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An investigation on physical limitations in vibration energy
harvesting
Rafael Rojas
Profesore guida: Antonio Carcaterra
Relatore: Aldo Sestieri
July 23, 2015

Index
1 Introduction
2 A survey on Krotov's method
3 Application to Piezo electric EH.
4 Fluttering Wing
5 Capacitor EH.
6 Conclusions
An investigation on physical limitations in vibration energy harvesting July 23, 2015 2 of 40

Outline
1 Introduction
4 Fluttering Wing
5 Capacitor EH.
6 Conclusions

Introduction
EH: Extracting environmental energy using non conventional
devices or sources .
Thermal.
Solar.
Electromagnetic.
Vibrations.
EH can be use to supply dierent power requirements.
Low power electronics , from acoustics/vibrations, solar,
electromagnetic sources: 10
−3W.
High power electronics , sea waves, wind, etc. : 10
6W.

Motivation
Upper bounds to energy extraction is a key problem in history of
technology.
Optimal Control Theory (OCT) has good chances to produce
technological advances in this eld.
In fact, some adjustable parameters of the EH plant can be
requested to track the best energy storage performance, and this is
possible with actual technologies.
Our scope is to control of device-environment interaction to
maximize energy extraction.

Prototype Equation
System Dynamics.
n DOF linear oscillator.
Internal dissipation.
u, controllable parameters.
Controllable dissipation term.
M¨q + C0 ˙q + Kq + C(u)˙q = f
Key Performance Index.
Running Cost: Power + Control
eort term
Final cost.
I =
T
0
˙q C(u)˙q − u Ru dt
I = I + F(T)
Objective
Control the EH process, to maximize our KPI.

Traditional OCT Approaches
Pontraying Maximum Principle (1960's)
Lagrange Equations + Constrains + Maximum principle.
LQR.
LQG.
Hamilton Jacobi Bellman (1960's)
Hamilton-Jacobi equation of CV with constraints.
Direct Methods
Algebraic Optimization.
Model Predictive Control, (MPC).

Key Problems
Local max− min (PMP and Direct Methods).
Mixed boundary conditions.
Dimensionality Curse (HJB).
Closed-Open loop dichotomy.
Possible singularities / degeneracy (singular arcs in PMP).

Krotov's Method
No engineering applications developed, but this OCT method
combined with EH has advantages:
Absolute maximum energy storages is attacked.
Benchmarking of operating EH devices in actual use.
No mixed boundary conditions (we only solve i.v. ODE's).
Absolute min− max (very good).
Elimination of certain types of singularities/degeneracies (singular
arcs).

Outline
1 Introduction
4 Fluttering Wing
5 Capacitor EH.
6 Conclusions

Krotov Iteration Method
Using an arbitrary rst guest control law u0 we will generate the
state trajectory x0 starting at z.
System
u0 x0

Krotov Iteration Method
Using an arbitrary rst guest control law u0 we will generate the
state trajectory x0 starting at z.
The trajectory x0 and control u0 will construct an equivalent
problem where we can easily construct a new control u1 better
than u0.
System
u0 Krotov's
Algorithm
x0 u1

Krotov's Method
The new control u(t, x) is constructed using an algebraic formula
that uses our bounding function:
u(t, x) = arg max
u∈U
∂ϕ(t, x)
∂x
f (t, x, u) − f 0
(t, x, u)
System
u0 Krotov's
Algorithm
x0 u1

Krotov's Method
u(t, x) = arg max
u∈U
∂ϕ(t, x)
∂x
f (t, x, u) − f 0
(t, x, u)
In other words, from an arbitrary control law u0 we will construct a
control u1(t, x) such that the process (x1, u1) is better than (x0, u0)
for the particular initial condition z
I(u0, z) I(u1, z)
System
Krotov's
Algorithm
x0 ui+1ui

Krotov's Method
u(t, x) = arg max
u∈U
∂ϕ(t, x)
∂x
f (t, x, u) − f 0
(t, x, u)
In other words, from an arbitrary control law u0 we will construct a
control u1(t, x) such that the process (x1, u1) is better than (x0, u0)
for the particular initial condition z
I(u0, z) I(u1, z)
This sequence is always improving and converges to the global
maximum.
System
Krotov's
Algorithm
x0 ui+1ui

Krotov's Method
At the end of the iterative process we will obtain a bounding
function ¯ϕ(t, x) that allows to compute the optimal controller
¯u(t, x) for the initial condition z.

Variation of the Initial Conditions
Once obtained the solving bounding function ¯ϕ we wish to know
how the value of I(u, z) behaves.
Theorem: Region on Improvement of a control u0
There exist a neighbourhood of z where the bounding function
¯ϕ(t, x) allows to compute a control that is improving w.r.t. the
control u0.

Constraints in the Control
The constrains in the control are common all engineering
applications.
u ∈ U(t, x)
The Krotov's method perfectly introduce time depended control
constraints by the same way that PMP does.
u = arg max
u∈U(t)
R(t, x, u)
To introduce time and state depended control constraints
u = arg max
u∈U(t,x)
R(t, x, u)
the only requirement is that U(t, x) must be wide enough to
guarantee improvements at each iteration.

Constraints in the State
Also a set of constraint in the state variables can be of interest in
engineering problems
x ∈ X(t)
This kind of constraints cannot be directly imposed as the previous
case.

Denition: Reachability Set
Given
˙x = f (t, x, u) u ∈ U(t, x) (*)
The reachability set is
A(x0, T, U) = {x(T) ∈ Rn : x(t) is solution of (*)}

We propose to solve the following problem to translate the state
constraints into control constraints:
Find ¯U(t, x) s.t. A(x0, T, ¯U) = X
Proposed Methodology
We have solved this problem in an intuitive way. Constructing upper
bounds of the variables of interest, we make an approximation of
R(t, x).

Outline
1 Introduction
4 Fluttering Wing
5 Capacitor EH.
6 Conclusions

Piezo electric device
Mass
x
Piezo R
i
y(t)
Equation of the system:
¨x + ω2
n(x − y (t)) − g ωn ˙χ = 0
¨χ +
1
R
˙χ + g ωn ˙x = 0
Harvested Energy
KPI =
T
0
i
2
R dt

Mass
x
Piezo R
i
y(t)
¨x + ω2
n(x − y (t)) − g ωn ˙χ = 0
¨χ +
1
R
˙χ + g ωn ˙x = 0
Harvested Energy
KPI = max
R∈U
T
0
i
2
R dt
R ∈ [Rmin, Rmax]
Problem: Maximize KPI controlling
R

Mass
x
Piezo R
i
y(t)
¨x + ω2
n(x − y (t)) − g ωn ˙χ = 0
¨χ +
1
R
˙χ + g ωn ˙x = 0
Harvested Energy
KPI =
T
0
i
2
R dt
Our Control variable is
dened as u = 1
R

Piezo electric EH. y(t) = Rand(
ω
PDF
)
Iteration 0
As a rst control guess we use the constant value R wich maximises our KPI
t
Harvested Energy ( KPI)
t
Control u = 1/R

ω
PDF
)
Iteration 1
t
t
Control u = 1/R

ω
PDF
)
Iteration 10
t
t
Control u = 1/R

ω
PDF
)
Iteration 25
t
t
Control u = 1/R

Piezo electric EH. Constrains on the state
If the harvested energy is too high the current owing on the circuit
can have an unacceptable magnitude.
ι = −
1
R
˙χ
To translate the constraints on ι to U we have considered the
following inequality
|ι| ιmax ⇐⇒ |−
1
R
˙χ| ιmax

Piezo electric EH. Behaviour of the Current.
Iteration 0
t
Current on the circuit

Iteration 1
t

Iteration 10
t

Iteration 25
t

Iteration
Desired Bouds
t

Piezo electric EH. Constrains on the Current
We have computed a new admissible control set ¯U that limit the
current, and not deteriorate the KPI in a excessive way.
t
Current
t
Harvested Energy

Piezo electric EH. Constrains on the Current
We have computed a new admissible control set ¯U that limit the
current, and not deteriorate the KPI in a excessive way.
Desired Bouds
t
Current
t
Control Shape

Piezo Electric EH. Variation of the Initial Condition
We have computed ¯ϕ(t, x) for the specic initial condition z0 = 0
t
Displacement
t
Harvested Energy

Now we present how varies teh nbehaviour of the system for dierent
initial conditions z ∈ D
t
Displacement
t
Harvested Energy

We observe better performance becaouse z0 = 0 is a position of
minimum energy.
t
Displacement
t
Harvested Energy

We observe that after a transitory the trajectory of the system
converge to the optimal limit cycle for z0
t
Displacement
Parallel
t
Harvested Energy

We observe that after a transitory the trajectory of the system
converge to the optimal limit cycle for z0
t
Displacement
t
Harvested Energy

Outline
1 Introduction
4 Fluttering Wing
5 Capacitor EH.
6 Conclusions

Fluttering Wing

Fluttering Wing: System Dynamic
We want to maximize the energy dissipated by the generator G
m¨z + CA ˙z + KAz = LAer − FG
J¨θ + CB ˙θ + KBθ = LAerbCS cos(θ) − FGbHS cos(θ)
Lift Force LAer =
1
2
ρU2
(t)SW CW θrel
Angle of attack θrel = −θ + atan2 ˙z − ˙θbLES, U(t)
Generator Force FG = β ˙z + ˙θbHS cos(θ)

This energy is KPI =
T
0 FGvH
Lift Force LAer =
1
2
ρU2
(t)SW CW θrel

Our Control Variables in the gain of the motor G u = β
Lift Force LAer =
1
2
ρU2
(t)SW CW θrel

KPI =
T
0
FG ˙z + ˙θbHS cos(θ)
Lift Force LAer =
1
2
ρU2
(t)SW CW θrel

max
u
T
0
u ˙z + ˙θbHS cos(θ)
2
dt
u ∈ [βmin, βmax]
Lift Force LAer =
1
2
ρU2
(t)SW CW θrel

Fluttering Wing: KPI

Simulation U = Const
Iteration 0
t
t
Control u = β

Iteration 2
t
t
Control u = β

Iteration 8
t
t
Control u = β

Iteration 11
t
t
Control u = β

Symulation U = Const + Rand(
ω
PDF
)
Iteration 0
t
t
Control u = β

ω
PDF
)
Iteration 2
t
t
Control u = β

ω
PDF
)
Iteration 21
t
t
Control u = β

ω
PDF
)
Iteration 33
t
t
Control u = β

Outline
1 Introduction
4 Fluttering Wing
5 Capacitor EH.
6 Conclusions

Energy harvesting from MEMS
V
R1
i1
R2
i3i2
mass
y(t)

V
R1
i1
R2
i3i2
mass
y(t)
Power from the source V P1 = R1i1

V
R1
i1
R2
i3i2
mass
y(t)
Power from R2 P1 = R2i3

V
R1
i1
R2
i3i2
mass
y(t)
Control Variables R1 and R2

System Dynamic
Liner oscillator ¨x + 2ζωn (˙x − ˙y(y)) + ω2
n(x − y(t)) =
q2
εA
Circuit ˙q =
1
R1
V −
1
R1
+
1
R2
q
(d − c)
Aε

System Dynamics
Power from the main source P1 = i2
1 R1.
Power from the resistor R2 P2 = i2
3 R2
Our KPI is the dierence between P1 and P2
KPI =
T
0
P2 − P1 dt
Our Control variables are the resistance R1 and R2

Symulation V = Const
Iteration 0
t
Harvested Energy
t
Control 1 u1 = 1/R1
Control 2 u2 = 1/R2

Iteration 1
t
Harvested Energy
t
Control 1 u1 = 1/R1
Control 2 u2 = 1/R2

Iteration 3
t
Harvested Energy
t
Control 1 u1 = 1/R1
Control 2 u2 = 1/R2

Iteration 27
t
Harvested Energy
t
Control 1 u1 = 1/R1
Control 2 u2 = 1/R2

Outline
1 Introduction
4 Fluttering Wing
5 Capacitor EH.
6 Conclusions

Conclusion
Using a tunable parameter we have proposed a control law that
improves the harvested energy in a considerable way, w.r.t. the
classical passive approach. This is the best possible performance
for this device for a particular initial condition z.
In a neighbourhood of z these controllers are suboptimal but
improving w.r.t. an initial control u0.
We have successfully introduced constraints in both, the control
and the state variables.
We have found upper bounds to the energy harvesting process for
our particular piezo-electric model that reveals an encouraging
horizon in using OCT in EH.

Perspectives
Direc Use , Krotov's methods for implementation of MPC's. This
requires
Stability.
Robustness.
Tracking accuracy.
Indirect Use Give upper bound for new proposed technologies and
benchmarking of current used technologies of EH.

Thank you

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