Analysis and Design of Linear Control System Part2-...Bode diagram Fig. 12.6: PI controller ( ) 1 (...

Analysis and Design of Linear

Control System –Part2-

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

Spring, 2015

12.1 Modeling Uncertainty

9.3 Stability Margins

2nd Lecture

Modeling Uncertainty, Stability Margin

Robust Stability

Keyword :

12 Robust Performance

(12.3 Performance in the Presence of Uncertainty)

Complementary Sensitivity Function

Small Gain Theorem

Keyword :

12.2 Stability in the Presence of Uncertainty

(11.5 Fundamental Limitation)

(9.2 The Nyquist Criterion)


(pp. 347--352)

(pp. 278--282)

(pp. 352--358)

(pp. 358--361)

(pp. 331--340)

(pp. 270--278)


• Parametric uncertainty

parameters describing the system are unknown

Mass of a car changes with the number of passengers

Fig. 3.3Disturbance response

(3rd) (4th) (5th)

mass

gear ratio

20001600 m

14 ,12 ,10 speed 4010 ev

3

Modeling Uncertainty

• Parametric uncertainty

parameters describing the system are unknown

• Unmodeled dynamics

neglected mechanisms such that the simple

model does not include.

• detailed model of the engine dynamics

• slight delays that can occur in electronically

controlled engines

The design based on a simple nominal

model will give satisfactory control. Ex. 12.1

4

122 2

2.05.0~

i iii ss

s

sP

Unmodeled Dynamics

Vibration mode

Frequency response

Nominal System

(Rigid body mode)

vibration modeUnmodeled dynamics

http://mobile.jaxa.jp/gallery_list/

index.php?category=iss

P~

P

P

sP

5.0

P

02.0,10

2,5.0,2.0

4

321

i

Unmodeled Dynamics

Additive perturbations Multiplicative perturbations

: nominal model

: unmodeled

dynamics

: actual model

PP~

P

PP )1(~

P

P/:

PP~

,

Frequency response

P~

P

[Ex. 12.2] Similar in Open Loop but Large Differences in Closed Loop

(a) Step response (open loop) (b) Step response (closed loop)

open loop

Fig. 12.3(a)

(12.1)

[When Are Two Systems Similar ?]

1)(1

s

ksP

22 )1)(1()(

sTs

ksP 025.0T

100kur y

)(sC )(sPe

1

r y)(sP 1)( sC

closed loop

[Ex. 12.3] Different in Open Loop but Similar in Closed Loop

(a) Step response (open loop) (b) Step response (closed loop)

open loop

Fig. 12.3(b)

(12.2)

[When Are Two Systems Similar ?]

closed loop

1)(1

s

ksP 100k

)1()(2

s

ksP

r y)(sP

ur y)(sC )(sP

e

1

1)( sC

Nyquist Criterion (§9.2)

Re

1

O

Im

)( jL

Fig. 11.1

ur y)(sC )(sP

e

1

d n

PCL

Loop Transfer Function

Nyquist’s Stability Theorem

Theorem 9.1 and 9.2

)( jL

Stability Margin (§9.3)

Re

1

OP

G

Im

m

Fig. 9.9 (a)

mg/1

0 dB

180

gc

pcPhase

Gain

mg1 02 0 lo g

m

Fig. 9.9 (b)

)(/1 pcm iLg

Gain Margin

(9.5) 52mg

Phase Margin

)(a r g g cm iL (9.6) 6 03 0m

Stability Margin ms

8.05.0 mssm Ms /1

( : maximum sensitivity)ms sM /1

( 6-14 dB)

Fig. 11.6(b)

Robust Stability Using Nyquist’s Criterion

Additive Uncertainty

: stable perturbations

Perturbed Nyquist curve

doesn’t reach -1 (i.e. perturbed

closed loop is stable) when

Fig. 12.5(b)

Nyquist plot (additive uncertainty)

PP

PCL

CPCCP )(

LC 1C

Im

ReO

Loop Transfer Function

CL

P

C

Fig. 12.5(b)

C


Perturbed Nyquist curve doesn’t reach -1

when

LC 1C

PC

1

CS

1

PCS

1

1

TPC

PC

P

11

PC

PCT

1

P/:

Im

ReO

Additive Uncertainty

Multiplicative Uncertainty

P

C

P

C


small big is allowed

big small is allowed

)( jT

)( jT

)( j

)( j

Sufficient condition for robust stability

(12.6)0

)(

1)(

jTj

Uncertainty

P

C

HW

un

P

T

PC

CGun

1 P

dPT

G

dG

un

un (12.13)

Performance in the Presence of Uncertainty (§12.3)

Fig. 11.1

ur y)(sC )(sP

e

1

d n

Measurement noise typically has high frequencies

dP

noise

Bode’s Integral Formula (§11.5)

Complementary sensitivity function:

where the summation is over all right

half-plane zeros.

(11.20)

(11.19)

Re

Im

Zero

slow fast

izd

jT 1)(log

0 2

kpdjS

0 )(log

0||log T1|| T

T

1|| T

0||log T

RHP zeros fast (big): better

slow (small): worse

Sensitivity function: (1st lecture)

Re

Im

××Pole

slow fast


)()(1

)()()(

sCsP

sCsPsT

1

11

1

PC

PC

PCTS

: Complementary Sensitivity

Bandwidth Frequency bT

])dB[3( 2

1)( jT

])dB[2( 25.1TM

)( jT

]dB[3bT

: Maximum Peak

Magnitude of

)(max

jTMT

])dB[ 2( 25.1TM

)( jTTM

Robust Stability

(sufficient condition)

Fig. 12.6Bode diagram

: PI controller

)(

1)(

jTj

0142.0

38.1)(

ssP

ssC

18.072.0)(

[Ex. 12.5] Cruise Control

40

20

0

-20

-40 210 110 010 110 210

should be smaller than

most sensitive

T

13.1TM

T/1

T/135.0mt

21.1bT

around the gain crossover frequencies

small is required

Fig. 12.6 ’Bode diagram

A simple model that describes

the process dynamics well

around the crossover frequency

is often sufficient for design

Robust Stability (sufficient condition)

1

T

)( j

T

T/1

[Ex. 12.5] Cruise Control

Robust Stability Using Small Gain Theorem

sufficient condition for robust stability

(12.6)

another interpretation by using small gain theorem

Theorem 9.4 Small Gain Theorem (§9.5)

Consider the closed loop system shown in Fig. 9.15,

where and are stable systems and the signal

spaces are properly defined. Let the gains of the

systems and be and . Then the closed

loop system is input/output stable if .

Fig. 9.15

1H 2H

1H 2H 1 2

121

2H

1H

0 )(

1)(

jT

j

Robust Stability Using Small Gain Theorem

sufficient condition for robust stability

Small Gain Theorem

Closed loop system is input/output stable

Fig. 12.7

P

C

1

T

1 T

T

PC

PCT

1

for small

for large

: with integral action

Load Sensitivity Function* (§11.3)

(11.8, 11.9)C

TPS

PC

PGyd

1

)(sCs

ki

Fig. 11.1

ur y)(sC )(sP

e

1

d n

C

1

1

1

sP

P

iyd k

sCC

TG 1 )1( T　

PPSGyd )1( S　

10-2 10-1 100 101 102

for small

for large

Noise Sensitivity Function* (§11.3)

(11.10)

PID + 2nd-order

noise filter

Roll off

P

TCS

PC

CGun

1

Fig. 11.1

ur y)(sC )(sP

e

1

d n

C

P

1

PPTGun

1 )1( T　

CCSGun )1( S　

3)1(1

s

P

12/1

22

sTsTsk

s

kkC

ff

di

p

10-1 100 101 102


9.3 Stability Margins

2nd Lecture

Modeling Uncertainty, Stability Margin

Robust Stability

Keyword :

12 Robust Performance

(12.3 Performance in the Presence of Uncertainty)


Small Gain Theorem

Keyword :


(11.5 Fundamental Limitation)

(9.2 The Nyquist Criterion)


(pp. 347--352)

(pp. 278--282)

(pp. 352--358)

(pp. 358--361)

(pp. 331--340)

(pp. 270--278)

Reading Assignment: 3rd Lecture

11.4 Feedback Design via Loop Shaping

Loop Shaping

Bode’s Relations

Keyword :

11 Frequency Domain Design

(9.4 Bode’s Relations and Minimum Phase Systems)

11.5 Fundamental Limitations

Right Half-Plane Poles and Zeros

Gain Crossover Frequency Inequality

Keyword :

(pp. 326--331)

(pp. 283--285)

(pp. 331--340)

Analysis and Design of Linear Control System Part2-...Bode diagram Fig. 12.6: PI controller ( ) 1 (...

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