Shinji HaraThe University of Tokyo, Japan

When is a Linear System When is a Linear System Easy or Difficult Easy or Difficult

to Control in Practice ?to Control in Practice ?

MTNS’06, Kyoto (July, 2006)

Outline・・ Motivation & Background:Motivation & Background:・・ H2 Tracking Performance Limits:H2 Tracking Performance Limits:

new paradigm

Explicit analytical solutions with examples

・・ Concluding remarksConcluding remarks

Explicit analytical solutions with examples・ H2 Regulation Performance Limits:

・・ Phase Property vs Achievable Robustness Performance

H_inf loop shaping procedure -

Begin with ...

Motivation &Background

-New paradigm on -Control theory -

New Paradigm on Control TheoryNew Paradigm on Control Theory

Finde(t) y(t)r(t)

d(t)

GivenP(s)

GivenP(s)-

u(t)BestK(s)

e(t) y(t)r(t)

d(t)

-BestK(s)BestK(s)

u(t) Desirable

P(s)

Characterize

Assumption： L(s)=P(s)K(s): stable, r.d. >1

Bode Integral RelationBode Integral Relation

0|))(log(|0

=∫∞

ωω djSClosed-loop system: stable

ω

|)(| ωjS

∑ ipπ

)()(11:)(

sKsPsS

+=

・・ Question !Question !Is any stable & MP plant always easy to controlunder physical constraints in practice ?

Characterization of easily controllable plants in practical situations

・・ Aim of researches on control Aim of researches on control perfperf. limits:. limits:

control input energymeasurement accuracy

sampling periodchannel capacity etc.

Answer: NO !Answer: NO !

3-Disk Torsion System

All 3 TFs are marginally stable & MP, but the achievable performances are different.

1k

2k

1J

2J

3J

1c

2c

3c

�����

�����

�����

�����

T

1θ

3θ

2θ

3J

2J

1θ

3θ

2θ

1c

3c

2c

1k

2k

u

1J

-2.5 -2 -1.5 -1 -0.5 0-80

-60

-40

-20

0

20

40

60

80

Re

Im

極disk1の零点disk2の零点

Disk 1 poles

Disk 2

0 0.2 0.4 0.6 0.8 1-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Step Response

Time (sec)

Am

plitude

(a)(b)

0 0.2 0.4 0.6 0.8 1-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Step Response

Time (sec)A

mpl

itude

(a)(b)

Disk1 is better than Disk2. Why ?

Disk1 Disk2

Step responses

・・ Question !Question !Is any stable & MP plant always easy to controlunder physical constraints in practice ?

Characterization of easily controllable plants in practical situations

・・ To provide guidelines of plant design To provide guidelines of plant design from the view point of controlfrom the view point of control

・ From Controller Design to Plant Design

・・ Aim of researches on control Aim of researches on control perfperf. limits:. limits:

New ParadigmNew Paradigm

First topic ...

H2 Tracking Performance Limits

- explicit analytical solutions& applications -

“Best Tracking and Regulation Performance under Control Energy Constraint” by J. Chen, S. Hara & G. Chen, IEEE TAC (2003)

“Optimal Tracking Performance for SIMO Feedback Control Systems: Analytical closed-form expressions and guaranteed accuracy computation

” by S. Hara, M. Kanno & T. Bakhtiar, CDC’06 (submitted)

Control Performance LimitationsControl Performance Limitations

・ Time-response performance・ Tracking performance (H2 norm)

・ H-inf norm performance

・ Bode Integral Relation・ SISO stable/unstable・MIMO ・ Discrete-time/Sampled-data・ Nonlinear

・ Regulation performance (H2 norm)

Special issue in IEEE TAC, Aug. ,2003 Seron et. al. “Fundamental Limitations in Filtering and Control “

HH22 Optimal Tracking ProblemOptimal Tracking Problem

Performance Index:

control effort

unit step input

tracking error

SIMO plant

G（ｓ）w(t)

u(t)y(t)

z(t)

P（ｓ）P（ｓ）

K(s)K(s)

1/s

-

−

−=

)(

)(

/10/1

)(sP

WsP

s

ssG u

Analytic solution

（closed-form）

Riccati & LMI

X1

uW

e(t)

SISO marginally stable plantSISO marginally stable plant

NMP zeros

Plant gain

11 =z

11 −=z1.0

2

*J

)( 21=uW

a

Numerical ExampleNumerical Example

Application to 3Application to 3--disk torsion systemdisk torsion system

uW

Disk 2Disk 3

Disk 1

J*

DiscreteDiscrete--time casetime case

NMP zeros

Plant gain

Delta Operator

Continuous-time result

General SIMO Case General SIMO Case Numerator:

Unstable poles & NMP zeros:

Stable terms:NMP zeros

Plant gain

Unstable terms: Unstable poles

Unstable pole / NMP zeros

RemarksRemarks::Several cases where the computation of

・ SIMO marginally stable ・ SISO non control input penalty・ SIMO

・ SIMO unstable: common unstable poles：Jcu=0many applications

is not required.

Optimal length of Inv. Pend. ?

0 0.5 1 1.5 23

3.5

4

4.5

5

5.5

6

l (m)

J* c2

Tracking performance limit

DiscreteDiscrete--time casetime case

NMP zeros

Plant gain

Delta Operator

Continuous-time result

Second topic ...H2 Regulation

Performance Limits-explicit analytical solutions

& an application -

“H2 Regulation Performance Limits for SIMO Feedback Control Systems” by T.Bakhtiar & S.Hara, MTNS’06

“Best Tracking and Regulation Performance under Control Energy Constraint” by J.Chen, S.Hara & G.Chen, IEEE TAC (2003)

HH22 Optimal Regulation ProblemOptimal Regulation Problem

Performance Index :

Impulseinput

SIMO plant

control effort performance on disturbance rejection

SISO MP plantSISO MP plant

unstable poles

Plant gain

Numerical ExampleNumerical Example

−1 0 1 2 3 4 50

500

1000

1500

2000

2500

3000

3500

p

Ec*

via Theorem 1via Toolbox

p

*cE

SIMO NMP plantSIMO NMP plant

MP case

CommonNMP zeros

Application to Application to a Magnetic Bearing Systema Magnetic Bearing System

Normalized state-space equation:

one unstable pole at p・current sensor:

・position sensor:

・multiple sensors:

NMPNMP

MPMP

MPMP

SISO MP discreteSISO MP discrete--time planttime plant: : r.d.=1

Delta Operator

Continuous-time result

Magnetic bearing systemMagnetic bearing system::

caused by discretizedNMP zeros

Last topic ...

Phase Property vs Achievable Robustness Performance

- H_inf loop shaping procedure -

“Finite Frequency Phase Property Versus Achievable Control Performance in H_inf Loop Shaping Design” by S. Hara, M. Kanno & M. Onishi,

SICE-ICCAS’06 (to be presented)

“Dynamical System Design from a Control Perspective: Finite frequency positive-realness approach” by T. Iwasaki & S. Hara, IEEE TAC (2003)

FFPRFFPR (Finite Frequency Positive Realness)

ー Finite Frequency Positive Realness ー

+<

−

DDBB

CIA

CIA

TT

T 00 0 　　

　

XY

Y 20ω

X

,)(

= DC

BAsG

(LMI condition)

≤>+ ∀ || ,0)()( * ωωω jGjG 0ω

00 >ωgiven

..,0 tsYYXX TT =>=∃

HHinfinf LSDPLSDP(Hinf Loop-Shaping Design Procedure)

Good Phase Property

2nd order plant

Characterization of good plants

10-2

10-1

100

101

102

103

-60

-40

-20

0

20

40

60Bode Diagram

Magnitude (dB)

10-2

10-1

100

101

102

103

-270

-180

-90

0

90

Frequency (rad/sec)

Phase (deg)

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5Nyquist Diagram

Real Axis

Imaginary Axis

Numerical ExampleNumerical ExampleP(s)

L(s)=P(s)K(s)K(s)

Nyquist plotsBode diagrams

Concluding RemarksConcluding Remarks

・・ H2 tracking performance limitsH2 tracking performance limitsExplicit analytical solutions for

・・ H2 regulation performance limitsH2 regulation performance limits

Characterizations of easily controllable plants in practical situations, which provide guidelines of plant design from the view point of control

Finite frequency phase property vs achievable robustness performance in H_inf LSDP