Tao Adaptive

8/2/2019 Tao Adaptive

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Adaptive ControlBasics and Research

Gang Tao


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Feedback Control System

-

Plant

-

Controller

Feedback

6

--r(t) e(t) u(t) y(t)

w(t)

Reference System- -

r(t) ym(t)

Goal of feedback control: limt(y(t) ym(t)) = 0


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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 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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Direct Adaptive Control System

Adaptive law

(t)

PlantController

C(s;(t))

Reference model

-

-

-

6

-

66 6

?

?u(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))- -

6

-

66

?u(t)ym(t) y(t)

c(t)

p(t)


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Control System Dynamic Models

Nonlinear models

x = f(x, u, w), y = h(x, u, v)

state vector x Rn, input u, output y, disturbances w, v; or

x = f(x) + g(x)u + d(x)w, y = h(x, u) + v

Linear state-variable model

x = Ax +Bu +Bww, y = Cx +Du + v

Linear time-invariant input-output model

y(t) = G(s)[u](t) + d(t)

G(s) = G0(s)(1 +m(s)) +a(s), G0(s) = kpZ(s)

P(s)

a(s), m(s): additive, multiplicative unmodeled dynamics.


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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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Linearized longitudinal equations

u

w

q

=

Xu Xw W0 g0 cos0Zu 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 cos0Nv 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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Adaptive Control versus Fixed Control

System

y(t) = (ap +)y(t) + u(t)

Reference model

yr(t) = aryr(t) + r(t), ar > 0

Ideal controller for = 0

u(t) = ky(t) + r(t), k = ap ar

Ideal performance for = 0

y(t) = ary(t) + r(t), limt

(y(t) yr(t)) = 0

Fixed controller for [1,2]

u(t) = ky(t) + r(t), k< ap 2


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Closed-loop system

y(t) = ary(t) + (ap ++ k+ ar)y(t) + r(t),

e(t) = y(t) yr(t) =ap ++ k+ ar

s + ar

1

s ap k[r](t)

Tracking performance (for r(t) = 1)

ess = limt

e(t) = ap ++ k+ arar(ap ++ k)

Adaptive controller

u(t) = k(t)y(t) + r(t)

k(t) = e(t)y(t), > 0

with k(0) being arbitrary, leading to limt e(t) = 0.

Observation: an adaptive controller ensures desired stability and

tracking, despite any large parameter uncertainty .


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Our Recent Work

G. Tao and P. V. Kokotovic, Adaptive Control of Systems with

Actuator and Sensor Nonlinearities, John Wiley & Sons, 1996.

G. Tao and F. L. Lewis, eds., Adaptive Control of Nonsmooth

Dynamic Systems, Springer, London, 2001.

A. Taware and G. Tao, Control of Sandwich Nonlinear Systems,Springer, Berlin, 2003.

G. Tao, Adaptive Control Design and Analysis, John Wiley & Sons,

Hoboken, New Jersey, 2003.

G. Tao, S. H. Chen, X. D. Tang and S. M. Joshi, Adaptive Control of

Systems with Actuator Failures, Springer, 2004.


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Adaptive Control of Aircraft with Synthetic Jet Actuators


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Synthetic Jets for Aircraft Flight 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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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 32.07560.0199 0.1209 0.1628 0

0.0024 0.0526 0.0252 0

0 1 0.0768 0

, B =

00.0431

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 8.261714.980 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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0 50 100 150 20010

010

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 2: 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/se

c

0 50 100 150 2000.1

0

0.1

deg/sec


0 50 100 150 2001

0.5

0

deg


Figure 3: State tracking errors with a fixed inverse.

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Adaptive Actuator Failure Compensation

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=1bjuj, 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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Block Diagram

Controller System

--

11

..

.

...

-?

1-

-

1m

-- ?-

-

-

m-

ru

m

u1

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) toimprove reliability and survivability.

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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)

ym

(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 90 100

0

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 failures

dr2 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

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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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Research Interests

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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Some On-Going Research Projects

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

Adaptive compensation control for synthetic jet actuators

Adaptive failure compensation for space robots

Adaptive compensation of sensor failures

Adaptive control of spacecraft with fuel slosh.

Documents

Tao Adaptive