nonlinear power flow control designwindpower.sandia.gov/other/088055.pdf · nonlinear power flow...

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

rdrobin@sandia.gov dwilso@sandia.gov

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

−12W

1212 WEE −=−

[Ref. Gyftopoulos]: E.P. Gyftopoulos and G.P. Beretta, Thermodynamics,

Foundations and Applications, MPC, NY, 1991.

- 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 −=

−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

Reservoir, Reservoir,oT

Reversible Expansion

Adiabatic Expansion – Irreversible

– Available work:

– Entropy:

– Adiabatic processes: Reversible:

Irreversible:

12121212 )(;)( WWWrevrev ≥−=Ω−Ω

)( Ω−Ε≡ ddcdS R

0)( =REVdS

0)( ≥irrevdS

– 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

dQdS =

etemperaturT −

∫==T

– Irreversible process [Ref. Prigogine]:

- Entropy change due to exchange of energy and matter

(entropy flux)

- Entropy change due to irreversible processes

SdSddS ie +=

[Ref. Prigogine] D. Kondepudi and I. Prigogine, Modern Thermodynamics: From

Heat Engines to Dissipative Structures, John Wiley & Sons, New York, N.Y., 1999.

Figure 1- Entropy changes

0≥=∑ kk

i dXFSd

k kF − thermodynamics force

thermodynamics flowth

k kX −

• 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

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.

VpdTdS

VpdTdSdU

∫∫∫∫∫

1Work Done (area)

- A uniform temperature exists throughout the system.

= internal energy

volumeVpressurep −− ,;

- Reversible Cycle Processes:

• 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

pekehmWQE &&&&

∑∑

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:

Force Vector

Position Vector

– Power:

Velocity Vector

– Kinetic Energy: (Newton’s 2nd law)

rdFdW ⋅=−W

dP &⋅=⋅==

−r&−P

rmF &&=

−⋅=

⋅=⋅=⋅=⋅=

dTrrmdrdrmrdrmrdFdW

&&&&&&

Kinetic Energy

Mechanics (cont.)

– Potential Energy:

Conservative and path independent force field

⋅=−

−=⋅

rdrFrVrV

)()()(

Potential Energy

==+⇒

−=⋅=

=⋅∫

Constant

dV0)( =+VT

Total Energy (stored energy)

Mechanics (cont.)

– Time Dependent Potential Field: −=⋅ rdtrF ),(

∂=+ )(

),( trdV

Mechanics (cont.)

-Non-conservative Forces:

Conservative Force (Potential Field)

Non-Conservative Force

NCc FFF +=

−NCF

&⋅+∂

∂=+= )(

Mechanics (cont.)

• Energy Diagrams and Phase Planes

– Mass, Spring, Damper System

1xmT &=

1kxV =

xcuF &−=

−u Control

−xc& Damping

xcukxxm &&& −=+

Mechanics (cont.)

a) Set conservative system, :oxcu == & =+= VTE Constant

1kxV =

1xmT &=

Mechanics (cont.)

The system path is constrained to the energy storage surface along

a constant energy orbit

1oo kxxmE += &

E= constant

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

∂−

),,(),,(1

tqqHtqqLqq

=−∂

∂≡ ∑

Mechanics (cont.)

– Canonical Momentum:

– Hamilton’s Canonical Equations of Motion:

– Time Derivative of the Hamiltonian:

),,(),,(1

tqqLqptpqH ii

&& −= ∑=

∂−=

LqpqpH

∂−=

∂−

∂−+=

&&&&&&

Mechanics (cont.)

– For most natural systems:

Power (work/energy) flow equation

&&& ∑=

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:

then and

dtSSSdSddS

∫∫∫

∫∫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

⋅=−=−=Ξ

∑∑

+== 11

ioST &

≥== ∑∑ kk

XFS &&&

Mechanics (cont.)

d) Assumptions applied to exergy rate equation:

e) A conservative system is equivalent to a reversible system when

then and

≅−

∑ FLOW

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

TdSVpd

dWTdSdE

VpdTdSdU

∫∫∫∫∫∫∫∫∫∫

dtqQdtqQ

dtSTdtW

dtSTWdtH

∑∫∑∫

∫∫∫∫

Mechanics (cont.)

– Sorting Power Terms:

a) Conservative Terms

Potential functions

b) Generator Terms

c) Dissipator Terms

⇒=⋅∫ 0dtqF c&

>∫⇒>⋅∫ ∑=

dtqQdtqF jj

cNCc&&

<∫⇒<⋅∫ ∑+

dtqQdtqF kk

cNCc&&

Mechanics (cont.)

– Limit Cycles:

dtqQqQ

dtSTdtW

∑∫∑∫

∫∫+

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

Statically unstable if

And Neutrally stable if

0)( >xV

ee xxH

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)

<= ∫c

AVE dtHHτ

>= ∫ dtHHc

== ∫ dtHHc

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 xVxmEH

uxfxgxm

+−=+

Stability (cont.)

a) Linearized system

b) Assume , Eigenvalue Problem

Undamped natural frequency (eigenvalue) of a statically stable and

dynamically neutral stable system

xxcuxkxxmH

uxckxxm

kcmxcxfkxxV

&&&&&&

0,,;)(;02

−=+=

+−=+

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:

xKcxkxxmH

D&&&&&

+−=+=

xKu D&−= 0>c

0>+ DKc0<+ DKc

0=+ DKc

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

=−=+= ∫∫ dtxxfudtxxgxmHcc

AVE&&&&&&

ττ ττ

0)( =+ xgxm &&

0)]([1

<−= ∫ dtxxfuHc

AVE&&&

0)]([1

>−= ∫ dtxxfuHc

AVE&&&

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

1)( kxxV −= ;0>k

>+=−==

xkxxkxxm

kxxmVTL

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

∫ dtxxfu

ave&&&

τVVVV

( ) 0)]([1

∫ dtxxfu

ave&&&

τVVVV

[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

1)( xkkxxV NC+−= 0, >NLkk =u

xkxkKxmH

xkxkKxm

xKuxkkxxm

+−+=

=+−+

−==+−

,xK p− 0=c

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

Tension in String

Mass/unit length

=τ=)(xσ

0),(),0( == tLwtw

b) Non-conservative Distributed Force

workwdxtxFWL

==⇒ ∫ ),(0

c) Extended Hamilton’s Principle

∂−

+−Τ=Ι

∫ ∫

∫∫

dxdtFwww

;Fww xx =−⇒ τσ && 0),(),0( == tLwtw

d) Static Stability

e) Dynamic Stability

1 22 >+=+= ∫ dxwwVTH x

oτσ &

0, >τσ

dxwwwwH

=⇒ ∫∫ dtdxwFH

&&ττ

Limit Cycle

• Controller: wKwdtKwKF DIxxp&−−= ∫

dxwwKwwH

wKwdtKwKw

dxwKwVTH

0])([2

−−=+−

>++=+=

dxwwKwdtK

dxwwKw

−−=

∫∫

∫ τσ

• 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

∞→=

−∑

])[()1(

=+Γ )1( j

)1()2)(1()(

)()()1(lim)(

rqqqqq

rhtfhtfD q

+−⋅⋅⋅−−=

−−= ∑=

→−= 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.

c) First Order Fractional Difference approximation

and for

d) Inhomogeneous Bagley-Torvik Equation

)()(~ )(

khtfwhtfD q

to −= ∑=

k )1()(

,...2,1,0=k

1)( =q

qw −

+−= ,...3,2,1=k

0),()()()( 2/3 >∀=++ ttutCytyDBtyA to&&

0)0()0( == yy &

e) Generalized to

f) First-Order Approximation/numerical algorithm

0),()()()( >∀=++ ttutKxtxDKtxM q

0)0()0( == xx &

01 == xxo

2112 )2()(q

xxMKxuhx

−−−−

−+−=

,...3,2=m

g) Sort power terms:

xxKxdtK

xxKKxMH

xKKxMH

xKxdtKxKu

uKxxCxM

−∫−=

−∫−−=

Storage: and

Dissipator:

Generator:

Storage Dissipator Storage Generator

2 1 0 -1

Coupled

h) Lag – Stabilized Controller [Ref. Robinett]

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.

The Fractional Calculus Approach

• Fractional calculus compared with PID type controllers

0 5 10 15 20-4

Time (sec)

Position (m)

yPD Control

yFC with α=1.0

0 5 10 15 20-8

Time (sec)

Position (m)

yP Control

yFC with α=0.0

0 5 10 15 20-25

Time (sec)

Position (m)

yI Control

yFC with α=-1.0

0 5 10 15 20-4

Time (sec)

Position (m)

yPD Control

yFC with α=0.5

0 5 10 15 20-20

Time (sec)

Position (m)

yPI Control

yFC with α=-0.5

• Optimal [Ref. Ogata] -

Asymptotically Stable to x=0

);,( uxfx =&

→= ))(,( xgxfx&

)],([min),(~

0),(),(~

uxLuxH

dtuxLJ

= ∫∞

VVVV &= 0),(1

=+= uxLuu

1uu=⇒VVVV & ),( 1uxL−=

dttutxLxxo

))(),(())0((0))(( 1∫∞

−==∞ VVVVVVVV

dttutxLxo

))()),(())0(( 1∫∞

)(xgu =

[Ref. Ogata] K. Ogata, Modern Control Engineering, Prentice-Hall, Inc., N.J., 1970.

a) Example #1: ukxxcxm =++ &&&

1 2222

oo kxxmkxxmHJ o +=+−=−= ∞ &&

dtxxcudtxkxxmoo

&&&&& ][][ −−=+−= ∫∫∞∞

xcuxxcu

<⇒<−=

b) Example #2: ukxxm =+&&

])4([2

⟩+⟨−±−=∗

++=−

=+=⇒+=

−−−=−=

++= ∫∞

xkxkkxxk

ukxkxkxu

xuxkxxmkxxm

ukxkxkuxL

dtukxkxkJo

&&&&&&

VVVVVVVV

uxuku &&3

1,0)( −==+

– Fixed Final Time: [ ] dtuuxJft

1+= ∫

TTxfLHuxfxuI ),(;

~);,(; θθλτθ &&&& =+====

BuxAuIx

−−=

∂∂−

∂∂−=

+−=⇒++=∂

uxuuxH

I, inertia

Horizontal Link

torque,

angle,

• Homotopy from i) min. energy to ii) hybrid min. energy plus min.

power, to iii) min. power cost functions

dtuWxuWJft

−+== ∫&

)()1())((2

21 ξαξαξφ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-4

Time (sec)

Horizontal Link Slew POWER opt

Angle Rate

Torque

0 0.2 0.4 0.6 0.8 1-4

Time (sec)

Horizontal Link Slew POWER opt

Torque

=−=Ψ

desiredf

θθξ

Advanced Design Control (cont.)

• Robust / Tracking

– Nonlinear Model:

a) Hamiltonian:

;)()( uxfxgxm +−=+ &&&

RR xxVxxmH −+−= &&

[ ] )()()( RRR xxxxgxxmH &&&&&&& −−+−=

[ ]uuu

xxuxfxgxxgxm

−+−−−+−= )()()()( &&&&

);(ˆ)(ˆ)(ˆˆ xfxgxxgxmu RRREF&&& ++−−=

)()()(ˆ RDRIR xxKdtxxKxxgu && −−−∫−−−=∆

)(ˆ)(ˆ

Robustness/Tracking addressed in the RLC example

– 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 &&&& −−−−∫−

[ ] []

RRRRIAVERD

xxxfxf

xgxgxxgxxgxmmdtxxKxxK

)()()(ˆ

)()(ˆ)(ˆ)(ˆ)()( 2

−⟩−⟨+

⟩−⟨+⟩−−−⟨+⟩−⟨+−∫−>−

Adaptive/Tracking: Simple RLC Model

∫ =+ 0])([ dtqqCqqL &&&&

iiRviqqRvqqCqLH

dyyCqLHq

)()]([)]([

−=−=+=

+= ∫&&&&&&

qRqL Ω==++ cos)(1

)( 0&&&

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

Hamiltonian and time derivative:

0)( >qRq && for 0≠q&

0)( >qCq for 0≠q

• 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

CqL NL

&&&& −−=

( ) ( )422 1

RCAPR qqC

qqLH −+−

++−= &&

ΦΓΦ+ − ~~

Icap VVVT +++=

Adaptive PID Control Strategy: Improve Performance

vqReKqC

ΦΓΦ++−+−−=

ΦΓΦ+++=

1ˆˆ ++=

Controller

selected as:

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≠=

22 1Ω==

eKedKeKvt

dissgencap&∫ −−−=∆

Incorporate into Hamiltonian Time Derivative

∫ ΦΓ+Φ+−−= −t

dissgen eYeeKedKH0

&&&& τ

]ˆ1ˆ[ˆ~

)]ˆ()11

()ˆ([~

)ˆ()11

()ˆ(~

qRRqCC

=Φ=Φ

−−−=Φ

−+−+−=Φ

Identify Adaptive Parameter Update Equations

Perfect Parameter Matching: Nonlinear Stability Boundary

dteKdteedK diss

gen ∫∫∫ <′−ττ

0][ &&

Results in asymptotically stable

(passivity) solution

Robustness/Tracking vs Adaptive/Tracking

5e65e6K_gen

10500K_diss

75010000K_cap

Gain Robustness Adaptive

Higher

controller

achieved

Robustness

at expense of

increased

power usage

Linear RLC Network w/ PID Numerical Results

Linear RLC Network w/ PID/Adaptive Numerical Results

Linear RLC Network Energy & Power Flow Results

/Adaptive

PID /adaptive

information

flow more

efficient than

PID with

physical flow

Next Model Includes Nonlinearities

vqsignRqC

qRqL NL

=++++ )(11 3 &&&&

ADDED: Nonlinear Capacitance and Nonlinear Resistance

)(ˆˆ)(11ˆ

qqeqqe

eqsignR

qsignRqReqC

−=−=

−−=

++−++=

Nonlinear RLC Network w/ PID Numerical Results

Nonlinear RLC Network w/ PID/Adaptive Numerical Results

Nonlinear RLC Network Energy & Power Flow Results

/Adaptive

PID /adaptive

information

flow more

efficient than

PID with

physical flow

PID/Adaptive Parameter Update Responses

Nonlinea

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:

1) Sinusoid Damping/ Nonlinear Feedback

i.) Model

Limit Cycles:

)sin(xukxxm &&& ==+

0)sin( =+− xxx &&&

⇒>+= 02

1 22 xxH & Statically Stable

[ ] ( )[ ]xxxxxmH &&&&&& sin=+=

0)sin( == ∫ dtxxHcyclic&&&

πnx ±=&,...3,1=n

Case Study #1:

ii.) Nonlinear second-order model – responses for initial limit cycle

3D Hamiltonian 2D Phase Plane

Case Study #1:

• Dynamically unstable, neutral stable, stable

Case Study #1:

• Nonlinear second-order model – responses for second limit cycle

Case Study #1:

• Dynamically unstable, neutral stable, stable

(second limit cycle)

Case Study #1:

• Nonlinear second-order model – responses for BOTH limit cycles

Case Study #1:

2) Multiple Input Multiple Output (MIMO)

i.) model:

1xmH &=

Case Study #1:

=1H& 111 xxm &&& [ ] 1121 xFF &−=

=2H& 222 xxm &&& [ ] 2122 xFF &+=

=12H 2

1xmxm && +

=12H& [ ] [ ] =+ 222111 xxmxxm &&&&&& )( 12122211 xxFxFxF &&&& −++

xm &∑=

=INH& ii

xF &∑=1

[ ]jjjj

xxF && −+ ++

∑ 1)1(

Case Study #1:

ii.) Decoupled (Neutral Stability) – Linear System Controller Design

Decoupling controller defined as:

−−−−=

+−−=

FxKxdKxKu

uxCKxFF

Case Study #1:

• Decoupled (Neutral Stability)

2,12 =+

= iforKc

Case Study #1:

• Decoupled

(Asymptotically Stable)

Case Study #1:

• Decoupled (Unstable)

Case Study #1:

• 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:

iii.) Coupled – Linear System Controller Design from combined Hamiltonian

Which matches up eigenvalues of system when (coupled integral gain matrix)

Eigenvalues and eignevectors:

[ ][ ] [ ]

[ ] [ ]

[ ][ ] [ ][ ] 0

=Φ+−=Φ−+

−+−=++=

KCKMKK

KKMKCK

xdKxKCxxKKxMxH

xKKxxMxH

τ&&&&&&

• System effectively decoupled into eigenspace:

Case Study #1:

Decoupled in

phase plane:

Case Study #1:

• Similar results (Dissipative):

Case Study #1:

• Similar results (Generative):

Case Study #1:

iv.) Coupled – Linear System Controller Design where

With a single PID controller

The coupled Hamiltonian

Resulting time derivative

( ) ( )

( ) ( )2121210

xxcxdxKxKcH

xxkxKkxmxmH

xKdxKxKu

Fanduumm

−−

−+−=

−++++=

−−−=

Case Study #1:

• Coupled (Neutral)

Case Study #1:

• Coupled (Asymptotically Stable)

Case Study #1:

• Coupled (Unstable)

Case Study #1:

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

&&&&&&

( )( )

−=∆

∆∆−+−

∆∆−−−=

121222

121211

xxKKxxKK

xxKKxxKKxxf

Case Study #1:

• Two-body dynamically neutral stability coupled control van der Pol system

•Positions

•Energy and

power flow

Case Study #1:

• 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:

)()()()(

21212121222

121122112111111

+−−−−=

+−−−−−−=

uxxfxxgxm

uxxfxxgxfxgxm

=1H )(2

11 xVxm +&

=1H& [ ] [ ] 11211221121111111 )()()()( xuxxgxxfxfxxgxm &&&&&&& +−−−−−=+

=2H& [ ] [ ] 2212121212222 )()( xuxxgxxfxxm &&&&& +−−−−=

Case Study #1:

=12H ( ) ( )211211

1xxVxVxmxm −+++ &&

=12H& [ ] [ ] ))(()( 21211222211111 xxxxgxxmxxgxm &&&&&&&& −−+++

[ ] [ ] 2121222121121111 )()()( xxxgxmxxxgxgxm &&&&&& −++−++=

[ ] [ ] 22121211211211 )()()( xuxxfxuxxfxf &&&&&&& +−−++−−−=

[ ] [ ] [ ] )()()( 212112221111 xxxxfxuxuxf &&&&&&& −−−+++−=

Case Study #1:

ii.) Design Controller

A) Pick and

Drive to !!

Statically Stable

0)( 11 >xV 0)( 2112 >− xxV

Case Study #1:

B) Pick

−∫−−=

222222

111111

xKdtxKxKu

=12H )()(2

1211211

11 11xxVxVxKxKxmxm pp −+++++ &

=12H& [ ] [ ] ))(()( 2111122222111111 21xxxxgxxKxmxxKxgxm pp&&&&&&&& −−+++++

[ ] [ ] 221212221121121111 21)()()( xxKxxgxmxxKxxgxgxm pp

&&&&& +−+++−++=

[ ] [ ] 22211111 221)( xdtxKxKxdtxKxKxf IDID

&&&&& ∫−−+∫−−−=

[ ] )()( 212112 xxxxf &&&& −−−+

Collocated

Control

Case Study #1:

C) Pick

;2221 111xKdtxKxKu DIp&−∫−−= 02 =u

=12H )()(2

1211211

11 xxVxVxmxm −+++ &&

=12H& [ ] [ ] 2121222121121111 )()()( xxxgxmxxxgxgxm &&&&&& −++−++

[ ] [ ][ ] [ ][ ] [ ] )()()(

)()()(

212112122211

2121121111

2121211211211

111xxxxfxxKdtxKxKxf

xxxxfxuxf

xxxfxuxxfxf

DIp&&&&&&&

&&&&&&

&&&&&&&

−−−+−∫−−−=

−−−++−=

−−++−−−=

Noncollocated

Control

Case Study #1:

* ⇒−=∆ 12 xxx xxx ∆+= 12

=12H [ ] 111111 )()()()(11

xxxKdtxxKxxKxf DIp&&&& ∆+−∆+∫−∆+−−

[ ] xxf && ∆∆−+ )(12

[ ] [ ] xxfxxKdtxKxKxf DIp&&&& ∆∆−+−∫−−−= )()( 12111111 111

[ ] 1111xxKxdtKxK DIp&&∆−∆∫−∆−+

=⇒ 12H 2

1211211

1xKxxVxVxmxm p+−+++ &

=12H& 11111 11111)( xxKxdtKxKxKxKxf

DIpID&

44444 344444 21&&&

∆+∆∫+∆−∫−−−

∆∆

[ ] xxf && ∆∆−+ )(12How big is this?

Case Study #1:

iii.) Collocated vs. NonCollocated (Unstable)

0 10 20 30 40-2

Position (m)

Time (sec)

0 10 20 30 40-100

Power Flow (W) Energy (J)

Time (sec)

H12 and Hdot 12

Dissdot

Gendot

0 10 20 30 40-10

Position (m)

Time (sec)

0 10 20 30 40-400

400Power Flow (W) Energy (J)

Time (sec)

H12 and Hdot 12

Dissdot

Gendot

Case Study #1:

0 10 20 30 400

Time (sec)

Energy (J)

0 10 20 30 40-100

Time (sec)

Power Flow (W)

Hdot12

0 10 20 30 400

Time (sec)

Energy (J)

0 10 20 30 40-80

Time (sec)

Power Flow (W)

Hdot12

Case Study #1:

0 10 20 30 40-4

Time (sec)

PID Control Effort (N)

0 10 20 30 40-100

Time (sec)

PID Control Effort for ∆ x (N)

∆ u∆ x

0 10 20 30 40-10

Time (sec)

0 10 20 30 40-80

Time (sec)

∆ u∆ x

Case Study #1:

iii.) NonCollocated (Neutral) vs. NonCollocated (Stable)

0 10 20 30 40-4

Position (m)

Time (sec)

0 10 20 30 40-150

Time (sec)

H12 and Hdot 12

Dissdot

Gendot

0 10 20 30 40-4

Position (m)

Time (sec)

0 10 20 30 40-50

Time (sec)

H12 and Hdot 12

Dissdot

Gendot

Case Study #1:

0 10 20 30 400

Time (sec)

Energy (J)

0 10 20 30 40-80

Time (sec)

Power Flow (W)

Hdot12

0 10 20 30 400

Time (sec)Energy (J)

0 10 20 30 40-80

Time (sec)

Power Flow (W)

Hdot12

Case Study #1:

0 10 20 30 40-4

Time (sec)

0 10 20 30 40-80

Time (sec)

∆ u∆ x

0 10 20 30 40-4

Time (sec)

0 10 20 30 40-80

Time (sec)

∆ u∆ x

Case Study #1:

4) Stable / Unstable Systems

i.) Model : Nonlinear Linear Oscillator

)()( xfuxgxm &&& −=+

1 2 xVxmH += &

[ ] ( )[ ]xxfuxxgxmH &&&&&& −=+= )(

Equation of motion:

Case Study #1:

ii.) Pick Functions:

=)(xV42

1xkkx NL+

=)(xf & )(xsigncxc NL&& +

=u xKxdtKxK DIp&−∫−−

( ) ( ) xdtKxsigncxKcxkxKkxm INLDNLp ∫−−+−=+++ )(3 &&&&

Case Study #1:

iii.) Evaluate Control Design

A. Static Stability:

B. Dynamic Stability:

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:

0>+ pKk

:0<+⇒ pKk Unstable at )0,0(

Stable for⇒

Case Study #1:

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=

)(),()(

qGqqCqqM

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

qKdqKqKqCqH

qqCqqH

DIPrefref

−−−+=

~~~ˆˆ

τEOM and PID Control Law:

Case Study #1: Lyapunov Function/

Hamiltonian Based on Error Energy

[ ] [ ]

aveioave

dqKqqKq

qKqqHqH

]~~[]~~[

∆=∆

−−=

+=∆=

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

Time (sec)

Position (rad)

q2 0 1 2 3 4 5

Time (sec)

Diss\Gen Exergy (J) and Exergy Rate (W)

To ∆ SDOT

∆ WDOT

To ∆ S

0 1 2 3 4 5-40

Time (sec)

To ∆ SDOT

∆ WDOT

To ∆ S

DISSIPATIVE

0 1 2 3 4 5-3

Time (sec)

Position (rad)

0 1 2 3 4 5-300

Time (sec)

To ∆ SDOT

∆ WDOT

To ∆ S

0 1 2 3 4 5-50

Time (sec)

To ∆ SDOT

∆ WDOT

To ∆ S

NEUTRAL STABILITY

0 1 2 3 4 5-10

Time (sec)

Position (rad)

0 1 2 3 4 5-6

Time (sec)

To ∆ SDOT

∆ WDOT

To ∆ S

0 1 2 3 4 5-6

Time (sec)

To ∆ SDOT

∆ WDOT

To ∆ S

GENERATIVE

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 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

Quadratic FitConcentration

Case Study #2:

• 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

Exact Fit

Least squares

# of robots

Surface

Case Study #2:

• 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:

• 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

Functions

Chemical

Kinetics

Chemical

Potential

Calculus of

Variations

Equations

of Motion

Shannon

Information

Fisher

Information

Case Study #2:

• 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

1ˆˆ),(ˆ +++=

kk xccxx

c+−=⇒=

∂ −+ 1

21ˆˆ0

2ˆˆ* ccx −−=

1ˆˆˆˆ

1)( 211

21 >++== −i

ici xcxxccccxVHi

Case Study #2:

And collective robots

4-Design the power flow, time derivative of the Hamiltonian, to meet the

dynamic stability requirements for the feedback controller

>== ∑∑==

[ ] iiii xxxccxcccx −=−−=+−= −− *ˆˆˆˆˆ1

2& (Linear tracker)

[ ] [ ] [ ]

0ˆˆˆˆˆˆˆ

<++−=+=

xcccxccxccxH

[ ] **

11xxxx

Nx cmi

cm +−=+−=⇒= ∑∑==

Case Study #2:

• Kinetic Control – dynamics included:

1-Shape the Hamiltonian surface to meet the static stability requirements for the

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

icii VxMxVTH &&

[ ] 011

>+== ∑∑==

∂−=

iDiI xKdtxK &−∫−

<−∫−=

xKdtxKxx

VxMxH i

&&&&&&

Case Study #2:

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

( )sec1.080log

;805.0/40/

etemperaturTbitsTTTn

LOWHIGH

−==∆−=COMM. Link = 50

Kbits/sec

Case Study #2:

aveHbits

n &==sec

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

xpdxxp

∂∫=

−= )()(2 xpxq

Case Study #2:

d) ‘Mean Kinetic Energy” interpretation of Fisher Information

From Quantum Mechanics in the “Classical Limit” of the expectation of

the momentum squared

44 2 ∫=∫= &

222 ),(

Ψ∂∫=

Case Study #2:

- Dirac’s constant

- wave function

- expectation operator

The classical limit is reached when

Is equivalent to

),( txΨ

=−==

Case Study #2:

classical

classicalclassical xF

dm =−=

Which occurs when

is small

This occurs when

which leads to

( ) ( ) ( )xFxxxFxF ′−+≈

( )xF≈

Case Study #2:

xxclassical =

( ) ( )22xxx −=∆

( ) ( ) ( ) ( ) ( ) ...!2

2+′′−+′−+= xFxxxFxxxFxF

e) Fisher Information Equivalency and Exergy

VVVTH1~~~~~

==∑=

potential

control potential

==∑=

information potential

Case Study #2:

( )[ ]dtVVVTdtHJI Ic

~8 +++∫=∫=+

>=+ HJI &&

<=+ HJI&&&&&

( ) =++∫= dtVVVJ Ic

~~~8Where

(Static stability)

(Dynamic stability)

Bound Fisher Information

Case Study #2:

[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

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:

• Wind tunnel model experienced stall flutter over its entire flight envelope M=0.4-0.9;

-10oto -3

oand 10

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 aerodynamic and structural model

– Nonlinear stall flutter model

– Single DOF model

Case Study #3:

- Equations of motion derived from Lagrange’s equation:

αταα

ααα

Ipcontrol

NLdamp

controlaerodamp

KdKKuQ

signCCQ

−−−==

−−=

∂−

Case Study #3:

ααα α

- The aerodynamic moments are generated based on the following hysteresis logic:

for the return hysteresis cycle

- 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

Torsional flutter occurs when

,stallαα < 0== NLNL CK

KCM ≥α

ˆ for 0=u

[ ] [ ] uCKCCI MM =−+−+ ααααα

ˆˆ &&&&

1ˆ VAKqAKC MMM ρααα

CCM ≥α&

ˆ for 0ˆ >−αMCK

and 0=u

.ˆ qAdKC MM αα &&=

Case Study #3:

B) Nonlinear Stall Flutter

When the motion reaches the model becomes nonlinear where

which produces

,stallαα >

),()( αααααα αα&&&&

&MMKCI +=++

1αα KIH += &

[ ] [ ]αααααααα αα&&&&&&&

& ),()( MMCKIH ++−=+=

[ ] [ ] dtCdtMM ααααααταατ

&&&&& ∫∫ =+ ),()(

Case Study #3:

Nonlinear Stall Flutter Linear Dynamics Results

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

α dot (deg/sec)

α (deg)

Phase Plane

Case 1

Case 2

Case 3

Case Study #3:

0 5 10 15 20-2

angle (deg) angle rate (deg/sec)

Time (sec)

CASE 1 Passive

α dot

0 5 10 15 20-40

Time (sec)

CASE 3 Generative

α dot

0 5 10 15 20-15

Time (sec)

CASE 2 Neutral

α dot Resp

0 5 10 15 20-0.03

-0.025

-0.015

-0.005

Time (sec)

Power Flow (W) and Energy (J)

Case 1 Dissipative

0 5 10 15 20-1.5

Time (sec)

Case 2 Neutral

er Flo

0 5 10 15 20-6

Time (sec)

Case 3 Generative

Case Study #3:

Aero Moments

-2 0 2 4 6-0.2

α (deg) and α dot (deg/s)

Aero Moments (N-m)

Case 1 Dissipative: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

-20 -10 0 10 20-2

Aero Moments (N-m)

Case 2 Neutral: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

-40 -20 0 20 40-10

Aero Moments (N-m)

Case 3 Generative: Nonlinear Hysteresis

Stall @ ± 10o

Mα dot

C) The nonlinear stall flutter can be further modified by adding the nonlinear

stiffness and damping

which produces a limit cycle when

),()()( 3 αααααααα αα&&&&&

&MMKKsignCCI NLNL +=++++

[ ] [ ] dtsignCCdtMM NL αααααααταατ

&&&&&& )(),()( +=+ ∫∫

1ααα NLKKIH ++= &

[ ] [ ]αααααααααα αα&&&&&&&&

& ),()()(3 MMsignCCKKIH NLNL ++−−=++=

Case Study #3:

Nonlinear Stall Flutter NonLinear Dynamics Results

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

α dot (deg/sec)

α (deg)

Phase Plane

Case 1

Case 2

Case 3

Case Study #3:

Nonlinear Stall Flutter NonLinear Dynamics Results

er Flo

0 5 10 15 20-1

Time (sec)

CASE 1 Passive

α dot

0 5 10 15 20-15

Time (sec)

CASE 2 Neutral

α dot

0 5 10 15 20-30

Time (sec)

CASE 3 Generative

α dot

0 5 10 15 20-0.025

-0.015

-0.005

Time (sec)

Case 1 Dissipative

0 5 10 15 20-1.5

Time (sec)

Case 2 Neutral

0 5 10 15 20-4

Time (sec)

Case 3 Generative

Case Study #3:

Aero Moments

-2 0 2 4 6-0.6

Aero Moments (N-m)

Stall @ ± 10o

Mα dot

-20 -10 0 10 20-2

Aero Moments (N-m)

Stall @ ± 10o

Mα dot

-40 -20 0 20 40-6

Aero Moments (N-m)

Stall @ ± 10o

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

which produces a limit cycle when

[ ] [ ] τααααααααα αα dKMMsignCKCKKKIt

INLDNLp ∫−++−+−=+++0

3 ),()()( &&&&&&

[ ] 422

1ααα +++= pKKIH &

[ ]αααα &&&& 3)( NLp KKKIH +++=

αταααααα αα&&&&

−++−+−= ∫ dKMMsignCKC

),()()()(

( )[ ] dtsignCKCdtdKMM NLD

I αααατααααταατ

&&&&&& )(),()( 1

−+ ∫∫∫

Case Study #3:

-60-40

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

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

Time (sec)

CASE 1 Passive

α dot

0 5 10 15 20-40

Time (sec)

CASE 2 Neutral

α dot

0 5 10 15 20-100

Time (sec)

CASE 3 Generative

α dot

Case Study #3:

Dynamic Stall Control System ResultsPow

er Flo

0 5 10 15 20-4

Time (sec)

Case 1 Dissipative

0 5 10 15 20-15

Time (sec)

Case 2 Neutral

0 5 10 15 20-40

Time (sec)

Case 3 Generative

Aero Moments

Case Study #3:

Dynamic Stall Control System Results

-40 -20 0 20 40-5

Aero Moments (N-m)

Stall @ ± 10o

Mα dot

-40 -20 0 20 40-10

Aero Moments (N-m)

Stall @ ± 10o

Mα dot

-100 -50 0 50 100-15

Aero Moments (N-m)

Stall @ ± 10o

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

•The limit cycles were shown to be stability boundaries

•Flutter suppression controllers were initially assessed

Case Study #3:

Case Study #4

Power Engineering

Case Study #4:

Power Engineering - OMIB

I. Model (Simple one-machine infinite bus swing equation):

II. Controller Design:

( )[ ]

( ) δδδ

ωδω

−−=

( )[ ]

( )[ ] [ ]

<++−⇒

++−=+=

∫ δδ

δδδδδ

&&&&&&

uPDPMH

0547.1

)sin(0

−−−−−= ∫

τδδδδδ &

Case Study #4:

Power Engineering- OMIB Trajectories

-8 -6 -4 -2 0 2 4 6 8-10

0.5 8/(100 3.1415) (ω)2+1.2647 (1-cos(δ))

• 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

111131212

sinsin1

ωδδω

−−−=

[ ] [ ] [ ]

[ ] [ ]( )

( ) 212122222322

121121111311

2222111112

2112223113

sinsin

cos1cos1cos12

uCDPCM

uDPuDPH

CCCMMH

+−−−=+

+−++−=

−+−+−++=

δδδδδ

δδδδ

δδδδδ

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:

generators

Generation

function (sun)

Loads setup similar to

Lanai “like” model

Case Study #4:

Lanai “Like” Control Model

0 0.1 0.2-0.4

Charge (mA-sec)

Time (sec)

0 0.1 0.2-100

Charge rate (mAmp)

Time (sec)

0 0.1 0.2-100

Vin (Volts)Time (sec)

videal

0 0.05 0.1 0.15 0.2-5

Rhat (Ohm)

Time (sec)

0 0.1 0.2-0.4

Energy (mJoules)

Time (sec)

∫ Pdiss d τ

∫ Pgen d τ

0 0.1 0.2-0.4

Power Flow (Watts)

Time (sec)

• 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:

nonlinear power flow control designwindpower.sandia.gov/other/088055.pdf · nonlinear power flow...

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