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08 T

he M

athW

orks

, Inc

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Model-Based Design—Unpolished

591 /* Logic: ' <S3>/either ' */592 rtb_either = power_window_con_B.passenger_control_b 593 || power_window_con_B.passenger_control_a; 594 595 /* Logic: ' <S13>/allow_action ' incorporates:596 * Inport: ' <Root>/driver_up '597 * Logic: ' <S13>/overrule '598 */599 rtb_temp34 = power_window_con_U.driver_up 600 && (!(rtb_either));

Pieter J. Mostermanpieter.mosterman@mathworks.com

Senior Research ScientistThe MathWorks, Inc.

2

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Agenda

� Requirements engineering� Feedback control

� Control law design� Evaluation criteria

� Adaptive control� Parameter adaptation� Switched control

� Hierarchical control� Design

� Functional control design flow� Implementation� Platform based design

� Wrap up

2

3

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Elevator redundancy management

PFCU1

PFCU2

LOactuator

LIactuator

Hydraulics 2

LDL RDL

LIO RIO

Hydraulics 1

Elevator

RIactuator

ROactuator

Hydraulics 3

Elevator

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Reconfiguration requirements

If multiple failures are detected in left hydraulic pressure and actuator positions, isolate the fault by switching both the Left Outer actuator and the Left Inner actuator to the isolated mode. The following combinations trigger this condition:

• Failures in hydraulic pressure 1, and hydraulic pressure 2

• Failures in Left Outer actuator position sensor, and Left Inner actuator position sensor

� Failures in hydraulic pressure 1, and Left Inner actuator position sensor

• Failures in hydraulic pressure 2, and Left Outer actuator position sensor

2.1.15 Multiple failures on Left hydraulic and actuators

If multiple failures are detected in hydraulic pressure 1 or 2 systems and either the Left Outer actuator position sensor or the Left Inner actuator position sensor, isolate the fault by switching both the Left Outer actuator and the Left Inner Actuator to the isolate mode.

2.1.15 Multiple failures on Left hydraulics and actuators

If a failure is detected in both the hydraulic pressure 2 system and the Left Inner actuator position sensor, while there are no other failures, isolate the fault by switching the Left Inner actuator to the off mode.

2.1.12 Hydraulic pressure 2 and Left Inner actuator position failures

If a failure is detected in both the hydraulic pressure 1 system and the Left Outer actuator position sensor, while there are no other failures, isolate the fault by switching the Left Outer actuator to the off mode.

2.1.11 Hydraulic pressure 1 and Left Outer actuator position failures

reengineer

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Requirements traceability

2.1.5 Hydraulic pressure 3 failureIf a failure is detected in the hydraulic pressure 3 system, while there are no other failures, isolate the fault by switching the Right Outer actuator to the off mode.

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Agenda

� Requirements engineering� Feedback control

� Control law design� Evaluation criteria

� Adaptive control� Parameter adaptation� Switched control

� Hierarchical control� Design

� Functional control design flow� Implementation� Platform based design

� Wrap up

4

7

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Control design architectures

controller plant

controller plantyur

controller plant

∆

yur e

yur e

en

plant

controller

environment

context

Feedforward control (experiment in pool room)

Feedback control (experiment in wine cellar)

Robust feedback control

system

reference

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A hydraulic cylinder

Rvalve

vpiston

ppump

Ccylinder

mpiston

Rcylinder

inherently (open loop) stable

second order model of dynamics

ideal picture model

5

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Feedback control system

rplant

pressure displacement

controller

error pressure

Scope

)()()(

)()()( sU

sGsHI

sGsHsY

+=

)(sH)(sG

pKsG =)(

s

KsKKs

sKs

KKsG

ipd

di

p

++=

++=

2

)(

straightforward:

position control

performance:

position-integral-derivative (PID) control

closed loop analysis

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Compensator pole placement

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+2%

-2%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Performance criteria

rise time

overshoot

settling time

10%

90%

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Stability classes

Lyapunov stable and

))()0()(0)(0)(0( εδδε <⇒<≥∀>∃>∀ txxt

)0)(lim)0()(0( =⇒<>∃∞→

txxt

δδ

))0()()0()(0)(0,,( textxxt βαδδβα −≤⇒<≥∀>∃

Lyapunov stable

Asymptotically stable

Exponentially stable

7

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Robustness analysis

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Agenda

� Requirements engineering� Feedback control

� Control law design� Evaluation criteria

� Adaptive control� Parameter adaptation� Switched control

� Hierarchical control� Design

� Functional control design flow� Implementation� Platform based design

� Wrap up

8

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Adaptive control

controller plant

identification

yur

controller plant

adjustment

yur

referencemodel

2

2

1)( eJ =θ

θγ

θγ

θγθ

∂∂−=

∂∂

∂∂−=

∂∂−= e

ee

e

JJ

dt

d

model identification adaptive control

model reference adaptive control

my

θ

performance criterion:

adaptation strategy:

process

adapt: slow

feedback: fast

Principle

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First order system

buaydt

dy +−=

cmmmm ubya

dt

dy +−=

)()()( 21 tytutu c θθ −=

b

aab

b

m

m

−==

==

022

011

θθ

θθ

myye −= cubas

by

2

1

θθ++

=

ybas

bu

bas

be

ubas

be

c

c

22

2

12

2

21

)( θθθ

θ

θθ

++−=

++−=

∂∂

++=

∂∂

masbas +≈++ 2θ

eyas

a

dt

d

euas

a

dt

d

m

m

cm

m

+=

+−=

γθ

γθ

2

1

plant

reference model

controller

ideal parameters

error

approximate

adaptation

reference model

uc ym

rplant

u y

monitor

control

th2

th1

uc

y

u

adaptation

uc

ym

y

th1

th2

25.0

12

45.0

2

2

1

=−=

==

θ

θ

9

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Switched control

control1: stable

plantyur e

supervisor chooses control law

control1

control2

control2: stable switched control: unstable

supervisor

the act of switching may cause unstable compound behavior

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Agenda

� Requirements engineering� Feedback control

� Control law design� Evaluation criteria

� Adaptive control� Parameter adaptation� Switched control

� Hierarchical control� Design

� Functional control design flow� Implementation� Platform based design

� Wrap up

10

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A B

S P T

Hierarchy in control

man machine interface

supervisory control

control law

plant

data

analysis

data

analysis

sensor actuator

PFCU1

PFCU2

LOactuator

LIactuator

Hydraulics 2

LDL RDL

LIO RIO

Hydraulics 1

Elevator

RIactuator

ROactuator

Hydraulics 3

Elevator

deflection

valve

opening

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Agenda

� Requirements engineering� Feedback control

� Control law design� Evaluation criteria

� Adaptive control� Parameter adaptation� Switched control

� Hierarchical control� Design

� Functional control design flow� Implementation� Platform based design

� Wrap up

11

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Control synthesis

),,()(

),,(0

),,()(

tuxhty

tuxg

tuxftx

==

=&

)()( txKtu iii =

+=+=

+=+=

)()()(

)()()(

)()()(

)()()(

00000

00000

tuDtxCty

tuBtxAtx

tuDtxCty

tuBtxAtx

nnnnn

nnnnn&

M

&

−=−=

)()()(

)()()(

txKDCty

txKBAtx

iiiii

iiiii&

Rvalve

vpiston

ppump

Ccylinder

mpiston

Rcylinder

model phenomena of interest

derive representation amenable to synthesis

design control law

validate control

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Functional control design stages

+=+=

)()()(

)()()(

tDutCxty

tButAxtx&

−−

−−+

−−

−−

11

11

)2

(2

)2

(

)2

(4

)2

)(2

(

AT

IDT

AT

ICT

AT

IBTT

kA

TIA

TI

+=

+=+

)(~

)(~

)(

)(~

)(~

)( 1

kkk

kkk

tuDtxCty

tuBtxAtx

discretize

0.0 …

output:

update: Du…

FoEoDoAoBo FoEoAo

0.0 0.1 0.2 …

output:

update: FuEu

FoEoAo

…FuEu

1 se

con

dbl

ocks

0.1

seco

nd

bloc

ks

FuEu

Co

task assignment

2.1.15 Multiple failures on Left hydraulics and actuatorsIf multiple failures are detected in hydraulic pressure 1 or 2 systems and either the Left Outer actuator position sensor or the Left Inner actuator position sensor, isolate the fault by switching both the Left Outer actuator and the Left Inner Actuator to the isolate mode.

verif

y

validate

12

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Specifications become requirements

2.1.15 Multiple failures on Left hydraulics and act uatorsIf multiple failures are detected in hydraulic pressure 1 or 2 systems andeither the Left Outer actuator position sensor or the Left Inner actuator position sensor, isolate the fault by switching both the Left Outer actuator and the Left Inner Actuator to the isolate mode.

PFCU1

PFCU2

LOactuator

LIactuator

Hydraulics 2

LDL RDL

LIO RIO

Hydraulics 1

Elevator

RIactuator

ROactuator Hydraulics 3

Elevator

plantyur e

control1

control2

supervisor

+=+=

)()()(

)()()(

tuDtxCty

tuBtxAtx

iiiii

iiiii&

+=

+=+

)(~

)(~

)(

)(~

)(~

)( 1

kkk

kkk

tuDtxCty

tuBtxAtx

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® ®

Agenda

� Requirements engineering� Feedback control

� Control law design� Evaluation criteria

� Adaptive control� Parameter adaptation� Switched control

� Hierarchical control� Design

� Functional control design flow� Implementation� Platform based design

� Wrap up

13

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What can we learn?

� Standard input performance excitation� Step input

� Standard architectures and terminology� Standard design representations

� Root locus� Bode plots� Practitioners can relate to theory (relatively) easy

� Well-defined analyses� Stability� Robustness

� Focused design stages� Strict separation

� …

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Wrap up

� Offline design is very involved� Some aspects highlit

� Adaptation does some design online� Which aspects of design can be automated?

� To be continued …