Controller And Observer Form, Lecture Notes - Linear System And Control, Study notes of Production Planning and Control

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Linear Control Systems

Lecture # 11

Controller and Observer Forms

Controller Form For Multi-input Systems

x

Ax

Bu

A, B

) is controllable

A

is

n

×

n,

B

is

n

×

m,

m >

B

has full rank

The controllability matrix has

n

linearly independent

columns

B

= [

b

1

, b

2

,... , b

m

]

C

= [

b

1

,... , b

m

, Ab

1

, Ab

m

,... , A

n

−

1

b

1

,... , A

n

−

1

b

m

]

We search for the

n

linearly independent columns of

C

from

left to right

Definition:

For

j

,... , m

, the controllability index

μ

j

is

the least integer such that the column

A

μ

j

b

j

is linearly

dependent on the columns to its left in the controllabilitymatrix

μ

1

μ

m

n

Example:

A

B

Define an

n

×

n

nonsingular matrix

C

by

C

= [

b

1

, Ab

1

,... , A

μ

1

−

1

b

1

,... , b

m

, Ab

m

,... , A

μ

m

−

1

b

m

]

C

−

1

M

1

M

μ

1

M

μ

1

M

μ

1

μ

2

M

μ

1

···

μ

m

−

1

M

n

Q

M

μ

1

M

μ

1

A

M

μ

1

A

μ

1

−

1

M

n

M

n

A

M

n

A

μ

m

−

1

It can be shown that

Q

C

is nonsingular

Q

is nonsingular

B

c

A

c

and

B

c

can be written as

A

c

A

c

B

c

A

m

B

c

B

c

B

m

A

c

= Block diag

μ

i

×

μ

i

, i

,... , m

B

c

= Block diag

μ

i

×

1

, i

,... , m

Example:

Reconsider the pair

A, B

of the previous

example

μ

1

, μ

2

, μ

3

Matlab Calculations:Cbar =

[B(:

A

B(:

A

2

B(:

B(:

A

B(:

B(:

3)]

M = inv(Cbar); Q = [M(

:); M(

A; M(

A

2

M(

:); M(

A ; M(

:)]

P = inv(Q); PI = Q;Ac = PIAP; Bc = PI*B;

A

c

B

c

rank [

sI

A

c

B

c

]

[

I

n

A

m

I

m

]

n,

s

rank [

sI

A

c

B

c

] =

n,

s

because

[

I

n

A

m

I

m

]

is nonsingular

A

c

B

c

) is controllable

Hence,

A

c

, B

c

is controllable if and only if

A

c

B

c

is

controllable. It can be shown that

A

c

B

c

is controllable

Remark:

Controllers forms are not unique. Different

choices of linearly independent columns of thecontrollability matrix lead to different controller forms Remark:

There is no loss of generality in assuming that

B

has full rank.

Suppose

rank

B

r < m

There exists an

m

×

m

nonsingular matrix

R

such that

BR

= [

B

1

0]

where

B

1

is

n

×

r

and

rank

B

1

r

Bu

BRR

−

1

u

= [

B

1

0]

[

u

1

u

2

]

B

1

u

1

Use the

r

-dimensional vector

u

1

as the control input

Define the polynomial matrices

s

and

S

s

by

s

s

μ

1

s

μ

2

s

μ

m

m

×

m

S

s

) = Block diag

1 s^ ...

s

μ

i

−

1

μ

i

×

1

, i

,... , m

Verify that

sI

A

c

S

s

B

c

s

Define the polynomial matrices

D

s

and

N

s

by

D

s

B

−

1

m

[Λ(

s

A

m

S

s

)]

N

s

C

c

S

s

D

c

D

s

D

s

is

m

×

m

and

N

s

is

p

×

m

. Verify that

det(

D

s

is a polynomial of degree

n

Claim :

sI

A

c

S

s

B

c

D

s

LHS = (

sI

A

c

B

c

A

m

S

s

B

c

s

B

c

A

m

S

s

RHS = ¯

B

c

B

m

B

−

1

m

[Λ(

s

A

m

S

s

)] = ¯

B

c

[Λ(

s

A

m

S

s

)]

Corollary :

sI

A

c

−

1

B

c

S

s

D

−

1

s