MTNS 06, Kyoto (July, 2006) Shinji Hara The University of Tokyo, Japan

MTNS 06, Kyoto (July, 2006) Shinji Hara The University of Tokyo, Japan

Outline Motivation & Background: H2 Tracking Performance Limits: new paradigm 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

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

Bode Integral Relation Assumption L(s)=P(s)K(s): stable, r.d. >1 0 log( Closed-loop system: stable S( jω ) ) dω = 0 π pi S( jω ) ω S( s) : = 1+ 1 P( s) K( s)

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! Aim of researches on control perf.. limits: Characterization of easily controllable plants in practical situations

3-Disk Torsion System ª ªª ª ªª ª ªª J 3 J 1 θ 3 θ J 1 2 J 2 θ 2 θ 2 J 3 J 1 θ θ 1 3 ªªª ªª c 3 c k 1 2 k 1 c 2 c k 2 1 k 2 T u c 3 c 1 poles Disk 1 Disk 2 All 3 TFs are marginally stable & MP, but the achievable performances are different.

Step responses Disk1 Disk2 Disk1 is better than Disk2. Why?

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

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 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

H2 Optimal Tracking Problem unit step input SIMO plant Performance Index: tracking error control effort

G w(t) u(t) y(t) z(t) P P K(s) K(s) 1/s - = ) ( ) ( 1/ 0 1/ ) ( s P W s P s s s G u Analytic solution closed-form Riccati & LMI 1 W u e(t)

SISO marginally stable plant NMP zeros Plant gain

Numerical Example 1 ( = ) 2 W u J 2 * a 1.0 z 1 = 1 z1 = 1

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

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

General SIMO Case Numerator: Unstable poles & NMP zeros:

Stable terms: NMP zeros Plant gain

Unstable terms: Unstable poles Unstable pole / NMP zeros

Remarks: 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 J * c2 4.5 4 3.5 3 0 0.5 1 1.5 2 l (m)

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

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

H2 Optimal Regulation Problem Impulse input Performance Index : SIMO plant performance on disturbance rejection control effort

SISO MP plant unstable poles Plant gain

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

SIMO NMP plant Common NMP zeros MP case

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

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

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

Magnetic bearing system: caused by discretized NMP zeros

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)

Finite Frequency Positive Realness + < D D B B C I A C I A T T T 0 0 0 X Y Y 2 ω 0 X, ) ( = D C B A G s (LMI condition) > + 0, ) ( ) ( * ω ω jω G j G 0 ω ω 0 > 0 given.., 0 t s Y Y X X T T = > =

Hinf LSDP (Hinf Loop-Shaping Design Procedure)

Good Phase Property

2 nd order plant

Characterization of good plants

Numerical Example P(s) K(s) L(s)=P(s)K(s) Nyquist plots Bode diagrams

Explicit analytical solutions for H2 tracking performance limits H2 regulation performance limits Finite frequency phase property vs achievable robustness performance in H_inf LSDP Characterizations of easily controllable plants in practical situations, which provide guidelines of plant design from the view point of control