Linear Optimal Control by Jeffrey B. Burl 1999

(Addison-Wiesley)

● A review of multivariable linear systems (2)

● Vector random processes (3)

● Cost function presentation (4)

● Robustness –unstructured uncertainty (5)

● The linear quadratic regulator (6)

● The Kalman filter (7)

● Linear quadratic Gaussian control (8)

1

1. Introduction ●A control system：

－stability

transient performance

tracking performance

robustness

disturbance rejection

control force

● Classical Control (～1960)

－root locus (Freq. domain, transfer function.)

Bode plots

Nichols charts

Nyquist plots

－gain margin

phase margin

－iterative method

－cause –and-effect bet. controller and performance

－cumber and ambiguous insight for high-order systems and MIMO

plants

● Linear quadratic Gaussian (LQG) optimal control

(time domain, state space model) 1960～

－cost function：combined performance specs.

＝＞ the single best controller

－no intuitive insight (the effect of controller parameter modification)

－high-order controller

2

2. Multivariable Linear Systems

●Mathematical models：

state models , Transfer function models

2.2 The continuous-time state model：

s t a t e m a t r i x . i n p u t m a t r i x

o u t p u t m a t r i x . I t o O c o u p l i n g

( ) ( ) ( ) ( 2 . 4 a )( ) ( ) ( ) ( 2 . 4 b )

x t A x t B u ty t C x t D u t

= +⎧⎨ = +⎩

LTIs：

( )

0( ) ( 0 ) ( ) , t > 0 ( 2 .5 )

tA t A tx t e x e B u dτ τ τ−= + ∫The state transition matrix：

2 2 3 31 1( ) . . . 2 ! 3 !

A tt e I A t A t A tφ = = + + + +

( )

0( ) (0) ( ) ( ) (2.6)

tAt A ty t Ce x Ce Bu d Du tτ τ τ−= + +∫The impulse response matrix：

1 1 1

1

( ) ( )

( ) ( ) ( )

u

y y u

n

n n n

g t g t

g tg t g t

⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥⎣ ⎦

( ) ( )i j ig t y t=

where ( ) fu n c t io n

( )0 ,o th e rw is ej

t im p u ls eu t

δ =⎧= ⎨⎩

(0) 0x =

3

( ) , t 0

( ) (2.7)0 , t < 0

AtCe B D tg t

δ⎧ + ≥= ⎨⎩

0( ) ( 0 ) ( ) ( )

( 0 ) ( ) ( ) ( 2 . 8 )

tA t

A t

y t C e x g t u d

C e x g t u t

τ τ τ= + −

= + ⊗

∫

: convolution⊗

Ex. 2.1 The state model

1 1 1

2 2 2

1 1

2 2

1

2

( ) ( ) ( )1 2 1 0;

( ) ( ) ( )0 3 0 4

( ) ( )1 1;

( ) ( )1 1with the input

( ) 5 ( ),

( ) 2 1( )where 1(t) is the unit step func

x t x t u tx t x t u t

y t x ty t x t

u t tu t t

δ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

2 2

2

3

3

ton. The state-transition matrix is

( ) ( ) ( 3 )1 ( ) ... 1 1 ( ) ( 3 ) ...2! 2! 2!=

( 3 )0 1 ( 3 ) ...2!

,0

whe the power series for the exponetial function

At

t t t

t

t tt t te

tt

e e ee

− − −

−

⎡ ⎤− −+ − + + − + − − − + − +⎢ ⎥

⎢ ⎥−⎢ ⎥

+ − + +⎢ ⎥⎣ ⎦⎡ ⎤−

= ⎢ ⎥⎣ ⎦

3

3 3

( ) ( )1

( ) ( ) 3( )02

is used .The impulse response is

1 1 1 0 4g(t)=

1 1 0 40 4Figure2.1

( ) 5 ( )4( ) 2 1( )4 8

t t t t t

t t t t

t tt

t t t

e e e e ee e e e

y t te ed

y t te e e

τ τ

τ τ τ

δ

− − − − −

− − − −

− − − −

− − − − − −

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

⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦ ⎣ ⎦∫

2t−

8

3

8 3 .8 163

3 3

t

t t

e

e e

τ

−

− −

⎡ ⎤−⎢ ⎥= ⎢ ⎥− +⎢ ⎥⎣ ⎦

4

♦

2.2 The Discrete-Time State Model

－Computer simulation：high order system, non-analytic solution, …

－differential eqs. －＞difference eqs.

1. Euler’s method：

( ) ( ) ( ) ( ) ( )x KT T x KT x KT Ax KT Bu KTT

+ −≈ = +

difference eqs. for the state：

{ }( ) ( ) ( ) ( ) (2 .9 )x K T T x K T T A x K T B u K T+ = + +

2. the zero-order hold approximation：LTIs.

Assuming： [ )( ) ( ) ,u t u KT t KT KT T= ∀ ∈ +

5

( ) ( )

0

( )

( )

0

0

( ) (0) ( )

( ) ( )

( ) ( )

( ) ( )

(

K T TA K T T A K T T

K T TAT A K T T

K Tt K T TAT A T

T t TAT A

x K T T e x e Bu d

e x K T e Bu d

e x K T e Bdt u K T

e x K T e Bd u K T

x K T

τ

τ

ττ

ττ

τ τ

τ τ

τ

++ + −

+ + −

← −−

← −

+ = +

= +

= +

⎡ ⎤⎡ ⎤= +⎣ ⎦ ⎢ ⎥⎣ ⎦+

∫∫

∫

∫) ( ) ( ) (2.10)T x K T u K Tφ= + Γ

－ ( ) ( ) ( ) (2.11)y KT Cx KT Du KT= +

－sampling time

2.3 Transfer function－LTIs ( ) ( ) ( )x t Ax t Bu t= +

－Laplace transform

L:

1 1

( ) (0) ( ) ( )( ) ( ) (0) ( ) ( )

sX s x AX s BU sX s sI A x sI A BU s− −

− = +

= − + −

{ }1 1

( ) ( ) ( )

( ) (0) ( ) ( ) (2.12)

Y s CX s DU s

C sI A x C sI A B D U s− −

= +

= − + − +

The transfer function (matrix): G(s) the matrix gains bet. U(s) and Y(s)

1( ) ( ) (2.13)G s C sI A B D−= − +

Assuming (0) 0x =

1111 1

11

1

( ) ( ) ( ) (2.14)( )( ) ( )( )

( ) ( )

( )( )

u

y y u

n

n n n

Y s G s U sN sG s G sD s

G sN s

G a sD s

=

⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

…

6

( ) : : ( ( )) ( ( ))( )

: : ( ) ( ) : 0

N s proper order N s order D sD s

strictly proper Q N Q Dwhen no input to output coupling D

≤

<− − =

1 1

( ) { ( )} { ( ) } (0) { ( ) } { ( )} (0) ( ) ( )At

y t Y sC sI A x C sI A B D U s

Ce x g t u t

− −

=

= − + − + ⊗

= + ⊗

LL L

-1

-1 -1 -1L

1 1

1 1 1

{( ) } (2.15)( ) { ( ) } { ( )} (2.16)( ) { ( )}

Ate sI Ag t C sI A B D G sG s g t

− −

− − −

= −

= − + ==

LL LL

Ex. 2.2 The state model 1 1 1

2 2 2

1 1

2 2

( ) ( ) ( )1 2 1 0 ;

( ) ( ) ( )0 3 0 4

( ) ( )1 1 ;

( ) ( )1 1The transfer function matrix is

(

x t x t u tx t x t u t

y t x ty t x t

G

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ ⎦

111 12

21 22

( ) ( ) 1 1 1 2 1 0)

( ) ( ) 1 1 0 3 0 4

1 41 1 .

1 4( 1)1 ( 1)( 3)

the impulse response matrix:

G s G s ss

G s G s s

s ss

s s s

−+ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥− +⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤⎢ ⎥+ +⎢ ⎥=

− −⎢ ⎥⎢ ⎥+ + +⎣ ⎦

-13

4 g(t)= { ( )} .

4 8

t t

t t t

e eG s

e e e

− −

− − −

⎡ ⎤= ⎢ ⎥−⎣ ⎦

L ♦

2.4 Frequency response

— 0( ) stu t u e= s: complex freq.

The steady-state output : (stable system)

[ ]( )( ) ( ) ( ) ( )

tA ty t Ce x g t u dτ τ τ− −∞

−∞= −∞ + −∫

0( )t sg t u e dττ τ−∞

= −∫

7

1

1( )1 10

( )t t

s t tg t e dt uτ= − ∞ −= ∫ 0

11 1 00

( ) st stg t e dt u e∞ −= ∫

0( ) (2.17)stG s u e=

When 0( ) j tu t u e ω= : a pure tone

. .

0( ) ( ) (2.18)s s

j ty t G j u e ωω⇒ =

( ):G j Fourier transfer functonω

2.4.1 Frequency response for SISO Systems { }

{ }( )0( ) ( ) j t j G jy t G j u e ω ωω + ∠=

( )0( ) ( ) j t j G jy t G j u e ω ωω + ∠=

( )G jω : the magnitude

{ }( )G jω∠ : the argument

the system freq. response the magnitudethe argument ⎧⎨⎩

{ }( ) cos( )

( ) ( ) cos ( )

u t t

y t G j t G j

α ω θ

ω α ω θ ω

= +

⎡ ⎤= + +∠⎣ ⎦

2.4.2 Frequency response for MIMO Systems

0 0

00

( )( ) ( )( )

pure j ttone

j t

G j u ey t G j uGain

u t uu e

ω

ω

ω ω= = =

2 21 2 unu u u u= + + ⋅⋅⋅+ 2

1 1cos( )

( )cos( )

u un n

t

u tt

α ω θ

α ω θ

+⎡ ⎤⎢ ⎥

= ⎢ ⎥⎢ ⎥+⎣ ⎦

8

1 1cos( )( )

cos( )y yn n

ty t

t

β ω φ

β ω φ

⎡ ⎤+⎢ ⎥

= ⎢ ⎥⎢ ⎥+⎣ ⎦

1

1

y

u

n

n

G ain

β

β

α

α

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦=⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0 0 0 0

0 00 01 1

0 0

( ) ( )( ) min ( ) min max max ( ) ( )

u u u u

G j u G j uG j u Gain G j u

u uω ω

σ ω ω ω σ ω= =

= = ≤ ≤ = =

SVD : singular value decomposition

{ }1

( ) ( ) ( ) ( ) (2.26)

min ,

1,

0,

p

i i ii

y u

i j i j

G j U V

p n n

i jU U V V

i j

ω σ ω ω ω+

=

+ +

=

=

⎧ =⎪= = ⎨≠⎪⎩

∑

1 2

,

0 ,u

p y

y u

n

if n n

if n n

σσ σ σ σ σ

⎧ ≥⎪= ≥ ≥ ≥ = = ⎨u

<⎪⎩

( ) : the principal gainsiσ ω

Ex 2.3 the state model

1 1

2 2

( ) ( ) ( )1 2 1 0( ) 0 3 ( ) 0 4 ( )

1 41 1

( )1 4( 1)

1 ( 1)( 3)

1

2

x t x t u tx t x t

j jG j

jj j j

ω ωω

ωω ω ω

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡ ⎤⎢ ⎥+ +⎢ ⎥=

− −⎢ ⎥⎢ ⎥+ + +⎣ ⎦

u t

9

the frequency response is 1 4

1 1( )

1 4( 1)1 ( 1)( 3)

j jG j

jj j j

ω ωω

ωω ω ω

⎡ ⎤⎢ ⎥+ +⎢ ⎥=

− −⎢ ⎥⎢ ⎥+ + +⎣ ⎦

and the principal gains in Fig 2.2

2.5.1 Poles and zeros for SISO systems 1G( ) ( )

( ) det(

det( )

( ) ( )

s C sI A B D

Cadj sI A B D sI AsI A

N sD s

−= − +

− + −=−

=

)

The zeros: iz ( ) 0iG z =

The poles: ip ( )iG p = ∞

10

The minimal transfer function : ( )G s

No common factors in num. and den. polynomials.

The minimal state-space realization:

det( )sI A− = den. of the minimal ( )G s

The zeros: ( ) 0iN z =

The poles: ( ) det( ) 0i iD p p I A= − =

Ex. : Poles and zero

[ ]

1 0 00 2 1

0 1

x x u

y x

−⎡ ⎤ ⎡= +⎢ ⎥ ⎢−⎣ ⎦ ⎣=

⎤⎥⎦

1 (( )( 1)( 2) ( )

s NG ss s D s

)s+= =

+ +

( 1) ( 1) ( 2) 0N D D− = − = − =

The pole: - 2 det( ) ( 1)( 2)sI A s s− = + +

2.5.2 Poles and zeros for MIMO systems For a minimal realization:

The poles: det( ) 0ip I A− =

The zeros: [ ] { }( ) min , i yrank G z n n< u

For square transfer function

The zeros: det[ ( )] 0iG z =

11

( ) 0 ( ) ( ) ( ) ( )

( ) ( ) ( )

0( )( )

0

det 0 (2.30)

Y s G s U s CX s DU s

sX s AX s BU s

sI A B X sC D U s

sI A Bzeros

C D

= = = +

= +

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

− −⎡ ⎤⇔ = ⇒⎢ ⎥

⎣ ⎦

Ex. 2.4 The state model

1 1

2 2

1 1

2 2

( ) ( ) ( )1 2 1 0( ) ( ) ( )0 3 0 4

( ) ( ) ( )1 1 1 0( ) ( ) ( )1 1 0 1

1

2

1

2

x t x t u tx t x t u t

y t x t u ty t x t

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

−⎡ ⎤ ⎡ ⎤ ⎡⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣u t

⎤⎥⎦

The transfer function is

2

41 1

( )1 8

1 ( 1)( 3)

ss s

G ss s

s s s

−⎡ ⎤⎢ ⎥+ +⎢ ⎥=

− − +⎢ ⎥⎢ ⎥+ + +⎣ ⎦

1

The poles:

1 2det ( 1)( 3) 0

0 3s

s ss

+ −⎡ ⎤= + + =⎢ ⎥+⎣ ⎦

Then 1 1p = − and 2 3p = − . The zeros:

12

2

2

41 1 7 12det ( ) det 0

1 8 1 ( 1)( 3)1 ( 1)( 3)

ss s s sG s

s s s ss s s

−⎡ ⎤⎢ ⎥+ + + −⎢ ⎥= =

− − + + +⎢ ⎥⎢ ⎥+ + +⎣ ⎦

=

Then and 1 8.42z = − 2 1.42z = . The direct computation of this

determinant yields a numerator and denominator of order 3. Cancellations always result in the determinant of the transfer function being a ratio of polynomials with orders less than or equal to the order of the state model; that is, the number of zeros is less than or equal to the order of the state model.

The zeros can also be found using:

2

1 2 1 00 3 0 4

det 7 12 01 1 1 01 1 0 1

ss

s s

+ − −⎡ ⎤⎢ ⎥+ −⎢ ⎥⎢ ⎥ = + − =⎢ ⎥−⎢ ⎥⎢ ⎥− −⎣ ⎦

2.5.3 Modes

For simple poles: ip te

For complex poles:

[ ]

[ ]

Re( )

Re( )

cos Im( ) ,

sin Im( )

i

i

p ti

p ti

e p

e p

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

t

t

For a pole with multiplicity k

k associated (natural) modes:

13

{ }1, , , i i ip t p t p tke te t e−

Ex. 2.5 The impulse response of the system

2

1( )( 2)( 2 5

sG ss s s

+=

+ + + )

As a linear combination of the modes:

2 (1 2 ) (1 2 )

1 2 3 4

21 5 6 4

( ) ( )

cos(2 ) sin(2 ) ( )

t j t j t

t t t

g t c e c e c e c t

c e c e t c e t c t

δ

δ

− − − − +

− − −

= + + +

= + + +

This expression, even without solving for the weights, contains most of the information about the impulse response that is required for control system design. Note that the weight on the impulse function equals zero whenever the transfer function is strictly proper, as in this example.

2.6 Stability

Bounded Input / Bounded Output – stable :

1. initial conditions = 0

2. 1 2( ) ( )i ju t M y t M< ⇒ <

-Causal LTIs :

0

1 0

1

1

( ) ( ) ( )

( )

t

j

t

u

u

ii

i

n

ji

n

ji

y t g u t d

M g d

τ τ τ

τ τ

=

=

= −

≤

∑∫

∑∫

-- ( )jy t bounded iff

14

0

( ) (2.33)jig dτ τ∞

< ∞∫ absolutely integrable.

-- a stable system :

( ) 0jig ∞ →

－ ( s.t. a temporary disturbance. ( ) 0y t → )t ↑

－

1 11

( ) ( )

n nt j tt j tji n ji

a i

g t e e e e D tσ ωσ ωα α δ= + + +

t j te eσ ωα is absolutely integrable.

0σ⇔ <

∴ A causal LTIs is stable iff Re( ) 0λ < .

-Internal Stability

BIBO stable with internal, unstable pole-zero cancellations

unbounded internal signals.

⇒

Ex. 2.6

F. 2.3 A system with pole-zero cancellation.

( ) 12( ) 1 2 11

1

sY s s ss

sR s ss s

−= = =− + ++

−s

: stable.

15

2 2( 1)

( ) 2( 1)2( ) 1 2 ( 1)1

1

sU s ss s

sV s s s ss s

− − −− −

= = =− + ++

−

: unstable.

2.6.1 Internal stability: All possible bounded inputs →all internal signals and all possible outputs bounded.

Figure 2.4

1 11 1

1 12 2

1 11

2

1 2 1 2

2 1 2 2

( ) ( ) (2.34a)

( ) ( )

( ) ( )

y u y u

y u y u

G Gy I GK G I GK GK u u

y I KG KG I KG K u G G u

e I KG I KG K

e

− −

− −

− −

⎡ ⎤⎡ ⎤+ − +⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

+ − +⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦

1 1

1 12 2

1 1 1 2

2 1 2 2

(2.34b)( ) ( )

e u e u

e u e u

G Gu u

I GK G I GK u G G u− −

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

1

2

Def.: Internally stable The eight transfer functions are stable. ⇔

1 1 2

2 2 1

1 1 1

2 2 2

1 1 1 2 1 1 1 2

2 1 2 2 2 1 2 2

0 0 .

0 0

0 0.

0 0y u y u u u

y u y u u u

e e

e e

e u ye u y

y e uI Iy e uI I

G G G GI IG G I G G I

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤

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

∵

Simplification : internally stable ⇔

,1 1 1 2 2 2 2 1, , and u u ue e e eG G G G u are stable.

i.e., The transfer function from the inputs

16

u1 and u2 to the errors e1 and e2 are stable. Alternatively : Internally stable ⇔

The transfer functions from the inputs u1 and u2

to the outputs y1 and y2 are stable. 2.7 Similarity Transformations

( ) ( ) ( )( ) ( ) ( )

x t Ax t Bu ty t Cx t Du t

= += +

Define a new state vector (change of basis):

1

1 1

( ) ( ) ( ) ( ) ( ) ( ) ( ) (2.36)

( ) ( ) ( ) ( ) ( ) (2.37a) ( ) ( ) ( ) ( ) (2.37b)

x t T x t x t Tx tTx t ATx t Bu tx t T AT x t T B u ty t CT x t Du t

−

− −

= ⇒ =

= +

⇒ = += +

The transfer function , poles and zeros are invariant. Ex. 2.7

[ ]

1 1

2 2

1

2

1 1

2 2

( ) ( )3 1 2( );

( ) ( )1 3 0

( ) ( ) 4 2 3 ( ).

( )

( ) ( )1 1.

( ) ( )1 1

x t x tu t

x t x t

x ty t u t

x t

x t x tx t x t

−⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡ ⎤= − +⎢ ⎥

⎣ ⎦⎡ ⎤ ⎡ ⎤⎡ ⎤

=⎢ ⎥ ⎢ ⎥⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ ⎦

[ ]

1

11

22

1

2

1 11 1 2 2 ; .1 1 1 1

2 2( )2 0 2( )

( );( )0 4 2( )

( ) ( ) 1 3 3 ( ).

( )

T T

x tx tu t

x tx t

x ty t u t

x t

−

⎡ ⎤⎢ ⎥⎡ ⎤

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

= +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦⎡ ⎤

= +⎢ ⎥⎣ ⎦

2.8.1 Controllability Def. : Controllable ⇔

0 By means of the input , any initial state 0( )x t x= can be

transferred to any other state ( )f fx t x= in a finite time.

17

LTI : Controllable ⇔

－Arbitrary pole assignment by state feedback.

－The controllability matrix

X( 1)n - AB A (2.39)B B⎡ ⎤= ⎣ ⎦C

has full rank, . un

－ 1det 0 , if xn =≠C , i.e., a single input.

2.8.2 Observability

Def. ：Observable ⇔

0( )x t at any can be determined 0t

from the input and output , ( )u t ( )y t 0 ft t t≤ ≤ < ∞

LTIs ：Observable ⇔

－observer poles can be placed at any location.

－control the rate of convergence of the estimates.

－the observability matrix

( 1 )

( 2 .4 0 )

xn

CC A

C A −

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

O

has full rank, xn

－ 1det 0 , if yn =≠O , i.e., a single output.

18

Ex. 2.8 Given the system described by the following state model：

1 1

2 1

( ) ( ) ( )1 2 1 0( ) 0 3 ( ) 0 4 ( )

1

2

x t x t u tx t x t

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦u t

1 1

2 1

( ) ( )1 1( ) 1 1 ( )

y t x ty t x⎡ ⎤ ⎡⎡ ⎤

=⎢ ⎥ ⎢⎢ ⎥−⎣ ⎦⎣ ⎦ ⎣ t⎤⎥⎦

⎤⎥

⎥

⇒ full rank ； 1 0 1 80 4 0 12

−⎡= ⎢ −⎣ ⎦

C

1 1 1 1

1 11 5

⎡ ⎤⎢ −⎢=⎢ ⎥− −⎢ ⎥−⎣ ⎦

O ⎥ full rank (a rank of 2).

● pole placement to meet specific transient performance like settling

time and damping ratio：

Re

Im

－root locus method , state feedback.

－greatly exceeding the transient performance Specifications.

19

⇒ unnecessarily large actuator inputs.

2.9 Observer Feedback State feedback design and observer design.

2.9.1 state feedback

( ) ( ) (2.42)u t Kx t= −

1 2det( ) ( )( ) ( )xnSI A BK S p S p S p− + = − − −

Ex. 2.9 pole-assignment

0 1 0( ) ( ) ( )

0 1 1x t x t

⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦

u t

A state feedback controller is desired that yields the following

closed-loop poles：

1 2 2p j= − + ； 2 2 2p j= − −

[ ]

( )( )( )

1 2

22 2

2

1 0 0 1 0det

0 1 0 1 1

1

2 2 2 2

4 8

s k

s k s k

s j s j

s s

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

= + + +

= + − + +

= + +

k

⇒ 1K =8 and =3 2K

∴ [ ]( ) 8 3 ( )u t x t= −

20

Figure 2.5

Ex. 2.10 (Multiple inputs and controllable)

11 12 131

21 22 232

1 2 33 2

12 23

12 23 23 12

0 1 0 0 0( ) 0 2 1 ( ) 1 0 ( ).

3 0 1 0 1

( )( ) ( ) ( )

( )2 3 , 2 3 , and 4.

det( ) 8 29 52.3 8

2

x t x t u t

k k ku tu t Kx t x t

k k ku tP j P j P

sI A BK s s sk kk k k k

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= − +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤= − = = − ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦= − + = − − = −

− + = + + ++ + =+ + + 11

11 11 23 22 22 13 13 21 13 21

2 293 3

kk k k k k k k k k k

+ =+ + − − − + + = 52

－An infinite number of gain matrices exit.

Extra freedom : to achieve design criteria other than pole placement.

－Full states measurement : expensive and even impossible.

21

2.9.2 Observers

－Estimate the plant states not measured :

( ) ( ) ( )x t Ax t Bu t= + Given the measured output :

( ) ( ) ( )m t Cx t Du t= +

－Reduce noise on measured state.

－The state estimate :

ˆ ( )x t ˆ( ) ( ) ( ) (2.45)Fx t Gm t Hu t= + +

Define the error :

ˆ( ) ( ) ( )

ˆ( ) ( ) ( )ˆ ( ) ( ) ( ) ( ) ( ) ( )

ˆ ( - ) ( ) ( ) ( ) ( ) (2.46) ( - ) ( )setting - and - .

e t x t x t

e t x t x tAx t Bu t Fx t GCx t GDu t Hu tA GC x t Fx t B GD H u tA GC e tF A GC H B GD

= −

= −= + − − − −= − + − −=

= =

Observability :

. 01 02 0det( ) ( )( ) ( ) (2.51)nsI A GC s p s p s p− + = − − −x

－A unique observer gain for SISO systems.

－Multiple gain matrices for observable MO systems.

Ex. 2.11 The ac motor :

0 1 0( ) ( ) ( )

0 1 1x t x t⎡ ⎤ ⎡ ⎤

= +⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦u t

The angular position is measured :

[ ] ( )( ) 1 0 .

( )t

m tt

θθ⎡ ⎤

= ⎢ ⎥⎣ ⎦

The desired observer poles : 01 02 8.p p= = −

The observer gain

22

[ ]1

2

21 1 2

2

0 0 1det( 1 0 )

0 0 1

(1 ) ( ) ( 8)( 8)

16 64

gsgs

s g s g g s s

s s

⎡ ⎤⎡ ⎤ ⎡ ⎤− + ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= + + + + = + +

= + +

∴G＝ ⎥⎦

⎤⎢⎣

⎡4915

F＝A－GC＝ ⎥⎦

⎤⎢⎣

⎡−−

−149

115

H＝B－GD＝ ⎥⎦

⎤⎢⎣

⎡10

The observer equation is then 15 1 15 0

ˆ ˆ ( ) ( ) ( ) ( )49 0 49 1

x t x t m t−⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦u t

ˆ(0)x should be the best guess at x(0)

Simulation:

[ ](0) 1 2 Tx = − and [ ]ˆ(0) 0 0 Tx = :

Figure 2.6

The plant state does not converge to zero since the plant is unstable.

23

Ex. 2.12 Observer Feedback An ac motor:

＝ ＋ )(.

tx ⎥⎦

⎤⎢⎣

⎡−1010

)(tx ⎥⎦

⎤⎢⎣

⎡10

)(tu

The measurement output:

＝ )(tm [ ]01 ⎥⎦

⎤⎢⎣

⎡

)(

)(.

t

t

θ

θ

The observer is designed with ＝ ＝－8: 1oP 2oP

15 1 15 0

ˆ ˆ( ) ( ) ( ) ( )49 0 49 1

x t x t m t−⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦u t

The state feedback using the state estimate:

[ ] ˆ( ) 8 3 ( )u t x t= −

The observer feedback control is simulated with [ ]ˆ(0) 0 0 Tx =

in Fig. 2.8

24

•Observer Feedback Control

?( ) ( ) ( ) ( )x t Fx t Gm t Hu t= + +

ˆ( ) ( )u t Kx t= −

The closed-loop system:

( ) 0 ( ) 0

( ) ( )ˆ0 ( )ˆ( )

x t A x t Bu t m t

F x t H Gx t⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

∵ ˆ( ) ( ) ( )e t x t x t= −

∴ ( ) 0 ( )ˆ( ) ( )

x t I x te t I I x t⎡ ⎤ ⎡ ⎤ ⎡

=⎢ ⎥ ⎢ ⎥ ⎢−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎤⎥ :similarity transformation.

∵ ( )x t ＝A ＋B )(tx )(tu

＝A －BK)(tx ˆ( )x t

＝(A－BK) ＋BK )(tx ( )e t

And ( ) ( ) ( )e t A GC e t= −

Then ( ) ( )( ) 0 ( )

x t A BK BK x te t A GC e t

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

The closed-loop poles:

det0

sI A BK BKsI A GC

− + −⎡ ⎤=⎢ ⎥− +⎣ ⎦

det( )det( ) 0sI A BK sI A GC− + − + =

•The deterministic separation principle:

The closed-loop poles＝the poles of the state feedback system＋

the poles of the observer.

The xn state feedback controller poles and xn observer poles are

designed independently with 4 ~ 10 ioip p= Disturbance rejection and measurement noise filtering:

Compromised pole locations.

25