THE THEORY OF FAST AND ROBUST ADAPTATION. 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.
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THE THEORY OF FAST
AND ROBUST ADAPTATION
Naira Hovakimyan
Department of Aerospace and Ocean EngineeringVirginia 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
18571918
G. Zames
19341997
Outline
Motivation
First Flight Test in 1967
The Crash of the X15A3 (November 15, 1967)
X15A3 on its B52 mothership
X15A3
Crash due to stable, albeit nonrobust adaptive controller!
Crash site of X15A3
Objectives of Feedback
Historical Background
Landmark Achievement: Adaptive Control in Transition
Lessons Learned: limited to slowlyvarying uncertainties, lack of transient characterization
Boeing question: How fast to adapt to be robust?
Stability, Performance and Robustness
The Theory of Fast and Robust Adaptation
Implementable nom. cont.
Smallgain theorem gives sufficient condition for boundedness:
Result: fast and robust adaptation via continuous feedback!
One Slide Explanation of the New Paradigm
System:
Nominal controller in MRAC:
Desired reference system:
This is the nominal controller in the L1 adaptive control paradigm!
Hurwitz, unknown
System dynamics
Ideal Controller
Adaptive Controller
Adaptive law:
Bounded Barbalat’s lemma
State predictor
Same error dynamics
independent of control choice
Closedloop state predictor with u
Two Equivalent Architectures of Adaptive Control
Enables insertion of a lowpass filter
Lowpass filter: C(s)
Implementation Diagrams
Reference system
MRAC
u cannot be lowpass filtered directly.
State predictor based reparameterization
No more reference
system upon
filtering!!!
Lowpass filter: C(s)
bounded
Barbalat’s lemma
Stability
nonadaptive predictor:
Stability via smallgain theorem
Plant with
unknown parameter
Reference
system of MRAC
Reference system
Closedloop Reference System
Guaranteed Transient Performance
Virtual Plant with
clean output
Plant with
unknown parameter
LTI System for Control Specifications
Design system
for defining the control specs
Reference system
achieved via fast adaptation
Independent of the
unknown parameter
Achieving Desired Specifications
Large adaptive gain Smaller stepsize Faster CPU
Guaranteed Performance Bounds
Desired design objective
Timedelay Margin and Gain Margin
Point Timedelay occurs
Plant
Lower bound for the timedelay margin
Projection domain defines the gain margin
Main Theorem of L1 Adaptive Controller
Fast adaptation, in the presence of ,
leads to guaranteed transient performance and
guaranteed gain and timedelay margins:
where is the timedelay margin of .
Tracking vs robustness can be analyzed analytically.
The performance can be predicted a priori.
Design Philosophy
Adaptive gain – as large as CPU permits (fast adaptation)
Fast adaptation leads to improvedperformance and improvedrobustness.
Lowpass filter:
Increase the bandwidth of the filter:
Robot Arm with Timevarying Friction and Disturbance
Control design parameters
Parametric uncertainties in statespace form:
Compact set for Projection
Disturbances in three cases:
Simulation Results without any Retuning of the Controller
System output
Control signal
With timedelay 0.1
Verification of the TimeDelay Margin
With timedelay 0.02
Application of nonlinear L1 theory
MRAC and L1 for a PI controller
MRAC
L1
The openloop transfer functions in the presence of timedelay
Timedelay margin
Timedelay margin
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
System with unmodeled dynamics:
Nominal values
of plant parameters:
plant
dynamics
Unmodeled
dynamics
Reference system dynamics:
Control signal:
Adaptive laws from Rohrs’ simulations:
Instability/Bursting due to Unmodeled Dynamics
Instability
Parameter drift
System output
Bursting
Parameters
System output
Rohrs’ Explanation for Instability: OpenLoop Transfer Function
135
180
At the frequency 16.1rad
the phase reaches 180degrees
in the presence of unmodeled dynamics,
reverses the sign of highfrequency gain,
the loop gain grows to infinity
instability
At the frequency 8rad
the phase reaches 139 degrees
in the presence of unmodeled dynamics,
no sign reversal for the highfrequency gain
the loop gain remains bounded
bursting
Rohrs’ Example with Projection for Destabilizing Reference Input
Improved knowledge of uncertainty
reduces the amplitude of oscillations
Predictor model
System
Adaptive Law
Control Law
Rohrs’ Example with L1 Adaptive Controller
Design elements
Adaptive laws:
No Instability/Bursting with L1 Adaptive Controller
Pref(s)
+
r(t)
y(s)
+
u

+
Pum(s)
P(s)
+
+
Ppr (s)
+
y(s)
r(t)
u
+
kg

+
+
Pum(s)
P(s)
L1
Difference in OpenLoop TFs
MRAC
MRAC cannot alter the phase in the feedforward loop
Adaptation on
feedforward gain
Continuous adaptation on phase
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
UAV with
Autopilot
Path Following
Kinematics
Cascaded system
(in collaboration withIsaac Kaminer, NPS)
Path following kinematics
Path following kinematics
Autopilot
UAV
Very poor performance
Exponentially stable
Path following controller
Path following controller
Path following kinematics
Autopilot
UAV
Output feedback
L1 augmentation
L1 adaptive controller
Path following controller
Augmentation of an Existing Autopilot by L1 Controller
Flight Test Results of NPS (TNT, Camp Roberts, CA, May 2007)
L1Adaptive Controller Augmented
L1Adaptive Controller
Outerloop pathplanning, navigation
With adaptation, errors reduced to 5m
Autopilot
UAV
Innerloop Guidance and Control
Without adaptation
Errors 100m
Guaranteed stability of the complete
system: use L1to maintain
individual regions of attraction
L
adaptation
1
Path
Onboard A/P
Path following
desired
Pitch rate
Generation
+ UAV
(Outer loop)
path
Yaw rate
(Inner loop)
commands
Network
Normalized
Velocity
Coordination
info
Path lengths
command
L
adaptation
1
Timecritical Coordination of UAVs Subject to Spatial Constraints:
HardwareintheLoop 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.
Assumptions:
Flight Control Design (Models provided by the Boeing Co.)
Tailless Unstable Aircraft
X45A
Autonomous Aerial Refueling
Elevon/yaw vectoring control mixing
MRAC
X45A: Transient Performance vs Robustness
TDM=0.045
Vertical acceleration
TDM=0.1
Aerial Refueling
[1] S.P Fears et. al., Lowspeed windtunnel 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
 Uniform transient and
steady state response
 Scaled control efforts
thrust channel
elevator channel
aileron channel
AirBreathing Hypersonic Vehicle (Boman model WP AFRL)
Desired path flight angle
desired speed
Control signal: Elevator (deg)
Velocity profile
System response
Control Signal: Equivalent ratio
Current Status and Open Problems
Conclusions
With very short proofs!
Acknowledgments
My group:
Collaborations:
Sponsors: AFOSR, AFRL, ARO, ONR, Boeing, NASA