Post on 10-Aug-2020
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