Observer State observer / State estimator状態観測器 ／ 状態推定器

∫A

Tcb

Tk

v u+

+

++

x yxState feedback u=kTxPossible only ifall the state variables

are measured and available

1( )

( )( )

n

x t

x tt

=

x

Design of state observer whichestimate x(t) from u(t) and measured y(t)

It is possible to design an observer whose output converges to the state x(t) with arbitrary convergence speedif the system (cT, A) is observable.

Simple simulator follows the original ?0( )= ( )+ ( ), (0)=t t u tx Ax b x x ( )= ( )Ty t txc Original system

0( )= ( )+ ( ), (0)=t t u tA bz z z z ( )= ( )Ty t tc z Simulator( ) ( ) ? or ( ) ( ) ?t t t t= →z x z x

( ) ( ) ( )t t t≡ −z xe State estimation error

( ) ( )( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

t t u t t u t

t t t

= + − +

= − =

A b A b

A A

e z xz x e

0 0( ) 00 = − ≠z xe

( )ob 0( ) ( ) ( ) ( ))( ) , (0Tyt u tt tt −= + + =A b kz z zz zc

A: unstable

( )→ + ∞

− →∞z x

z x

Necessary to compensate estimation error

( ) ( ) T t y t→ ⇒ →z x zcObserver design = Selection of suitable kob

( )t →∞e

Full order observer order of observer = order of plant

( )ob 0( ) ( ) ( ) ( ))( ) , (0Tyt u tt tt −= + + =A b kz z zz zc0( ) ( ) ( ), (0)t t u t= + =x Ax b x x

( ) ( ) ( )t t t= −z xe( ) ( )

( )ob

ob 0 0

( ) ( ) ( ) ( ) ( ) ( ), (0)

T T

T

t t t t tt

= − + −= − = −

kA xA k

e z x ze z xe

c cc

Estimation error = difference

( )( )ob

ob : stable ( ) 0, 0

( ) ( ), 0T

T t tt t tσ −− ⊂

→ →⇒ →

−→CA k

A kc

ez x

c

ob obDetermine so that is stable T−k A k c( ) ( )ob ob

T TTσσ − −= AA kk cc

( ) obPole assignment for by state feedback gain TT ,− kA c

Controllability/Observability of Dual systems

( ) : controllable T ,−A c1 - -( ) : full rank- n TT − ⇔ c AA cc

11

: full rank : full rank

T

T

T n

T

T

T n −−

−

⇔

cc A

c A

cc A

c A

( ) ( )= ( )+ ( ): observable

( )= ( )observ : a l b e

T

T t t u t

y t t,

⇔

x Ax b

xcAc

( ) ( )( ) ( )

: controllable : observable

: observab

le : controllable

T T

T T

, ,

, ,

⇔

⇔

A A

A A

b b

c c

( )= ( ) ( ): controllable

( ) ( )

T

T

t t u ty t t

− =

x A x cb x

Observer based state feedback system

v

A

Ab

obk

∫ Tc

Tc −++

++

++

x

x

y

y

u

∫

Tk

++

ˆ( ) ( ) ( )Tu t t v t= +k x

Example 1 Observer design

[ ]

0 2 1 11 1 0 2

0 1

( ) ( ) ( ), (0)

( ) ( )

t t t

t t

u

y

− −

−

= + =

=

x x x

x1. Check the observability of the system.

2. Design a full-order observer whose characteristic roots areσ={-5,-6}.

3. Draw the graph of the free response of the system with z(0)=0.

0 1 0,

2 1 1: Controllable ?

T − −

−= =A c

( )

0

ob

+ ( )

( )=

( )= ( ) , (0)=( )=

( )

( )

( )= ( ) , (0)=0

+ ( ( 0))

T

T

t ty t t

t

u t

uyu ttt t

t−+

b

b k

x Ax x xx

A

c

z zzz c

Example 1 Observer Design

Controllable canonical form

State feedback to assign observer poles at *ob { 5, 6} σ = − −

Characteristic polynomial 2 2s s+ +

0 1

2 1

01

( )T u u− −

=

= +−+x A x xc

[ ]ob 10T k ku = =x xk

2( 5)( 6) 11 30s s s s+ + = + +Ideal characteristic equationIdeal poles

State feedback gain for controllable canonical form system[ ] [ ]ob 2 30 1 11 28 10T = − − = − −k

( )[ ] ( )

ob ob

28 0 1100 2 0 301 1 1 11 { 5, 6 }

Tσ σ

σ σ− −−

− −− −

= =−

= −

= − −

A k c

Exercise 1 Observer design0 1 0 1⎡ ⎤ ⎡ ⎤ ⎡ ⎤

[ ]

0 1 0 12 1 1 2

( ) ( ) ( ), (0)

( ) ( )

t t t

t t

u

y

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= + =x x x

x[ ]1 0( ) ( )t ty = xFor the above linear system, follow the instructions below:

1. Check the stability and the observability of the system.2. Design a full-order observer whose characteristic roots are

{-4,-5}. 3. Show the block diagram of the system with the observer.4 Draw the graph of the free response of the system with an4. Draw the graph of the free response of the system with an

observer initial state z(0)=0. 5. Repeat 4. for the step response.p p p

( ) 0 2 1T T − −⎡ ⎤ ⎡ ⎤+A bk0( )= ( )+ ( ), (0)=

( ) ( )Tt t u tt tx Ax b x x

( ) 0 2 1,

1 1 0⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

−=+ =A bk c( )ob

( )= ( )( )= ( )+ ( ) , (0)=0( ) ( )T

Ty t tt t yt tu t+ −

xb kA

cz zzz c

Exercise 1-1 Stability /Observability 0 1 0 12 1 1 2

( ) ( ) ( ), (0)t t tu⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= + =x x x

[ ]1 0( ) ( )t ty⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= x1 Ch k h bili d h b bili f h

controllable canonical form

1. Check the stability and the observability of the system.

Characteristic equation 2

Free response state

2

1 7

2 0j

s s− ±

+ + =

→

q

0

1

2

1 72

js ±→ =

stable -2

-1

Observability matrix Uo=J1 0Nstable0 2 4 6 8

observable

Observability matrix UoJ0 1Possible to design an observer

Exercise 1-2 Observer Design 0 1 0⎡ ⎤ ⎡ ⎤0 1 02 1 1

1 0( ) ( ) ( ), ( ) ( )t t t t tu y⎡ ⎤ ⎡ ⎤⎡ ⎤⎣ ⎦⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

= + =x x xObserver poles σ b

0 2

1 1

1,

0T −=

−

⎛ ⎞−⎡ ⎤⎡ ⎤= = − =⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠A A b c

Observer poles σob

⎡ ⎤

( ) ( )ob ob obT T Tσ σ σ= =− +A ck A bk

( ) ( ) ( )

11 1

1 0, ( ) ( )σ σ −− −

=−

⎡ ⎤ =⎢ ⎥⎣ ⎦T M T MT

( ) ( ) ( )1 1 1 1ob ob= T TT Tσ σ σ− − − −+− = = +T T TTT A T c T A b Ak kT bk

1 11 1 0 1 0 TT T− − − −=⎡ ⎤⎡ ⎤= = = ⎢ ⎥⎢ ⎥ = = − =T AT T A T T b T c k TA b k

Controllable canonical form Characteristic polynomial

bo2 1 1, , =

− −= = = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

= = − =T AT T A T T b T c k TA b k

⎡ ⎤C c e s c po y o

2 2s s+ +0 1

2 1

01

u u=− −

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

= + +bx x xAT ⎡ ⎤k

State feedback to assign observer poles at *ob { 4, 5} σ = − −

10T k ku = ⎡ ⎤⎣ ⎦=x xk

Exercise 1-3 Observer Design (continued)2

Ideal poles *ob { 4, 5} σ = − −

Characteristic polynomial 2 1 2s s+ +

2( 4)( 5) 9 20s s s s+ + = + +Ideal characteristic equation

State feedback gain for controllable canonical form system

[ ] [ ]TT kk [ ] [ ]ob 2 20 1 9 18 8TT = = − − = − −TkkobTT = k Tk

[ ] [ ]1 1ob 18 8 8 10T T − −= = − − =T Tk k

ob

[ ] [ ]ob( )

( )ob ob

80 1 8 1

Tσ σ= −

⎛ ⎞⎡ ⎤⎡ ⎤ ⎡ ⎤

A k cCheck

[ ] ( )8 1 0100 1 8 12 1 12 1 { 4, 5}σ σ −− − − −

= =−⎛ ⎞⎡ ⎤⎡ ⎤ ⎡ ⎤ = − −⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦⎝ ⎠

Exercise 1-4 Block-diagram of Observer System

[ ]0 1 01 0

2 1 1( ) ( ) ( ), ( ) ( )t t t t tu y⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦= + =x x x

b 1 s

Tc++

x yu

As+

A1 T ++ x y1 s

Tc −++

++x y

Akobk

Exercise 1-5 State Observation (Free response)00 1⎡ ⎤ ⎡ ⎤ [ ]

( )

01

0

0 11 0

2 1

0 1 8

( ) ( ) , ( )

(

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ))ˆ ˆ ˆˆ ˆ

t t t ttu y⎡ ⎤⎢ ⎥− −⎣ ⎦⎡ ⎤ ⎡ ⎤

⎡ ⎤⎢ ⎥⎣ ⎦⎡ ⎤

= =+x x x

( ) [ ]01

0 1 81 0

2 1 10(( ) ( ) ( ) ( ) , ( ) ( ))ˆ ˆ ˆˆ ˆt t t tt t ty y yu⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥⎡ ⎤⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎣ ⎦

= + =+ −x x x

12

0

(0) ⎡ ⎤⎢ ⎥⎣ ⎦⎡ ⎤

=x2( )ˆ tx

00

(0)ˆ ⎡ ⎤⎢ ⎥⎣ ⎦

=x1( )tx

1( )ˆ tx

2( )tx

Exercise 1-6 State Observation (Step response)0 1 0⎡ ⎤ ⎡ ⎤ [ ]

( )

0 1 01 0

2 1 1

0 1 0 8

( ) ( ) ( ), ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆˆ ˆ

t t t t tu y⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + =x x x

( ) [ ]0 1 0 81 0

2 1 1 101( ) ( ) ( ) ( ) , ( ) ( )ˆ ˆ ˆˆ ˆt t t t t ty y y×

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= + + − =x x x

12

0

(0) ⎡ ⎤⎢ ⎥⎣ ⎦⎡ ⎤

=x2( )ˆ tx

00

(0)ˆ ⎡ ⎤⎢ ⎥⎣ ⎦

=x1( )tx

1( )ˆ tx

2( )tx

State Observation (for sine curve input)0 1 0⎡ ⎤ ⎡ ⎤ [ ]

( )

0 1 01 0

2 1 1

0 1 0 8

( ) ( ) ( ), ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )ˆ ˆ ˆˆ ˆ

t t t t tu y⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + =x x x

( ) [ ]0 1 0 81 0

2 1 1 105sin2( ) ( ) ( ) ( ) , ( ) ( )ˆ ˆ ˆˆ ˆtt t t t t ty y y×

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= + + − =x x x

12

0

(0) ⎡ ⎤⎢ ⎥⎣ ⎦⎡ ⎤

=x1( )tx2( )ˆ tx

00

(0)ˆ ⎡ ⎤⎢ ⎥⎣ ⎦

=x 1( )

1( )ˆ tx

2( )tx

Observer Design DetailsD i i f f i i f ll bl i l fDerivation of transformation matrix for controllable canonical form

( )bT Tσ σ= −A ckWe have to stabilize .( )ob o σ σ A ckWe have to stabilize .

So should be transformed0 2

1 1

1,

0T −=

−⎡ ⎤⎡ ⎤= = − = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦A A b c

into a controllable canonical form. The controllability matrix for is obtained as

1 1 0−⎢ ⎥⎣ ⎦ ⎣ ⎦

( )A bThe controllability matrix for is obtained as

from .

( ), A b

0 2 1 0− − ⎡ ⎤⎡ ⎤⎡ ⎤= = ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦AbC

1 0−⎡ ⎤⎡ ⎤= =⎣ ⎦ ⎢ ⎥⎣ ⎦U b Ab o .

1 1 0 1− −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦C 0 1−⎣ ⎦ ⎢ ⎥⎣ ⎦

Using the bottom row of , we have[ ]1C 2, 0 1T− = −U l

2

1 10 1 1 11 1 1 0

T

T

− −− − −

− −

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

ll A

T 1 0 11 1

− −

−

⎛ ⎞⎡ ⎤=⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

T2 1 1 1 0⎣ ⎦ ⎣ ⎦⎣ ⎦l A 1 1⎣ ⎦⎝ ⎠

Dynamic Compensator 動的補償器

G l f t t f db k t ll hi h f ti

( ) ( ) ( ) (0) ( ) qt t t t+ξ Dξ E ξ ξ ξ R

General-form output feedback controller, which may function as a State feedback + Observer

0( ) ( ) ( ), (0) , ( )( ) ( ) ( )

qt t t tt t t= + = ∈= +

ξ Dξ Ey ξ ξ ξ Ru Fξ Gy

SF DC( ) ( ), ( ) ( ) ( )t t t t t= = +u Kx u Fξ GySF DC( ) ( )t t≡u u

( )( ) ( ) ( ) ( )t t t t= − = −Fξ Kx Gy K GC xSF DC( ) ( )

( ) ( ) q nt t ∃ ×= ∈ξ Ux U R⎡ ⎤

( ) ( ), t t= ∈ξ Ux U R∴ = −FU K GC [ ]⎡ ⎤ =⎢ ⎥⎣ ⎦

UF G KC( ) ( ) ( ) ( ) ( )t t t t t+ +ξ Dξ E DU EC

( )( ) ( )t t= −FUx K GC x( ) ( ) ( ) ( ) ( )t t t t t= + = +ξ Dξ Ey DUx ECx

( ) ( )( ) ( ) ( ) ( ) ( )t t t t t= = + = +ξ Ux U Ax Bu U A BK x

⎡ ⎤( )∴ + = +DU EC U A BK [ ] ( )⎡ ⎤ = +⎢ ⎥⎣ ⎦UD E U A BKC

Statefeedback-equivalent Dynamic Compensator0( ) ( ) ( ), (0)

( ) ( ) ( )t t tt t t= + == +

ξ Dξ Ey ξ ξu Fξ Gy ( ) ( )t t=≡ u Kx( ) ( ) ( )ξ y

( )⎡ ⎤⎡ ⎤ ⎡ ⎤

Pole assignmentOptimal regulatorServomechanism( )⎡ ⎤⎡ ⎤ ⎡ ⎤ = ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

U A+ BKD E UF G C K

Servomechanism:

0 0=ξ Ux Impossible if x0 is unknown

S ti ThSeparation Theorem( ) ( ) ( ) ( ) ( ( ) ( )) ( )

( ) ( ) ( ) ( ) ( )( )t t t t t t t

t t t t tt⎡ ⎤ + + +⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

x Ax Bu Ax B Fξ GCx A+ BGC BF xDξ Ey Dξ ECx EC D ξξ ( ) ( ) ( ) ( ) ( )( ) t t t t tt⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ Dξ Ey Dξ ECx EC D ξξ

1

σ σ σ−⎛ ⎞⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎝ ⎠

A+BGC BF I O A+BGC BF I O A+BK BFEC D U I EC D U I O D UBF

( ) ( ) σ σ

⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟− −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎝ ⎠= +

EC D U I EC D U I O D-UBF

A+BK D-UBF D-UBF must be sufficiently stable

Pole assignment for DC ( )σ D-UBFA li i f G i h’ b d i h dApplication of Gopinath’s observer design method

B. Gopinath: On the Control of Linear Multiple Input-Output Systems, The Bell Technical Journal, Vol. 50, No. 3, pp.1063-1081, 1971

[ ]rank such that :n nmm ∃ ×= ⇒ ∈ = =C M R C CM I O

1−⎡ ⎤U V I M V bi ⎡ ⎤

, , , pp ,

( ) 1 1q q q

− −⎡ ⎤ ⎡ ⎤ ⎡ ⎤− − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = =U A+BKD E UI I II UB UI UBD UBF UA

1: q−= ⎡ ⎤⎣ ⎦U V I M , V: arbitrary : q= = ⎡ ⎤⎣ ⎦U UM V I

1 11 121 1

1 q

q q

q q−−

−−

⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎛ ⎞⎛ ⎞ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎣ ⎦ ⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤⎢ ⎥= = = A A

I UB F G C CK

V IU U V I

I UBO O O

IIMUM AM A

D UBF UA

M U M1

q−⎡ ⎤⎢ ⎥I

21 22q

m⎢ ⎥ ⎢ ⎥⎣⎡ ⎤⎜ ⎟ ⎣ ⎦ ⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎢ ⎥⎦ ⎣⎦ ⎦⎣

= = = V IC C IAO OO AMUM AM AM U M

11 12 11 12 1222 12

qmq q q

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥⎡

⎢ ⎥⎣ ⎦

⎡ ⎤⎡ ⎤ ⎡ ⎤ =⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦= = = +O IV I V I

O

I V IIA A A A AO

IOV A VA21 22 21 22 22

22 12q

q q qq

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣−⎣ ⎦⎦ ⎦+V I V I V IIA A IOV A A A

A VA

T T) )( , ): observable ( , ): observable

( b bl ( t ll bl⇔C A C A

A A A AT T12 22 22 12, ) , ) ( : observable ( : controllable ⇔ ⇔A A A A

Pole assignment by V is possible for ( ) ( )22 12σ σ= −+V DA A UBF