Analysis and Design of Linear Control System –Part2-

Spring, 2015

Schedule: 9, 16, 23, 30 June7, 14, 21 July

Instructor: Assoc. Prof. Takeshi Hatanaka (S5-204B)

)(sF

Basic Feedback Loop

: Process output: Measured signal

: Measurement noise

: Load disturbance : Control variable: Reference signal

: Feedback block: Process Fig. 11.1

: Output errorrdn

euηy

)(sC)(sPController Process

nd

ur

y

yη)(sC )(sPe ν

∑∑∑

)(sF( : Feedforward block)

1−1=F

Control System Transfer Functions

nduPnP ++=+= )(ν=y n+η

nPdPCryPC ++=+ )1( nPC

dPC

PrPC

PCy+

++

++

=1

111

nPdPCe ++=

+−

+−

+

+−

+−

+

+−

++

+−

++

+++

=

ndr

PCPCP

PC

PCC

PCPC

PCC

PCC

PCPCC

PCPC

PCP

PCPC

PCPCP

PCPC

eu

y

11

111

111

111

1

111

11

11

νη

(11.1)

Fig. 11.1Controller Process

nd

ur

y

yη)(sC )(sPe ν

∑∑∑

1−1=F

nPdyrPC ++−= )(

Gang of Four

Gang of FourComplementarySensitivityFunction

LoadSensitivityFunction

NoiseSensitivityFunction

SensitivityFunction

(11.3)

PCS

d +=

→ 11

)( ν PCPCT

r +=

→ 1)( η

PCPPS

d +=

→ 1)( η PCCCS

un +=

→ 1)(

Fig. 11.1

Controller Process

nd

ur

y

yη)(sC )(sPe ν

∑∑∑

1−

General Representation* General representation

Fig. 11.2

u y

=

rnd

w

=

e

vz η

Fig. 11.1Controller Process

nd

ur

y

yη)(sC )(sPe ν

∑∑∑zw

P

C

C=C

−−−=

PPPPP

1100

1001

Fig. 11.2

)(sP ∑∑

)(sC

∑

)( 1wd )( 1zν )( 2zη )( 2wn

)( 3wr)( 3ze

y

u

1−1=F

[Ex. 11.1]

Process

ControllerStable ?

Fig. 11.1

nd

ur yη)(sC )(sP∑ ∑ ∑

1−

sasksC )()( −

=

0>k

assP

−=

1)( ?=yrG

0 , )()( >+

== kks

ksTsGyr

e ν

yr →

Step response

Complementary Sensitivity Function

Load Sensitivity Function

Sensitivity Function

Noise Sensitivity Function

can be unstable

Gang of Four

1−=a

1=aunstable

ksssS+

=)(ks

ksT+

=)(

))(()(

asksssPS

−+=

ksasksCS

+−

=)()(

TS

PS

CS

Frequency response

S T

PS CS

Internal Stability

All of the “Gang of Four” are stable. Internal Stability :

ComplementarySensitivityFunction

LoadSensitivityFunction

NoiseSensitivityFunction

SensitivityFunction

(11.3)Well-posed: 0)()(1 ≠∞∞+ CP

Fig. 11.1

nd

ur yη)(sC )(sP∑ ∑ ∑

PCS

d +=

→ 11

)( ν PCPCT

r +=

→ 1)( η

PCPPS

d +=

→ 1)( η PCCCS

un +=

→ 1)(

1−

e ν

Youla Parameterization (stable process) (§12.2)

“All” stabilizing controllers

(12.8)

: Stable process: Stable transfer

function (parameter)

PQQC−

=1

Fig.12.8 (a)

)(sP∑

∑)(sP−

)(sQ

1−

Gang of FourAll of these 4 transfer functions are stable.

All stabilizing controllers areparameterized by Q

IMC : Internal Model Control

PQS −=1)1( PQPPS −=

QCS =PQT =

)(sP)(sQ

: Stable

Youla Parameterization (unstable process)

: Stable

)()()(

sBsAsP =

Coprime Factorization

)(),( sBsA

100 =+ BGAF )(, )( 00 sGsF

)2)(1(1)(

−−=

sssP

22 )1()2)(1()(,

)1(1)(

+−−

=+

=s

sssBs

sA[Ex.]

[Ex.]

16)(,

11119)( 00 +

+=

+−

=sssG

sssF

22 )1()2)(1()(,

)1(1)(

+−−

=+

=s

sssBs

sA

Youla Parameterization (unstable process)

Gang of Four

: Stable : “All” stabilizingcontrollers(12.9)

[Ex.]

QAGQBFC

−+

=0

0

)0( 61119)( =

+−

= QsssC)2)(1(

1)(−−

=ss

sP

00

0 )(BGAFQAGBS

+−

=00

0 )(BGAFQBFAT

++

=

00

0 )(BGAFQAGAPS

+−

=00

0 )(BGAFQBFBCS

++

=

)(sQ

Exercise: Internal Stability

1st Lecture

Basic Feedback Loop, Gang of FourInternal Stability, Youla Parameterization

Keyword :

11 Frequency Domain Design

11.1 Sensitivity Functions(12.2 Youla Parameterization)

Sensitivity FunctionKeyword :

11.3 Performance Specifications(12.3 Performance in the Presence of Uncertainty)

(11.5 Fundamental Limitation)

(pp. 315--319)

(pp. 352--358)

(pp. 322--326)

(pp. 358--361)

(pp. 331--340)

Disturbance Attenuation ~ Open Loop vs. Closed Loop ~ Open Loop

Closed Loop(feedback)

Disturbances are attenuated

The sensitivity function tells how the variations in the outputs are influenced by feedback

yd →PGyd =

yd →

?=ydGPC

PGyd +=

1PS=

)( ωjS

1)( <ωjS

11.3 Performance Specifications

Process Variations (§12.3)

The response to load disturbances is insensitive to process variations for frequencies where is small

HW

yd →

)( ωjS

PSPCPGyd =+

=1 P

dPSGdG

yd

yd = (12.11)

Fig. 11.1

ur yη)(sC )(sPe ν

∑∑∑

1−

d ndP

HW

yr →

TPC

PCGyr =+

=1 (12.15)

• Insensitivity to Plant Variations

Benefits of Feedback• Disturbance Attenuation

Fig. 11.1

ur yη)(sC )(sPe ν

∑∑∑

1−

d n

• Stabilization

PdPS

GdG

yr

yr =

Process Variations (§12.3)

dP

Waterbed Effect (§11.5)

… Waterbed Effect

If the loop transfer function has no right half-plane poles…

… Conservation Law

There exists a range of freq.such that

SS ↔> 0log SS ↔< 0log

1>S

1> 1<

0)(log

0 =∫

∞ωω djS

Assume that the loop transfer function of a feedback system goes to zero faster than as , and let be the sensitivity function. If the loop transfer function has poles in the right half-plane, then the sensitivity function satisfies the following integral:

Bode’s Integral Formula (§11.5)

Theorem 11.1 Bode’s Integral Formula

(11.19)∑∫∫ =

+=

∞∞

kpdjL

djS πωω

ωω

0

0 )(11log )(log

s/1)(sL∞→s )(sS

kp

Fundamental Limitation

Re

Im

××Pole

slow fastRHP poles fast (big): worse

slow (small): better

[Ex.]

(unstable)

: Sensitivity Bandwidth Frequency

: Maximum Steady-state Tracking Error Type 1:

Sensitivity function )( ωjS

• Insensitivity to Plant Variations

Benefits of Feedback

• Disturbance Attenuation

])dB[ 3( 2

1)( −=ωjS

0=AA

bsω Absω

2<sM )( ωjS

: Maximum Peak Magnitude of )( ωjS

2<sM)(max ωω

jSM s =

sM

[dB]

: maximum sensitivity

: stability margin

: the shortest distancefrom the Nyquist curveto the critical point (-1)

1)( <ωjS

+=

LS

11 1)(1 >+↔ ωjL

ms

ms

ms sM /1=

8.05.0 << ms

Sensitivity function )( ωjS

2<sM

Fig. 11.6(b)

1st Lecture

Basic Feedback Loop, Gang of FourInternal Stability, Youla Parameterization

Keyword :

11 Frequency Domain Design

11.1 Sensitivity Functions(12.2 Youla Parameterization)

Sensitivity FunctionKeyword :

11.3 Performance Specifications(12.3 Performance in the Presence of Uncertainty)

(11.5 Fundamental Limitation)

(pp. 315--319)

(pp. 352--358)

(pp. 322--326)

(pp. 358--361)

(pp. 331--340)

12.1 Modeling Uncertainty

9.3 Stability Margins

Reading Assignment: 2nd Lecture

Modeling Uncertainty, Stability MarginRobust Stability

Keyword :

12 Robust Performance

(12.3 Performance in the Presence of Uncertainty)

Complementary Sensitivity FunctionSmall Gain Theorem

Keyword :

12.2 Stability in the Presence of Uncertainty

(11.5 Fundamental Limitation)

(9.2 The Nyquist Criterion)

12.2 Stability in the Presence of Uncertainty

(pp. 347--352)

(pp. 278--282)(pp. 352--358)

(pp. 358--361)(pp. 331--340)

(pp. 270--278)