SLIDE 1 Shinji Hara
The University of Tokyo, Japan
W h e n i s a L i n e a r S y s t e m W h e n i s a L i n e a r S y s t e m E a s y
D i f f i c u l t E a s y
D i f f i c u l t t
t r
i n P r a c t i c e ? t
t r
i n P r a c t i c e ?
MTNS’06, Kyoto (July, 2006)
SLIDE 2
Outline
・ ・ Motivation & Background: Motivation & Background: ・ ・ H2 Tracking Performance Limits: H2 Tracking Performance Limits:
new paradigm Explicit analytical solutions with examples
・ ・ Concluding remarks Concluding remarks
Explicit analytical solutions with examples
・ H2 Regulation Performance Limits: ・ ・ Phase Property vs Achievable Robustness Performance
H_inf loop shaping procedure -
SLIDE 3 B e g i n w i t h . . . M
i v a t i
& B a c k g r
n d
e w p a r a d i g m
t r
t h e
y
SLIDE 4 New Paradigm on Control Theory New Paradigm on Control Theory
Find e(t) y(t) r(t) d(t) Given
P(s)
Given
P(s)
Best
K(s)
e(t) y(t) r(t) d(t)
K(s)
Best
K(s)
u(t)
Desirable
P(s)
Characterize
SLIDE 5 Assumption： L(s)=P(s)K(s): stable, r.d. >1 Bode I ntegral Relation Bode I ntegral Relation
|) ) ( log(| =
∫
∞
ω ω d j S
Closed-loop system: stable
ω
| ) ( | ω j S
∑
i
p π
) ( ) ( 1 1 : ) ( s K s P s S + =
SLIDE 6 ・ ・ Question ! Question !
Is any stable & MP plant always easy to control under physical constraints in practice ? Characterization of easily controllable plants in practical situations
・ ・ Aim of researches on control Aim of researches on control perf
. limits:
control input energy measurement accuracy sampling period channel capacity etc.
Answer: NO !
SLIDE 7 3-Disk Torsion System
All 3 TFs are marginally stable & MP, but the achievable performances are different.
1
k
2
k
1
J
2
J
3
J
1
c
2
c
3
c
1
θ
3
θ
2
θ
3
J
2
J
1
θ
3
θ
2
θ
1
c
3
c
2
c
1
k
2
k
u
1
J
. 5
. 5
5
2 4 6 8 R e I m 極 d i s k 1 の零点 d i s k 2 の零点
Disk 1 poles Disk 2
SLIDE 8 . 2 . 4 . 6 . 8 1
6
4
2 . 2 . 4 . 6 . 8 1 S t e p R e s p
s e T i m e ( s e c ) A m p l i t u d e ( a ) ( b ) . 2 . 4 . 6 . 8 1
6
4
2 . 2 . 4 . 6 . 8 1 S t e p R e s p
s e T i m e ( s e c ) A m p l i t u d e ( a ) ( b )
Disk1 is better than Disk2. Why ?
Disk1 Disk2
Step responses
SLIDE 9 ・ ・ Question ! Question !
Is any stable & MP plant always easy to control under 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 control from the view point of control ・ From Controller Design to Plant Design ・ ・ Aim of researches on control Aim of researches on control perf
. limits:
New Paradigm New Paradigm
SLIDE 10 F i r s t t
i c . . .
H 2 T r a c k i n g P e r f
m a n c e L i m i t s
x p l i c i t a n a l y t i c a l s
u t i
s & a p p l i c a t i
s
- “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)
SLIDE 11 Control Performance Limitations Control 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 “
SLIDE 12 H H 2
2 Optim al Tracking Problem
Optim al Tracking Problem Performance Index:
control effort unit step input tracking error SIMO plant
SLIDE 13 G（ ｓ ）
w(t) u(t) y(t) z(t)
P（
ｓ ）
P（
ｓ ）
K(s) K(s)
1/s
− − = ) ( ) ( / 1 / 1 ) ( s P W s P s s s G
u
Analytic solution （ closed-form） Riccati & LMI
X
1
u
W
e(t)
SLIDE 14
SISO marginally stable plant SISO marginally stable plant
NMP zeros Plant gain
SLIDE 15 1
1 =
z
1
1
− = z
1.0
2
*
J
) (
2 1
=
u
W
a
Numerical Example Numerical Example
SLIDE 16 Application to 3 Application to 3-
disk torsion system
u
W
Disk 2 Disk 3 Disk 1 J
*
SLIDE 17 Discrete Discrete-
time case
NMP zeros Plant gain
Delta Operator
Continuous-time result
SLIDE 18
General SI MO Case General SI MO Case Numerator: Unstable poles & NMP zeros:
SLIDE 19
Stable terms:
NMP zeros Plant gain
SLIDE 20
Unstable terms:
Unstable poles Unstable pole / NMP zeros
SLIDE 21
Rem arks Rem arks: : Several cases where the computation of
・
SIMO marginally stable
・
SISO non control input penalty
・
SIMO
・
SIMO unstable: common unstable poles： Jcu=0 many applications is not required.
SLIDE 22
Optimal length of Inv. Pend. ?
SLIDE 23 0.5 1 1.5 2 3 3.5 4 4.5 5 5.5 6 l (m) J*
c2
Tracking performance limit
SLIDE 24 Discrete Discrete-
time case
NMP zeros Plant gain
Delta Operator
Continuous-time result
SLIDE 25 S e c
d t
i c . . .
H 2 R e g u l a t i
P e r f
m a n c e L i m i t s
x p l i c i t a n a l y t i c a l s
u t i
s & a n a p p l i c a t i
- n
- “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)
SLIDE 26 H H 2
2 Optim al Regulation Problem
Optim al Regulation Problem Performance Index : Impulse input SIMO plant control effort performance on disturbance rejection
SLIDE 27
SISO MP plant SISO MP plant
unstable poles Plant gain
SLIDE 28 Numerical Example Numerical Example
−1 1 2 3 4 5 500 1000 1500 2000 2500 3000 3500 p Ec* via Theorem 1 via Toolbox
p
* c
E
SLIDE 29
SIMO NMP plant SIMO NMP plant
MP case Common NMP zeros
SLIDE 30 Application to Application to a Magnetic Bearing System a Magnetic Bearing System
Normalized state-space equation:
SLIDE 31
・ current sensor: ・ position sensor: ・ multiple sensors: NMP NMP MP MP MP MP
SLIDE 32
SLIDE 33 SISO MP discrete SISO MP discrete-
time plant: : r.d.=1
Delta Operator
Continuous-time result
SLIDE 34
Magnetic bearing system Magnetic bearing system: :
caused by discretized NMP zeros
SLIDE 35 L a s t t
i c . . .
P h a s e P r
e r t y v s A c h i e v a b l e R
u s t n e s s P e r f
m a n c e
_ i n f l
s h a p i n g p r
e d u r e
- “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)
SLIDE 36
FFPR FFPR (Finite Frequency Positive Realness)
SLIDE 37 ー Finite Frequency Positive Realness ー
+ < − D D B B C I A C I A
T T T
X Y Y
2
ω X
, ) ( = D C B A s G
(LMI condition)
≤ > +
∀
| | , ) ( ) (
*
ω ω ω j G j G ω
0 >
ω
given
. . , t s Y Y X X
T T
= > =
∃
SLIDE 38
H Hinf
inf LSDP
LSDP
(Hinf Loop-Shaping Design Procedure)
SLIDE 39
Good Phase Property
SLIDE 40
2nd order plant
SLIDE 41
Characterization of good plants
SLIDE 42 1
1
1 1
1
1
2
1
3
2 4 6
B
e D i a g r a m M a g n i t u d e ( d B )
1
1
1 1
1
1
2
1
3
7
8
9
F r e q u e n c y ( r a d / s e c ) P h a s e ( d e g )
. 5
5 . 5 1 1 . 5
. 5
5 . 5 1 1 . 5
N y q u i s t D i a g r a m R e a l A x i s I m a g i n a r y A x i s
Numerical Example Numerical Example
P(s) L(s)=P(s)K(s) K(s)
Nyquist plots Bode diagrams
SLIDE 43 C
c l u d i n g R e m a r k s C
c l u d i n g R e m a r k s
・ ・ H2 tracking performance limits H2 tracking performance limits
Explicit analytical solutions for
・ ・ H2 regulation performance limits H2 regulation performance limits Characterizations of easily controllable plants in practical situations, which provide guidelines
- f plant design from the view point of control
Finite frequency phase property vs achievable robustness performance in H_inf LSDP
