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Problem 7.06PP
A certain system with state x is described by the state matrices
- [ = - ] ■ - = [ 1 ] -
C = [ 1 0 ] , D = 0 .
Find the transformation T so that if x = Tz, the state matrices describing the dynamics of z are in 
controi canonical form. Compute the new matrices A, B, C, and D.
Step-by-step solution
step 1 of 2 
Step 1 of 2
(a)
Consider the state description matrix.!l.(1)
B
■[:;;]
C = [l 0] (3)
D = 0 ......(4)
Write the general formula for ^matrix.
A = T 'A T ..... (5)
Where,
X is transformation matrix.
Write the general formula for entire transformation matrix,
t,A 
t
T"' = . ......(6)
Consider the general formula for inverse of the transformation matrix, . 
f . = [0 0 ... l]C -‘ (7)
Consider the general formula for controllability matrix. C ■
C = [ b AB ... A - ’b ] ..... (8)
Calculate AB^rom equations (1) and (2).
k ;
- [ ' J
oil 3
AB = | . I ......(9)
Substitute equations (2), (9) in equation (8).
^̂ =[3 J
Calculate t^from equations (7) and (10).
- [ I t ]
Write the general formula for t,- 
t ,= t jA ......(12)
Calculate t|from equations (1), and (11) in equation (12).
'■[I t K ;i
- [ ? I] (13)
Calculate entire transformation matrix, j - t from equation (6). (11) and (13). 
3 "
5 5
3 -1
L5 5.
Calculate transformation matrix, x ftoid equation (14).
(14)
T = L J ......(15)
Substitute equations (1), (14) and (15) in equation (5).
d 1
K ® :i
(16)
Consider the general formula for q .
B s T ' B ...... (1^)
Substitute equations (2) and (14) in equation (17).
B s
l l I
5 5
3 ^
5 5J
(18)
Write the general formula for q .
C = C T ..... (19)
Step 2 of 2
Substitute eauations (3) and (15) in eauation (19).
"i[! 3
C = [l 3] ......(20)
Write the general formula for ^ .
D ^ D ...... (21)
Substitute equations (4) in equation (21).
D ^O
Hence, the value of new matrices are
A =
-2 -2 ] 
.1 oJ
B =
i]
c = > 3] 
0