In the talk, we discuss the development of the homogenized energy model (HEM) and the surrogate model dynamical mode composition for a PZT bimorph actuator used for micro-air vehicles including Robobee. HEM quantifies the nonlinear, hysteretic, and rate-dependent behavior inherent to PZT in highly dynamic operating regimes. Due to computation complexity of HEM, we must develop a surrogate model. The surrogate model must be parameter and control dependent to be able to perform inverse problems or uncertainty quantification in different driving regimes. In the literature, DMD can be adapted to handle different control inputs. We will discuss using interpolation over the parameters to adapt the DMD to include parameter dependence. Finally, we will discuss control design using the surrogate model and quantifying the uncertainty in the controls.

1. Parameter-dependent surrogate model
development and control design for PZT
bimorph actuators employed for micro-
air vehicles
Nikolas Bravo
Dr. Ralph Smith
May 15, 2019
Supported by AFOSR Grant FA9550-15-1-0299 and
SAMSI Model Uncertainty: Mathematical and Statistical Program 1

2. Motivation
Left image from: R. Wood, E. Steltz, and R. Fearing, Nonlinear performance limits for high energy density
piezoelectric bending actuators, Proceedings of the 2005 IEEE International Conference on Robotics and
Automation, (2005), pg. 3633-3640.
• Micro-air vehicle
• Autonomous pollination
• Search and rescue
• Hazardous areas
• PZT Bimorph
2

3. Behavior of PZT
Right image from: R. Smith and Z. Hu, The homogenized energy model for characterizing polarization and strains in
hysteretic ferroelectric materials: Material properties and uniaxial model development, Journal of Intelligent Material
System and Structures, 23 (2012), pp. 1833-1867.
• Hysteretic
• Rate dependent
• Creep, saturation, and stress-
based effects
3

4. Homogenized Energy Model
↵ = ±180
↵ = 90
Domain level:
P↵ = d↵ + E + P↵
R
"↵ = sE
↵ + d↵E + "↵
R
Grain level:
P(E, ) = E +
X
↵
(d↵ + P↵
R)x↵(E, )
"(E, ) = sE +
X
↵
(d↵E + "↵
R)x↵(E, )
4
P(E, ) = d(E, ) + E + Pirr(E, )
"(E, ) = sE + d(E, )E + "irr(E, )
Constitutive relations:
Pb
O
Ti
Unit Cell:
PZT Patch

5. Beam Model
• Euler-Bernoulli cantilever beam:
⇢(x)
@2w(x, t)
@t2
+
@w(x, t)
@t
@2M(x, t)
@x2
= 0,
w(0, t) =
@w
@x
(0, t) = M(L, t) =
@M
@x
(L, t) = 0
5
x1 = 1 mm
x2 = 5.5 mm
x3 = 6.5 mm
x4 = 10 mm
hcf = .1778 mm
hs = 31.75 µm
hpzt = 0.127 mm
b0 = 1.5 mm
b1 = 1 mm

6. Nominal Parameters
Air damping coe cient 0.0015
⇢cf Density of the CF layer (kg/m3) 2.2743e3
⇢s Density of the S2 Glass (kg/m3) 1.8133e3
⇢pzt Density of the PZT actuators (kg/m3) 1.2158e3
Ycf Elastic modulus of CF (Pa) 4.5715e11
Ys Elastic modulus of S2 Glass (Pa) 8.6469e11
ccf Damping coe cient for CF 1.2479e4
cs Damping coe cient for S2 Glass 6.6649e3
cpzt Damping coe cient for PZT 1.3216e3
sE Elastic compliance (1/Pa) 1.1159e-11
d± Piezoelectric coupling coe cient for ↵ = ± (m/V) 8.7867e-10
"±
R Remanent strain for ↵ = ± (%) 0.1771
"90
R Remanent strain for ↵ = 90 (%) -8.1179e-13
P±
R Remanent polarization for ↵ = ± (C/m2) 0.1208
Ferroelectric susceptibility (F/m) 1.2373e-6
⌧90 Relaxation time for 90 switching (s) 1.5013e-6
⌧180 Relaxation time for 180 switching (s) 9.2519e-13
pzt Inverse of relative thermal energy (m3/J) 0.0853
6

7. Model Fit
Field [MV/m]
-1.5 -1 -0.5 0 0.5 1 1.5
TransverseDisplacement[µm]
-300
-200
-100
0
100
200
300
Data
Model
Data: Wood, R., Steltz, E., and Fearing, R., “Nonlinear performance limits for high
energy density piezoelectric bending actuators,” Proceedings of the 2005 IEEE
International Conference on Robotics and Automation , 3633–3640 (April 2005).
7

8. Dynamic Mode Decomposition
• Koopman operator:
• Construct
• The columns of 𝑋"
#$"
are a Krylov subspace, so
• We can write
8
xk+1 = Axk
XM 1
1 = [x1, x2, ..., xM 1] = [x1, Ax1, ..., AM 2
x1]
xM =
M 1X
k=1
skxk + r
XM
2 = XM 1
1
˜A + re(M 1)
Schmid, P. Dynamic mode decomposition of numerical and experimental data. Journal
of Fluid Mechanics, Cambridge University Press (CUP), 2010, 656 (August), pp.5-28.

9. Dynamic Mode Decomposition with Control
• Practically:
• For control problem,
9
A ⇡ XM
2 (XM 1
1 )†
xk+1 = Axk + Buk :
XM
2 = [A B]

XM 1
1
U
=) [A B] ⇡ XM
2

XM 1
1
U
†
Time [s]
0 0.5 1 1.5 2 2.5 3 3.5
TransverseDisplacement[µm]
-150
-100
-50
0
50
100
150
Model
DMD
Proctor, J. L., Brunton, S. L., and Kutz, N. J., 2016. “Dynamic mode decomposition with control”. SIAM J. Applied
Dynamical Systems, 15(1), pp. 142–161.

10. DMD Extension
xk+1 = A(q)xk + B(q)uk
10
• Consider:
• Let:
where 𝐴" = 𝐴(𝑞") and 𝑞" ∈ 𝑹,
• For a single parameters with 𝑞- ≤ 𝑞∗ ≤ 𝑞-0",
A = {A1
, A2
, A3
. . . AN
}
A⇤
ij ⇡ fkij
(q⇤
) =
3X
h=0
↵
kij
h
✓
q⇤ qk
qk+1 qk
◆h

11. DMD Extension
• Used MATLAB's interpn and griddatan functions to
interpolate over multiple parameters.
• Clenshaw-Curtis Grid:
• Total number of nodes for tensored grids is
11
qr
` =
1
2
✓
1 cos
✓
⇡(r 1)
2` 1
◆◆
, r = 1, . . . , 2` 1
+ 1
(2` 1
+ 1)p

12. Sparse Grid
12
0 0.2 0.4 0.6 0.8 1
p
2
0
0.2
0.4
0.6
0.8
1
p
1
Lobatto Sparse Grid (5th level-401 nodes): Latin Hyper Cube (200 nodes):
-1 -0.5 0 0.5 1
p1
-1
-0.5
0
0.5
1
p
2

13. DMD with Parameters Results
13
0 0.05 0.1 0.15 0.2
Time [s]
0
50
100
150
200
250
TransverseDisplacement[µm]
Model
DMD
DMDpInpol
0.072 0.074 0.076 0.078 0.08 0.082 0.084 0.086 0.088 0.09
Time [s]
60
80
100
120
140
160
180
200
TransverseDisplacement[µm]
Model
DMD
DMDpInpol
Parameters:
pzt, P±
R , d±, "±
R

14. DMD with Parameters Results
14
0.1 0.12 0.14
P
R
±
0
0.05
0.1
0.15
0.2
||yHEM
-yDMDip
||∞
/||yHEM
||∞
1 1.5 2 2.5
ϵ
R
±
×10-3
0
0.05
0.1
0.15
0.2
||y
HEM
-y
DMDip
||
∞
/||y
HEM
||
∞
0.5 1 1.5
d
± ×10
-9
0
0.05
0.1
0.15
0.2
||yHEM
-yDMDip
||∞
/||yHEM
||∞
0.06 0.08 0.1 0.12
γ
pzt
0
0.05
0.1
0.15
0.2
||y
HEM
-y
DMDip
||
∞
/||y
HEM
||
∞
0.1 0.12 0.14
P
R
±
0.05
0.1
0.15
0.2
||y
HEM
-y
DMDip
||
∞
/||y
HEM
||
∞
1 1.5 2 2.5
ϵ
R
±
×10-3
0.05
0.1
0.15
0.2
||y
HEM
-y
DMDip
||
∞
/||y
HEM
||
∞
0.5 1 1.5
d
± ×10
-9
0.05
0.1
0.15
0.2
||y
HEM
-y
DMDip
||
∞
/||y
HEM
||
∞
0.06 0.08 0.1 0.12
γ
pzt
0.05
0.1
0.15
0.2
||y
HEM
-y
DMDip
||
∞
/||y
HEM
||
∞
0.1 0.12 0.14
P
R
±
0.05
0.1
0.15
0.2
||y
HEM
-y
DMDip
||
∞
/||y
HEM
||
∞
1 1.5 2 2.5
ϵ
R
±
×10
-3
0.05
0.1
0.15
0.2
||yHEM
-yDMDip
||∞
/||yHEM
||∞
0.5 1 1.5
d
± ×10
-9
0.05
0.1
0.15
0.2
||y
HEM
-y
DMDip
||
∞
/||y
HEM
||
∞
0.06 0.08 0.1 0.12
γ
pzt
0.05
0.1
0.15
0.2
||yHEM
-yDMDip
||∞
/||yHEM
||∞
Clenshaw-Curtis :
Sparse: Latin:

15. DMD with Parameters Results
15
0.1 0.12 0.14
PR
±
0
0.01
0.02
0.03
||yDMD
-yDMDip
||∞
/||yHEM
||∞
1 1.5 2 2.5
ϵR
±
×10-3
0
0.01
0.02
0.03
||yDMD
-yDMDip
||∞
/||yHEM
||∞
0.5 1 1.5
d± ×10-9
0
0.01
0.02
0.03
||yDMD
-yDMDip
||∞
/||yHEM
||∞
0.06 0.08 0.1 0.12
γpzt
0
0.01
0.02
0.03
||yDMD
-yDMDip
||∞
/||yHEM
||∞
Clenshaw-Curtis :
0.1 0.12 0.14
PR
±
0
0.005
0.01
0.015
||y
DMD
-y
DMDip
||
∞
/||y
HEM
||
∞
1 1.5 2 2.5
ϵR
±
×10
-3
0
0.005
0.01
0.015
||yDMD
-yDMDip
||∞
/||yHEM
||∞
0.5 1 1.5
d± ×10
-9
0
0.005
0.01
0.015||y
DMD
-y
DMDip
||
∞
/||y
HEM
||
∞
0.06 0.08 0.1 0.12
γpzt
0
0.005
0.01
0.015
||yDMD
-yDMDip
||∞
/||yHEM
||∞
Latin:
0.1 0.12 0.14
PR
±
0
0.005
0.01
0.015
||yDMD
-yDMDip
||∞
/||yHEM
||∞
1 1.5 2 2.5
ϵR
±
×10
-3
0
0.005
0.01
0.015
||y
DMD
-y
DMDip
||
∞
/||y
HEM
||
∞
0.5 1 1.5
d± ×10
-9
0
0.005
0.01
0.015
||yDMD
-yDMDip
||∞
/||yHEM
||∞
0.06 0.08 0.1 0.12
γpzt
0
0.005
0.01
0.015
||y
DMD
-y
DMDip
||
∞
/||y
HEM
||
∞
Sparse:

16. Timing of DMDip
Method Time (s)
Model 65.4
DMD 0.2
DMDip-Uniform 1.1
DMDip-Sparse Grid 24.5
Latin Hypercube 18.6
16

17. Optimal Tracking Problem
• Cost functional:
• Solution:
17
J0 =
1
2
(CxN rN )T
P(CxN rN )+
1
2
N 1X
k=1
(Cxk rk)T
Q(Cxk rk)+ukRuk
xk+1 = Axk + Buk,
uk = F1xk + Fv
1vk+1,
F1 = (BT
S1B + R) 1
BT
S1A,
Fv
1 = (BT
S1B + R) 1
BT
,
S1 = AT
S1A AT
S1B(BT
S1B + R) 1
BT
S1A + CT
QC,
vk = (A BF1)vk+1 + CT
Qrk.
0 0.05 0.1 0.15 0.2
Time [s]
0
20
40
60
80
100
120
TransverseDisplacement[µm]
tracking function
yobs

18. Optimal Tracking Plus PI Control
• Control Design:
18
uk = uopt
k (105
)ei (108
)
Z t
0
ei(s)ds

19. Conclusion
• We are able to use DMD to develop a robust control for
the actuator.
• We extended DMD to be parameter dependent.
• Quantify the uncertainty of the model and parameters
using the DMD surrogate model.
• Investigate sparse grid algorithms to improve
performance. 19
Future Work

20. Acknowledgements
• I was supported by the Air Force Office of Scientific
Research on AFOSR Grant FA9550-15-1-0299.
• This material was based upon work partially supported
by the National Science Foundation under Grant DMS-
1638521 to the Statistical and Applied Mathematical
Sciences Institute. Any opinions, findings, and
conclusions or recommendations expressed in this
material are those of the author(s) and do not
necessarily reflect the views of the National Science
Foundation.
20