Linear Systems Ch13 [Compatibility Mode]
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Transcript of Linear Systems Ch13 [Compatibility Mode]
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