The Role of Infinite Dimensional Adaptive Control Theory in Autonomous Systems

(or When Will SkyNet Take Over the World)

Mark J. BalasDistinguished Professor

Aerospace Engineering Department Embry-Riddle Aeronautical University

Daytona Beach, FL

1

Mark’s AutonomousControl Laboratory

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Autonomy:Greek autonomía =independence

2

(philosophy) the doctrine that the individual human will is or ought to be governed only by its own principles and laws

AutonomousAir Traffic Control

Will Autonomous SystemsTake Over the World?

My brain is a Neural Net Processor

Artificial Intelligence Controlled Network Defense ComputerSystem

3

Autonomous Systems vs Autonomic Systems

4

Adaptive ControlSystems

Humans usually do not have to think about the operation of these autonomic systems and so they can do other higher level things, eg sex, murder, presidential elections

involuntary or spontaneous

5

F-16 Flexible Structure Model:Fluid-Structure Interaction

USAF-Edwards AFBFlight Test Center

Flutter

Hypersonic Aircraft X51A Wave Rider

6

The X-51A WaveRider is an unmanned, autonomous supersonic combustion, ramjet-powered hypersonic flight-test demonstrator for the U.S. Air Force.The X-51A demonstrates a scalable, robust endothermic hydrocarbon-fueledscramjet propulsion system in flight, as well as high temperature materials, airframe/engine integration and other key technologies within the hypersonic range of Mach 4.5 to 6.5.

6 Minutes at Mach 5.1

Reality

AFRL-Wright Patterson AFB

7

Evolving Systems= Autonomously Assembled Active Structures

Or Self-Assembling Structures,which Aspire to a Higher Purpose; Cannot be attained by Components Alone

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

GuidanceEvolving Systems

NASA-JPL

Susan FrostIntelligent SystemsNASA Ames Research Center

Wind Energy

1979: 40 cents/kWh

210 MW Lake Benton Wind Farm 4 cents/kWh

2006: 3 - 5 cents/kWh

2000: 4 - 6 cents/kWh

• R & D Advances

• Increased Turbine Size

• Manufacturing Improvements

• Large Wind Farms

Who Needs Control Anyway?

NREL-NWTC

Flow Control of Wind Turbine Aerodynamics

u

Adaptive Controller

yG

Fondest Hopes Wildest Dreams:Control the Whole Wind Farm as One Turbine!

Smart Grids: Virtual Interconnecting Forces

11

“It is surprising how quickly we replace a human operator with an algorithm and call it SMART”

Power System Perturbed with a Wind Farm

• When a wind farm is placed at

a distance of α, the perturbed power system becomes :

with

• Power flow at a distance u is :

Inter-Area Oscillations

Cyber Security of Electric Power Grids

13

Adaptive Disturbance Tracking Control for Large Horizontal Axis Wind Turbineswith Disturbance Estimator in Region II

OperationMark J. Balas. Kaman S. Thapa Magar and Qian Li

ASME 2014

Universal Infinite Dimensional Example: Heat Diffusion

14

( )

2

2

2

0

;

( ) ( ) / smooth and BC: 0 0 ( )

with , ( ) ( )

(0) ( )( , ); ( ) ( )

Ax

x x but z

b z D A x x(t, ) x(t,l)

X L

x y x t y t dt

x x D Ay c x c z D A

Ω

∂ ∂= + ∂ ∂

∈ ≡ = =

⊂ ≡ Ω

≡

= ∈ = ∈

∫

)(),0(0 zfzxx ==

0)0,( =tx 0),( =ltx

zΩ

)(ty)(tu

Symmetric Hyperbolic Systems

15

0;0),()(:

);()(; 2

constant 0

1symmetric

constant

≥Ω∂∈∀=Λ

ℜΩ≡⊂∈+∂

∂=

∂

∂∑=

tztzzConditionsBoundary

LXADxAz

At

l

A

lxl

n

i ilxli

ϕ

ϕϕϕ

ϕ

xAxxxxx

x

xxx

A

z

A

tt

z

tzztt

≡

−

+

=⇒

≡⇔

−=

01

000

00

:nsOscillatio Area-Inter :GridSmart

2

ηνν

ην

23

41

Dirac Equation: ( ) ( )ii iPauli

SpinMatrices

mcc A i It xφ φ φ

=

∂ ∂= − +

∂ ∂∑

≡

−−−

+∂∂

−

−

+∂∂

=⇔

+∂∂

+∂∂

=∂∂

∆

t

z

z

AAA

t

x

xx

xx

xxzx

zxx

xzx

zx

tx

2

1

021

where

000101100010010

0010010110000100

0001011001001000

)(equation wavedim-2

21

22

2

12

2

2

2

γ

γ

“Simplicity” via Infinite Dimensional Spaces

16

[ ]0;)(),(

),(...),(),(

)()0(

00

21

0

1

≥∀=⇒

==

⊂∈=

+=+=∂∂ ∑

=

txtUwtx

xcxcxcCxy

XADxx

ubAxBuAxtx

XinEvolutionT

m

m

iii

“Boil Away” all the special properties of Nℜ

=→

==

=+

−

→ 0)at t continuous ( )(

) generates ( )()()(

property) (semigroup )()()(:)( Operators Bounded of Semigroup

000

0

xxtU

U(t)AAtUtAUtUdtd

sUtUstUtUC

t

J. Wen & M.Balas, “Robust Adaptive Control in Hilbert Space ”,J. Mathematical. Analysis and

Applications, Vol 143, pp 1-26,1989.

J. Wen & M.Balas ,"Direct Model Reference Adaptive Control in Infinite-Dimensional Hilbert Space," Chapter in Applications of Adaptive Control Theory, Vol.11,K. S. Narendra, Ed., Academic Press, 1987

The Devil Lurks in the Details

17

y

Semigroups

18

XxADt

xxtUAx

tXXtU

C

xtUtxADxx

Axtx

tt in dense exists lim/)( with )(lim :Generator

0 operators bounded :)(

Semigroup

)()()()0(

Solve

00

0

0

0

+→+→ ≡−

=

≥→

−

=⇒

∈=

=∂∂

=

≡−=−

−

)()),((

OperatorResolvent ),()())(( Transform LaPlace

1

1

tUARL

ARAItUL

λ

λλ

Closed LinearOperator

!)(dim

0 ktAetUX

k

k

kAt ∑∞

=

≡=⇒∞<

Spectrum of A

19

C

point

Resolvent Set ( ) / ( , ) : bounded linear op on

Spectrum ( ) ( ) ( ) ( ) ( )

cont residual

A R A X X X

A A A A A

ρ λ λ

σ ρ σ σ σ

≡ →

≡ = ∪ ∪

XAIA

XAIAAAIA

residual

cont

po

of subspaceproper a is rangebut 1,-1 is /)( in denseonly is range itsbut 1,-1 is /)(

0/1-1 NOT is /)(int

−≡−≡

=∋≠∃=−≡

λλσλλσ

φλφφλλλσ

When is a Semigroup Exponentially Stable ?

20

semigroup) stablelly exponentia ( 0;

onto is real and

0 ),(),Re(

, spaceHilbert complex in dense )(:1993) Rogers &(Renardy

**

2

≥≤⇒

−∋−>∃

>∈∀−≤

−

− teU(t)IA

ADxxxAx

XADPhillipsLumer

tα

λαλ

αα

λλρλλ

complex such allfor ,),( and )(0Re stablelly exponentia is )(

. space Hilbert aon )( semigp-C a generates Assume :)inerPruss,&GreGearhart,(

0

∞<≤

∈⇒>⇔

MARAtU

XtUATheorem

Resolvent

21

Direct Adaptive Model Following Control (Wen-Balas 1989)

0→ ∞→tyePlant

Reference Model

Gm

Gu

Ge

um ym

xm

yu

+

AdaptiveGain Laws

SignalsKnown ),,( ymm exu

Infinite Dimensional

The Godfather

22

Direct Adaptive Persistent Disturbance Rejection (Fuentes-Balas 2000)

0→ ∞→tye

Plant

Reference model

Ge

um ym

xm

yu

+

AdaptiveGain Laws

SignalsKnown ),,,(Basis eDisturbanc

Dymm exu φ

DisturbanceGenerator

23

Stability via Lyapunov-Barbalat

0

0

allfor as 0)(0),(*0)0(

0 when0)()( : Functionlike- EnergyFind

)0(),(

DynamicsNonlinear

xttxxtfgradVVV

xxVxV

xxxtfx

N

∞→→⇒<≡

=≠>

ℜ∈=

=

bounded are )( ies trajectorAll 0 txV ⇒≤

∞→→⇒≤ ttVtV as 0)(continuousuniformly and 0)(

:lemma sBarbalat' From

Often does Not happen

24

Infinite-Dimensional Lyapunov-Barbalat Theory: PDE & Delay Systems

≥∈=

∆∆+≡∆ −

0;)()(with

)(21),(),,(

Let 0

1

tXxtUtx

GGtrxtVGxtV Tγ

SpaceBanach or Hilbert X

var

( ( , ) ) ( , , ) ( ( , ) )Theorem: If

( , , ) ( ) 0( ( )) ( )and ( ) is bounded, then ( ( )) 0 and bounded.

If is coercive in the partial state

t

FrechetDeri ive

x G V t x G x G

V t x G W xdW x t W x t W x t G

dt x t

W(x)

α β

→∞

∆ ≤ ∆ ≤ ∆

∆ ≤ − ≤∂ ∂

= → ∆∂ ∂

x ,or ( ) ( ), then ( ) 0.tW x x x tγ →∞≥ →

Linear or Nonlinear Evolution

Lyapunov-Balas

25

Linear System Strict Dissipativity

0)0(

0;0),()( :Function StorageEnergy

=≠∀>≡

VxPxxxV

PowerDissipatedInternallyPower

Externaluyx

RateStorageEnergy

xuyPBuxxPxdtdV 2

),(

),(),(),Re(21

2

αα

−≤+=⇒−≤

A Linear Dynamic Infinite-Dimensional System is STRICTLY DISSIPATIVE when

DISSIPATIVE when α=0

=

∈∀−≤+≡

∋≤≡≤

→∃−

*

)(

2

2max

2min

int

)(;)],(),[(21),Re(

),()(

:

CPB

ADxxPAxxxPAxxPAx

xpxPxxVxp

XXP

xW

PositiveAdjoSelfOpLinear

α

Phillips-Lumer When ⇒≡ IP

For Finite & Infinite Dimensions

All Roads Lead To Rome

)( gains adaptive bounded with0)( produces

0; Controller Adaptive *

tGtx

yyGGyu

Direct

t →

>−=

=⇒

∞→

σσ

26

[ ]

(ASD) eDissipativStrictly Almost ),,(with

),(...),(),(

)()0(

21

0

1

CBA

xcxcxcCxy

XADxx

ubAxBuAxtx

Tm

m

iii

==

⊂∈=

+=+=∂∂ ∑

=

Control Porno

Finite- Dimensional LINEAR ASD: Two Simple Open-Loop Properties

High Frequency Gain is Sign-Definite (CB>0)

Open-Loop Transfer Function is Minimum Phase(i.e. Transmission Zeros are all stable)

)( gains adaptive bounded with0)( produces

0; n RegulatioAdaptive *

tGtx

yyGGyu

t →

>−=

=

∞→

σσ

27

Almost Strictly Dissipative

28

Transmission Zeros: Infinite Dimensional Systems

0 ,0for when,),,( of

zero ) blocking-ssionor transmi (on transmissi a is )0( ;

For

*0

*

0

≡⇒=≡

==

+=∂∂

yweuxCBA

xxCxy

BuAxtx

tλ

λ

operator linear closed)(

:0

)( where

0))((when when),,( of

zeroion transmissa is

*

*

MM XxxAD

CBIA

H

HNCBA

ℜ→ℜ

−≡

≠

λλ

λ

λ

Normal Form

[ ]

dynamics" zero" thecalled ),,(

0

0

&y: FormNormal

122122

2221

1211

2221

1211

2M

AAA

zy

Iy

uCB

zy

AAAA

zy

zAyAzCBuzAyAy

lz

m

=

+

=

⇔

+=

++=

∈ℜ∈

nonsing CBCxy

BuAxx⇒

=+=

Via Non-Orthogonal Projections

N(C)PIPR(B)CCBBP

onto onto )(

22

11

−≡≡ −

)Aσ(Z(A,B,C) 22

Zeros onTransmissi

:Result = Also (A,B,C) ASD if and only ifNormal Form is ASD

lorthomorma,....,,...,, 2121

)()( rnonsingula θθspbbbsp

CNBRXCBm

⊕=⇒

Zero Dynamics (Normal Form)

30

zv

y

Zero Dynamics

h

z CBuvyAy ++= 11

211

22122122

12 )()(or AAsIAsPyAzAz

zAvz

z −−≡

+=

=

CBu h

M- Dimensional

Nonsingular

Note: Zero Dynamics are Invariant under Output Feedback

My Infinite-Dimensional Version of the “Two Simple Open Loop Properties” Theorem

31

[ ] semigroup) stablelly exponentia generates (i.e.,

"" 0))( /Zeros onTransmissi and 0),(

ifonly and if issipativeStrictly DAlmost is ),,(

operator linear closed)(:0

)( where

0))(( when),,( of zero nransmissioa t is : Def:

22

22mxm

**

A

stable)A(σN(H(A,B,C)bcCB

CBA

XxxADC

BIAH

HNCBACTheorem

pij

MM

=≠≡>=

ℜ→ℜ

−≡

≠∈

λλ

λλ

λλ

Pretty Close !!

[ ]

∈==

⊂∈=

+=+=∂∂ ∑

=

)(,;),(...),(),(

)()0(

semigroup a generates ;

*

21

0

10

ADcbxcxcxcCxy

XADxx

CAubAxBuAxtx

jim

m

iii

Adaptive Augmentation

32

Fixed Gain Controller

Nonlinear System

Adaptive Regulator

0;][

>−=

=

γγTe

e

yyG

yGu

AutonomicSystem !!!

NASA Space Launch System SLS

33

SLS 130 Metric TonEvolved Configuration

NASA MSFC

Adaptive Control=Risk Management

Adaptive Control in Quantum Information Systems

34

Quantum Computing

35

A Quantum computer will operate differently from a Classical one.It will be involved w physical systems on an atomic scale, eg atoms, photons, trapped ions, or nuclear magnetic moments

Quantum Gate

Unitary Reversible

Could be improved with Adaptive ControlSo that Quantum Error Correctiion can work!!!

Quantum Basics (Dirac & Von Neumann)

36states" pure of ncombinatiolinear a is state mixed A"

c1&c1or 1),(

:Spacelbert complex Hi State Mixed

2

1kk

2

1kk

∴

===⇒==

∈

∑∑∞

=

∞

=

ϕϕϕϕϕϕ

ϕ

k

X

A

xAx

XXA

k

xP

kkk

k

adjoselfbounded

k

of ionseigenfunct :States Pure

),(Resolvent Compact

: Observable

1

int

ϕ

ϕϕλ∑

∞

=

−

=⇒

→OrthonormalEigen –Basis for X

Schrodinger Wave Equation

37

∞==∴ 10 )( Spectrum Discrete kkiH λσ

∞−MarginallyStable

kllkkkk

ti ketU

XXtU

δφφφφϕϕ λ ==

→⇒

∑∞

=

, with ,)( and

)reversible ( Group Unitary :)(

10

0

ϕϕϕ )(

ResolventCompact

AdjointSelfSkew 0 uHH

ti C+=

∂∂

−

Quantum Measurement

38

S M

Back Action

whXXX

lk

Ml

Skkl

MS

⊗≠⊗=

⊗=

∑,

)(

ntEntangleme

ϕϕαϕ

Ontology ( what is) vs Epistemology ( What is measured)Existence =Interaction

Small Quantum Systems We can begin to experiment with just one

electron, atom or small molecule Need:

Precise control Isolation from the environment

Simple small systems : single particles or small groups of particles

…… David Wineland NIST

39

Physics Nobel Prize 2012S. Haroche & D. Wineland

40

Adaptive Quantum Model Tracking to ReduceDecoherence

yu

DD Lu φθ=

Adaptive Gain Law

[ ] ),,,( DmmyDmue yuehGGGGG φ−==

Adaptive Quantum Controller

0→−≡ ∞→tmy yye

Desired Hamiltonian

*H

Reference Model: Closed System

my

=

++=∂∂

−

),(

)(

Re

int0

ϕ

ϕϕϕϕ

cy

HuHHt

iesDisturbanc

Linblad

Control

C

solventCompact

AdjoSelf

Open Physical System

QND Measurement ???? &Quantum Error Correction

“No intelligent idea can gain general acceptance unless some stupidity is mixed

in with it” ….. Fernando Pessoa, The Book of Disquiet

Famous Lisbon Poet