The Role of Adaptive Control in

Quantum Information Systems

Mark J. Balas

Distinguished Professor

Aerospace Engineering Department

Embry-Riddle Aeronautical University

Daytona Beach, FL

1

Mark’s Autonomous

Control Laboratory

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidance

Application

Theory

iain-HamiltonControl

ResolventCompact

AdjointSelf

0 )(

uHHt

i C

It’s not theories about stars;

it’s the actual stars that count.”

……… Freeman Dyson

Plant

Parameter

Estimation

+

Gain

Re-calculator

State Estimator

State

Feedback

Gain

ur

Indirect Adaptive Control

Note: Called “Self-Organizing” System (Kalman)

& “Self-Tuning” Regulator ( Astrom)

u y

3

4

Direct Adaptive Model Following Control

(Wen-Balas 1989)

0tye

Plant

Reference

Model

Gm

Gu

Ge

um ym

xm

y u

+

Adaptive

Gain Laws

SignalsKnown ),,( ymm exu

Infinite

Dimensional

5

Direct Adaptive Persistent Disturbance Rejection

(Fuentes-Balas 2000)

0tye

Plant

Reference

model

Ge

um ym

xm

y u

+

Adaptive

Gain Laws

SignalsKnown ),,,(

Basis eDisturbanc

Dymm exu

Disturbance

Generator

Persistent Disturbance Example

6

D

BA

utu

1

0)(

1

0

00

10

D

D

t

taaatataau

t

tlltAu

L

D

D

D

L

DDDD

2

210

2

210

21

1

cos

sin )sin(

esDisturbanc

Known

Basis

Known

Basis

Unknown

7

)(

)()(Plant Nonlinear

xCy

uxBxAt

x

Adaptive Regulation

u

Gy

Nonlinear Adaptive Gain Law

0; TyyG

y

G

Controller

0t

x

Adaptive Control Is Not Complicated !

Use ONLY

Outputs &

Know Almost

NOTHING

about the Plant

“Simplicity” via Infinite

Dimensional Spaces

8

0;)(),(

)(

SpaceHilbert or Banach )0(

)()(

000

txtUwtx

xCy

Xxx

uxBxAt

x

XinEvolution

NonlinearorLinear

“Boil Away” all the special properties of N

Infinite-Dimensional Space

Viewpoint: Linear Semigroups

9

0;)()()(),(

operatorlinear defineddensely closed, )(:

)0(

000

0

tXdFtUxtUwtx

XXADA

xx

FAxt

x

t

0)at t continuous ( )(

) generates ( )()()(

property) (semigroup )()()(

:)( Operators Bounded of Semigroup

000

0

xxtU

U(t)AAtUtAUtUdt

d

sUtUstU

tUC

t

Hilbert Space

with inner

product ),( yx

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

Example: Heat Diffusion

10

2

2

2

0

;

( ) ( ) / smooth and BC: 0 0

( )

with , ( ) ( )

(0) ( )

( , ); ( ) ( )

Ax

x xbu

t z

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

X L

x y x t y t dt

x x D A

y c x c z D A

)(),0(0 zfzxx

0)0,( tx 0),( ltx

z

)(ty)(tu

Euler-Bernoulli Beam

11

4

4

00

( )( )0t t

A

Iw w

u tEIw w b zt

z

Symmetric Hyperbolic Systems

12

0;0),()(:

);()(; 2

constant

0

1

symmetricconstant

tztzzConditions

Boundary

LXADxAz

At

l

A

lxl

n

i ilxl

i

resolvent.compact has and 0),(),(

)( Coercive are ConditionsBoundary )4

)(dim)3

02)

0rnonsingula is)(:Symbol )1

1967) Phillips-Lax from help w1974, Balas(:

1

NormSobolev

*

0

1

AAA

A

AN

AA

AA

Theorem

o

nn

i

ii

Examples

13

t

z

z

AAA

t

x

x

x

x

x

xxz

x

z

xx

xz

x

z

x

t

x

2

1

021

where

000

1011

0001

0010

0010

0101

1000

0100

0001

0110

0100

1000

)(equation wavedim-2

21

2

2

2

1

2

2

2

2

Wave Equation

23

4

1

Dirac Equation: ( ) ( )i

i iPauliSpinMatrices

mcc A i I

t x

Relativistic Fields ( Mandl & Shaw 2010)

14

Stability via Lyapunov-Barbalat

0

0

allfor as 0)(0)(*

0)0(

0 when 0)(

)( :Function like-Energy Find

)0(

)(DynamicsNonlinear

xttxxfgradVV

V

xxV

xV

xx

xfx

N

bounded are )( ies trajectorAll 0 txV

ttVtV as 0)(continuousuniformly and 0)(

:lemma sBarbalat' From

Often does

Not happen

15

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

0;)()(with

)(2

1),(),,(

Let

0

1

tXxtUtx

GGtrxtVGxtV T

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 x

dW x t W x tW x t G

dt x t

W(x)

x ,or ( ) ( ), then ( ) 0.t

W x x x t

Linear or Nonlinear

Evolution

16

Cxy

uBuAxx D

Adaptive Model Tracking

in the Presence

of Persistent Disturbances

y

u

DD Lu

Adaptive Gain Law

),,,( DmmyDmue yuehGGGGG

Controller

),0(or 0 *RNety

mmm

mmmmm

xCy

uBxAx

Reference Model

my

),0(or 0 ** RNxxxt

Adaptive Control Law

where

ionStabilizatRejection eDisturbancTracking Model

T

yye

T

DyD

T

mym

T

myu

yeDDmmmu

eeG

eG

xeG

ueG

eGGwGuGu

Gain Adaptation

Laws

17

Using Ideal Trajectories

Solar Power System Satellite

Long Ago

and Far Away

18

NASA-Johnson

Hubble, Bubble, Toil and Trouble

19

NASA MSFC

Deployable

Optical Telescope

DOT

Primary Mirror

Supports

AFRL-Kirtland

Deployable Optical Telescope Experiment R. Fuentes, M. Balas, K. Schrader, and R.S. Erwin "Direct Adaptive Disturbance Rejection and Control for a Deployable

Space Telescope, Theory and Application”, Proceedings of ACC, Arlington, VA, June 2001.

Steering

Mirrors

22

Evolving Systems=

Autonomously

Assembled

Active Structures

Or Self-Assembling

Structures,

which Aspire to a

Higher Purpose;

Cannot be attained

by Components Alone

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidance

Control

EstimationModeling

& DynamicsGuidanceEvolving Systems

NASA-JPL & MSFC

23

Genetics of Evolving

Systems: Inheritance

of

Component Traits

Controllability/Observability

Stability

Optimality

Robustness

Disturbance Rejection/Signal Tracking

Source: CNN.com

Stability is Necessary During the

Entire Evolution Process

Irony: Synthetic Biology ???

Composability in Synthetic Biology

“It is difficult to define signal exchanges between

biological units unambiguously”

25

F-16 Flexible Structure Model:

Fluid-Structure Interaction

USAF-Edwards AFB

Flight Test Center

Flutter

One Possible Solution

Aerodynamically

Shaped Graduate

Student

Smart Grids:

Virtual Interconnecting Forces

27

“It is surprising how quickly we replace a human operator

with an algorithm and call it SMART”

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

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 :

Adaptive Control in

Quantum Information Systems

31

Merde

Quantum Computing

32

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 Control

So Quantum Error Correctiion can work!!!

Q

Quantum Information Systems

Quantum Basics

(Dirac & Von Neumann)

34

states" pure ofn combinatioconvex a is stateA "

c1&c1or 1),(

:SpaceHilbert complex State

2

1k

k

2

1k

k

k

X

A

xAx

XXA

k

xP

kk

k

k

adjoselfbounded

k

of ionseigenfunct :States Pure

),(Resolvent Compact

: Observable

1

int

Orthonormal

Eigen –Basis for X

Schrodinger Wave Equation

35

10 )( Spectrum Real Discrete

kkH

kllkkk

k

ti ketU

XXtU

, with ,)( and

e)(reversibl GroupUnitary :)(

1

0

0

)(

SpaceHilbert complex

ian-Hamilton

Control

ResolventCompact

AdjointSelf

0

uHHt

i

X

C

Marginally

Stable

OLD

NEW

itt

:Time Imaginary

with Diffusion

Quantum Measurement

36

S M

Back

Action

wh

XXX

lk

M

l

S

kkl

MS

,

)(

ntEntangleme

Ontology ( what is) vs Epistemology ( What is measured)

YOW!!

!

Uncertainty Principle

37

yx(x,y) 0

Inequality Schwarz-Cauchy

todueProperty SpaceHilbert A

BAABBABATrBA

BA

TrAATrΔA

ATrA

XXA

],[ commutator ;]),[(2

)()(

& oft Measuremen usSimultaneo :Principlety UncertainHeisenberg

)1)(&0( operatordensity or state a is where))(( Dispersion

)(Mean

Hilbert : Observable

22

2

adjoint-selfbounded

2

2

22 )2

()],[()()( Recall

i

pzTrpz

In fact ALL of Quantum Mechanics is based on a Hilbert space

of states and a (C*) algebra of bounded linear self adjoint observables

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

38

Control of Individual Quantum

Systems: Quantum Feedback Loop

39

Purpose:

Use information from weak QND measurements to prepare photon number

(Fock) states of a cavity field and

protect them against decoherence.

Method:

Quantum feedback realized by atoms as QND probes and

small coherent field injections into the cavity mode as an

actuator.

Physics Nobel Prize 2012

S. Haroche & D. Wineland

40

Adaptive Quantum

Model Tracking to Reduce

Decoherence

y

u

DD Lu

Adaptive Gain Law

),,,( DmmyDmue yuehGGGGG

Adaptive Quantum Controller

),0(or 0 *RNety

Desired

Hamiltonian

*H

Reference Model:

Closed System

my

),(

)(

Re

int

0

cy

HuHHt

i

esDisturbanc

Linblad

Control

C

solventCompact

AdjoSelf

Open Physical System

QND Measurement

&Quantum Error Correction

“Physics is like sex: sure, it may give

some practical results, but that's not why

we do it.”

― Richard P. Feynman

In a tile motif on the back of the Ross Dress For Less building

on Lake Ave, Pasadena, CA

41

Infinite-Dimensional Adaptive

Control Theory