k() s g() l() n...−controllability / observability concept. −performance criteria / nominal...

Nano System Control Lab.Chosun University 235

Summary

K ( )N s G ( )N s 0L ( )s

Plant Output

1max 0 min( L ( ) I ) I G ( )K ( ) ,N Nj j j

1max 0 min(L ( ) I ) I G ( )K ( ) ,N Nj j j

or


K ( )N s G ( )N s1L ( )s

Plant Iutput

1max 1 min( L ( ) I ) I K ( )G ( ) ,N Nj j j

1max 0 min(L ( ) I ) I K ( )G ( ) ,N Nj j j

or


Quiz #2 : 3-problem, 1-bonus

Some True – False questions

− Multivariable Transmission Zeros : GEP solving

− Controllability / Observability Concept.

− Performance Criteria / Nominal Stability

Understand MIMO Nyquist Criterion

• SV & SVD

• perf. Spec. via s-plane

− Bonus – Simple Eigenstructure Assignment.


1 2U [u ,u ] u i

1 2V [ v , v ] vi

− right eigenvector ofHA A

HA A− right eigenvector of

1u A vi ii

H H T H H T

H

A U V , A V U , A U V1 1v A u , u A vi i i i

i i

1.

2.min min max( I A ) (A) 1 (A) 1 ,

1 1min min

max

1( I A ) (A ) 1 1(A)

min max

1 1 ,( I A ) ( A ) 1

max1

min max

( A )1( I A ) (A ) 1

3.max min1.0 , 0.0000474 '' ''depends on untis

Hw #5 solution


4.

45

45

1 1 1

2,3 1 1.02j

1 2 2B , u 0 ,b b 12

1 0 0C

0 0 1cc

1q 1,2,3

0ii

1 1p ( I A ) Bq ( I A ) b ,i i i Q 1 1 1

1

1

A bS

c 0

q 1i


Design of MIMO Feedback System

Standard form

K( )sr ( )s e( )s u ( )s

d( )s

y( )sL( )sG( )s

n ( )s

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

e ( ) S( ) r ( ) d ( ) C( ) n ( )t s s s s s s

Error bound

max maxL ( ) I e ( ) ,j all


Sensitivity

disturbance

Sensor noise

max S( )j

min S( )j


Closed Loop

HighLoop gain

Sensornoise

max C( )j max

1 ( )e j

min C( )j


Issue

T( ) , C( ) , S( )s s s Time Domain

T( )sr( )s e( )sy( )s

y T( ) es

1x A x Bu y C(sI A ) Bey C x

1T( ) C(sI A ) Bs

u e


1y C ( ) rs

e r y Cx r

1x [A BC]x Br y C(s I A BC) B ry C x

11C ( ) C(s I A BC) Bs

e ( ) S( ) r ( )t s s s

1x [A BC]x Br C(s I A BC) B I re C x I r e

1S( ) C(s I A BC) B Is


• can analyze given MIMO design

• can’t design compensator

− Nyquist diagram does not help !

− Eigenstructure assignment ; Standard form

• How about applying SISO methods ?

1r

rm

1u

u m

MIMOG( )sm m

1y

y m

11K

K mm

K ( )s

So far , Discussion ;


• Dumb way

- Set Design

- Set Design

(1) Must decide which control for which output.

For TITO

1 11 1 12 2y ( ) g ( ) u ( ) g ( ) u ( )s s s s s

2 21 1 22 2y ( ) g ( ) u ( ) g ( ) u ( )s s s s s

12g ( ) 0s

21g ( ) 0s 11K ( )s

22K ( )sWatch out for

stability of TITO

• Closing the loop one at a time.


• Sensible way

Close Loop #2, first

2r ( )s e( )s 2u ( )s

2y ( )s22K ( )s 22g ( )s

21g ( )s

1u ( )s

− Ignore Design 21g ( )s 22K ( )s

2u ( ) ;s 2 22 2u K e 22 2 22 21 1 22 2K r K (g u g u )

22 22 2 22 2 22 21 1(1 K g )u K r K g u

22 22 212 2 1

22 22 22 22

K K gu ( ) r u1 K g 1 K g

s 2 2 2 21 1

h ( ) r h ( )g ( ) us s s


Impact on closed – loop #1

1 11 1 12 2y g ( ) u g ( ) us s 11 1 12 2 2 2 21 1g ( ) u g (h r h g u )s

11 12 2 21 1 12 2 2[g g h g ]u g h r

First Loop

1r ( )s 1e ( )s 1u ( )s

1y ( )s11K ( )s 11f ( )s

12 2g h

2r ( )s

Design for

− Good command following

− Good dist. rej. of

11K ( )s

12 2g h

11 1 12 2 2f u g h r


1K

2K

1r

2r

1u

2u

1y

2y

G

d irec t con tro l

Interaction of control Loops and the Relative Gain Array (RGA) (E. Bristol, 1966)

− Tool for Static loop interaction analysis

• Cannot design each loop separately

• MIMO Loop should be stable

Which is the “best” coupling 1 1 1 22 2 2 1

y u y uor

y u y u

ind irec t con tro l


RGA analysis (Relative Gain Array)

Step 1 : static gain between

(other loop gain)

1 1y & u

1u

2u 0

1y

2y2

1

1 u 0

yu


Step 2 : static gain between

(other loop closed )

1 2y & u

1u

2u

1y

2y 02r2K

2

1

1 y 0

yu

2

2

1 1 u 011

1 1 y 0

( y u )( y u )

1 1( y & u )forRelative gain

Relative gain Array :11 12

21 22


e.g. (=exempli gratia)

1 11 1 12 2y g u g u

2 21 1 22 2y g u g u 1

2 22 21 1u g g u 2( y 0)for

11 11 12 22 21 1y g g g g u

11 11 2211 1

11 12 22 21 11 22 12 21

g g gg g g g g g g g

11 12 1

21

1

g g gg

G

g g

n

n nn

u 0;

y 0;

( y u )( y u )

k

k

i j k jij

i j k i


Note :

u 0;( y u ) g [G]ki j k j ij ij

1y 0;

y 0;

1 ( u y ) [G ]( y u ) k

k

j i k i jii j k i

T 1G (0) (G (0) )ij ij ij 0RGA evaluated at s

TITO case

11 2211

11 22 12 21

g gg g g g

, y G u

• For Static Interaction (s=0)

11 12 21: 1 g g 0 '' ''complete decoupling1 00 1


• Properties of RGA

1.1 1

1n n

ij ijj i

11 0

11 1

11 22g g 0

11 22g g 0

11 22 12 21g g g g

'' ''bad decoupling

11 2211

11 22 12 21

g gg g g g

'' ''holding effect 11 121 , 0

11 0 '' ''wrong direction


e.g.

1 11 0.1 1G

0.2 0.80.5 1 1

s s

s s

T 111 11 111G (0) (G (0) ) 0.8

1.25

0.8 0.20.2 0.8

1 1 1 2

2 2 2 1

y u y uy u y u

or

21 1 u ( ) 0 11[ y ( ) u ( ) ] g ( )ss s s

2

11 1 y ( ) 0 11 12 22 21[ y ( ) u ( ) ] g g g gss s

This condition is impossible for holding


Property

2. RGA is invariant under input or output scaling

RGA (G(0) ) If

1 2RGA ( G (0) )S S

Then

3. If G is diagonal or triangular

Then I


Relative Gain

SISO yMIMO y

i

i

Normal measure output responseof decoupling output response

0 cross coupled

1 completely decoupled

with all the output = 0

0y 0 r

0

11 12

21 22

g gG

g g

;


Nominal Stability & RGA

r

yk Is

K( )s G( )s

T ( ) G ( )K ( )s s s

Thm 1. Iff det[T (0) ] 0

Then is integrally stabilizableT( )s

RGA[G (0) ]


Thm 2. If some of at least one of the following is true.

(a) MIMO C.L. is unstable.

(b) Loop is unstable with all the other loops open.

(c) MIMO C.L. is unstable even with loops open.

0,ii

ithi

If for all0,ii 1, ,i n

Then MIMO C.L. system is stabilizable.


Sensitivity to Modeling Error

be actual static gain matrix.

be nominal static gain matrix.

G (0)A

G (0)N

condition

1

G (0) G (0) 1 1G (0) [G ]G G

A N

N NN N

max min[ ] [ ] [ ]N N NG G G

1* min 2max ,


Regulator Design

Nominal Plant Dynamic

x ( ) A x ( ) Bu ( ) : y ( ) C x( )t t t t t

Feedback Law

u ( ) G x( ) 0t t : r ( ) 0t


o

B 1(sI A) G

C

u ( )s

Gyy( )t

B 1( sI A )

Gr

y( )t

x r

u

x ( )t

y( )t

G x ( )t

y( )x ( )

x ( )r

tt

t

yu ( ) G y G xr rt

o


• Linear Quadratic Regulator (LQR) Problem ( R. E. Kalman )

− How do we obtain the value of ?G

Optimal Control Problem

T T

0[ x ( )Q x( ) u ( )R u( ) ]t t t t dt

: minimize

x ( ) A x( ) Bu( )t t t

- Symmetric & Positive definite, real & positive Eigenvalues.

Optimal Control - Find that minimizes

constrained to x ( ) A x( ) Bu( )t t t Ju ( )t

Q, R

J

Q = min. state size , R = min. control size

J


• LQR Solution

− There exists a unique solution which can be realized in

feedback form, assuming full state feedback.

Feedback Law u ( ) G x ( )t t

LQR1 TG R B K

Where is matrix

− symmetric & positive semidefinite

− solution of CARE ( Control Algebraic Riccati Equation )

K n n

T 1 T0 K A A K Q K BR B K

Re (A BG ) 0i A, B : GivenQ, R : Design


• LQR Variant 1

− Cross - Coupled cost functional

T T T

0[ x Q x 2 x Su u R u ]J dt

control u G x TQ S

MS R

1 T TG R [S B K ]

is solution of CAREK

T 1 T T0 K A A K Q [K B S]R [B K S ]

Guarantee : Re[ (A BG )] 0i : Nominal stability

, : positive semi-definite


• LQR Variant 2

− Exponential weighting

T T

0[ x Q x u R u ]J ate dt

0a j

0

a


Control : u G x 1 TG R B K

where is solution of CARE K

T 1 T0 K A I A I K Q K B R B Ka a

• LQR problem

x A x Bu

Control : u G x 1 TG R B K

T T[ x Q x u R u ]J fi

t

tdt

is a solution of CARE K

T 1 T0 Q KA A K KBR B K


• Optimal Control – LQR Problem

minimize V L( x , x , )fi

t

tt dt

L( x , x , )t is continuous to second partial

ˆ ˆV x ( ) V x ( ) x x xt t for

Find forV x ( )t

V V(x x ) V(x)

22

2x( ) x( )

V 1 VV(x) x x V(x)x 2! xt t

21V V ;2!

V

first ordervariation of

x( )

VVx t

x


• Differential Calculus

x x 0 0as t

x 0dx dxtdt dt

.cond for extremum

2 2

2 2max 0 , min 0d x d xif ifdt dt

• Variational Calculus

V 0 x 0as

x( )

VV V xx t


x( )

V 0x t

.cond for extremum

2

2x( )

Vmax 0x t

if

2

2x( )

Vmin 0x t

if

V L( x , x , )fi

t

tt dt

V L( x x , x x , ) L ( x , x , )fi

t

tt t dt

L LV x xx x

f

i

t

tdt


ˆx x x

ˆ( x ) x x xd d ddt dt dt

( x ) xd dt

Second term of V

L Lx x xx x x

ff f

i ii

tt t

t tt

d Ldt dtdt

L L LV x xx x x

ff

ii

tt

tt

d dtdt

Boundary Condition


Necessary condition

L L 0x x

ddt

Euler equation

comment

If L L( x , u )

①

②

and are independent x , u Apply Euler eq. directly

and are dynamically constrained by x , u

x ( x , u , )f t

− change to incorporate the constraint

Lagrange Multiplier

L


Example : maximize area of rectangle with fixed parameter

b

l

,a l b 2 2p l b constraint

( 2 2 )c l b l b p

2 0c bl

2 0c lb

2b 2l ,,

2 2 0

4 4

8

c l b p

pp

① ② ③

from ① , ② b l

③ ① ②, ,4 4p pl b

, ,


Method of Pontryagin

min J L( x , u , )fi

t

tt dt

subject to constraint x ( x , u , ) 0f t

create TJ* L( x , u , ) p ( ) ( x )fi

t

tt t f dt

p ( )t Vector of Lagrange multiplier 1n

Design Parameter

p ( ) K x ( )t t 1n ( ) ( 1)n n n


• Resulting Euler Eq.

T Tx : [L p ( x ) ] L p ( x ) 0x x

f ft

①

define TH L p f • Hamiltonian control

• State function of Pontryagin

TH p x 0x xddt

② TH px


T Tu : L p ( x ) L p ( x ) 0u u

df fdt

⑤

⑥

③

④H 0u

LQR

T TL x Q x u R u , A x Buf

T T TH x Q x u R u p (A x Bu )

②⑤

④⑤

T T 1 Tu R p B 0 : u R B p

0

Tp Q x A p T T Tx Q p A p :


p ( ) K x ( )t t

Tp ( ) K x K x Q x A pt

TK (A x Bu ) K x Q x A p

1 T TK A x K B( R B p) K x Q x A p

T 1 TK x Q x A K x K A x K BR B K x

T 1 TK Q A K K A K BR B K

'' . ''general eq of CARE

K : const. gain matrix ; K 0

p K x


Example

1) 1st Order

x x ua 2 2

0

x uJx um m

dt

'' ''Bryson Method

2

1Qxm

21R

um

when max. des. value for xm x

max. des. value for um u

CARE

T 1 T0 K A A K Q K BR B K

2 22

10 2 K K uxm m

a

,

,


2 2ux

2

( )K

u

mm

m

a a

positive

Finally 1 TG R B K

2 2ux( )m ma a

Closed – Loop system

2 2uxx (A BG ) x ( ) xm ma

x ( ) x (0) tt e

If unstable open-loop poles 0 ,a

2G a '' ''Expensive Control

0


2) 2nd Order

1 2x x x ua a

2 2 2

0

x x uJx x um m m

dt

'' ''Bryson Method

21

2

1 0x

Q10

x

m

mn

n n

21

2

1 0u

R10

u

m

mm

m m


Normalizing

2 2 2 2

0u J x x um P R dt

where 2ux

mP

m

2ux

mR

m

After Some Algebra

x xx

u G Gx

2x 2 2G Pa a

2 2x 1 1 2 2G 2( )R Pa a a a

,


Closed – Loop : x (A BG ) x

20 0x 2 x x 0

12 4

0 2( )Pa 2

1 22

2

21 22

R

P

a aa

If is fixed , R P 2

2

If is fixed , P R

( )a ( )c( )b

,

0 0 0


Summary

x A x Bu State Dynamics

T T

0J ( x Q x u R u )dt

LQR Solution

u G x State feedback

-1 TG R B K : K T 1 T0 K A A K Q K BR B K

; cost function


o

B1( I A)s

G( )s

1LQT ( ) G (s I A) Bs

Loop TFM

Sensitivity TFM

Closed - Loop TFM

1

LQ LQS ( ) I T ( )s s

1

LQ LQ LQC ( ) I T ( ) T ( )s s s


o

B G( )s

E

Stability Robustness

1

max min LQ[E ( ) ] [ I T ( ) ]j j

or

max LQ max[C ( ) ] 1 [E ( ) ]j j


KFDE ( Kalman Freq. Domain Equality )

From T T 1 T0 K A A K N N K BR B K

After extensive algebra manipulate

T TLQ LQI T ( ) R I T ( ) R N ( )B N ( )Bs s s s KFDE

R I , 0

T TLQ LQ1KFDE : I T ( ) I T ( ) I N ( )B N ( )Bs s s s

Freq. Domain s j

2LQ1I T ( ) 1 N ( )Bi ij j

− Important for synthesis

design parameter , N

T; N N Q


If 1

Then , 21 11 N ( )B N ( )Bi ij j

LQ 1T ( ) N ( )Bi ij j

Small , large ( loop gain )

good C.F. , Disturbance Reduction.

LQT ( )i j

For Low Frequency Approximation

1( ) (s I A )s

s 0 1(0) A

At DC ( 0)

1LQ 1T (0) N A Bi i

: cheap control


For identical SV at DC

1NA B I

1T 1 TN B A B B

1T TN B B B A or

LQ 1T (0)i

1


KFDE

2LQ1I T ( ) 1 N ( )Bi ij j

For 1

LQ1T ( ) N ( )Bi ij j

(1) Low Frequency

T 1 TLQ

1T ( ) N (B B) B Ai j

(2) High Frequency

1( ) I Aj j

For 1( ) Ijj

,

,


KFDE

LQ1T ( ) N ( )Bi ij j

1 N Bi

20dbdec

maxc


Design

want T 1 TNB I N (B B) B

To estimate maxc

max maxT ( ) 1cj

LQ1T ( )i j

(0 )db

max1 11 c


Summary

(1) Cheap control ( 0)

− Control weighting is negligible compared to state weighting

High gain , High bandwidth

Main Result

If is minimum phaseN ( )Bs

Then 0

lim G W N

T1G B K

TW W I orthonorm alsquare m atrix


(2) Expensive control

− Control weighting is huge compared to state weighting

Low gain , Low bandwidth

Main Result

If is stable

Then lim G 0

A

( )


Disturbance rejection in LQR Design

System : x A x Bu ; y C x

T T T T T

0 0J y y u u x C C x u udt dt

1LQS ( ) I G ( )Bs s

As 0 1

LQS ( ) I G ( )Bs s

1I WC ( )Bs

0 LQ; : S ( ) 0in cheap control s

0 0

W C


Ex )

o

u( )t

y( )t

u( )t

d( )t

x x ( )tB

L

A

C

G

CL Dynamics

x (A BG) x Ld( ) : y C x ( )t t


y( ) S d( )ds s

1S ( ) C(s I A BG ) Ld s

For good distur. rejection

max S ( ) 1d s

As ( cheap control ) 0

1 1S ( ) C( ( ) BG) Ld s s

1 1C( ( ) B G) Ls

1C(BWC) L

0

W C


• LQ – Servo − How to adapt LQR based design to do command following

u G x r

• LQR − Full state Feedback

C

B G( )so

d ( )s r ( )s

y( )s e( )s


If output is part of the states

B ( )su ( )s

y p

x r

x ( )t entire state

System

x A x Bu

y C x ( )p p t

x ( ) D x ( )r Pt t

( )C Ip m m m n mO

( ) ( ) ( )D Ip n m m n m n mO

,

,


• Mod. LQR

y( )to G y B ( )s

Gr

y px r

u ( ) G y G xy p r rt

x A BG C BG D xy p r p CL :


• LQ - Servo ( Definition )

e( )tr ( )tG y B ( )s

Gr

y px r

d

y

Loop TFM , for LQ - Servo T ( )s

1T ( ) C (s I A BG D ) BGp r p ys

1LQT ( ) G (s I A ) Bs

LQT ( ) T ( )s s


LTR ( Loop Transfer Recovery ) Method

K ( )s G ( )sr ( )s

e( )s u ( )s y( )s

compensator plant

H C( )sr ( )s

e( )s

y( )s

( )with Filter Loop Tagret Loop


MBC ( Model – Based Compensator )

Plant : 1G ( ) C(s I A ) B C ( )Bs s

MBC : 1K ( ) G (s I A BG HC) Hs

Where Control gain matrixG u G z

x : x A x Bu

z : z A BG HC z H e

u G z

: y C x

: y re

H Filter gain matrix n m

m n


Block Diagram

r ( )sI H ( )s

C

B

G B ( )s C

z ( )su( )s

y( )s

K ( )s

G ( )s

xx

x x zc

2 1n

A BG BG 0: x x r

0 A HC Hc c

y C 0 xcAc


det ( I A ) det ( I A BG ) det ( I A HC)c

triangular matrix

※ Nominal stability

Re (A BG ) 0i all i

Re (A HC) 0i all i

Loop TFM

11T ( ) G ( )K ( ) C ( )BG ( ) BG HC Hs s s s s

C ( )Hs

I


Step. 1.

Fix Filter matrix H s.t.

Re [A HC] 0 ,i i

C ( )Hs

Step. 2.

Apply cheap control of LQR

Re [A BG ] 0i

T

0lim G W C ; W W I

•

•

•

• Desired Loop TFM


C ( )Hs

Main Result ( LTR )

As 0 , T( ) C ( )Hs s : Target Loop

r ( )s y ( )s

r ( )s y ( )s

Hdetermine

K ( )s G ( )s


What is happening ?

The LTR causes the MBC based design to cancel

the dynamics of the plant , and substitute

the Filter ( Target ) Loop dynamics.

− example of inverting the plant in a stable manner.

Observ.

(1) Some poles of must cancel zeros of

the rest of the poles must be very “fast”.

(2) Zeros of must be zeros of Target loop,

(3) and has same poles.

G ( )s

G ( )sK ( )s

K ( )s C ( )H.s

C ( )Hs C ( )Bs


References

'' '' , ,Dynamics of physical system Cannon HcGraw Hill

'' '' ,Linear system Fundamentals Reid

'' ; '' , &system Dynamics A unified Approach Karnopp Rosenberg

k() s g() l() n...−controllability / observability concept. −performance criteria / nominal...

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