Controller Form for Multi-Input

Systems

EE-826 Linear Control System

College of Electrical & Mechanical Engineering

National University of Sciences & Technology (NUST)

Chapter-13

3

Controller Form

Fact

1 1 1

1 2 1 2 1 2[ ...... ...... .............. ...... ]

n n n

m m mW B B B AB AB AB A B A B A B− − −= ⋮ ⋮ ⋮

If is dependent on columns to its left then is also

dependent on columns to its left

q

rA B 1q

rA B+

Controllability Index

The controllability index is the least integer such that the

column is linearly dependent on the columns to its left

jρ

j

jA Bρ

1 2 11 1

1 1 1 2 2 2[ ...... ...... ......... ...... ]m

A m m mW B AB A B B AB A B B AB A Bρρ ρ −− −= ⋮ ⋮ ⋮

4

Controller Form

1

2

n

M

MM

M

=

⋮

1

AM W−=

5

Controller Form

1

1

1

1

1 2

1 2

2

1 2

1 2

1 2

1

1 2

1

1

1

. . . .

. . . .

1

. . . .

m

m

m

m

M

M A

M A

M

M A

PM A

M

M A

M A

ρ

ρ

ρρ

ρ ρ

ρ ρ

ρρ ρ

ρ ρ ρ

ρ ρ ρ

ρρ ρ ρ

−

−

+

+

−

−+

+ +

+ +

−+ +

=

⋮

⋯

⋮

⋯

⋮

⋯

⋮

6

Controller Form

7

Controller Form

8

Controller Form

9

Controllability and Observability

Proof

Part 1: If controllability gramian is invertible

10

Controllability and Observability

Proof

Part 1: If the system is controllable

Contradiction argument

11

Controllability and Observability

Proof

12

Controllability and Observability

13

Controllability and Observability

Proof

14

Controllability and Observability

15

Controllability and Observability

Example 9.6

16

Controllability and Observability

Observability

17

Controllability and Observability

Observability

Discrete-time Reachability and

Observability

Chapter-25

19

Reachability and Observability

20

Reachability and Observability

21

Reachability and Observability