Modern Control Systems
Dr. Mohammad Al Janaideh
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Eigenvalues and Diagonalization in Modern Control Systems, Slides of Automatic Controls
Automatic Control Automatic Control
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2021/2022
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Modern Control Systems
Dr. Mohammad Al Janaideh
Eigenvalues
- Eigenvalues of an n×n matrix
- The eigenvalues of an n×n matrix A are the roots of the
characteristics equation: I A 0
A ,
I A
1 2 3 0
(^36 )
( )( )( )
1 1 , 2 2 , 3 3
Diagonalization
1 2 1
0 0 0 1
0 0 1 0
0 1 0 0
a a a a
A n (^) n n ...
... ....... ...
... X Pz
1 ^1 2 ^13 ^1 ^1
121 222 233 2
1 1 1 1
n n n nn
n
n P
...
... ....
...
...
...
A has n distinct eigenvalues
n
P AP
...
... ....
...
...
...
0 0 0
0 0 0
0 0 0
0 0 0 3
2
1 1
Example (1)
A u
x
x
x x
x
x
3
2
1
3
2
x
x
x y
(1) The Eigenvalues matrix A are : λ 1 = - 1 , λ 2 = - 2, λ 3 = - 3 (2) The matrix P are : λ 1 = - 1 , λ 2 = - 2, λ 3 = - 3
2 12 22 3
1 2 3
1 1 1
P
1 4 9
1 2 3
1 1 1 P
Example (1)
0 0 3
0 2 0 1 1 0 0 P AP
CP 1 1 1
P B
The Eigenvalues (^) P ^1 AP identical to the eigenvalues of the A. (Proof)
Proof:
The Eigenvalues identical to the eigenvalues
of the A.
P ^1 AP
6 11 6 3 26 G ( s ) (^) s s s
3
3 2
6 1
3 G ( s ) (^) s s s
0 0 3
0 2 0
1 0 0 A
B C 1 1 1
(^) 1 2 s 3 C s
B s
A
Transformation of System Models with Matlab Using Matlab we can transform the system model from transfer function to state space.
transfer function to state space :^ We use the following Matlab command to transform the
UY^ (( ss ) ) numden
[A,B,C,D]=tf2ss(num,den) We use the following Matlab command to transfer the state space to transfer function: [num,den]=ss2tf(A,B,C,D)
Example (2)
a
A
A=[-2 -8 -8 -2;1 0 0 0; 0 1 0 0; 0 0 1 0]; >>B=[1;0;0;0]; C=[0 2 >>D=0; 1 9] ;
>>[num,den]=ss2tf(A,B,C,D)
num = 0 -0.0000 2.0000 1.0000 9. den = 1.0000 2.0000 8.0000 8.0000 2.
B
C 0 2 1 9