time is obtained 0=τ .
Therefore, the equation for the final portion of the movement trajectory of the multilayer
electroelastic engine with a longitudinal piezoeffect is obtained on the phase plane in the form of the
switching line:
Figure 2. The control function in the optimal system.

Appl. Syst. Innov. 2020,3, 53 5 of 7
The solution for the system of Equation (8) for u=const has the form









ξ1=kU
33ut −TnB2e−t
Tn+B1
ξ2=kU
33u+B2e−t
Tn
, (13)
when the initial location of the system is at the origin and when the time is
τ=−t
, the constants are
found in the form 






B1=−kU
33uTn
B2=−kU
33u, (14)
then, the system of equations is determined in the form







ξ1=−kU
33uτ+kU
33uTne
τ
Tn−kU
33uTn
ξ2=kU
33u1−e
τ
Tn.. (15)
From (15), the equation has the form
e
τ
Tn=1−ξ2
kU
33u. (16)
Accordingly, the time is found in the form
τ=Tnln




1−ξ2
kU
33u




. (17)
At the parameters ξ1=0 and ξ2=0, the time is obtained τ=0.
Therefore, the equation for the final portion of the movement trajectory of the multilayer
electroelastic engine with a longitudinal piezoeffect is obtained on the phase plane in the form of the
switching line:
ξ1=−kU
33uTnln




1−ξ2
kE
33u




+kU
33uTn




1−ξ2
kE
33u





−kU
33uTn. (18)
From (8), the control function is found in the form
u=−u0sign ξ2. (19)
The switching line or the last section of the trajectory divides the phase plane into two regions at
s>
0 and
s<
0, When each area of management is constant, a change in the control occurs at
s=
0.
Accordingly, we choose a control for each region so that the point on the phase plane moves toward
the switching line.
Then we obtain the expression (19) for the equation into Equation (18) for the last section of the
movement trajectory of the multilayer electroelastic engine with a longitudinal piezoeffect. Accordingly,
the equation for the last section of the movement trajectory is obtained in the following form:
s(ξ1,ξ2)=ξ1−kU
33u0Tnsignξ2ln




1+|ξ2|
kU
33u0





+Tnξ2=0. (20)
Therefore, the equation for the last section of the trajectory has the form
ξ1=kU
33u0Tnsignξ2ln




1+|ξ2|
kU
33u0






−Tnξ2. (21)

Appl. Syst. Innov. 2020,3, 53 6 of 7
Then the inequality is obtained in the form
d

s(ξ1,ξ2)


dt ≤0. (22)
Therefore, the inequality is written as
ds(ξ1,ξ2)
dt signs(ξ1,ξ2)≤0. (23)
Substituting expression (21) for the function
s(ξ1,ξ2)
into expression (23) and considering system
(8), the condition has the form
ξ2kU
33u0
kU
33u0+|ξ2|signs+|ξ2|kU
33
kU
33u0+|ξ2|usigns≤0 (24)
The second term of expression (24) depends on the control. Accordingly, condition (24) is satisfied
under the control of the form
u=−u0signs(25)
Therefore, in this system with an ordinary second-order differential equation, the optimal control
takes only two values and changes once. The expressions of the control function and switching line
are obtained with using the Pontryagin maximum principle for the optimal control system of the
multilayer electroelastic engine at a longitudinal piezoeffect with an ordinary second-order differential
equation of system.
3. Conclusions
The optimal control of the multilayer electroelastic engine with a longitudinal piezoeffect and
a minimal control time is obtained for nanomechatronics systems. The expressions of the control
function and switching line on the phase plane are determined for the optimal control of this multilayer
electroelastic engine. Using the Pontryagin maximum principle, the control law is obtained for
the transition of the multilayer electroelastic engine from any initial state to a given end point in a
minimal amount of time. The optimal control of the multilayer electroelastic engine with an ordinary
second-order differential equation takes only two values and changes once.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
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