Naira Hovakimyan
Department of Aerospace and Ocean Engineering
Virginia Polytechnic Institute and State University
This talk was originally given as a plenary at
SIAM Conference on Control and Its Applications
San Francisco, CA, June 2007
A. M. Lyapunov
1857-1918
G. Zames
1934-1997
Outline
 Historical Overview
 Limitations and the V&V Challenge of
Existing Approaches
 Paradigm Shift
 Speed of Adaptation
 Performance and Robustness
 Rohrs’ Example
 Various aerospace applications
 Future Directions of Research
Motivation
 Early 1950s – design of autopilots operating at a wide
range of altitudes and speeds
 Fixed gain controller did not suffice for all conditions
 Gain scheduling for various conditions
 Several schemes for self-adjustment of controller parameters
 Sensitivity rule, MIT rule
 1958, R. Kalman, self-tuning controller
 Optimal LQR with explicit identification of parameters
 1950-1960 - flight tests X-15 (NASA, USAF, US Navy)
 bridge the gap between manned flight in the atmosphere and space
flight
 Mach 4 - 6, at altitudes above 30,500 meters (100,000 feet)
 199 flights beginning June 8, 1959 and ending October 24, 1968
 Nov. 15, 1967, X-15A-3
First Flight Test in 1967
The Crash of the X-15A-3 (November 15, 1967)
X-15A-3 on its B52 mothership
X-15A-3
Crash due to stable, albeit nonrobust adaptive controller!
Crash site of X-15A-3
Objectives of Feedback
 Stabilization and/or Tracking
 Robustness (measure of relative
stability)
 Performance (comparison with
desired system behavior)
Historical Background
 Sensitivity Method, MIT Rule, Limited Stability Analysis (1960s)
 Whitaker, Kalman, Parks, et al.
 Lyapunov based, Passivity based (1970s)
 Morse, Narendra, Landau, et al.
 Global Stability proofs (1970-1980s)
 Morse, Narendra, Landau, Goodwin, Keisselmeier, Anderson, Astrom, et al.
 Robustness issues, instability (early 1980s)
 Egard, Ioannou, Rohrs, Athans, Anderson, Sastry, et al.
 Robust Adaptive Control (1980s)
 Ioannou, Praly, Tsakalis, Sun, Tao, Datta, Middleton,et al.
 Nonlinear Adaptive Control (1990s)
 Adaptive Backstepping, Neuro, Fuzzy Adaptive Control
 Krstic, Kanelakopoulos, Kokotovic, Zhang, Ioannou, Narendra, Ioannou, Lewis
Polycarpou, Kosmatopoulos, Xu, Wang, Christodoulou, Rovithakis, et al.
 Search methods, multiple models, switching techniques (1990s)
 Martenson, Miller, Barmish, Morse, Narendra, Anderson, Safonov, Hespanha, et al.
Landmark Achievement: Adaptive Control in Transition
 Air Force programs: RESTORE (X-36 unstable tailless
aircraft 1997), JDAM (late 1990s, early 2000s)
 Demonstrated that there is no need for wind tunnel
testing for determination of aerodynamic coefficients
(an estimate for the wind tunnel tests is 8-10mln
dollars at Boeing)
Lessons Learned: limited to slowly-varying
uncertainties, lack of transient characterization
 Fast adaptation leads to high-frequency oscillations in
control signal, reduces the tolerance to time-delay in
input/output channels
 Determination of the “best rate of adaptation” heavily
relies on “expensive” Monte-Carlo runs
Boeing question: How fast to adapt to be robust?
Stability, Performance and Robustness
 Absolute Stability versus Relative Stability
 Linear Systems Theory
 Absolute stability is deduced from eigenvalues
 Relative stability is analyzed via Nyquist criteria: gain and
phase margins
 Performance of input/output signals analyzed
simultaneously
 Nonlinear Systems Theory
 Lack of availability of equivalent tools
 Absolute stability analyzed via Lyapunov’s direct method
 Relative stability resolved in Monte-Carlo type analysis
 No correlation between the time-histories of input/output
signals
The Theory of Fast and Robust Adaptation
 Main features
 guaranteed fast adaptation
 guaranteed transient performance
 for system’s both signals: input and output
 guaranteed time-delay margin
 uniform scaled transient response dependent on changes in
 initial condition
 value of the unknown parameter
 reference input
 Suitable for development of theoretically justified Verification
& Validation tools for feedback systems
One Slide Explanation of the New Paradigm
System:
x (t )  Am x(t )  b(u(t )   (t ) x(t ))
Nominal controller in MRAC: unom (t )   (t ) x(t )  k g r (t )
Desired reference system: xref (t )  Am xref (t )  bk g r (t )
Implementable nom. cont.
unomL1 (s)  C(s)  (t ) x(t )  k g r (t )
This is the nominal controller in the L1 adaptive control paradigm!
Small-gain theorem gives sufficient condition for boundedness:
xref L  , if
1
Result:
(1  C (s))( sI  Am ) 1 b  max  1
L1
fast and robust adaptation via continuous feedback!
Two Equivalent Architectures of Adaptive Control
x  Am x  bu   T x  A
m Hurwitz,  unknown
T
yc x
System dynamics
xm  Am xm  bk g r
Ideal Controller u   x  k g r k g  c A b
*
T
T
1
m
xm
State predictor
~
~
V  eT Pe  T1  V  0
error
Barbalat’s lemma

xˆ  Am xˆ  b u  ˆT (t ) x
yˆ  cT xˆ
Closed-loop state
predictor with u
xˆ  Am xˆ  bk g r
yˆ  c xˆ
T

T
ˆ
e  Ame  b    x, e  xm  x
Adaptive law: ˆ(t )   x(t ) eT (t ) Pb
 


Bounded
ym  cT xm

Adaptive Controller u  ˆT (t ) x  k g r
regressor
1

e0


T
ˆ
e  Ame  b    x
e0
V  0
Same error dynamics
independent of control choice
Implementation Diagrams
Reference system
r
xm  Am xm  bk g r
u  ˆx  k g r
u
xm
e
x
x  Am x  b(u   T x)
MRAC
ˆ ˆ(t )   x(t )eT (t )Pbdt

u cannot be low-pass
filtered directly.
State predictor based reparameterization
Enables insertion of
a low-pass filter
u
xˆ  Am xˆ  b(u  ˆT (t ) x)
x̂
Low-pass filter: C(s)
r
x  Am x  b(u   x)
T
u  ˆ(t ) x  k g r
ˆ ˆ(t )   x(t )eT (t )Pbdt

x
No more reference
system upon
filtering!!!
e
Stability
u
xˆ  Am xˆ  b(u  ˆT (t ) x)
x̂
Low-pass filter: C(s)
r
x
x  Am x  b(u   T x)
u  ˆ(t ) x  k g r
e
ˆ ˆ(t )   x(t )eT (t )Pbdt


 Closed-loop sxˆ ( s)  Am xˆ ( s)  b (C ( s)  1){ˆT (t ) x(t )}  C ( s)k g r




 Solving: xˆ ( s)  ( sI  Am ) 1 b (C ( s)  1) ˆT (t ) xˆ (t )  e(t )   C ( s)k g r

Ho (s)
bounded
 Sufficient condition for stability via small-gain theorem
1  C (s)sI  Am 
1
b max  1
L1
lim ( x(t )  xˆ (t ))  0
t 
Barbalat’s lemma

Closed-loop Reference System
uref (s)  C(s)u* (s), u* (t )   T xref (t )  k g r (t )
 Filtered ideal controller:
 Closed-loop system with
filtered ideal controller =
non-adaptive predictor:

x( s )  b(C ( s )  1)

sx ( s )  Am x( s )  b uref ( s )   T x( s ) 
 Am
T
x( s )  C ( s )k g r ( s )
Stability via small-gain theorem
r
Plant with
unknown parameter
kg

C (s )
uref

( sI  Am ) 1 b
xref
c
y ref
1  C (s)H o s  L max
1
1

C ( s)  1

Reference system
Reference
system of MRAC

Guaranteed Transient Performance
 System state:
 Control signal:
x  xref
u  uref
L
L

1

2

 C ( s)  1 (MRAC)

lim x  xref
 
lim u  uref
 
2 
L
0
L
0
LTI System for Control Specifications
r
Plant with
unknown parameter
kg

C (s )
uref

1
( sI  Am ) b
r
xref
c
y ref

C (s )
uref

( sI  Am ) 1 b
xref
c
yr


Design system
for defining the control specs
Reference system
achieved via fast adaptation

yref (s)  c I  (C (s)  1)( sI  Am ) b
1
1
ydes (s)  c (sI  Am ) bC (s)k g r (s)
T

C (s )

T
Virtual Plant with
clean output
kg

T 1
(sI  Am ) 1 bC (s)k g r (s)
Independent of the
unknown parameter
Achieving Desired Specifications
yref  ydes
uref  udes
L
L



1 

1 
cT
L1
k g H o ( s )C ( s )
C ( s ) T
L1
L1
r
k g H o ( s )C ( s )
 Sufficient condition for stability
  1  C ( s) H o s  L  max  1
1
 Performance improvement
  min
L
L1
r
L
Guaranteed Performance Bounds
 Use large adaptive gain
c
 1 
 1 
, u t   uref t   O
, t  0
 y t   yref t   O

  

 c
 c
 Design C(s) to render   1  C ( s) H o s  L  max sufficiently small
1

yref t   ydes t   O( ), uref t   udes t   O( ), t  0
Desired design objective
 1 
 1 
, u t   udes t   O( )  O
, t  0
yt   ydes t   O( )  O
  
  
c


 c
Large adaptive gain
Smaller step-size
Faster CPU
Time-delay Margin and Gain Margin
rb (t )


1
1  C ( s)
C (s)  1  unbounded rb (t )
 (t )

Point Timedelay occurs
C ( s )1   T H ( s ) 
Plant

C ( s) 1   T H ( s)
H ( s) 
1  C ( s)


H ( s )  sI  Am  b T

1
b
lim Tmargin  Tm ( H ( s )) Lower bound for the time-delay margin
 
Gm  [1 , 2 ] Projection domain defines the gain margin
Main Theorem of L1 Adaptive Controller
Fast adaptation, in the presence of (1  C ( s)) H o (s) L max  1 ,
leads to guaranteed transient performance and
guaranteed gain and time-delay margins:
1
lim u (t )  uref (t )   0;
 
lim x(t )  xref (t )   0;
 
lim Tmargin  Tm ;
 
Gmargin  Gm ,
where
Tm
is the time-delay margin of


C ( s) 1   T H ( s)
H ( s) 
.
1  C ( s)
Design Philosophy
Adaptive gain – as large as CPU permits (fast adaptation)
 Fast adaptation ensures arbitrarily close tracking of the virtual
reference system with bounded away from zero time-delay margin
Fast adaptation leads to improved performance and improved robustness.
Low-pass filter:
 Defines the trade-off between tracking and robustness
Increase the bandwidth of the filter:
 The virtual reference system can approximate arbitrarily closely the
ideal desired reference system
 Leads to reduced time-delay margin
Tracking vs robustness can be analyzed analytically.
The performance can be predicted a priori.
Robot Arm with Time-varying Friction and Disturbance
MgL cosq(t ) 
Iq(t ) 
 F (t )q (t )  F1 (t )q(t )   (t )  u (t )
2
1
Control objective : q( s)  2
r ( s)
s  1.4s  1
Control design parameters
1
D( s)  , k  60,   10000
s
Parametric uncertainties in state-space form:
 1 ,
2  cos(t )


 t   

2

0
.
3
sin(

t
)

0
.
2
cos(
2
t
)


Compact set for Projection
  [10;10]
Disturbances in three cases:  1 t   sin( t )
 2 t   cos( x1 (t ))  2 sin( 10t )  cos(15t )  3 t   cos( x1 (t ))  2 sin( 100t )  cos(150t )
Simulation Results without any Retuning of the Controller
1 t   sin( t )
 2 t   cos( x1 (t )) 
2 sin( 10t )  cos(15t )
 3 t   cos( x1 (t )) 
2 sin( 100t )  cos(150t )
System output
Control signal
Verification of the Time-Delay Margin

C ( s) 1   T H ( s)
H ( s) 
1  C ( s)

With time-delay 0.02
Tm  0.0256
With time-delay 0.1
MRAC and L1 for a PI controller
L1
MRAC
The open-loop transfer functions in the presence of time-delay

H ( s) 
e  s
s( s  1)
Time-delay margin
 * : 

 j *
e
 1
j ( j  1)
 ( ) 
*
H ( j )

 (e  s  1 )
H ( s)  C ( s) 2
s s
 * : 
Time-delay margin
(e  j  1)
 1
2
   j   ( j  1)
*


lim  * () 
 

2
 1.57
Application of nonlinear L1 theory
C ( s) 
1

, Tm   1.57 sec
s 1
2
lim Tmargin  Tm
 
Miniature UAV in Flight: L1 filter in autopilot
The Magicc II UAV is flying in 25m/h wind,
which is ca. 50% of its maximum flight speed
Courtesy of Randy Beard, BYU, UTAH
Rohrs’ Example: Unmodeled dynamics
kp
229
System with unmodeled dynamics: y ( s) 
u ( s)
2
s  a s  30s  229
Nominal values
k p  2, a  1
plant
of plant parameters:
Unmodeled
dynamics
Reference system dynamics: ym ( s ) 
dynamics
3
r (s)
s3
Control signal: u (t )  k y (t ) y (t )  kr (t )r (t )
Adaptive laws from Rohrs’ simulations:
ky (t )   k y (t )( ym (t )  y (t )), k y (0)  0.65, k *y  1
kr (t )   r r (t )( ym (t )  y (t )), k r (0)  1.14,
k r*  1.5
Instability/Bursting due to Unmodeled Dynamics
r (t )  0.3  1.85 sin( 16.1t )
r(t)=0.3+1.85*sin(16.1*t)
4
3
r(t)=0.3+1.85*sin(16.1*t)
15
Kr
10
Ky
2
5
Instability
-1
-2
Parameters
0
-5
-10
-3
-4
0
0
5
10
Time (sec)
15
-15
20
0
5
10
Time (sec)
15
20
Parameter drift
System output
r(t)=0.3+2*sin(8*t)
r (t )  0.3  2 sin( 8t )
5
r(t)=0.3+2*sin(8*t)
4
Kr
Ky
3
0
2
1
Bursting
0
-1
Parameters
Output
Output
1
-5
-2
-10
-3
0
5
10
Time (sec)
15
System output
0
5
10
Time (sec)
15
20
Parameters
20
Rohrs’ Explanation for Instability: Open-Loop Transfer Function
Bode Plot (Rohr's Ex) System: P0
Frequency (rad/sec): 16.1
Magnitude (dB): -24.2
0
-20
System: P
Frequency (rad/sec): 16.1
Magnitude (dB): -30.6
-40
-60
Magnitude (dB)
Magnitude (dB)
-20
P0=1/(s+1)
-80
P=P0*229/(s2+30s+229)
-100
-120
-60
P0=1/(s+1)
-80
P=P0*229/(s2+30s+229)
-100
-140
0
System: P0
Frequency (rad/sec): 16.1
Phase (deg): -86.4
-45
-90
System: P
Frequency (rad/sec): 16.1
Phase (deg): -180
-135
-180
-180
System: P0
Frequency (rad/sec): 8
Phase (deg): -82.8
-45
Phase (deg)
Phase (deg)
System: P
Frequency (rad/sec): 8
Magnitude (dB): -20.3
-40
-120
-140
0
-90
-135
-135
-225
-270
-2
10
System: P0
Bode Plot (Rohr's Ex)
Frequency (rad/sec): 8
Magnitude (dB): -18.1
0
System: P
Frequency (rad/sec): 8.01
Phase (deg): -139
-180
-225
-1
10
0
1
10
10
Frequency (rad/sec)
2
10
3
10
r (t )  0.3  1.85 sin( 16.1t )
-270
-2
10
-1
10
0
1
10
10
Frequency (rad/sec)
2
10
r (t )  0.3  2 sin( 8t )
At the frequency 16.1rad
the phase reaches -180 degrees
in the presence of unmodeled dynamics,
reverses the sign of high-frequency gain,
the loop gain grows to infinity
At the frequency 8rad
the phase reaches -139 degrees
in the presence of unmodeled dynamics,
no sign reversal for the high-frequency gain
the loop gain remains bounded
instability
bursting
3
10
Rohrs’ Example with Projection for Destabilizing Reference Input
With Projection operator(3) r(t)=0.3+1.85*sin(16.1*t)
With Projection operator(3) r(t)=0.3+1.85*sin(16.1*t)
0.9
3
0.8
 3  ky  3
0.6
Output
0.5
0.4
 3  kr  3
0.3
0.2
0.1
2
5
10
Time (sec)
15
Ky
0
-1
-2
0
-0.1
0
Kr
1
Parameters
0.7
-3
20
0
5
10
Time (sec)
r (t )  0.3  1.85 sin( 16.1t )
15
20
With Projection operator(1.5) r(t)=0.3+1.85*sin(16.1*t)
2.5
With Projection operator(1.5) r(t)=0.3+1.85*sin(16.1*t)
0.7
2
0.6
Output
0.5
 1.5  k y  1.5
0.4
0.3
 1.5  k r  1.5
0.2
0.1
0
0
5
10
Time (sec)
15
20
Parameters
1.5
1
Kr
0.5
Ky
0
-0.5
-1
-1.5
-2
-2.5
0
Improved knowledge of uncertainty
reduces the amplitude of oscillations
5
10
Time (sec)
15
20
Rohrs’ Example with L1 Adaptive Controller
u
Design elements
System
1
1
D( s)  , k 
s
3
 k   r  100
|| G ( s )(1  C ( s )) || L1 L  0.88  1
Tm  0.91
Predictor model
ŷ
y 
Adaptive Law
r
kˆy , kˆr
kˆr u  k y y  3r
u
D(s)  (1 / s)
u(t )  3 (t )
Control Law
Adaptive laws:
ky (t )   k y (t )( yˆ (t )  y (t )), k y (0)  0.65
kr (t )   r r (t )( yˆ (t )  y (t )), k r (0)  1.14

No Instability/Bursting with L1 Adaptive Controller
(L1) r1(t)=0.3+1.85*sin(16.1*t)
(L1) r1(t)=0.3+1.85*sin(16.1*t)
4
0.7
3
0.6
2
0.5
1
Parameters
0.3
0.2
0
-1
kr
-2
ky
0.1
r (t )  0.3  1.85 sin( 16.1t )
0
-3
-4
-0.1
-5
-0.2
0
5
10
Time (sec)
15
20
-6
0
5
10
Time (sec)
15
20
15
20
(L1) r2(t)=0.3+2*sin(8*t)
6
(L1) r2(t)=0.3+2*sin(8*t)
1.5
4
2
1
Parameters
0
0.5
Output
Output
0.4
0
r (t )  0.3  2 sin( 8t )
-0.5
-2
-4
kr
-6
ky
-8
-10
-12
-1
0
5
10
Time (sec)
15
20
-14
0
5
10
Time (sec)
Difference in Open-Loop TFs
ym
1
s3
Pref(s)
r(t)
Kˆ r (u, ~y )
+
+
u
229
s 2  30s  229
Pum(s)
2
s 1
+
y(s)
-
MRAC
~y
P(s)
MRAC cannot alter the phase in the
feedforward loop
Adaptation on
feedforward gain
Kˆ y ( y, ~y )
+
~
ˆ
K r (u, y )
+
Kˆ (u, ~y )
Continuous adaptation on phase
y
r(t)
L1
kg
+
+
+
k

s
Kˆ r (u , ~y )
u
229
s 2  30s  229
Pum(s)
Kˆ y (u, ~y )
2
s 1
P(s)
1
s3
Ppr (s)
+
y(s)
-
~y
Classical Control Perspective
L1
MRAC
Adaptation simultaneously
on both loop gain and phase
Adaptation only on loop gain
No adaptation on phase
Stabilization of Cascaded Systems with Application to a
UAV Path Following Problem
Ge
Gp
u
UAV with
Autopilot
y
y(s)  G p ( s)(u( s)  z (s))
u
x
Path Following
Kinematics
x  f ( x)  g ( x) y
Gp
Autopilot
y
UAV
Cascaded system
(in collaboration with Isaac Kaminer, NPS)
Augmentation of an Existing Autopilot by L1 Controller
Path following
kinematics
Autopilot
Exponentially stable
Path following
controller
Autopilot
UAV
UAV
Path following
kinematics
Very poor performance
Path following
controller
Path following
kinematics
Output feedback
L1 augmentation
L1 adaptive
controller
Path following
controller
Flight Test Results of NPS (TNT, Camp Roberts, CA, May 2007)
L1 Adaptive Controller Augmented
L1 Adaptive Controller
u
Autopilot
y
UAV
Inner-loop Guidance and Control
yc
Outer-loop path-planning, navigation
Without adaptation
Errors  100m
With adaptation, errors reduced to 5m
Architecture for Coordinated Path Following
• Path following: follow predefined trajectories using virtual path length
• Coordination: coordinate UAV positions along respective paths using
velocity to guarantee appropriate time of arrival
Guaranteed stability of the complete
system: use L1 to maintain
individual regions of attraction
Path
Generation
Network
info
desired
Path following
path
(Outer loop)
Normalized
Path lengths
Coordination
L1 adaptation
Onboard A/P
+ UAV
(Inner loop)
Pitch rate
Yaw rate
commands
Velocity
command
L1 adaptation
Time-critical Coordination of UAVs Subject to Spatial Constraints:
Hardware-in-the-Loop Flight Test Results
Performance comparison with and
without adaptation for one vehicle
Simultaneous arrival of two
vehicles with L1 enabled
Mission scenarios: Sequential autolanding, ground target suppression, etc.
 Arrival time is the same, but is not specified a priori
Assumptions:
 Vehicles dynamically decoupled
 Underlying communication network topology is fixed in time (currently relaxed)
 Each vehicle communicates only with its neighbors
Flight Control Design (Models provided by the Boeing Co.)
X-45ATailless Unstable Aircraft
Elevon/yaw vectoring control mixing
 Compensation for pitch break
uncertainty and actuator failures
Autonomous Aerial Refueling
 Uncertain dynamic environment
 Receiver dynamics due to
aerodynamic influences from
tanker
 Drogue dynamics due to
aerodynamic influences from
tanker and receiver
 Finite time to contact the drogue in
highly uncertain dynamic environment.
X-45A: Transient Performance vs Robustness
Case 2: ROB Elevon fail @ 1 sec, Pitch Break ON
•
•
1
Transient tracking not too sensitive to
changes in adaptive gain beyond certain
lower bound
Filter impacts the transient tracking
The filter can be selected to improve the
margin at the price of transient tracking
0
-1
-2
Az ft/sec/sec
•
-3
C (s )
-4
C1 ( s)
C(s),  =5000
C(s),  =50000
C1(s),  =5000
C1(s),  =50000
-5
-6
-7
-8
-9
TDM=0.045
0
5
10
15
20
25
30
Time (sec)
35
40
45
Vertical acceleration
Case 2: ROB Elevon fail @ 1 sec, Pitch Break ON
TDM=0.1
0
•
•
Transient tracking compared for
MRAC and L1 with two different filters
The performance of C1(s) (with better
margins) still better than for MRAC
Az, ft/sec/sec
-2
C1 ( s)
-4
-6
MRAC
C(s)=1/(0.05s+1)
C1(s)=C(s)*(-6s+1) 2/(8s+1)2
C (s )
-8
-10
MRAC
0
5
10
15
20
25
30
35
40
45
Aerial Refueling
 Control objective:
 Finite time to contact the drogue in highly
uncertain dynamic environment.
 Solution/Approach:
 An adaptive method without retuning adaptation gains
 for highly uncertain dynamic environment
 with guaranteed transient performance
 for finite time-to-contact
• Barron Associates Tailless Aircraft Model
• 6 DOF Flying-wing UAV, Statically unstable at low angle of attack
• Aerodynamic data primarily taken from [1]
• Vortex effect data from wind tunnel test [2] – ICE model
• Tanker: KC-135R, Receiver: Tailless 65 degree delta wing UAV
• Aerodynamic coefficients change as a function of relative separation
→ Varying more significantly with lateral and vertical separation
[1] S.P Fears et. al., Low-speed wind-tunnel Investigation of the stability and control characteristics of a series of flying wings with sweep angles of 50 degree
[2] W.B. Blake et. al., UAV Aerial Refueling – Wind Tunnel Results and Comparison with Analytical Predictions
Scaled Response
- Starting from different
initial positions
- Scaled response
No retuning!
- Twice the vortex –
model of a different
tanker
thrust channel
elevator channel
aileron channel
- Uniform transient and
steady state response
- Scaled control efforts
Air-Breathing Hypersonic Vehicle (Boman model WP AFRL)
 Control Challenge of AirBreathing Hypersonic Vehicles
 Vibration due to high speed and
structural flexibility
 Uncertainties caused by rapid
change of the operating conditions
 Integration of airframe dynamics
and propulsion system
 Coupling of attitude and velocity
Desired path flight angle
desired speed
Control signal: Elevator (deg)
Velocity profile
System response
Control Signal: Equivalent ratio
Current Status and Open Problems
 Adaptive Output Feedback Theory
 Partial results in this direction
 Numerical discretization issues due to fast adaptation
 Integration of stiff systems
 Adaptive Observer Design
 Control of distributed autonomous systems over ad-hoc
communication network topologies
 Performance evaluation on the interface of different
disciplines
 More work on its way towards final answers!!!
Conclusions
 What do we need to know?
 Boundaries of uncertainties
sets the filter bandwidth
 CPU (hardware)
sets the adaptive gain
 Fast adaptation
no more trial-error based gain tuning!
 performance can be predicted a priori
 robustness/stability margins can be quantified analytically
 Theoretically justified Verification & Validation tools for
feedback systems
at reduced costs
With very short proofs!
Acknowledgments
My group:










Chengyu Cao (theoretical developments)
Jiang Wang (aerial refueling)
Vijay Patel (UCAV, flight test support)
Lili Ma (vision-based guidance)
Yu Lei (hypersonic vehicle)
Dapeng Li (theoretical developments, vision-based guidance)
Amanda Young (cooperative path following)
Enric Xargay (cooperative path following)
Zhiyuan Li (discretization issues for flexible aircraft)
Aditya Paranjape (differential game theoretic approaches for VBG)
Collaborations:





The Boeing Co. (E. Lavretsky, K. Wise, UCAV model, aerial refueling)
Wright Patterson AFRL, CerTA FCS program (ICE vortex)
Raytheon Co. (R. Hindman, B. Ridgely)
NASA LaRC (I. Gregory, SensorCraft)
Wright Patterson AFRL (D. Doman, M. Bolender, hypersonic model)
 Randy Beard (BYU)
 Isaac Kaminer (NPS)
Sponsors: AFOSR, AFRL, ARO, ONR, Boeing, NASA