W h e n i s a L i n e a r S y s t e m W h e n - PowerPoint PPT Presentation

MTNS’06, Kyoto (July, 2006) 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 o r D i f f i c u l t E a s y o r D i f f i c u l t t o C o n t r o l i n P r a c t i c e ? t o C o n t r o l i n P r a c t i c e ? Shinji Hara The University of Tokyo, Japan

Outline ・ ・ Motivation & Background: Motivation & Background: new paradigm ・ ・ H2 Tracking Performance Limits: H2 Tracking Performance Limits: Explicit analytical solutions with examples ・ H2 Regulation Performance Limits: Explicit analytical solutions with examples ・ ・ Phase Property vs Achievable Robustness Performance H_inf loop shaping procedure - ・ ・ Concluding remarks Concluding remarks

B e g i n w i t h . . . M o t i v a t i o n & B a c k g r o u n d - N e w p a r a d i g m o n - C o n t r o l t h e o r y -

New Paradigm on Control Theory New Paradigm on Control Theory d(t) Find u(t) y(t) e(t) r(t) Best Given Given - K(s) P(s) P(s) Characterize d(t) u(t) y(t) e(t) r(t) Best Desirable Best - K(s) P(s) K(s)

Bode I ntegral Relation Bode I ntegral Relation Assumption ： L(s)=P(s)K(s): stable, r.d. >1 Closed-loop system: stable ∞ ∫ ω ω = ∑ log(| ( ) |) 0 S j d π p 0 i ω | ( ) | S j ω 1 = ( ) : S s + 1 ( ) ( ) P s K s

・ ・ Question ! Question ! Is any stable & MP plant always easy to control under physical constraints in practice ? control input energy measurement accuracy sampling period channel capacity etc. � Answer: NO ! � Answer: NO ! ・ ・ Aim of researches on control perf perf. limits: . limits: Aim of researches on control Characterization of easily controllable plants in practical situations

3-Disk Torsion System 8 0 J 3 poles 極 6 0 J d i s Disk 1 k 1 の零点 1 c θ ����� 3 3 4 0 d i s k 2 の零点 c Disk 2 k 1 θ 1 2 J k 2 2 0 1 J 2 0 c ����� θ m I 2 c 2 2 θ k 2 1 - 2 0 k 2 J J 1 3 - 4 0 ����� c θ c 1 3 θ u 1 - 6 0 3 T - 8 0 ������� - 2 . 5 - 2 - 1 . 5 - 1 - 0 . 5 0 R e All 3 TFs are marginally stable & MP, but the achievable performances are different.

Step responses Disk1 Disk2 S t e p R e s p o n s e S t e p R e s p o n s e 1 1 ( a ) ( a ) 0 . 8 ( b ) 0 . 8 ( b ) 0 . 6 0 . 6 0 . 4 0 . 4 e e d d u u t t i i 0 . 2 l 0 . 2 l p p m m A A 0 0 - 0 . 2 - 0 . 2 - 0 . 4 - 0 . 4 - 0 . 6 - 0 . 6 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 T i m e ( s e c ) T i m e ( s e c ) Disk1 is better than Disk2. Why ?

・ ・ Question ! Question ! Is any stable & MP plant always easy to control under physical constraints in practice ? ・ ・ Aim of researches on control perf perf. limits: . limits: Aim of researches on control Characterization of easily controllable plants in practical situations New Paradigm New Paradigm ・ From Controller Design to Plant Design ・ ・ To provide guidelines of plant design To provide guidelines of plant design from the view point of control from the view point of control

F i r s t t o p i c . . . H 2 T r a c k i n g P e r f o r m a n c e L i m i t s - e x p l i c i t a n a l y t i c a l s o l u t i o n s & a p p l i c a t i o n 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)

Control Performance Limitations Control Performance Limitations ・ Bode Integral Relation ・ SISO stable/unstable ・ MIMO ・ Discrete-time/Sampled-data ・ Nonlinear ・ H-inf norm performance ・ Time-response performance ・ Tracking performance (H2 norm) ・ Regulation performance (H2 norm) Special issue in IEEE TAC, Aug. ,2003 Seron et. al. “Fundamental Limitations in Filtering and Control “

H 2 2 Optim al Tracking Problem Optim al Tracking Problem H unit step input SIMO plant Performance Index: control effort tracking error

G （ ｓ ） 1/s 1 w(t) z(t) W - u u(t) P （ ｓ ） y(t) P （ ｓ ） e(t) K(s) K(s) X Riccati & LMI −   1 / ( ) s P s   = Analytic ( ) 0 G s W   u solution   −   1 / ( ) s P s （ closed-form ）

SISO marginally stable plant SISO marginally stable plant NMP zeros Plant gain

Numerical Example Numerical Example = 1 ( ) W u 2 * J 1 = 2 1 z 1.0 = − 1 z 1 a

Application to 3- -disk torsion system disk torsion system Application to 3 J * Disk 3 Disk 2 Disk 1 W u

Discrete- -time case time case Discrete NMP zeros Plant gain � Delta Operator Continuous-time result

General SI MO Case General SI MO Case Numerator: Unstable poles & NMP zeros:

Stable terms: NMP zeros Plant gain

Unstable terms: Unstable poles Unstable pole / NMP zeros

Rem arks: : Rem arks Several cases where the computation of is not required. ・ SIMO marginally stable ・ SISO non control input penalty ・ SIMO ・ SIMO unstable: common unstable poles ： Jcu=0 � many applications

Optimal length of Inv. Pend. ?

Tracking performance limit 6 5.5 5 c2 4.5 J * 4 3.5 3 0 0.5 1 1.5 2 l (m)

Discrete- -time case time case Discrete NMP zeros Plant gain � Delta Operator Continuous-time result

S e c o n d t o p i c . . . H 2 R e g u l a t i o n P e r f o r m a n c e L i m i t s - e x p l i c i t a n a l y t i c a l s o l u t i o n s & a n a p p l i c a t i o n - “Best Tracking and Regulation Performance under Control Energy Constraint” by J.Chen, S.Hara & G.Chen, IEEE TAC (2003) “H2 Regulation Performance Limits for SIMO Feedback Control Systems” by T.Bakhtiar & S.Hara, MTNS’06

H 2 2 Optim al Regulation Problem Optim al Regulation Problem H Impulse input SIMO plant Performance Index : control effort performance on disturbance rejection

SISO MP plant SISO MP plant unstable poles Plant gain

Numerical Example Numerical Example 3500 via Theorem 1 via Toolbox 3000 2500 2000 * E E c * c 1500 1000 500 0 −1 0 1 2 3 4 5 p p

SIMO NMP plant SIMO NMP plant Common NMP zeros MP case

Application to Application to a Magnetic Bearing System a Magnetic Bearing System Normalized state-space equation:

one unstable pole at p ・ current sensor: NMP NMP ・ position sensor: MP MP ・ multiple sensors: MP MP

SISO MP discrete- -time plant time plant: : r.d.=1 SISO MP discrete � Delta Operator Continuous-time result

Magnetic bearing system: : Magnetic bearing system caused by discretized NMP zeros

L a s t t o p i c . . . P h a s e P r o p e r t y v s A c h i e v a b l e R o b u s t n e s s P e r f o r m a n c e - H _ i n f l o o p s h a p i n g p r o c e d u r e - “Dynamical System Design from a Control Perspective: Finite frequency positive-realness approach” by T. Iwasaki & S. Hara, IEEE TAC (2003) “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)

FFPR (Finite Frequency Positive Realness) FFPR

ー Finite Frequency Positive Realness ー (LMI condition)   A B ω 0 > =   given ( ) , G s 0   C D ω ∀ ω + ω > ω ≤ * ( ) ( ) 0 , | | G j G j 0 ∃ = > = T T 0 , . . X X Y Y s t −  T       Y 0 X B A I A I <         ω + 2 T T     0 0     C C B D D Y X 0

H inf inf LSDP LSDP H (H inf Loop-Shaping Design Procedure)

Good Phase Property

2 nd order plant

Characterization of good plants

Numerical Example Numerical Example P(s) K(s) L(s)=P(s)K(s) Bode diagrams B o d e D i a g r a m Nyquist plots 6 0 N y q u i s t D i a g r a m 4 0 1 . 5 2 0 ) B d ( 0 1 e d u t i - 2 0 n g a M - 4 0 0 . 5 - 6 0 - 2 - 1 0 1 2 3 1 0 1 0 1 0 1 0 1 0 1 0 s i x A 0 9 0 y r a n i g a m 0 I - 0 . 5 ) g e d - 9 0 ( e s a - 1 h P - 1 8 0 - 2 7 0 - 1 . 5 - 2 - 1 0 1 2 3 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 1 0 1 0 1 0 1 0 1 0 1 0 R e a l A x i s F r e q u e n c y ( r a d / s e c )