Basics Adaptive Control DDEB

7/30/2019 Basics Adaptive Control DDEB

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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



deg/s

ec

0 50 100 150 2000.05

0

0.05

d

eg/sec



0 50 100 150 2000.5

0

0.5

deg



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


deg

/sec

0 50 100 150 2000.05

0

0.05

deg/sec


0 50 100 150 2000.5

0

0.5

deg


Figure 17: State tracking errors.

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Simulation II: Comparison with a fixed inverse

0 50 100 150 20010

5

0


ft/sec

0 50 100 150 2000.1

0

0.1


deg/s

ec

0 50 100 150 2000.1

0

0.1

deg/sec


0 50 100 150 2001

0.5

0

deg


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



ft/sec

0 50 100 150 2000.5

0

0.5



deg/s

ec

0 50 100 150 2000.1

0

0.1

deg/sec



0 50 100 150 2000.5

0

0.5

deg



Figure 19: Plant and reference states (with saturation).

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0 50 100 150 20010

0

10


ft/se

c

0 50 100 150 2000.2

0

0.2


de

g/sec

0 50 100 150 200

0.05

0

0.05

deg/sec


0 50 100 150 2000.5

0

0.5

deg


Figure 20: State tracking errors (convergent).

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(ii) non-convergent results

0 50 100 150 20050

0

50



ft/sec

0 50 100 150 2001

0

1



deg/s

ec

0 50 100 150 2000.2

0

0.2

deg/sec



0 50 100 150 2000.5

0

0.5

deg



Figure 21: Plant and reference states (with saturation).

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0 50 100 150 20020

0

20


ft/se

c

0 50 100 150 2000.5

0

0.5


de

g/sec

0 50 100 150 200

0.1

0

0.1

deg/sec


0 50 100 150 2000.5

0

0.5

deg


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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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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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

7

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


70/73

8 7

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.

65

9 6Conclusions


71/73

8 7

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.

66

9 6Research Interests


72/73

8 7

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

67

9 6Some On-Going Research


73/73

8 7

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

68

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Basics Adaptive Control DDEB

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