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Nonlinear Power Flow Control Design:Utilizing Exergy, Entropy, Static and Dynamic
Stability, and Lyapunov Analysis
Rush D. Robinett, III David G. WilsonEnergy, Resources & Systems Analysis Center
Sandia National Laboratories, P.O. Box 5800
Albuquerque, NM, 87185-1108 USA
[email protected] [email protected]
47th IEEE Conference on Decision and Control
Fiesta Americana Grand Coral Beach Hotel
Cancun, Mexico December 8, 2008
CDC ’08 Workshop
Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed
Martin Company, for the U.S. Department of Energy’s National Nuclear Security Administration under
contract DE-AC04-94AL85000.
Goal of Workshop
• Present innovative control system design process based on R&D at SNL in the area of renewable energy electric grid integration (future electric power grid control, wind turbine load alleviation control, and novel micro-grid control design).
• The main result of this research is a unique set of criteria to design controllers for nonlinear systems with respect to both performance and stability, instead of following the typical linear controller zero-sum design trade-off process between stability and performance.
• This control design process combines concepts from thermodynamic exergy and entropy; Hamiltonian systems; Lyapunov's direct method and Lyapunov optimal analysis; static and dynamic stability analysis, electric AC power concepts including limit cycles; and power flow analysis. The thermodynamic concepts combined with Hamiltonian systems provide the theoretical foundations necessary to view control design as power flow control problem of balancing the power flowing into the system versus the power being dissipated within the system subject to power being stored in the system.
• Emphasis is placed on necessary design steps for which the concepts are introduced and explained with examples and case studies.
Agenda
8:30 - 9:15 Welcome and Introduction Charlie Hanley
9:15 - 10:15 Theory Rush Robinett III
10:15 - 10:30 BREAK!
10:30 - 12:00 Continue Theory/Discussions Rush Robinett III
David Wilson
12:00 - 1:00 Lunch!
1:00 - 2:30 Case Studies 1 & 2 Rush Robinett III
David Wilson
2:30 - 2:45 BREAK!
2:45 - 4:15 Case Studies 3 & 4 Rush Robinett III
David Wilson
4:15 - 4:30 Open Discussions All
4:30 Adjourn!
Speakers
Energy Infrastructure Futures - Sandia National Labs
• Charles J. Hanley (Keynote Speaker) - Mr. Hanley is manager of Sandia’s Photovoltaic Systems Evaluation Laboratory and the Distributed Energy Technologies Laboratory. He manages research and development focused on new photovoltaic components and systems with improved performance, higher reliability, and lower overall cost, in support of the US Department of Energy’s efforts to make these technologies cost effective. This research includes; modernization of the electric grid through controls and communications, to support the eventual high penetration of renewable energy technologies into our infrastructure. Until 2002, Charlie managed Sandia’s international renewable energy programs, through which he oversaw the implementation of more than 400 photovoltaic, wind, and passive solar energy systems in Latin America. He received his B.S. degree in Engineering Science from Trinity University in San Antonio, TX, and his M.S. degree in Electrical Engineering from Rensselaer Polytechnic Institute, in Troy, N.Y.
• Rush D. Robinett III - has three degrees in Aerospace Engineering from Texas A&M University (B.S. - 1982, Ph.D.- 1987) and The University of Texas at Austin (M.S. - 1984). He has authored over 100 technical articles including two books and holds 7 patents. He began working on ballistic missile defense at Sandia National Laboratories in 1988. In 1995, he was promoted to Distinguished Member of Technical Staff. In 1996, he was promoted to technical manager of the Intelligent Systems Sensors and Controls Department within the Robotics Center. In 2002, he was promoted to Deputy Director of the Energy and Transportation Security Center where he is developing distributed power and transportation infrastructures focusing on exergy, entropy and information metrics.
• David G. Wilson - has three degrees in Mechanical Engineering from Washington State University (B.S. - 1982, M.S. 1984) and The University of New Mexico (Ph.D. - 2000). He has authored over 50 technical articles including two books. He is currently a Principal Member of Technical Staff at Sandia National Laboratories, Energy Systems Analysis Department. He has over 20 years of research and development engineering experience in energy systems, robotics, automation, and space and defense projects. His current areas of research include: design, analysis, and implementation of nonlinear/adaptive control; distributed decentralized control architectures; exergy/entropy and collective control for dynamical systems.
Workshop Notes and References
Viewgraph Handouts (booklet)
Several conference and journal papers included:1) R.D. Robinett III and D.G. Wilson, What is a Limit cycle?, International Journal of Control, Vol.
81, No. 12, Dec. 2008, pp. 1886-1900.
2) R.D. Robinett III and D.G. Wilson, Exergy and Irreversible Entropy Production Thermodynamic Concepts for Nonlinear Control Design, accepted for publication, International Journal of Exergy, February 2008.
3) R.D. Robinett III and D.G. Wilson, Collective Plume Tracing: A Minimal Information Approach to Collective Control, accepted for publication, International Journal of Robust and Nonlinear Control, Nov. 2008.
4) R.D. Robinett III and D.G. Wilson, Nonlinear Power Flow Control Applied to Power Engineering, SPEEDAM 2008, International Symposium on Power Electronics, Electrical Drives, Automation and Motion, Ischia, Italy, June 2008.
5) R.D. Robinett III and D.G. Wilson, Collective Systems: Physical and Information Exergies, SNL, SAND2007-2327 Report, March 2007.
6) R.D. Robinett III and D.G. Wilson, Exergy and Entropy Thermodynamic Concepts for Control System Design: Slewing Single Axis, AIAA Guidance, Navigation, and Control Conference and Exhibit, Keystone, Co., August 2006.
7) R.D. Robinett III and D.G. Wilson, Exergy and Entropy Thermodynamic Concepts for Nonlinear Control Design, Proceedings of IMECE2006-15205, 2006 ASME International Mechanical Engineering Congress and Exposition, Nov. 2006, Chicago, Illinois.
8) R.D. Robinett III and D.G. Wilson, Exergy and Irreversible Entropy Production Thermodynamic Concepts for Control System Design: Robotic Servo Applications, Proceedings of the 2006 IEEE International Conference on Robotics and Automation, Orlando, Florida, May 2006.
9) Robinett, III, R.D., Wilson, D.G., and Reed, A.W., Exergy Sustainability for Complex Systems, No. 1616, InterJournal Complex Systems, New England Complex Systems Institute, 2006.
Acknowledgments
• Ben Schenkman, Dept. 6336 – Power Engineering Modeling
•Val Weekly, Dept. 6330 – Workshop Coordinator
• Innovative Control of a Flexible, Adaptive Energy Grid, Sandia
National Laboratories, LDRD Project, No. 09-1125, Oct. 2006 –
Oct. 2009.
Sandia National Laboratories is a multiprogram laboratory operated by Sandia Corporation, a Lockheed
Martin Company, for the U.S. Department of Energy’s National Nuclear Security Administration under
contract DE-AC04-94AL85000.
Outline
• Theory (Morning)
– Thermodynamics
– Mechanics
– Stability
– Advanced Control Design
• Case Studies (Afternoon)
– Control Design Issues
– Collective Plume Tracing
– Nonlinear Aeroelasticity
– Power Engineering
THERMODYNAMICS
• First Law - Energy
• Second Law – Stability/Entropy
• Equilibrium Thermodynamics - Reversible/Irreversible Paths
• Local Equilibrium – Non Equilibrium Thermodynamics
• Rate Equations – Energy, Entropy, and Exergy
Thermodynamics (cont.)
• First Law [Ref. Gyftopoulos] - A statement of the existence of a
property called energy
– Energy is a state function: path independent; E
– Adiabatic work is path independent
Energy at state i of system
Work done by system between
states 1 and 2
−iE
−12W
1212 WEE −=−
[Ref. Gyftopoulos]: E.P. Gyftopoulos and G.P. Beretta, Thermodynamics,
Foundations and Applications, MPC, NY, 1991.
Thermodynamics (cont.)
- Corollary of First Law
change of state; path independent
flow of heat; path dependent
work done by the system; path dependent
(conservation of energy)
WQdE δδ −=
−dE
−Qδ
−Wδ
Thermodynamics (Cont.)
• Second Law – [Ref. Gyftopoulos]: A statement of the existence of stable equilibrium states and of special processes that connect these states together
– Equilibrium State: A state that does not change with time while the system is isolated from all other systems in the environment
– Stable Equilibrium State: A finite change of state cannot occur,regardless of interactions that leave no net effects in the environment
– Restate Second law: Among all the allowed states of a system with given values of energy, number of particles, and constraints, one and only one is a stable equilibrium state
[Ref. Gyftopoulos]: E.P. Gyftopoulos and G.P. Beretta, Thermodynamics,
Foundations and Applications, MPC, NY, 1991.
– Reversible and Irreversible Processes: A process is reversible if the system
and its environment can be restored to their initial states, except for
differences of small order of magnitude than the maximum changes that occur
during the process
Example 1: Expansion of Gas
Thermodynamics (cont.)
Reservoir, Reservoir,oT
FF
oT
Reversible Expansion
Adiabatic Expansion – Irreversible
Thermodynamics (cont.)
– Available work:
– Entropy:
– Adiabatic processes: Reversible:
Irreversible:
Ω
S
12121212 )(;)( WWWrevrev ≥−=Ω−Ω
)( Ω−Ε≡ ddcdS R
0)( =REVdS
0)( ≥irrevdS
Thermodynamics (cont.)
– Corollary: At a stable equilibrium state, the entropy will be at its maximum value for fixed values of energy, number of particles, and constraints
– Reversible process: Interacting systems quasi-statically pass only through stable equilibrium states
;)( TdSQdQ REV == δ
;T
dQdS =
etemperaturT −
∫==T
dQS 0
Thermodynamics (cont.)
– Irreversible process [Ref. Prigogine]:
- Entropy change due to exchange of energy and matter
(entropy flux)
- Entropy change due to irreversible processes
T
QdS
δ≥
SdSddS ie +=
Sde
Sd i
[Ref. Prigogine] D. Kondepudi and I. Prigogine, Modern Thermodynamics: From
Heat Engines to Dissipative Structures, John Wiley & Sons, New York, N.Y., 1999.
Thermodynamics (cont.)
Sdi
Sde
Figure 1- Entropy changes
0≥=∑ kk
k
i dXFSd
th
k kF − thermodynamics force
thermodynamics flowth
k kX −
Thermodynamics (cont.)
• Equilibrium Thermodynamics [Ref. Prigogine] – In classical
thermodynamics it is assumed that every irreversible transformation that
occurs in nature can also be achieved through a reversible process where
∫+=2
112
T
dQSS
irrev
1
2
rev
E
S
Reservoir
T=Constant
[Ref. Prigogine] D. Kondepudi and I. Prigogine, Modern Thermodynamics: From
Heat Engines to Dissipative Structures, John Wiley & Sons, New York, N.Y., 1999.
Thermodynamics (cont.)
VpdTdS
VpdTdSdU
dWVpd
dQTdS
dEdU
VpdTdSdU
∫∫∫∫∫
=⇒
−==
=
=
=
−=
0
4
3 2
1Work Done (area)
T
1
23
4
S
p
- A uniform temperature exists throughout the system.
= internal energy
volumeVpressurep −− ,;
- Reversible Cycle Processes:
V
Thermodynamics (cont.)
• Local Equilibrium – Thermodynamic quantities are well-defined concepts locally (i.e., within each elemental volume)
– Temperature is not uniform, but is well defined locally
– Non-equilibrium systems: define thermodynamic quantities in terms ofdensities
– Thermodynamic variables become functions of position and time
• Rate Equations
– Energy:
– Entropy:
– Exergy (Available work)
...)( +++++= ∑∑∑ llll
l
k
k
j
j
pekehmWQE &&&&
0≥=
++=+=
∑
∑∑
ll
l
i
ikk
kj
j
j
ie
XFS
SsmT
QSSS
&&
&&&
&&&
io
l
Flow
llok
k
j
j
o
j
o STmVpWQT
TSTE &&&&&&&& −+−+−=−=Ξ ∑∑∑ ζ)()1(
MECHANICS
• Work, Energy, and Power
• Energy Diagrams and Phase Planes
• Hamiltonian Mechanics
• Connections between Thermodynamics and Hamiltonian Mechanics
• Line Integrals and Limit Cycles
Mechanics (cont.)
• Work, Energy, and Power
– Work:
Work
Force Vector
Position Vector
– Power:
Power
Velocity Vector
– Kinetic Energy: (Newton’s 2nd law)
rdFdW ⋅=−W
−F
−r
rFrdt
dFW
dt
dP &⋅=⋅==
−r&−P
rmF &&=
−⋅=
=
⋅=⋅=⋅=⋅=
rrmT
dTrrmdrdrmrdrmrdFdW
&&
&&&&&&
2
1
2
1
Kinetic Energy
Mechanics (cont.)
– Potential Energy:
Conservative and path independent force field
−
⋅=−
−=⋅
∫V
rdrFrVrV
rdrF
r
r
2
1
)()()(
)(
21
Potential Energy
⇒
−
==+⇒
−=⋅=
=⋅∫
E
EVT
rFdt
dT
rdF 0
Constant
)(rdV
⇒dt
dV0)( =+VT
dt
d
Total Energy (stored energy)
Mechanics (cont.)
– Time Dependent Potential Field: −=⋅ rdtrF ),(
t
VVT
dt
d
∂
∂=+ )(
),( trdV
Mechanics (cont.)
-Non-conservative Forces:
Conservative Force (Potential Field)
Non-Conservative Force
NCc FFF +=
−cF
−NCF
rFt
VVT
dt
d
dt
dENC
&⋅+∂
∂=+= )(
Mechanics (cont.)
• Energy Diagrams and Phase Planes
– Mass, Spring, Damper System
2
2
1xmT &=
2
2
1kxV =
xcuF &−=
−u Control
Force
−xc& Damping
Force
k
c
xcukxxm &&& −=+
Mechanics (cont.)
a) Set conservative system, :oxcu == & =+= VTE Constant
2
2
1kxV =
x
2
2
1xmT &=
x&
xx
E E
E
Mechanics (cont.)
The system path is constrained to the energy storage surface along
a constant energy orbit
22
2
1
2
1oo kxxmE += &
⇒
E= constant
x&
x
Mechanics (cont.)
b) Set and
The system path is constrained to the energy storage surface as it
spirals down into the minimum energy state
0=u 0>c
⇒
Mechanics (cont.)
• Hamiltonian Mechanics
– Lagrangian:
Time Explicitly
N Dimensional Generalized Coordinate Vector
N Dimensional Generalized Velocity Vector
– Equations of Motion:
Generalized Force Vector
– Hamiltonian:
),(),,( tqVtqqTL −= &
−t−q
−q&
L
q
L
dt
d=
∂
∂−
∂
∂)(
&
−Q
),,(),,(1
tqqHtqqLqq
LH i
i
N
j
&&&&
=−∂
∂≡ ∑
=
Mechanics (cont.)
– Canonical Momentum:
– Hamilton’s Canonical Equations of Motion:
– Time Derivative of the Hamiltonian:
),,(),,(1
tqqLqptpqH ii
N
i
&& −= ∑=
j
j
j
j
j
Hp
p
Hq
+∂
∂−=
∂
∂=
&
&
t
LqQ
Lq
q
L
t
LqpqpH
jj
N
j
j
j
j
j
jjjj
N
j
∂
∂−=
∂
∂−
∂
∂−
∂
∂−+=
∑
∑
=
=
&
&&&
&&&&&&
1
1
)(
i
iq
Lp
&∂
∂=
Mechanics (cont.)
– For most natural systems:
Power (work/energy) flow equation
jj
N
j
qQqqH
t
L
&&& ∑=
=
=∂
∂
1
),(
0
Mechanics (cont.)
• Connections between Thermodynamics and Hamiltonian Mechanics
– Conservative Mechanical Systems:
a) and Constant
b) Conservative Force (storage device)
For any closed path
0=H& =H
0=⋅=⋅=⋅ ∫∫∫ dtqQdtrFrdF &&
Mechanics (cont.)
– Reversible Thermodynamic Systems:
– Irreversible Thermodynamic Systems:
for
then and
T
QS
dtSSSdSddS
T
dQdS
T
dQdS
e
ieie
&&
&&
=⇒
=+=+=
==
=
∫∫∫
∫∫0][][
0
(Second Law)and 0=iS&
0][ =+= ∫∫ dtSSdS ie&&
0≤eS& 0≥iS
&
Mechanics (cont.)
– Hamiltonian is stored exergy since potential and kinetic energies are available work; Exergy rate relationship:
a) Power flowing in (N generators)
b) - Power Dissipation (M Dissipators)
c) (Assuming Local Equilibrium)
qFqQqQSTW
qFqQH
NCll
NM
Nl
jj
N
j
io
NCkk
k
&&&&&
&&&
⋅=−=−=Ξ
⋅==
∑∑
∑+
+== 11
−W&
ioST &
01
≥== ∑∑ kk
ko
kk
k
i qQT
XFS &&&
Mechanics (cont.)
d) Assumptions applied to exergy rate equation:
e) A conservative system is equivalent to a reversible system when
and
then and
0
0
01
0
=
=
≅−
≅
∑ FLOW
kk
k
o
j
o
i
m
Vp
T
T
Q
ζ&
&
&
0=H& 0=eS&
0=iS& 0=W&
Mechanics (cont.)
• Line Integrals and Limit Cycles
– Cyclic Equilibrium Thermodynamics:
– Cyclic Non-Equilibrium Thermodynamics with Local Equilibrium
TdSdW
TdSVpd
dWTdSdE
VpdTdSdU
∫∫∫∫∫∫∫∫∫∫
=⇒
=⇒
=−=
=−=
0
0
dtqQdtqQ
dtSTdtW
dtSTWdtH
kk
NM
Nk
jj
N
j
i
io
][][
0][
11
0
&&
&&
&&&
∑∫∑∫
∫∫∫∫
+
+==
=⇒
=⇒
=−=
Mechanics (cont.)
– Sorting Power Terms:
a) Conservative Terms
Potential functions
b) Generator Terms
c) Dissipator Terms
⇒=⋅∫ 0dtqF c&
0][01
>∫⇒>⋅∫ ∑=
dtqQdtqF jj
N
j
cNCc&&
0][01
<∫⇒<⋅∫ ∑+
+=
dtqQdtqF kk
MN
Nk
cNCc&&
Mechanics (cont.)
– Limit Cycles:
dtqQqQ
dtSTdtW
kk
MN
Nk
jj
N
j
io
=
=
∑∫∑∫
∫∫+
+==
&&
&&
11
Mechanics (cont.)
van der Pol Oscillator: Limit Cycle Identification
-Dissipative
-Neutral
-Generative
-Hamiltonian
surface
Mechanics (cont.)
Linear Limit Cycles: Power Grid Applications
• Goal Power Engineering maximize real power flow: match frequency and phase of applied voltage and resulting current
• Exergy/Entropy Control Design identifies stability boundaries and improves system performance with nonlinear feedback by providing robustness to disturbances and adaptation to parameter changes
• Current Power Engineering solutions “open loop” –subject to changing load variations and delays due to manual operator set point updates and approvals
• Simple RLC circuit – Driven with 60 Hz sinusoidal input demonstrates variations and changes in load and frequency
Exergy/Entropy Distributed Control: Predicts
Performance and Stability for Power System
STABILITY
• Static and Dynamic Stability
• Eigenanalysis
• Lyapunov Analysis
• Energy Storage Surface and Power Flow: Necessary and Sufficient
Conditions for Stability
Stability (cont.)
• Static and Dynamic Stability
– Static Stability: If the forces and moments on a body caused by a disturbance tend initially to return (move) the body towards (away from) its equilibrium state, the body is statically stable (unstable) [Refs. Anderson, Robinett]
a) Equilibrium State- An unaccelerated motion wherein the sums of the forces and moments on the body are zero
b) Neutral Stability - Occurs when the body is disturbed and the sums of the forces and moments on the body remain zero
[Ref. Anderson] J.D. Anderson, Jr., Introduction to Flight, McGraw-Hill, 1978.
[Ref. Robinett] R.D. Robinett III, A Unified Approach to Vehicle Design, Control, and
Flight Path Optimization, PhD Dissertation, Texas A&M University, 1987.
),( ee xx&
Stability (cont.)
Statically
Stability (cont.)
c) The Energy Storage Surface is
d) A system is statically stable if
for
Statically unstable if
And Neutrally stable if
0)( >xV
0),(
0
=
>
ee xxH
H
&
0)( <xV
0)( =xV
EVTH =+=
ee xxxx && ≠≠∀ ,
Stability (cont.)
• Dynamic Stability: Defined in terms of the time history of the motion
of a body after encountering a disturbance
– A body is dynamically stable (unstable) if, out of its own accord, it
eventually returns to (deviates from) and remains at (away from) its
equilibrium state over a period of time [Refs. Anderson, Robinett1]
a) A dynamically neutral stable body occurs when a limit cycle
exists [Refs. Robinett2, Robinett3]
b) A dynamically stable body must always be statically stable, but
static stability is not sufficient to ensure dynamic stability [Refs.
Abramson, Anderson, Robinett1] Therefore, static stability is a
necessary condition for stability
[Ref. Anderson] J.D. Anderson, Jr., Introduction to Flight, McGraw-Hill, 1978.
[Ref. Robinett1] R.D. Robinett III, A Unified Approach to Vehicle Design, Control, and Flight Path Optimization, PhD
Dissertation, Texas A&M University, 1987.
[Ref. Robinett2] R.D. Robinett III, What is a Limit Cycle?, Int’l Journal of Control, Vol. 81, No. 12, Dec. 2008, pp.
1886-1900.
[Ref. Robinett3] R.D. Robinett III and D.G. Wilson, Collective Systems: Physical and Information Exergies, Sandia
National Laboratories, SAND2007-2327 Report, March 2007
[Ref. Abramson] H.N. Abramson, Introduction to the Dynamics of Airplanes, Ronald Press, 1958.
Stability (cont.)
c) The time derivative of the energy storage surface defines the power
flow into, dissipated within, and stored in the system
d) A system is dynamically stable if
dynamically unstable if
and dynamically neutral stable if (Limit Cycle)
01
<= ∫c
oc
AVE dtHHτ
τ&&
01
>= ∫ dtHHc
oc
AVE&&
τ
τ
01
== ∫ dtHHc
c
AVE&&
ττ
Stability (cont.)
• Eigenanalysis: The goal is to relate/ extend eigenanalysis of linear
systems to nonlinear systems via the energy storage surface, power flow,
and limit cycles
– Mass, spring, damper system
);(2
1
)()(
2 xVxmEH
uxfxgxm
+==
+−=+
&
&&&
x
xVxg
∂
∂=
)()(
Stability (cont.)
a) Linearized system
b) Assume , Eigenvalue Problem
Undamped natural frequency (eigenvalue) of a statically stable and
dynamically neutral stable system
xxcuxkxxmH
uxckxxm
kxxmH
kcmxcxfkxxV
&&&&&&
&&&
&
&&
][][
02
1
2
1
0,,;)(;02
1)(
22
2
−=+=
+−=+
>+=
>=>=
m
k
kxxm
=
=+
2
0
ω
&&
0,0 === Hxcu &&
Stability (cont.)
c) Assume and
This system is statically stable and dynamically stable if
Dynamically unstable
Dynamically neutral stable if
d) Extend results to nonlinear systems:
2
22
][][
02
1
2
1
xKcxkxxmH
kxxmH
D&&&&&
&
+−=+=
>+=
xKu D&−= 0>c
0>+ DKc0<+ DKc
0=+ DKc
04
1
2
1
2
1)(
2
1 4222 >++=+= xkkxxmxVxmH NL&&
Stability (cont.)
This system is statically stable and dynamically neutral stable (limit cycle) if
with nonlinear frequency content of
and dynamically stable if
and dynamically unstable if
0)]([1
)]([1
=−=+= ∫∫ dtxxfudtxxgxmHcc
cc
AVE&&&&&&
ττ ττ
0)( =+ xgxm &&
0)]([1
<−= ∫ dtxxfuHc
oc
AVE&&&
τ
τ
0)]([1
>−= ∫ dtxxfuHc
oc
AVE&&&
τ
τ
Stability (cont.)
• Lyapunov Analysis: The goal is to relate static and dynamic stability, energy storage surface, and power flow to Lyapunov Analysis
– The definition of static stability is equivalent to the requirement placed on a Lyapunov function: must be positive definite
– A statically unstable system is unstable in the sense of Lyapunov.
Assume and for pick Lagrangian
for which is unstable
0),( >= Hxx &VVVV
0== xcu & 2
2
1)( kxxV −= ;0>k
02][
02
1
2
1 22
>=+=
>+=−==
xkxxkxxm
kxxmVTL
&&&&&
&
VVVV
VVVV
0=−kxxm&&
Stability (cont.)
– The definition of dynamic stability is equivalent to the requirements
placed on a Lyapunov function. A system is asymptotically stable if
[Ref. Robinett1]
– And unstable if
The limit cycle defines the stability boundary between asymptotically
stable and unstable [Ref. Robinett2]
( ) 0)]([1
0
<−=
>=
∫ dtxxfu
H
c
oc
ave&&&
τ
τVVVV
VVVV
( ) 0)]([1
0
>−=
>=
∫ dtxxfu
H
c
oc
ave&&&
τ
τVVVV
VVVV
[Ref. Robinett2] R.D. Robinett III and D.G. Wilson, What is a Limit Cycle?,
International Journal of Control, Vol. 81, No. 12, Dec. 2008, pp. 1886-1900.
[Ref. Robinett1] R.D. Robinett III and D.G. Wilson, Exergy and Irreversible Entropy
Production Thermodynamic Concepts for Nonlinear Control Design, accepted for
publication, International Journal of Exergy, Feb. 2008.
Stability (cont.)
• Energy Storage Surface and Power Flow: Necessary and Sufficient
Conditions for Stability
– Energy Storage Surface: Total energy; stored exergy;
paths/trajectories are constrained to this surface; infrastructure
a) Determines static stability which is a necessary condition for
stability
b) Proportional feedback affects static stability; assume
for and 42
2
1
2
1)( xkkxxV NC+−= 0, >NLkk =u
422
3
3
4
1][
2
1
2
1
0][
xkxkKxmH
xkxkKxm
xKuxkkxxm
NCp
NLp
pNL
+−+=
=+−+
−==+−
&
&&
&&
,xK p− 0=c
Stability (cont.)
pKk >
Stability (cont.)
• Power Flow: Exergy Flow; 3 types– into, dissipated within, and stored in
the system; determines the path/trajectory across the energy storage
surface
a) Determines the dynamic stability which is a sufficient condition
for stability
b) Sort Power Terms:
c) Derivative Feedback:
d) Integral Feedback:
iSTWHo
&&& −=
2xKST Dio&& =⇒
xKu D&−=
xdtKu I ∫−=
xxdtKW I&& ∫−=⇒
Stability (cont.)
,0>c dtSTdtW iooocc && ττ ∫<∫
xckxxm &&& −=+
Dynamically Stable
Stability (cont.)
,0>c dtSTdtW iooocc && ττ ∫>∫
Dynamically Unstable
xckxxm &&& =+
Stability (cont.)
,0=c dtSTdtW iooocc && ττ ∫=∫
Dynamically Neutral Stable
0=+ kxxm &&
ADVANCED CONTROL DESIGN
•Distributed Parameter/PDE’s
•Fractional Calculus
•Optimal
•Robust/ Tracking
•Adaptive/Tracking
Advanced Control Design (cont.)
• Distributed Parameter/PDE’s
– Vibrating String
x)( 1txw
0 L
Tension in String
Mass/unit length
=τ=)(xσ
t
ww
∂
∂=&
x
wwx
∂
∂=
0),(),0( == tLwtw
Advanced Control Design (cont.)
a)
b) Non-conservative Distributed Force
dxwV
dxwxT
x
L
L
2
0
2
0
2
1
)(2
1
∫
∫
=
=
τ
σ &
workwdxtxFWL
==⇒ ∫ ),(0
Advanced Control Design (cont.)
c) Extended Hamilton’s Principle
0
0
)(2
1
][
0 0
22
=−+
=
∂
∂
∂
∂−
∂
∂
∂
∂−
∂
∂
=
+−=
+−Τ=Ι
∫ ∫
∫∫
∫
wwF
w
f
tw
f
xw
f
fdxdt
dxdtFwww
dtWV
xx
tx
t L
x
L
o
t
o
t
o
&&
&
στ
τσ
;Fww xx =−⇒ τσ && 0),(),0( == tLwtw
Advanced Control Design (cont.)
d) Static Stability
for
e) Dynamic Stability
0)(2
1 22 >+=+= ∫ dxwwVTH x
L
oτσ &
0, >τσ
dxwF
dxwww
dxwwwwH
L
o
xx
L
o
xx
L
o
&
&&&
&&&&&
∫
∫
∫
=
−=
+=
][
][
τσ
τσ
;01
=
=⇒ ∫∫ dtdxwFH
L
oc
AVEc
&&ττ
Limit Cycle
Advanced Control Design (cont.)
• Controller: wKwdtKwKF DIxxp&−−= ∫
dxwwKwwH
wKwdtKwKw
dxwKwVTH
xxp
L
c
DIxxp
xp
L
c
])([
)(
0])([2
1
0
22
0
&&&&&
&&&
&
++=
−−=+−
>++=+=
∫
∫
∫
τσ
τσ
τσ
dxwwKwdtK
dxwwKw
DI
L
xxp
L
&&
&&&
][
])([
0
0
−−=
+−=
∫∫
∫ τσ
Advanced Control Design (cont.)
• Fractional Calculus
– Differintegral: Derivative and Integral to Arbitrary order
a) Oldham and Spanier [Ref. Oldham]:
q= Arbitrary real number
Gamma Function
b) Podlubny [Ref. Podlubny]:
−
−+Γ
−Γ
−Γ
−
∞→=
−∑
−
=
−
])[()1(
)(
)(
][lim
)]([
1
0 N
axjxf
j
qj
q
N
ax
Naxd
fd N
j
q
q
q
=+Γ )1( j
!
)1()2)(1()(
)()()1(lim)(
r
rqqqqq
r
rhtfhtfD q
r
rn
or
q
oh
q
ta
atnh
+−⋅⋅⋅−−=
−−= ∑=
−
→−= h t== small change in
[Ref. Podlubny] I. Podlubny, Fractional Differential Equations, Academic Press, N.Y., 1999.
[Ref. Oldham] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, N.Y., 1974.
Advanced Control Design (cont.)
c) First Order Fractional Difference approximation
for
and for
d) Inhomogeneous Bagley-Torvik Equation
)()(~ )(
]/[
khtfwhtfD q
k
ht
ok
to −= ∑=
−
−=
k
qw q
k )1()(
,...2,1,0=k
1)( =q
ow)(
1
)( )1
1( q
k
q
k wk
qw −
+−= ,...3,2,1=k
0),()()()( 2/3 >∀=++ ttutCytyDBtyA to&&
0)0()0( == yy &
Advanced Control Design (cont.)
e) Generalized to
f) First-Order Approximation/numerical algorithm
For
0),()()()( >∀=++ ttutKxtxDKtxM q
toq&&
0)0()0( == xx &
01 == xxo
)2(
)(
1
)2(
)2(
2112 )2()(q
q
jm
q
j
m
j
q
q
q
q
mmmmm
hKM
xwhK
hKM
xxMKxuhx
−
−=
−
−−−−
+
−
++
−+−=
∑
,...3,2=m
Advanced Control Design (cont.)
g) Sort power terms:
xxKxdtK
xxKKxMH
xKKxMH
xKxdtKxKu
uKxxCxM
DI
p
p
DIp
&&
&&&&
&
&
&&&
][
])([2
1
0)(2
1 22
−∫−=
++=
>++=
−∫−−=
=++
Advanced Control Design (cont.)
Storage: and
Dissipator:
Generator:
2
2
dt
xdo
o
dt
xd
1
1
dt
xd
1
1
−
−
dt
xd
:q
Storage Dissipator Storage Generator
2 1 0 -1
Type:
Coupled
)21(
Advanced Control Design (cont.)
h) Lag – Stabilized Controller [Ref. Robinett]
For
Equivalent to positive proportional and negative derivative
feedback
Shifted Sinusoids
)(txDKu q
toqq −=
:2,0=q
:02 >> q
:10 −≥> q
Storage
Dissipator
Generator
)()( dp txKtuKxxM τ−==+&&
0, >dpK τ
⇒
⇒xKxKu Dp&−≅
[Ref. Robinett] R.D. Robinett III, B.J. Peterson, J.C. Fahrenholtz, Lag-Stabilized Force Feedback
Damping, Journal of Intelligent and Robotic Systems, Vol. 21, Issue 3, March 1998, PP. 277-285.
Advanced Control Design (cont.)
The Fractional Calculus Approach
• Fractional calculus compared with PID type controllers
q=α
0 5 10 15 20-4
-2
0
2
4
6
8
Time (sec)
Position (m)
yPD Control
yFC with α=1.0
0 5 10 15 20-8
-6
-4
-2
0
2
4
6
8
Time (sec)
Position (m)
yP Control
yFC with α=0.0
0 5 10 15 20-25
-20
-15
-10
-5
0
5
10
15
20
Time (sec)
Position (m)
yI Control
yFC with α=-1.0
0 5 10 15 20-4
-2
0
2
4
6
8
Time (sec)
Position (m)
yPD Control
yFC with α=0.5
0 5 10 15 20-20
-15
-10
-5
0
5
10
15
Time (sec)
Position (m)
yPI Control
yFC with α=-0.5
Advanced Control Design (cont.)
• Optimal [Ref. Ogata] -
Asymptotically Stable to x=0
);,( uxfx =&
→= ))(,( xgxfx&
)],([min),(~
min
0),(),(~
),(
uxLuxH
uxLuxH
dtuxLJ
uu
o
+=
≥+=
= ∫∞
VVVV
VVVV
&
&
VVVV &= 0),(1
=+= uxLuu
1uu=⇒VVVV & ),( 1uxL−=
dttutxLxxo
))(),(())0((0))(( 1∫∞
−==∞ VVVVVVVV
dttutxLxo
))()),(())0(( 1∫∞
=VVVV
)(xgu =
Note:
[Ref. Ogata] K. Ogata, Modern Control Engineering, Prentice-Hall, Inc., N.J., 1970.
Advanced Control Design (cont.)
a) Example #1: ukxxcxm =++ &&&
][2
1][
2
1 2222
oo kxxmkxxmHJ o +=+−=−= ∞ &&
dtxxcudtxkxxmoo
&&&&& ][][ −−=+−= ∫∫∞∞
22
2
1
2
1
0][
kxxm
xcuxxcu
+=
<⇒<−=
&
&&&&
VVVV
VVVV
Advanced Control Design (cont.)
b) Example #2: ukxxm =+&&
])4([2
1
:0*
][2
1
2
1
),(
][
2/12
2
2
13
2
3
2,1
21
2
3
2
2
2
1
22
2
3
2
2
2
1
2
3
2
2
2
1
⟩+⟨−±−=∗
==
++=−
=+=⇒+=
−−−=−=
++= ∫∞
xkxkkxxk
u
kk
ukxkxkxu
xuxkxxmkxxm
ukxkxkuxL
dtukxkxkJo
&&&
&&
&&&&&&
&&
&
VVVVVVVV
VVVV
xk
uxuku &&3
3
1,0)( −==+
Advanced Control Design (cont.)
– Fixed Final Time: [ ] dtuuxJft
o
22
2 )(2
1+= ∫
TTxfLHuxfxuI ),(;
~);,(; θθλτθ &&&& =+====
BuxAuIx
x
Iu
x
x
xx +=
+
=
=
=
=
/1
0
00
10
/ 2
12
2
1
θ
θ&&
&
&
&&
)1(
1
.0
/~/
~
)1(/1
~
0
2
1)(
2
1~
2
2
22
2
12
2
2
11
12
2
2
1
2
1
2
222
2
2
221
22
2
0
+−
==
−−=
==⇒
−−=
∂∂−
∂∂−=
+−=⇒++=∂
∂=
+++=
xIu
Ix
xu
const
xuxH
xH
xIuI
uuxu
H
I
uxuuxH
o
λ
λλ
λλ
λλ
λ
λλ
λλ
&
&&
&
τ
θ
I, inertia
Horizontal Link
torque,
angle,
Advanced Control Design (cont.)
• Homotopy from i) min. energy to ii) hybrid min. energy plus min.
power, to iii) min. power cost functions
[ ]
BuxAx
dtuWxuWJft
o
+=
−+== ∫&
)()1())((2
1)( 2
2
2
21 ξαξαξφ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4
-3
-2
-1
0
1
2
3
4
Time (sec)
Min
imu
m E
ffo
rt T
raje
cto
rie
s (
θ,
θd
ot,
τ )
Horizontal Link Slew POWER opt
Angle
Angle Rate
Torque
0 0.2 0.4 0.6 0.8 1-4
-3
-2
-1
0
1
2
3
4
Time (sec)
Po
we
r (W
) T
orq
ue
(N
-m)
Horizontal Link Slew POWER opt
Power
Torque
0)()(
0)()(
0)()(
00
2
1
==
==Ψ
=−=Ψ
tt
t
t
f
desiredf
θθ
θξ
θθξ
&
&
Advanced Design Control (cont.)
• Robust / Tracking
– Nonlinear Model:
a) Hamiltonian:
;)()( uxfxgxm +−=+ &&&
)()(2
1 2
RR xxVxxmH −+−= &&
[ ] )()()( RRR xxxxgxxmH &&&&&&& −−+−=
[ ]uuu
xxuxfxgxxgxm
REF
RRR
∆+=
−+−−−+−= )()()()( &&&&
);(ˆ)(ˆ)(ˆˆ xfxgxxgxmu RRREF&&& ++−−=
)()()(ˆ RDRIR xxKdtxxKxxgu && −−−∫−−−=∆
x
xVxg
∂
∂=
)()(
x
xVxg
∂
∂=
)(ˆ)(ˆ
Robustness/Tracking addressed in the RLC example
Advanced Design Control (cont.)
– Static Stability:
– Dynamic Stability:
– Robustness:
0)(ˆ)()(2
1ˆ 2 >−+−+−= RRR xxVxxVxxmH &&
[ ] )()(ˆ)()(ˆRRRR xxxxgxxgxxmH &&&&&&
&−−+−+−=
[ ] )()()()(ˆ)( RRRR xxuxfxgxxgxxgxm &&&&& −+−−−+−+−=
[ ]⟩−⟨+⟩−⟨+⟩−−−⟨+⟩−⟨= )()(ˆ)()(ˆ)(ˆ)(ˆ xfxfxgxgxxgxxgxmm RRR&&&&
] )()()( RRDRI xxxxKdtxxK &&&& −−−−∫−
[ ] []
AVER
RRRRIAVERD
xxxfxf
xgxgxxgxxgxmmdtxxKxxK
)()()(ˆ
)()(ˆ)(ˆ)(ˆ)()( 2
&&&&
&&&&
−⟩−⟨+
⟩−⟨+⟩−−−⟨+⟩−⟨+−∫−>−
Advanced Design Control (cont.)
Adaptive/Tracking: Simple RLC Model
∫ =+ 0])([ dtqqCqqL &&&&
iiRviqqRvqqCqLH
dyyCqLHq
)()]([)]([
)(2
1
0
2
−=−=+=
+= ∫&&&&&&
&
tvvqC
qRqL Ω==++ cos)(1
)( 0&&&
Goal:
Design applied voltage
that creates desired
limit cycle (e.g., 60 Hz)
∫ =− 0])([ dtiiRvi
∫ =+ 0])([ dtqqCqqL &&&&
Where storage terms matched
vi maximally used by load
EOM:
Hamiltonian and time derivative:
0)( >qRq && for 0≠q&
0)( >qCq for 0≠q
Advanced Control Design (cont.)
• Goal: 60Hz Limit Cycle
for matched storage terms and maximal real power
c) Specific Nonlinearities:
d) Tracking Hamiltonian with controller and information potentials:
[ ] [ ] 0)()( =−=+ ∫∫ dtqqRvdtqqCqLcc
&&&&&ττ
)(11 3 qsignRqRvq
Cq
CqL NL
NL
&&&& −−=
++
( ) ( )422 1
4
11
2
1)(
2
1R
NL
RCAPR qqC
qqKC
qqLH −+−
++−= &&
ΦΓΦ+ − ~~
2
1 1T
Icap VVVT +++=
Advanced Design Control (cont.)
Adaptive PID Control Strategy: Improve Performance
)(
~~]
1[
~~
2
1
2
11
2
1
2
1
1
1222
R
capRR
T
cap
Icap
qqe
vqReKqC
qLH
eKeC
eLH
VVVTH
−=
ΦΓΦ++−+−−=
ΦΓΦ+++=
+++=
−
−
&&&&&
&
RRR
R
qC
qRqLv
vvv
1ˆˆ ++=
∆+=
&&&
Controller
selected as:
Advanced Design Control (cont.)
One Wants to Design for Ideal 60 Hz Condition
• Ideal
• Then
• Requires feedback control if
• To maintain a power factor of 1
• Select feedback portion as
2qRqv RR&& =
22 1Ω≠=
LCω
22 1Ω==
LCω
eKedKeKvt
dissgencap&∫ −−−=∆
0τ
Advanced Design Control (cont.)
Incorporate into Hamiltonian Time Derivative
∫ ΦΓ+Φ+−−= −t
TT
dissgen eYeeKedKH0
1 ]~
[~
][&
&&&& τ
][
]ˆ1ˆ[ˆ~
)]ˆ()11
()ˆ([~
)ˆ()11
()ˆ(~
qqqY
RC
L
RRCC
LL
qRRqCC
qLLY
RR
TT
T
RR
&&&
&&
&&&
&&&
=
=Φ=Φ
−−−=Φ
−+−+−=Φ
Where
Advanced Design Control (cont.)
Identify Adaptive Parameter Update Equations
)(
ˆ
1
ˆ
3
2
1
R
R
R
qqe
eqR
eqC
eqL
&&&
&&&
&
&
&&&&
−=
−=
−=
−=
γ
γ
γ
Advanced Design Control (cont.)
Perfect Parameter Matching: Nonlinear Stability Boundary
dteKdteedK diss
t
gen ∫∫∫ <′−ττ
τ 2
0][ &&
Results in asymptotically stable
(passivity) solution
Advanced Design Control (cont.)
Robustness/Tracking vs Adaptive/Tracking
5e65e6K_gen
10500K_diss
75010000K_cap
Gain Robustness Adaptive
Higher
controller
gains
achieved
Robustness
at expense of
increased
power usage
Advanced Design Control (cont.)
Linear RLC Network w/ PID Numerical Results
Advanced Design Control (cont.)
Linear RLC Network w/ PID/Adaptive Numerical Results
Advanced Design Control (cont.)
Linear RLC Network Energy & Power Flow Results
PID
Contr
oller
PID
/Adaptive
PID /adaptive
with
information
flow more
efficient than
PID with
physical flow
alone
Advanced Design Control (cont.)
Next Model Includes Nonlinearities
vqsignRqC
qC
qRqL NL
NL
=++++ )(11 3 &&&&
ADDED: Nonlinear Capacitance and Nonlinear Resistance
)()(
)(ˆ
][1
)(ˆˆ)(11ˆ
5
33
4
33
RR
NL
NL
NL
NL
RRref
qqeqqe
eqsignR
eeqC
qsignRqReqC
qC
qLv
&&&
&&&
&&
&
&&&&
−=−=
−=
−−=
++−++=
γ
γ
Advanced Design Control (cont.)
Nonlinear RLC Network w/ PID Numerical Results
Advanced Design Control (cont.)
Nonlinear RLC Network w/ PID/Adaptive Numerical Results
Advanced Design Control (cont.)
Nonlinear RLC Network Energy & Power Flow Results
PID
Contr
oller
PID
/Adaptive
PID /adaptive
with
information
flow more
efficient than
PID with
physical flow
alone
Advanced Design Control (cont.)
PID/Adaptive Parameter Update Responses
Lin
ear
RLC
Model
Nonlinea
r R
LC
Model
Advanced Design Control (cont.)
Adaptive/Tracking Conclusions
•Applied new nonlinear power flow control design to power engineering
•Used to design adaptive PID control architecture
•The power flow and energy responses demonstrated the required energy consumption of each controller
•Adaptive PID control showed to be more efficient than just PID control alone
•Future work to extend apply to power grid control problems
Case Study #1:
Control Design Issues
Case Study #1:
Control Design Issues
1) Sinusoid Damping/ Nonlinear Feedback
i.) Model
Limit Cycles:
Where
for
)sin(xukxxm &&& ==+
0)sin( =+− xxx &&&
⇒>+= 02
1
2
1 22 xxH & Statically Stable
[ ] ( )[ ]xxxxxmH &&&&&& sin=+=
0)sin( == ∫ dtxxHcyclic&&&
τ
πnx ±=&,...3,1=n
Case Study #1:
Control Design Issues
ii.) Nonlinear second-order model – responses for initial limit cycle
3D Hamiltonian 2D Phase Plane
Case Study #1:
Control Design Issues
• Dynamically unstable, neutral stable, stable
Case Study #1:
Control Design Issues
3D Hamiltonian 2D Phase Plane
• Nonlinear second-order model – responses for second limit cycle
Case Study #1:
Control Design Issues
• Dynamically unstable, neutral stable, stable
(second limit cycle)
Case Study #1:
Control Design Issues
3D Hamiltonian 2D Phase Plane
• Nonlinear second-order model – responses for BOTH limit cycles
Case Study #1:
Control Design Issues
2) Multiple Input Multiple Output (MIMO)
i.) model:
12222
12111
FFxm
FFxm
+=
−=
&&
&&2
11 12
1xmH &=
2
2222
1xmH &=
Case Study #1:
Control Design Issues
=1H& 111 xxm &&& [ ] 1121 xFF &−=
=2H& 222 xxm &&& [ ] 2122 xFF &+=
=12H 2
21
2
112
1
2
1xmxm && +
=12H& [ ] [ ] =+ 222111 xxmxxm &&&&&& )( 12122211 xxFxFxF &&&& −++
=INH2
1 2
1ii
N
i
xm &∑=
=INH& ii
N
i
xF &∑=1
[ ]jjjj
N
j
xxF && −+ ++
−
=
∑ 1)1(
1
1
Case Study #1:
Control Design Issues
ii.) Decoupled (Neutral Stability) – Linear System Controller Design
Decoupling controller defined as:
=
=
=
−
−−−−=
−
−=
−
−=
=
+−−=
+
−==
∫
2
1
2
1
2
1
0
0,,
ˆ
ˆ
,,0
0
12
12
12
12
12
12
0
2212
1211
2212
1211
2
1
122
121
I
I
I
D
D
D
P
P
P
t
DIP
K
KK
Kc
cKK
Kk
kKK
F
FxKxdKxKu
cc
ccC
kk
kkK
m
mM
uxCKxFF
FFFxM
&
&&&
τ
Case Study #1:
Control Design Issues
• Decoupled (Neutral Stability)
2,12 =+
=+
= iforKc
K
m
Kk
i
ii
Dii
I
i
Pii
iω
Case Study #1:
Control Design Issues
• Decoupled
(Asymptotically Stable)
Case Study #1:
Control Design Issues
• Decoupled (Unstable)
Case Study #1:
Control Design Issues
• Decoupled Energy Responses (neutral, stable, unstable)
• Decoupled Power Flow Responses (neutral, stable, unstable)
Zero power flow for
dynamically neutral
stability decoupled
control case
Case Study #1:
Control Design Issues
iii.) Coupled – Linear System Controller Design from combined Hamiltonian
Which matches up eigenvalues of system when (coupled integral gain matrix)
Eigenvalues and eignevectors:
[ ]
[ ][ ] [ ]
[ ] [ ]
[ ][ ] [ ][ ] 0
2
1
2
1
22
1
012
12
=Φ+−=Φ−+
++=
−+−=++=
++=
−
∫
DIP
PDI
t
ID
T
P
T
P
TT
KCKMKK
KKMKCK
xdKxKCxxKKxMxH
xKKxxMxH
ωω
τ&&&&&&
&&
• System effectively decoupled into eigenspace:
Case Study #1:
Control Design Issues
Decoupled in
phase plane:
Case Study #1:
Control Design Issues
• Similar results (Dissipative):
Case Study #1:
Control Design Issues
• Similar results (Generative):
Case Study #1:
Control Design Issues
iv.) Coupled – Linear System Controller Design where
With a single PID controller
The coupled Hamiltonian
Resulting time derivative
[ ]
( ) ( )
( ) ( )2121210
11112
2
1212
2
11
2
22
2
1112
01111
2121
11
1
111
2
1
2
1
2
1
2
1
00
xxcxdxKxKcH
xxkxKkxmxmH
xKdxKxKu
Fanduumm
t
ID
P
t
DIP
T
&&&&&
&&
&
−−
−+−=
−++++=
−−−=
==>
∫
∫
τ
τ
Case Study #1:
Control Design Issues
• Coupled (Neutral)
Case Study #1:
Control Design Issues
• Coupled (Asymptotically Stable)
Case Study #1:
Control Design Issues
• Coupled (Unstable)
Case Study #1:
Control Design Issues
v.) van der Pol / Nonlinear Controller:
1. Nonlinear MIMO system
2. Van der Pol damping treated as controller
3. Combined Hamiltonian
4. Time derivative
5. Neutral stability condition
uxxfKxxM ∆+−=+ ),( &&&
[ ] [ ]
[ ]∫∫ −==
−=+=
+=
ττdtxxfxdtH
xxfxKxxMxH
KxxxMxH
T
TT
TT
),(0
),(
2
1
2
1
12
12
12
&&&
&&&&&&
&&
( )( )
−
−=∆
−=∆
∆∆−+−
∆∆−−−=
12
12
12
2
2
2
2
2
1
2
1
ˆ
ˆ
)(
)(),(
121222
121211
F
Fu
xxx
xxKKxxKK
xxKKxxKKxxf
DGDG
DGDG
&&
&&&
Case Study #1:
Control Design Issues
• Two-body dynamically neutral stability coupled control van der Pol system
•Positions
•Energy and
power flow
Case Study #1:
Control Design Issues
• Two-body dynamically neutral stability coupled control van der Pol system
•Phase plane
projections
•Power flow
partitioned into
dissipation and
generation for
coupled 12
system
3) Noncollocated Control
i.) Model
Case Study #1:
Control Design Issues
Case Study #1:
Control Design Issues
0)()(
)()(
)()()()(
2222
21212121222
121122112111111
==
+−−−−=
+−−−−−−=
xfxg
uxxfxxgxm
uxxfxxgxfxgxm
&
&&&&
&&&&&
=1H )(2
111
2
11 xVxm +&
=1H& [ ] [ ] 11211221121111111 )()()()( xuxxgxxfxfxxgxm &&&&&&& +−−−−−=+
=2H2
222
1xm &
=2H& [ ] [ ] 2212121212222 )()( xuxxgxxfxxm &&&&& +−−−−=
Case Study #1:
Control Design Issues
=12H ( ) ( )211211
2
22
2
112
1
2
1xxVxVxmxm −+++ &&
=12H& [ ] [ ] ))(()( 21211222211111 xxxxgxxmxxgxm &&&&&&&& −−+++
[ ] [ ] 2121222121121111 )()()( xxxgxmxxxgxgxm &&&&&& −++−++=
[ ] [ ] 22121211211211 )()()( xuxxfxuxxfxf &&&&&&& +−−++−−−=
[ ] [ ] [ ] )()()( 212112221111 xxxxfxuxuxf &&&&&&& −−−+++−=
Case Study #1:
Control Design Issues
ii.) Design Controller
A) Pick and
Drive to !!
Statically Stable
0)( 11 >xV 0)( 2112 >− xxV
⇒
⇒
)0,0(
Case Study #1:
Control Design Issues
B) Pick
−∫−−=
−∫−−=
222222
111111
2
1
xKdtxKxKu
xKdtxKxKu
DIp
DIp
&
&
=12H )()(2
1
2
1
2
1
2
1211211
2
2
2
1
2
22
2
11 11xxVxVxKxKxmxm pp −+++++ &
=12H& [ ] [ ] ))(()( 2111122222111111 21xxxxgxxKxmxxKxgxm pp&&&&&&&& −−+++++
[ ] [ ] 221212221121121111 21)()()( xxKxxgxmxxKxxgxgxm pp
&&&&& +−+++−++=
[ ] [ ] 22211111 221)( xdtxKxKxdtxKxKxf IDID
&&&&& ∫−−+∫−−−=
[ ] )()( 212112 xxxxf &&&& −−−+
Collocated
Control
Case Study #1:
Control Design Issues
C) Pick
;2221 111xKdtxKxKu DIp&−∫−−= 02 =u
=12H )()(2
1
2
1211211
2
22
2
11 xxVxVxmxm −+++ &&
=12H& [ ] [ ] 2121222121121111 )()()( xxxgxmxxxgxgxm &&&&&& −++−++
[ ] [ ][ ] [ ][ ] [ ] )()()(
)()()(
)()()(
212112122211
2121121111
2121211211211
111xxxxfxxKdtxKxKxf
xxxxfxuxf
xxxfxuxxfxf
DIp&&&&&&&
&&&&&&
&&&&&&&
−−−+−∫−−−=
−−−++−=
−−++−−−=
Noncollocated
Control
Case Study #1:
Control Design Issues
* ⇒−=∆ 12 xxx xxx ∆+= 12
=12H [ ] 111111 )()()()(11
xxxKdtxxKxxKxf DIp&&&& ∆+−∆+∫−∆+−−
[ ] xxf && ∆∆−+ )(12
[ ] [ ] xxfxxKdtxKxKxf DIp&&&& ∆∆−+−∫−−−= )()( 12111111 111
[ ] 1111xxKxdtKxK DIp&&∆−∆∫−∆−+
=⇒ 12H 2
1211211
2
22
2
11 12
1)()(
2
1
2
1xKxxVxVxmxm p+−+++ &
=12H& 11111 11111)( xxKxdtKxKxKxKxf
xu
DIpID&
44444 344444 21&&&
∆+∆∫+∆−∫−−−
∆∆
[ ] xxf && ∆∆−+ )(12How big is this?
Case Study #1:
Control Design Issues
iii.) Collocated vs. NonCollocated (Unstable)
0 10 20 30 40-2
0
2
4
Position (m)
Time (sec)
x1
x2
x12
0 10 20 30 40-100
-50
0
50
Power Flow (W) Energy (J)
Time (sec)
H12 and Hdot 12
Diss
Gen
Dissdot
Gendot
0 10 20 30 40-10
-5
0
5
10
Position (m)
Time (sec)
x1
x2
x12
0 10 20 30 40-400
-200
0
200
400Power Flow (W) Energy (J)
Time (sec)
H12 and Hdot 12
Diss
Gen
Dissdot
Gendot
Case Study #1:
Control Design Issues
iii.) Collocated vs. NonCollocated (Unstable)
0 10 20 30 400
20
40
60
Time (sec)
Energy (J)
H12
0 10 20 30 40-100
-50
0
50
Time (sec)
Power Flow (W)
Hdot12
0 10 20 30 400
10
20
30
40
50
Time (sec)
Energy (J)
H12
0 10 20 30 40-80
-60
-40
-20
0
20
Time (sec)
Power Flow (W)
Hdot12
Case Study #1:
Control Design Issues
iii.) Collocated vs. NonCollocated (Unstable)
0 10 20 30 40-4
-2
0
2
4
Time (sec)
PID Control Effort (N)
u1
u2
0 10 20 30 40-100
-50
0
50
Time (sec)
PID Control Effort for ∆ x (N)
∆ u∆ x
0 10 20 30 40-10
-5
0
5
10
Time (sec)
PID Control Effort (N)
u1
u2
0 10 20 30 40-80
-60
-40
-20
0
20
Time (sec)
PID Control Effort for ∆ x (N)
∆ u∆ x
Case Study #1:
Control Design Issues
iii.) NonCollocated (Neutral) vs. NonCollocated (Stable)
0 10 20 30 40-4
-2
0
2
4
Position (m)
Time (sec)
x1
x2
x12
0 10 20 30 40-150
-100
-50
0
50
100
150
Power Flow (W) Energy (J)
Time (sec)
H12 and Hdot 12
Diss
Gen
Dissdot
Gendot
0 10 20 30 40-4
-2
0
2
4
Position (m)
Time (sec)
x1
x2
x12
0 10 20 30 40-50
-40
-30
-20
-10
0
10
Power Flow (W) Energy (J)
Time (sec)
H12 and Hdot 12
Diss
Gen
Dissdot
Gendot
Case Study #1:
Control Design Issues
iii.) NonCollocated (Neutral) vs. NonCollocated (Stable)
0 10 20 30 400
10
20
30
40
50
Time (sec)
Energy (J)
H12
0 10 20 30 40-80
-60
-40
-20
0
20
Time (sec)
Power Flow (W)
Hdot12
0 10 20 30 400
10
20
30
40
50
Time (sec)Energy (J)
H12
0 10 20 30 40-80
-60
-40
-20
0
20
Time (sec)
Power Flow (W)
Hdot12
Case Study #1:
Control Design Issues
iii.) NonCollocated (Neutral) vs. NonCollocated (Stable)
0 10 20 30 40-4
-2
0
2
4
Time (sec)
PID Control Effort (N)
u1
u2
0 10 20 30 40-80
-60
-40
-20
0
20
Time (sec)
PID Control Effort for ∆ x (N)
∆ u∆ x
0 10 20 30 40-4
-2
0
2
4
Time (sec)
PID Control Effort (N)
u1
u2
0 10 20 30 40-80
-60
-40
-20
0
20
Time (sec)
PID Control Effort for ∆ x (N)
∆ u∆ x
Case Study #1:
Control Design Issues
4) Stable / Unstable Systems
i.) Model : Nonlinear Linear Oscillator
)()( xfuxgxm &&& −=+
)(2
1 2 xVxmH += &
[ ] ( )[ ]xxfuxxgxmH &&&&&& −=+= )(
Equation of motion:
Case Study #1:
Control Design Issues
ii.) Pick Functions:
=)(xV42
4
1
2
1xkkx NL+
=)(xf & )(xsigncxc NL&& +
=u xKxdtKxK DIp&−∫−−
( ) ( ) xdtKxsigncxKcxkxKkxm INLDNLp ∫−−+−=+++ )(3 &&&&
Case Study #1:
Control Design Issues
iii.) Evaluate Control Design
A. Static Stability:
B. Dynamic Stability:
422
4
1)(
2
1
2
1xkxKkxmH NLp +++= &
⇒ Stable for 0, >NLkm and 0>+ pKk
⇒ Pick :0<+ pKk Unstable at )0,0(
( )[ ]xxdtKxsigncxKcH INLD&&&& ∫−−+−= )(
( )[ ] [ ] dtxxdtKxxsigncxKc INLDcc
&&&& ∫−=++ ∫∫ ττ)(
Limit cycle:Gain Scheduling
Case Study #1:
Control Design Issues
0>+ pKk
:0<+⇒ pKk Unstable at )0,0(
Stable for⇒
Case Study #1:
Control Design Issues
5) Many popular coupled dynamical systems: (robots,
spacecraft, etc.) can be modeled as
External torque
Generalized position
Generalized velocity
Symmetric mass matrix
Centripetal/Coriolis vector
Gravitational vector=
=
=
=
=
=
++=
)(
),(
)(
)(),()(
qG
qqC
qM
q
q
qGqqCqqM
&
&
&&&
τ
τ
Dynamical
systems in
this form can
be shown to
be designed as
decoupled
SISO systems
Work rate of the gyroscopic or
centripetal/Coriolis terms are
zero [Ref. Slotine, Robinett]
[Ref. Slotine] J..-J.E. Slotine and W. Li, Applied Nonlinear Control, Prentice-Hall, Inc., N.J., 1991
[Ref. Robinett] R.D. Robinett, G.G. Parker, H. Schaub, J.L. Junkins, Lyapunov Optimal Saturated Control for
Nonlinear Systems, J. Guidance, Control, and Dynamics, Vol. 20, No. 6, pp. 1083-1088, Nov.-Dec., 1997
Case Study #1: MIMO 2 DOF
Planar Robot w/ PID Control System
qqq
qKdqKqKqCqH
qqCqqH
ref
t
DIPrefref
−=
−−−+=
=+
∫~
~~~ˆˆ
),()(
0
&&&&
&&&
ττ
τEOM and PID Control Law:
Case Study #1: Lyapunov Function/
Hamiltonian Based on Error Energy
[ ] [ ]
ave
t
I
T
aveD
T
aveioave
t
I
T
D
T
P
TT
dqKqqKq
STW
dqKqqKq
qKqqHqH
]~~[]~~[
~~~~
~~
2
1~~
2
1
0
0
∫
∫
−=
∆=∆
−−=
+=∆=
τ
τ
&&&
&&
&&&&
&&
VVVV
VVVV
Nonlinear stability boundary
Applied per DOF
(assumes perfect
parameter matching)
Case Study #1: Planar Robot w/ PID Control
Numerical Simulation Results - Case 1
0 1 2 3 4 5-2
-1
0
1
2
3
Time (sec)
Position (rad)
q1ref
q1
q2ref
q2 0 1 2 3 4 5
-25
-20
-15
-10
-5
0
5
Time (sec)
Diss\Gen Exergy (J) and Exergy Rate (W)
To ∆ SDOT
∆ WDOT
To ∆ S
∆ W
0 1 2 3 4 5-40
-30
-20
-10
0
10
Time (sec)
Diss\Gen Exergy (J) and Exergy Rate (W)
To ∆ SDOT
∆ WDOT
To ∆ S
∆ W
DOF 1
DOF 2
DISSIPATIVE
0 1 2 3 4 5-3
-2
-1
0
1
2
3
Time (sec)
Position (rad)
q1ref
q1
q2ref
q2
0 1 2 3 4 5-300
-200
-100
0
100
200
300
Time (sec)
Diss\Gen Exergy (J) and Exergy Rate (W)
To ∆ SDOT
∆ WDOT
To ∆ S
∆ W
0 1 2 3 4 5-50
0
50
Time (sec)
Diss\Gen Exergy (J) and Exergy Rate (W)
To ∆ SDOT
∆ WDOT
To ∆ S
∆ W
DOF 1
DOF 2
NEUTRAL STABILITY
Case Study #1: Planar Robot w/ PID Control
Numerical Simulation Results - Case 2
0 1 2 3 4 5-10
-5
0
5
10
15
Time (sec)
Position (rad)
q1ref
q1
q2ref
q2
0 1 2 3 4 5-6
-4
-2
0
2
4
6x 10
4
Time (sec)
Diss\Gen Exergy (J) and Exergy Rate (W)
To ∆ SDOT
∆ WDOT
To ∆ S
∆ W
0 1 2 3 4 5-6
-4
-2
0
2
4
6
8x 10
4
Time (sec)
Diss\Gen Exergy (J) and Exergy Rate (W)
To ∆ SDOT
∆ WDOT
To ∆ S
∆ W
DOF 1
DOF 2
GENERATIVE
Case Study #1: Planar Robot w/ PID Control
Numerical Simulation Results - Case 3
Case Study #2
Collective Plume Tracing:
A Minimal Information Approach
to Collective Control
• Problem: Track a chemical plume to its source; find buried landmines
• Characteristics: 1- Nonlinear time/spatially varying plume field
2-Complicated model-turbulence
3-Difficult to correlate model to data in time/space
4- Can’t count on wind or water flow direction
5- Sensor Latency
Solution ??
),( txC
Case Study #2:
Collective Plume Tracing
• Possible solution: Person with a chemical sensor and a GPS receiver
1. Time–spatial correlation issues
2. Sensor latency/dynamic range issues
3. No redundancy
4. Leads to exhaustive search
Case Study #2:
Collective Plume Tracing
• Collective Solution: Team of simple robots performing a collective search
– Distributed sensing
– Decentralized control (optimization/Newton update)
– Simple model of plume
– Redundant
– Independent of latency for stability (Performance?)
– Minimize processing, memory, and communications
– Beat down noise
x
x
x
x
x
x
r
Quadratic FitConcentration
Case Study #2:
Collective Plume Tracing
• Emergent Behavior: Quadratic fit w/o memory
• Evaluate Resource Utilization: Trade-off between processing,
memory, and communications
– Baseline: Minimize all three simultaneously
• 8 bit processor, no memory, broadcast 3 words
Minimum
Norm
Exact Fit
Least squares
# of robots
Surface
Fit
Index
6
Case Study #2:
Collective Plume Tracing
• Goal of DARPA Distributed Robotics [Ref. Byrne] Program: build smallest, dumbest robot that couldn’t do anything other than random motion
• As a team of like robots - could solve complicated problem
• N land-based robots whose task to localize a source that emits a measurable scalar field F(x)
• Robots evaluate the field with a sensor and communicate locations and sensor measurements to neighboring robots
• Robots shared sensor information, but individually decided course of action based on their own estimate of plume field
• Robot samples its environment, broadcasts information to others
0.75x0.71x1.6 inches
Onboard temperature sensor
8-bit RISC processor
CSMA radio network
50 Kbits/sec data rate
[Ref. Byrne] Byrne, et. al., Miniature Mobile Robots for Plume Tracking and
Source Localization Research, J. MicroMech., Vol. 1, No. 3, 2002.
Case Study #2:
Collective Plume Tracing
• Universal Mathematical Approach
– Nonlinear control design via exergy/entropy thermodynamic concepts
– Hamiltonian (universal function) is the key element
– Each column has the ability to work on collective systems
Hamiltonian
Lyapunov
FunctionalsStatistical
Mechanics
Nonlinear
Control
Theory
Equations
of State
Quantum
Mechanics
State
Functions
Chemical
Kinetics
Chemical
Potential
Calculus of
Variations
Equations
of Motion
Shannon
Information
Fisher
Information
Case Study #2:
Collective Plume Tracing
• Kinematic Control – No dynamics; assume tight autopilot loop; control velocity (speed)
1- Need to measure plume field: concentration at a point in time
2- Convert concentration measurements to range and bearing to target
Fit a quadratic surface to distributed sensor measurements
Perform decentralized Newton updates for feedback control
3- Shape the “virtual potential” with a minimum (maximum) at
for Hamiltonian to meet static stability requirements for the robot:
−),( txC
⇒⇒
),( txcxcxxcctxc T
o 21ˆ
2
1ˆˆ),(ˆ +++=
kk xccxx
c+−=⇒=
∂
∂ −+ 1
1
21ˆˆ0
ˆα
x1
1
2ˆˆ* ccx −−=
thi
0ˆ2
1ˆˆˆˆ
2
1)( 211
1
21 >++== −i
T
ii
TT
ici xcxxccccxVHi
Case Study #2:
Collective Plume Tracing
And collective robots
4-Design the power flow, time derivative of the Hamiltonian, to meet the
dynamic stability requirements for the feedback controller
011
>== ∑∑==
ic
N
i
i
N
i
VHH
[ ] iiii xxxccxcccx −=−−=+−= −− *ˆˆˆˆˆ1
1
221
1
2& (Linear tracker)
[ ] [ ] [ ]
0
0ˆˆˆˆˆˆˆ
1
21
1
22121
<=
<++−=+=
∑=
−
i
N
i
i
T
ii
T
ii
HH
xcccxccxccxH
&&
&&
[ ] **
11
11xxxx
Nxx
Nx cmi
N
i
cmi
N
i
cm +−=+−=⇒= ∑∑==
&
Case Study #2:
Collective Plume Tracing
• Kinetic Control – dynamics included:
1-Shape the Hamiltonian surface to meet the static stability requirements for the
robot
and the collective of robots
2- Design the power flow, time derivative of the Hamiltonian, to meet the
dynamic stability requirements for the feedback controller
iii uxm =&&
thi 02
1>+=+=
ii cii
T
icii VxMxVTH &&
[ ] 011
>+== ∑∑==
ici
N
i
i
N
i
VTHH
x
Vu ic
i∂
∂−=
iDiI xKdtxK &−∫−
[ ]
0
0
1
<=
<−∫−=
∂
∂+=
∑=
i
N
i
i
iDiI
T
i
c
ii
T
ii
HH
xKdtxKxx
VxMxH i
&&
&&&&&&
Case Study #2:
Collective Plume Tracing
Case Study #2: Collective Robots
Numerical Results
Convergence:
Dissipative > Generative
for the collective
Case Study #2: Collective Robots
Stability Boundary
• Collective robot oscillations at the limit cycle
• Information Theory Applied To:
1- Kinematic Control: Shannon information /entropy
a) Fundamental trade-off: processing, memory, and communications
minimize all three simultaneously
8 bit processor, no memory, 3 words to communicate
b) Stability: latency independent due to no dynamics; stop until next
update/command
c) Performance: the limitation is the sensor update rate; 60 sec
Channel/system capacity of an equiprobable source
⇒
⇒
⇒
=aveC
( )
( )sec1.080log
60
1
;805.0/40/
2
bitC
etemperaturTbitsTTTn
ave
LOWHIGH
<=⇒
−==∆−=COMM. Link = 50
Kbits/sec
Case Study #2:
Collective Plume Tracing
aveHbits
n &==sec
log1
2τ
informationtime
• Kinetic Control: Fisher Information
a) Fisher Information – Measure of how well the receiver can estimate the
message from the sender.
Shannon Entropy – Measure of the sender’s transmission efficiency
over a communications channel
b) Hamiltonian is exergy; portion of energy that can do work; physical and
information exergies
c) Fisher Information:
“Real Amplitude” function of the probability
density function
dxx
xqdx
x
xp
xpdxxp
x
xpI
222)(
4)(
)(
1)(
)(ln
∂
∂∫=
∂
∂∫=
∂
∂∫=
−= )()(2 xpxq
Case Study #2:
Collective Plume Tracing
d) ‘Mean Kinetic Energy” interpretation of Fisher Information
From Quantum Mechanics in the “Classical Limit” of the expectation of
the momentum squared
Tdtm
dtqI2
44 2 ∫=∫= &
dxx
tx
mp
222 ),(
2 ∂
Ψ∂∫=
h
Case Study #2:
Collective Plume Tracing
- Dirac’s constant
- wave function
- expectation operator
The classical limit is reached when
Is equivalent to
),( txΨ
h
tt
ttxF
dx
xdVx
dt
dmp
dt
d)(
)(2
2
=−==
Case Study #2:
Collective Plume Tracing
)()(
2
2
classical
classical
classicalclassical xF
dx
xdVx
dt
dm =−=
Which occurs when
and
is small
This occurs when
which leads to
( ) ( ) ( )xFxxxFxF ′−+≈
( )xF≈
Case Study #2:
Collective Plume Tracing
xxclassical =
( ) ( )22xxx −=∆
( ) ( ) ( ) ( ) ( ) ...!2
1)(
2+′′−+′−+= xFxxxFxxxFxF
e) Fisher Information Equivalency and Exergy
where
i
i
N
i
Ic Hm
VVVTH1~~~~~
1
∑=
=+++=
i
i
N
i
Tm
T1~
1
∑=
=
==∑=
j
j
N
j
Vm
V1~
1
==∑=
kc
k
N
k
c Vm
V1~
1
potential
control potential
==∑=
lI
l
N
l
I Vm
V1~
1
information potential
Case Study #2:
Collective Plume Tracing
( )[ ]dtVVVTdtHJI Ic
~~~~8
~8 +++∫=∫=+
0~
>=+ HJI &&
0~
<=+ HJI&&&&&
( ) =++∫= dtVVVJ Ic
~~~8Where
(Static stability)
(Dynamic stability)
Bound Fisher Information
Case Study #2:
Collective Plume Tracing
[Ref. Frieden] B.R. Frieden, Physics from Fisher Information: A Unification,
Cambridge University Press, 1999.
Case Study #2:
Summary and Conclusions
•Analyze and design “emergent” behaviors to enable a team
of simple robots to perform plume tracing with assistance of
information theory
•Demonstrated fundamental nature of Hamiltonian function
in design of collective systems
• Equivalences between physical and information-based
exergies shown for Shannon information, Fisher
information, and virtual fields
• Fisher Information Equivalency developed that can serve as
an ideal optimization functional to measure performance and
stability of collective system with respect to required
information resources
Case Study #3
Nonlinear Aeroelasticity
Aero-Servo-Elasticity
Coupling
Aerodynamics
Turbulence
Wind gusts
Flutter
Dynamic Stall
Structures
Multi-Dynamics
Elasticity
Flutter
Lightweight
Controls
Classical/Modern
Nonlinear
Safety
Reliability
Intelligent
Case Study #3:
Challenging Engineering Problem
Case Study #3:
Challenging Engineering Problem
Exergy/Entropy Nonlinear Power Flow Control:
• Rigid Torque Controller Region II & II ½
• Hybrid Pitch and Active Aero Controller Region III
• Include: Dynamic Stall, Classical Flutter, etc.
WINDPACT 1.5 MW WT:
Region III starts around 12 m/s
WT Plant
Wind Conditions
Sensor
Feedback• Tip Deflection
• Tip Rate
• Tip Accelerometer
• Other
Active Aero
Actuator Device
• MicroTabs
• Morphing Wing
• Conventional Flaps
Control
System
Case Study #3:
Active Aero Load Control Architecture Analysis/Design
Model
Reference
Control
System
Optimized Energy Capture
• Region II
• Region II ½
• Region III
• Conventional
(PD,PID,PIDA)
• Rate Feedback
• PPF
• SMC
• State-Space
• Nonlinear Power Flow
Case Study #3:
Nonlinear Aeroelasticity
• Wind tunnel model experienced stall flutter over its entire flight envelope M=0.4-0.9;
-10oto -3
oand 10
oto 3
oangle of attack [Ref. Guiterrez]
•The wings broke and cracked at the 1/3 span position due to
fatigue; first torsional mode ~ 450 Hz
[Ref. Guiterrez] W.T. Gutierrez, R. Tate, H. Fell, Investigation of a
Wind Tunnel Model High Aspect Ratio Wing Fracture, AIAA 94-
2558, 18th AIAA Aeropsace Ground Testing conference, June 1994,
Colorado Springs, CO.
Case Study #3:
Nonlinear Aeroelasticity
• Nonlinear aerodynamic and structural model
– Nonlinear stall flutter model
– Single DOF model
Case Study #3:
Nonlinear Aeroelasticity
- Equations of motion derived from Lagrange’s equation:
αταα
ααα
αα
αα
α
αα
αα
α
α
&
&
&&
&
&
&
D
t
Ipcontrol
aero
NLdamp
controlaerodamp
NL
KdKKuQ
MMQ
signCCQ
QQQQ
KKV
IT
VTL
where
QLL
dt
d
−−−==
+=
−−=
++=
+=
=
−=
=∂
∂−
∂
∂
∫0
42
2
),()(
)(
4
1
2
1
2
1
Case Study #3:
Nonlinear Aeroelasticity
=0
ˆ),(
ααα α
α
&& &
&MCM
=
0
0
ˆ
)(
α
αα
α
MC
M
stall
stall
αα
αα
>
<
- The aerodynamic moments are generated based on the following hysteresis logic:
for
for
for the return hysteresis cycle
for
for
stall
stall
αα
αα
>
<
- Applying Lagrange’s equation yields the EOM :
),()()(3 αααααααα αα&&&&&
&MMusignCCKKI NLNL +++−−=++
A) Linear Region
For the model is linear with or
which produces typical linear aeroelastic behavior. Divergence occurs when
where
Torsional flutter occurs when
where
,stallαα < 0== NLNL CK
KCM ≥α
ˆ for 0=u
[ ] [ ] uCKCCI MM =−+−+ ααααα
ˆˆ &&&&
== 2
2
1ˆ VAKqAKC MMM ρααα
CCM ≥α&
ˆ for 0ˆ >−αMCK
and 0=u
.ˆ qAdKC MM αα &&=
Case Study #3:
Nonlinear Aeroelasticity
B) Nonlinear Stall Flutter
When the motion reaches the model becomes nonlinear where
with
which produces
,stallαα >
),()( αααααα αα&&&&
&MMKCI +=++
22
2
1
2
1αα KIH += &
[ ] [ ]αααααααα αα&&&&&&&
& ),()( MMCKIH ++−=+=
[ ] [ ] dtCdtMM ααααααταατ
&&&&& ∫∫ =+ ),()(
Case Study #3:
Nonlinear Aeroelasticity
Case Study #3:
Nonlinear Stall Flutter Linear Dynamics Results
-40
-20
0
20
40
-40
-20
0
20
40
0
0.5
1
1.5
2
2.5
α (deg)
H = 0.5*I*α dot2 + 0.5*k*α
2 + 0.25*k
NL α4
α dot (deg/s)
Hamiltonian (J)
Dissipate
Neutral
Generate
-40 -20 0 20 40-40
-20
0
20
40
α dot (deg/sec)
α (deg)
Phase Plane
Case 1
Case 2
Case 3
Case Study #3:
Nonlinear Stall Flutter Linear Dynamics Results
0 5 10 15 20-2
0
2
4
6
angle (deg) angle rate (deg/sec)
Time (sec)
CASE 1 Passive
α
α dot
0 5 10 15 20-40
-20
0
20
40
angle (deg) angle rate (deg/sec)
Time (sec)
CASE 3 Generative
α
α dot
0 5 10 15 20-15
-10
-5
0
5
10
15
angle (deg) angle rate (deg/sec)
Time (sec)
CASE 2 Neutral
α
α dot Resp
onses
0 5 10 15 20-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
Time (sec)
Power Flow (W) and Energy (J)
Case 1 Dissipative
Pgen
Egen
Pdiss
Ediss
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
Time (sec)
Power Flow (W) and Energy (J)
Case 2 Neutral
Pgen
Egen
Pdiss
Ediss
Pow
er Flo
w
0 5 10 15 20-6
-4
-2
0
2
4
6
8
Time (sec)
Power Flow (W) and Energy (J)
Case 3 Generative
Pgen
Egen
Pdiss
Ediss
Case Study #3:
Nonlinear Stall Flutter Linear Dynamics Results
Aero Moments
-2 0 2 4 6-0.2
0
0.2
0.4
0.6
0.8
α (deg) and α dot (deg/s)
Aero Moments (N-m)
Case 1 Dissipative: Nonlinear Hysteresis
Stall @ ± 10o
Mα
Mα dot
-20 -10 0 10 20-2
-1
0
1
2
α (deg) and α dot (deg/s)
Aero Moments (N-m)
Case 2 Neutral: Nonlinear Hysteresis
Stall @ ± 10o
Mα
Mα dot
-40 -20 0 20 40-10
-5
0
5
10
α (deg) and α dot (deg/s)
Aero Moments (N-m)
Case 3 Generative: Nonlinear Hysteresis
Stall @ ± 10o
Mα
Mα dot
C) The nonlinear stall flutter can be further modified by adding the nonlinear
stiffness and damping
with
which produces a limit cycle when
),()()( 3 αααααααα αα&&&&&
&MMKKsignCCI NLNL +=++++
[ ] [ ] dtsignCCdtMM NL αααααααταατ
&&&&&& )(),()( +=+ ∫∫
422
4
1
2
1
2
1ααα NLKKIH ++= &
[ ] [ ]αααααααααα αα&&&&&&&&
& ),()()(3 MMsignCCKKIH NLNL ++−−=++=
Case Study #3:
Nonlinear Aeroelasticity
Case Study #3:
Nonlinear Stall Flutter NonLinear Dynamics Results
-40
-20
0
20
40
-30
-20
-10
0
10
20
30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
α (deg)
H = 0.5*I*α dot2 + 0.5*k*α
2 + 0.25*k
NL α4
α dot (deg/s)
Hamiltonian (J)
Dissipate
Neutral
Generate
-40 -20 0 20 40-30
-20
-10
0
10
20
30
α dot (deg/sec)
α (deg)
Phase Plane
Case 1
Case 2
Case 3
Case Study #3:
Nonlinear Stall Flutter NonLinear Dynamics Results
Resp
onses
Pow
er Flo
w
0 5 10 15 20-1
0
1
2
3
4
5
angle (deg) angle rate (deg/sec)
Time (sec)
CASE 1 Passive
α
α dot
0 5 10 15 20-15
-10
-5
0
5
10
15
angle (deg) angle rate (deg/sec)
Time (sec)
CASE 2 Neutral
α
α dot
0 5 10 15 20-30
-20
-10
0
10
20
30
angle (deg) angle rate (deg/sec)
Time (sec)
CASE 3 Generative
α
α dot
0 5 10 15 20-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
Time (sec)
Power Flow (W) and Energy (J)
Case 1 Dissipative
Pgen
Egen
Pdiss
Ediss
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5
Time (sec)
Power Flow (W) and Energy (J)
Case 2 Neutral
Pgen
Egen
Pdiss
Ediss
0 5 10 15 20-4
-2
0
2
4
6
Time (sec)
Power Flow (W) and Energy (J)
Case 3 Generative
Pgen
Egen
Pdiss
Ediss
Case Study #3:
Nonlinear Stall Flutter Linear Dynamics Results
Aero Moments
-2 0 2 4 6-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
α (deg) and α dot (deg/s)
Aero Moments (N-m)
Case 1 Dissipative: Nonlinear Hysteresis
Stall @ ± 10o
Mα
Mα dot
-20 -10 0 10 20-2
-1
0
1
2
α (deg) and α dot (deg/s)
Aero Moments (N-m)
Case 2 Neutral: Nonlinear Hysteresis
Stall @ ± 10o
Mα
Mα dot
-40 -20 0 20 40-6
-4
-2
0
2
4
6
α (deg) and α dot (deg/s)
Aero Moments (N-m)
Case 3 Generative: Nonlinear Hysteresis
Stall @ ± 10o
Mα
Mα dot
D) The nonlinear system can be modified by feedback control to meet several
performance requirements. A PID controller is implemented to show the
effects of feedback control. The model becomes
with
which produces a limit cycle when
[ ] [ ] τααααααααα αα dKMMsignCKCKKKIt
INLDNLp ∫−++−+−=+++0
3 ),()()( &&&&&&
[ ] 422
4
1
2
1
2
1ααα +++= pKKIH &
[ ]αααα &&&& 3)( NLp KKKIH +++=
αταααααα αα&&&&
&
−++−+−= ∫ dKMMsignCKC
t
INLD0
),()()()(
( )[ ] dtsignCKCdtdKMM NLD
t
I αααατααααταατ
&&&&&& )(),()( 1
0++=
−+ ∫∫∫
Case Study #3:
Nonlinear Aeroelasticity
-60-40
-200
2040
60
-100
-50
0
50
100
0
1
2
3
4
5
6
7
8
9
α (deg)
H = 0.5*I*α dot2 + 0.5*k*α
2 + 0.25*k
NL α4
α dot (deg/s)
Hamiltonian (J)
Dissipate
Neutral
Generate
-100 -50 0 50 100-100
-50
0
50
100
α dot (deg/sec)
α (deg)
Phase Plane
Case 1
Case 2
Case 3
• Nonlinear Power Flow Control Design Dynamic Stall: Limit Cycle Identification
Case Study #3:
Dynamic Stall Control System Results
0 5 10 15 20-30
-20
-10
0
10
20
30
angle (deg) angle rate (deg/sec)
Time (sec)
CASE 1 Passive
α
α dot
0 5 10 15 20-40
-20
0
20
40
angle (deg) angle rate (deg/sec)
Time (sec)
CASE 2 Neutral
α
α dot
0 5 10 15 20-100
-50
0
50
100
angle (deg) angle rate (deg/sec)
Time (sec)
CASE 3 Generative
α
α dot
Resp
onses
Case Study #3:
Dynamic Stall Control System ResultsPow
er Flo
w
0 5 10 15 20-4
-3
-2
-1
0
1
2
3
Time (sec)
Power Flow (W) and Energy (J)
Case 1 Dissipative
Pgen
Egen
Pdiss
Ediss
0 5 10 15 20-15
-10
-5
0
5
10
15
Time (sec)
Power Flow (W) and Energy (J)
Case 2 Neutral
Pgen
Egen
Pdiss
Ediss
0 5 10 15 20-40
-20
0
20
40
Time (sec)
Power Flow (W) and Energy (J)
Case 3 Generative
Pgen
Egen
Pdiss
Ediss
Aero Moments
Case Study #3:
Dynamic Stall Control System Results
-40 -20 0 20 40-5
-4
-3
-2
-1
0
1
2
α (deg) and α dot (deg/s)
Aero Moments (N-m)
Case 1 Dissipative: Nonlinear Hysteresis
Stall @ ± 10o
Mα
Mα dot
-40 -20 0 20 40-10
-5
0
5
10
α (deg) and α dot (deg/s)
Aero Moments (N-m)
Case 2 Neutral: Nonlinear Hysteresis
Stall @ ± 10o
Mα
Mα dot
-100 -50 0 50 100-15
-10
-5
0
5
10
15
20
α (deg) and α dot (deg/s)
Aero Moments (N-m)
Case 3 Generative: Nonlinear Hysteresis
Stall @ ± 10o
Mα
Mα dot
•Nonlinear power flow control was applied to a
nonlinear stall flutter problem (dynamic stall)
•Nonlinear structural and discontinuous aerodynamic
models were directly accommodated
•The limit cycles were found by partitioning power
flows
•The limit cycles were shown to be stability boundaries
•Flutter suppression controllers were initially assessed
Case Study #3:
Summary and Conclusions
Case Study #4
Power Engineering
Case Study #4:
Power Engineering - OMIB
I. Model (Simple one-machine infinite bus swing equation):
II. Controller Design:
( )[ ]
( ) δδδ
ωδ
ωδ
ωδω
ωδ
&&&
&
&&&
&
&
DPPM
DPPM
mMAX
MAXm
−=+
=
=
−−=
=
sin
sin1
( )[ ]
( )[ ] [ ]
[ ] 0
0
sin
cos12
1 2
<++−⇒
<⇒
++−=+=
−+=
∫ δδ
δδδδδ
δδ
τ
&&
&
&&&&&&
&
uPD
H
uPDPMH
PMH
m
mMAX
MAX
( )
0
0547.1
)sin(0
==
=
−−−−−= ∫
DP
s
t
sIDsP
KK
dKKKu
δ
τδδδδδ &
Case Study #4:
Power Engineering- OMIB Trajectories
-8 -6 -4 -2 0 2 4 6 8-10
-5
0
5
10
0
1
2
3
4
0.5 8/(100 3.1415) (ω)2+1.2647 (1-cos(δ))
δ
ω
• Dark blue (2.0865,0)
•Black (2.0865*1.2,0)
•Green (2.0865*1.2,0), K_I = 1.0
•Cyan (-4.197,0)
•Red (-4.196,0)
•Magenta (-4.197,0), K_I = 0.01
•Magenta (dash) (-4.197,0), K_I = 0.05
Case Study #4:
Power Engineering - 2MIB
I. Model (Two-machine infinite bus swing equations):
II. Controller Design:
[ ]
[ ]222232112
2
2
111131212
1
1
22
11
sinsin1
sinsin1
2
1
ωδδω
ωδδω
ωδ
ωδ
DCCPM
DCCPM
m
m
−−−=
−−−=
=
=
&
&
&
&
[ ] [ ] [ ]
[ ] [ ]( )
( ) 212122222322
121121111311
2222111112
2112223113
2
22
2
1112
sinsin
sinsin
cos1cos1cos12
1
2
1
2
1
21
uCDPCM
uCDPCM
uDPuDPH
CCCMMH
m
m
mm
+−−−=+
+−−−=+
+−++−=
−+−+−++=
δδδδδ
δδδδδ
δδδδ
δδδδδ
&&&
&&&
&&&&&
&&
For:
Case Study #4:
Overview of Lanai “Like” Model
• 12470 V, 5.5 MW peak load
• 4 Diesel Generators (2 – 2.2 MW, 2 – 1.0 MW) with controls
• 1.2 MW High Level PV controls
• Switchable Loads
• Distribution lines (series RL) calculated for 350kcmil under 5 miles
• 3-phase faults at each bus
Model Courtesy: Ben Schenkman
Lanai “Like” Model Parameter Characteristics:
• Real World Microgrid – “Lanai and Kauai”
• “Like” Model of Real World Microgrid
• Advanced Controls on Distributed Generation
Case Study #4:
Lanai ”Like” MicroGrid Model
Model Courtesy: Ben Schenkman
YELLOW – Electrical Power Output
PURPLE – Control Output
BLUE – Speed Error
Case Study #4:
Lanai “Like” Control Model
•Nonlinear Power Flow Control Design:
∑4
1
generators
PV
Generation
function (sun)
Loads setup similar to
Lanai “like” model
Case Study #4:
Lanai “Like” Control Model
0 0.1 0.2-0.4
-0.2
0
0.2
0.4
Charge (mA-sec)
Time (sec)
qref
q
0 0.1 0.2-100
0
100
200
Charge rate (mAmp)
Time (sec)
qdot
ref
qdot
0 0.1 0.2-100
-50
0
50
100
Vin (Volts)Time (sec)
v
vref
∆ v
videal
0 0.05 0.1 0.15 0.2-5
0
5
10
Rhat (Ohm)
Time (sec)
0 0.1 0.2-0.4
-0.2
0
0.2
0.4
Energy (mJoules)
Time (sec)
∫ Pdiss d τ
∫ Pgen d τ
0 0.1 0.2-0.4
-0.2
0
0.2
0.4
Power Flow (Watts)
Time (sec)
Pdiss
Pgen
• Investigated OMIB model – Limit cycles and control
design options
• Investigated 2MIB model
• Identified microgrid model based on actual Lanai
“like” electric power grid system
•Evaluated Exergy/entropy control design for
simplified Lanai “like” model
Case Study #4:
Summary and Conclusions