basics adaptive control ddeb
TRANSCRIPT
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Adaptive Control
Assistant Professor
IIT Guwahati
Dr. Dipankar Deb
Basics and Applications
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Feedback Control System
EPlantEController
Feedback '
T
EEr(t) e(t) u(t) y(t)
w(t)
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Issues of Automatic Feedback Control
System modeling
Control objectives
stability, transient, tracking, optimality, robustness
Parametric uncertainties
payload variation, component aging, condition change
Structural uncertainties
component failure, unmodeled dynamics
Environmental uncertainties
external disturbances
Nonlinearities
smooth functions and nonsmooth (hard) characteristics
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Adaptive Control Methodology
Adapting to parametric uncertainties
Robust to structural and environmental uncertainties
Aimed at both stability (signal boundedness) and tracking
Self-tuning of controller parameters
Systematic design and analysis
Real-time implementable
Effective for failures and nonsmooth nonlinearities
High potential for applications
Attractive open and challenging issues
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Aircraft Flight Control System Models
State variables
x,
y,
z = position coordinates = roll angleu,v,w= velocity coordinates = pitch anglep = roll rate = yaw angleq = pitch rate = side-slip angler = yaw rate = angle of attack
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Nonlinear equations of motion (in body axis)
Force equations:
m(u+qw rv) = Xmgsin+Tcos
m(v+ ru pw) = Y+mgcossin
m(w+ pvqu) = Z+mgcoscosTsin
T: engine thrust; : thrust angle; X,Y,Z: aerodynamic forces
Moment equations:
Ix p+Ixzr+ (IzIy)qr+Ixzqp = L
Iyq+ (Ix
Iz)pr+Ixz(r2
p
2
) = MIzr+Ixz p+ (IyIx)qpIxzqr = N
L,M,N: aerodynamic torques
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Linearized longitudinal equations
u
w
q
=
Xu Xw W0 g0 cos0
Zu Zw U0 g0 sin0
Mu Mw Mq 00 0 1 0
u
w
q
+
Xe
Ze
Me
0
e
output = : pitch angle perturbation
Linearized lateral equations
r
p
=
Yv U0 V0 g0 cos0
Nv Nr Np 0
Lv Lr Lp 0
0 tan0 1 0
r
p
+
Yr Ya
Nr Na
Lr La
0 0
r
a
output = r: yaw rate perturbation
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Direct Adaptive Control System
Adaptive law
(t)
PlantController
C(s;(t))
Reference model
E
E
E
T
E
TT T
'c
cu(t)r(t) y(t)
ym(t)
(t)
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Indirect Adaptive Control System
Design
equation
Parameter estimator
p(t)
Plant
G(s;p)
Controller
C(s;c(t))E E
T
E
TT
'
cu(t)ym(t) y(t)
c(t)
p(t)
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Part I: Adaptive Control Theory
Issues in Automatic Feedback Control
Adaptive Control Methodology
Direct Model Reference Adaptive Control
Indirect Adaptive Pole Placement Control
Multivariable Adaptive Control
Nonlinear Adaptive Control
Performance, Convergence and Robustness
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Summary
System uncertainties
common in control systems
challenges for system performance
Adaptive control
handles system uncertainties effectively
ensures desired asymptotic performance Adaptive control theory
mature with systematic design procedures
developing with new challenges
Adaptive control techniques
proved to be useful for many practical control problems
promising for new aerospace applications
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Part II: Adaptive Control for Aerospace Systems
Adaptive Control for Actuator and Sensor Nonlinearities
Adaptive Inversion Control for Synthetic Jet Actuators
Adaptive Actuator Failure Compensation
Design for Linearized Longitudinal B737 Model
Design for Linearized Lateral B737 Model
Open Research Problems
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Adaptive Control for Actuator and Sensor Nonlinearities
Actuator and sensor nonlinearities
Research motivation
Adaptive inverse compensation
Adaptive inverse control designs
Adaptive control of sandwich nonlinear systems
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Actuator and Sensor Nonlinearities
Dead-zones
hydraulic valves, servo motors
Backlash
gear-boxes, mechanical links
Hysteresis
piezoelectric materials
Piecewise-linearity
different operation conditions
Non-linear characteristics
non-uniform hardware properties
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y(t)
Spool positionv(t)
Piston
Load
M, f
Return
Return
Pressuresource
Figure 1: Dead-zone in a servo-valve.
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Ks(Ms+B)
E E EET
v yu
Figure 2: Block diagram of the servo-valve.
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v(t)
u(t)
c-crl
Figure 3: Backlash in mechanical links.
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u
vc
c
m
m
0
l
r
Figure 4: Backlash characteristic.
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v
u
d
u = B(v)
control
h(s) = G(s)(u(s) - d(s))
G(s) = k/s
h
gear-train
backlash
Figure 5: Backlash in the valve control mechanism of a liquid tank.
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-Controller
motorand
Amplifier
trainGear
y ym
Mirror
Figure 6: Output backlash in a positioning system.
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(c)(b)
(a)
backlash and flexibility
b b
uJJ
l m
ml
m
ml
l
b-bb-b
Figure 7: (a) Gear-train with backlash and flexibility; (b) Backlash for rigid gears:
l = B(m); (c) Dead-zone: = D(m l ).
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ffffc EEEEEEEEEEE''
TT T
1Jms
1s
D() k1
Jl s1s
bm bl
u m m v l l
Figure 8: Sandwich nonlinear system with feedback blocks.
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7Figure 9: Hysteresis characteristic in precision control actuators.15
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Research Motivation
Actuator and sensor nonlinearities limit performance
Actuator and sensor nonlinearities are uncertain
Adaptive compensation is a desirable choice
Algorithm-based compensation is aimed at
reduction of system component cost
improvement of system performance.
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Adaptive Inverse Compensation
i G(D)Na()NIa()
C2(D)
C1(D)
'
TE E E E E Er u
dyv u
Figure 10: Adaptive inverse control for actuator nonlinearity.
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Adaptive Inverse Control Designs
Parametrization of nonlinearity u(t) = N(v(t))
u(t) = N(v(t)) = N(; v(t)) = T(t) + as (t)
Parametrization of nonlinearity inverse v(t) = NI(ud(t))ud(t) =
T(t)(t) + as(t)
Feedback control law for ud(t) based on model reference, pole placement, PID, lead/lag
compensator, feedback linearization, or backstepping design
for G(D) known or G(D) unknown, SISO or MIMO
Adaptive law for (t)
(t) = (t)(t)
m2(t)+ f(t)
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An Illustrative Example
k 1s +
s
B()E E E E E
T
r v u y
PI controller Plant
Figure 11: One-integrator plant with input backlash and PI controller.
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-2 0 2 4-6
-4
-2
0
2
4
(a) e(t) vs. v(t).-2 0 2 4
-2
0
2
4
6
(b) u(t) vs. v(t).
0 10 20 30 40 50-6
-4
-2
0
2
4
(c) Tracking error e(t).
System response without backlash inverse.20
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2 0 2 46
4
2
0
2
(a) e(t) vs. v(t).2 0 2 4
1
0
1
2
3
4
5
(b) u(t) vs. v(t).
0 10 20 30 406
4
2
0
2
(c) Tracking error e(t).0 10 20 30 40
0
0.5
1
^(d) Parameter error c(t) c.
Figure 12: Adaptive backlash inverse control response.
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Adaptive Inverse Control of Sandwich Systems
Eyr NIo() Ezrg Eud NIi() Ev Ni() Eu G(s) Ez No() EyT
'gggy
' NIo()T2 b(s) 'z
T1 a(s)
Figure 13: Adaptive compensation of actuator/sensor nonlinearities.
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Adaptive Inverse Control for Synthetic Jet Actuators
Research motivation
Synthetic jets for aircraft control
Adaptive inverse approach
Adaptive inverse control design and evaluation
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Research Motivation
Virtual shaping of airfoils using synthetic jets
Synthetic jet actuators have unknown nonlinearities Adaptive inverse approach to cancel nonlinearities
Adaptive feedback control for desired performance
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Synthetic Jets for Aircraft Control
Physics of synthetic jet
piezo-electric sinusoidal voltage acts on diaphragm
diaphragm vibrations cause cavity pressure variations
ejection and suction of air, creating vortices
jet is synthesized by a train of vortices
lift is produced on the airfoilvirtual shaping.
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Tailless aircraft with jets (top view)
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Mathematical model of synthetic jet actuators
u(t) = 2 1
v(t)= N(v(t);)
u(t): equivalent virtual airfoil deflection (lift force)v = A2pp: peak-to-peak voltage amplitude
= [1,2]
T: unknown parameters dependent on many factors
Control objectiveadaptive compensation ofN(;)
tracking control for aircraft flight trajectory
Design strategyadaptive inversion ofN(;)
feedback control for aircraft dynamics
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Variation of deflection with actuation voltage (2 = 15)
0 2 4 6 8 10 1225
20
15
10
5
0
5
10
15
v(t), the input voltage in volts
u(t),
the
virtualde
flection
on
the
air
foil
*
1= 20
*
1= 30
*
1 = 40
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Adaptive Inverse Approach
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Nonlinearity parametrization
u(t) = N(; v(t)) = T(t), (t) = [1
v(t), 1]T
Nonlinearity inverse
v(t) = NI(ud(t)) = 1(t)2(t) ud(t)
ud(
t) =
T
(t)
(t)
, = [
1
,2]
T
ud(t): a desired feedback control signal
Control error
u(t) ud(t) = 2(t) ud(t)1(t)
(1(t) 1) (2(t) 2)= ((t) )T(t)
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Design Example: Linear Dynamics
System model
x(t) = Ax(t) +Bu(t)
Feedback control signal
ud(t) = Kx(t) + r(t)
Control erroru(t) ud(t) = ((t)
)T(t)
Reference model
xr(t) = (A BK)xr(t) +Br(t)
Error system
e(t) = (A BK)e(t) +B((t) )T(t), e(t) = x(t) xr(t)
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Adaptive laws
i(t) = gi(t) + fi(t)
g1(t) = 1eT(t)PB
2(t) ud(t)
1(t)
, 1 > 0
g2(t) = 2eT
(t)PB, 2 > 0
fi(t) =
0 if i(t) > ai , or
if i(t) = ai and gi(t) 0
gi(t) otherwise
initial estimates i(0) ofi : i(0)
ai > 0
Assumption (no saturation): ud(t) < 2(t)
Closed-loop system properties
boundedness ofx(t) and (t), and i(t) > ai
asymptotic tracking: limt(x(t) xr(t)) = 0.
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Simulation Results
System state variables
lateral velocity: x1(t) roll rate: x2(t)
yaw rate: x3(t) roll angle: x4(t)
System model
A =
0.0134 48.5474 632.3724 32.0756
0.0199 0.1209 0.1628 0
0.0024 0.0526 0.0252 0
0 1 0.0768 0
, B =
0
0.0431
0.0076
0
D. L. Raney, R. C. Montgomery, L. L. Green and M. A. Park, Flight Control
using Distributed Shape-Change Effector Arrays, AIAA paper No.
2000-1560, April 3-6, 2000
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Control gain K
LQR design with Q = I4, R = 10
K=
1.0113 77.1793 115.8959 9.1691
P =
0.751 14.980 159.812 8.2617
14.980 27181.878 138979.668 7843.345
159.813 138979.668 723352.800 40670.052
8.262 7843.345 40670.052 2301.187
Reference signal:
r(t) = 1.5sin(t) 0 t 60
1.5sin(t) + 3sin(2t) t 60
Adaptation gains: 1 = 1, 2 = 2
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Simulation I: Adaptive inverse performance
0 50 100 150 20010
0
10
Plant state x1(t) (solid), reference state x
m1(t) (dotted) vs. time (sec)
ft/sec
0 50 100 150 2000.1
0
0.1
Plant state x2(t) (solid), reference state x
m2(t) (dotted) vs. time (sec)
deg/s
ec
0 50 100 150 2000.05
0
0.05
d
eg/sec
Plant state x3(t) (solid), reference state x
m3(t) (dotted) vs. time (sec)
0 50 100 150 2000.5
0
0.5
deg
Plant state x4(t) (solid), reference state x
m4(t) (dotted) vs. time (sec)
Figure 16: Plant and reference states.
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0 50 100 150 20010
0
10
Tracking error e1(t) vs. time (sec)
ft/sec
0 50 100 150 2000.2
0
0.2
Tracking error e2(t) vs. time (sec)
deg
/sec
0 50 100 150 2000.05
0
0.05
deg/sec
Tracking error e3(t) vs. time (sec)
0 50 100 150 2000.5
0
0.5
deg
Tracking error e4(t) vs. time (sec)
Figure 17: State tracking errors.
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Simulation II: Comparison with a fixed inverse
0 50 100 150 20010
5
0
Tracking error e1(t) vs. time (sec)
ft/sec
0 50 100 150 2000.1
0
0.1
Tracking error e2(t) vs. time (sec)
deg/s
ec
0 50 100 150 2000.1
0
0.1
deg/sec
Tracking error e3(t) vs. time (sec)
0 50 100 150 2001
0.5
0
deg
Tracking error e4(t) vs. time (sec)
Figure 18: State tracking errors with a fixed inverse.
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Simulation III: Effect of saturation
A possible modification for control signal
ud
(t) = Kx(t) + r(t)
ud(t) =
2 if ud(t) 2
ud(t) otherwise
> 0 is a small constant
Simulation signals
(i) r1(t) = 5r(t) (convergent)
(ii) r2(t) = 6r(t) (non-covergent)
both leading to saturation ofud(t).
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(i) convergent results
0 50 100 150 20050
0
50
Plant state x1(t) (solid), reference state x
m1(t) (dotted) vs. time (sec)
ft/sec
0 50 100 150 2000.5
0
0.5
Plant state x2(t) (solid), reference state x
m2(t) (dotted) vs. time (sec)
deg/s
ec
0 50 100 150 2000.1
0
0.1
deg/sec
Plant state x3(t) (solid), reference state x
m3(t) (dotted) vs. time (sec)
0 50 100 150 2000.5
0
0.5
deg
Plant state x4(t) (solid), reference state x
m4(t) (dotted) vs. time (sec)
Figure 19: Plant and reference states (with saturation).
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0 50 100 150 20010
0
10
Tracking error e1(t) vs. time (sec)
ft/se
c
0 50 100 150 2000.2
0
0.2
Tracking error e2(t) vs. time (sec)
de
g/sec
0 50 100 150 200
0.05
0
0.05
deg/sec
Tracking error e3(t) vs. time (sec)
0 50 100 150 2000.5
0
0.5
deg
Tracking error e4(t) vs. time (sec)
Figure 20: State tracking errors (convergent).
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(ii) non-convergent results
0 50 100 150 20050
0
50
Plant state x1(t) (solid), reference state x
m1(t) (dotted) vs. time (sec)
ft/sec
0 50 100 150 2001
0
1
Plant state x2(t) (solid), reference state x
m2(t) (dotted) vs. time (sec)
deg/s
ec
0 50 100 150 2000.2
0
0.2
deg/sec
Plant state x3(t) (solid), reference state x
m3(t) (dotted) vs. time (sec)
0 50 100 150 2000.5
0
0.5
deg
Plant state x4(t) (solid), reference state x
m4(t) (dotted) vs. time (sec)
Figure 21: Plant and reference states (with saturation).
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0 50 100 150 20020
0
20
Tracking error e1(t) vs. time (sec)
ft/se
c
0 50 100 150 2000.5
0
0.5
Tracking error e2(t) vs. time (sec)
de
g/sec
0 50 100 150 200
0.1
0
0.1
deg/sec
Tracking error e3(t) vs. time (sec)
0 50 100 150 2000.5
0
0.5
deg
Tracking error e4(t) vs. time (sec)
Figure 22: State tracking errors (non-convergent).
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Discussion
Synthetic jet actuators are for novel aircraft
Compensation of actuator nonlinearity is crucial
Actuator parameters are highly uncertain
Algorithm-based adaptive inverse is promising
It is applicable to jet arrays and nonlinear dynamics
Actuator saturation is an important issue
Actuator failure is another important issue adaptive failure compensation
adaptive nonlinearity compensation.
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Adaptive Actuator Failure Compensation
Research motivation
Systems with actuator failures
Research goals and technical issues
Adaptive failure compensation techniques
Study of aircraft flight control applications
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Research Motivations
Actuator failures
common in control systems
uncertain in failure time, pattern, parameters
undesirable for system performance
Adaptive control
deals with system uncertainties
ensures desired asymptotic performance
is promising for actuator failure compensation
has potential for critical applications
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Effective methods for handling system failures
multiple-model, switching and tuning
indirect adaptive control
fault detection and diagnosis
robust or neural control Direct adaptive failure compensation approach
use of a single controller structure
direct adaptation of controller parameters
no explicit failure (fault) detection
stability and asymptotic tracking
Potential applications include
aircraft flight control
smart structure vibration control
space robot control
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Systems with Actuator Failures
System Models
x = f(x) +m
j=1
gj(x)uj, y = h(x)
x = Ax +m
j=1
bjuj, y = Cx
state variable vector: x(t) Rn
output: y(t)
input vector: u = [u1, . . . , um]T
Rm
whose components mayfail during system operation
f(x), gj(x), h(x), A, bj, C with unknown parameters.
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Actuator Failures
Loss of effectiveness
uj(t) = kj(t)vj(t), kj(t) (0, 1), t tj
Lock-in-place
uj(t) = uj, t tj, j {1, 2, . . . , m}
Lost control
uj(t) = uj +k
djkjk(t) +j(t), t tj, j {1, . . . , m}
Failure uncertainties
the failure values kj, uj and djk, failure time tj, pattern j, and
components j(t) are all unknown.
How much, how many, which and when the failures happen??
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Examples
aircraft aileron, stabilizer, rudder or elevator failures
their segments stuck in unknown positions
their unknown broken pieces (including wings)
satellite motion control actuator failures
MEM actuator/sensor failures on fairing surface
heating device failures in material growth
generator failures in power systems
transmission line failures in power system
power distribution network failures
cooperating manipulator failures
bioagent distribution system failures
etc.
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System Input (for lost control failures uj(t))
System input in the presence of actuator failures is
u(t) = v(t) +(u(t) v(t))
v = [v1, v2, . . . , vm]T: a designed control input, and
u(t) = [u1(t), u2(t), . . . , um(t)]T
= diag{1,2, . . . ,m}
j =
1 if the jth actuator failed, i.e., uj(t) = uj(t)
0 otherwise
Actuation error u(t) v(t) = (u(t) v(t)) is uncertain.
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Block Diagram
Controller
SystemE
E11
...
...
Ec1E
E
1m
E
E cE E
E
mE
rum
u
1
yu1...
um
v1...
vm
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Research Goals
Theoretical framework for adaptive control of systems with uncertainactuator (sensor, or component) failures
Guidelines for designing control systems with guaranteed stability
and tracking performance despite parameter and failure uncertainties
Solutions to key issues in adaptive failure compensation: controller
structures, design conditions, adaptive laws, stability, robustness
New adaptive control techniques for critical systems (e.g., aircraft) to
improve reliability and survivability.
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Technical Challenges
Redundancy
necessary for failure compensation
problematic for control: up to m q failures
Failure uncertainties
parametric, structural, and environmental
Robustness and transient performance
Application issues
system modeling and control implementation
aircraft, robot, smart structure, power system, satellite
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Adaptive Failure Compensation
Control Objective
Stability and asymptotic tracking for up to mq failures.
Basic Assumption
The system is so constructed that for any up to m q (0 < q m) failedactuators, the alive actuators can still achieve some desired performance.
Key Task
Adaptively adjust the remaining actuators (controls) to achieve desiredperformance when system and failure parameters are unknown.
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Design Tools
State feedback for state tracking
u = KTx + krr+ kc, limt
(x(t) xr(t)) = 0
State feedback for output tracking
u = KTx + krr+ kc, limt
(y(t) yr(t)) = 0
Output feedback for output tracking
u = T1 1 +T2 2 +3r+ kc, 1 = F1(s)u, 2 = F2(s)y
limt
(y(t) yr(t)) = 0.
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Designs for linear systems
model reference for minimum phase systems
pole placement for nonminimum phase systems
decoupling for MIMO systems
Designs for nonlinear systems
feedback linearizable systems
parametric-strict-feedback systems
output-feedback systems
output feedback for state-dependent systems.
Aircraft flight control applications
lateral: Boeing 737, Boeing 747, DC-8
longitudinal: Boeing 737, Twin Otter, hypersonic.
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9 6Example: Boeing 737 Landing
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Example: Boeing 737 Landing
System model
x(t) = Ax(t) +Bu(t), y(t) = , B = [b1, b2]T
x = [Ub,Wb, Qb,]T: forward speed Ub, vertical speed Wb, pitch angle
, pitch rate Qb; u = [dele1, dele2]T: elevator segment angles
Study of an aircraft with two elevator segments Output feedback output tracking design
One elevator segment fails during landing at t = 30 sec.
Simulation results
response with no compensation (fixed feedback)
response with adaptive compensation.
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0 10 20 30 40 50 60 70 80 90 1000
0.02
0.04
0.06
0.08
time (sec)
y(t),ym(t)(rad) y(t)y
m(t)
0 10 20 30 40 50 60 70 80 90 1000
0.005
0.01
0.015
time (sec)
e(t)(rad)
0 10 20 30 40 50 60 70 80 90 1004
3
2
1
0
time (sec)
v(
t)(deg)
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0 10 20 30 40 50 60 70 80 90 1000
0.02
0.04
0.06
0.08
time (sec)
y(t),ym(t)(rad)
y(t)ym
(t)
0 10 20 30 40 50 60 70 80 90 1000.01
0
0.01
0.02
time (sec)
e(t)(rad)
0 10 20 30 40 50 60 70 80 90 1004
3
2
1
0
time (sec)
v
(t)(deg)
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9 6Example: Boeing 737 Lateral Motion
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Example: Boeing 737 Lateral Motion
MIMO system model
x = Ax +Bu, y = Cx
x = [vb,pb, rb,,]T: lateral velocity vb, roll rate pb, yaw rate rb, roll
angle , yaw angle
y = [,]T
: roll angle , yaw angle u = [dr, da]
T: rudder position dr, aileron position da,
segmented into: dr1, dr2, da1, da2
Actuator failuresdr2 fails at t = 50, da2 fails at t = 100 seconds
Simulation results
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0 20 40 60 80 100 120 140 160 180 2000
1
2
3
4
5
6
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Roll angle (t): , reference outputm(t):
deg
0 20 40 60 80 100 120 140 160 180 2000
2
4
6
8
10
Yaw angle(t): , reference outputm(t):
deg
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9 6Discussion
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Effective failure compensation has major interest
Direct adaptive compensation is aimed at
automatic tuning of controller parameters
handling of large class of failures
guaranteed system tracking performance
A solution framework has been developed for parametrization of actuator failures
failure compensation conditions
adaptive compensation designs
Desired performance was verified on numerous aircraft models
There is high potential for other applications.
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9 6Conclusions
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Adaptive control is a mature control methodology
Adaptive controllers adjust themselves to system uncertainties
Effective uncertainty compensation is a key for aerospace systems
Adaptive control technologies have high potential.
Adaptive compensation techniques are developed for practical actuator and sensor nonlinearities
actuator failures (and sensor failures, in progress)
Applications to aerospace systems have been formulated
Various designs were verified on aircraft system models
Further research and more advances are critical.
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9 6Research Interests
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Adaptive control theory
actuator/sensor/component failure compensation
multivariable and nonlinear systems
actuator and sensor nonlinearity compensation
Adaptive control applications
aircraft flight control
fairing structure vibration reduction
space robot cooperative and compensation control
synthetic jet actuator compensation control
satellite motion control
high precision pointing systems
dynamic sensor/actuator networks
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9 6Some On-Going Research
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Rudder failure compensation by engine differentials
aircraft model with engine differentials
adaptive failure compensation control Adaptive compensation control for aircraft damages
dynamic modeling of aircraft damages
direct adaptive damage compensation control
Applications to aircraft and UAVs
Adaptive compensation control for synthetic jet actuators
Adaptive failure compensation for space robots Adaptive compensation of sensor failures
Adaptive control of power systems with failures
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