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Problem 7.40PP
The linearized dynamic equations of motion for a satellite are 
i = Ax + Ba, 
y = c*.
Where
■ 0 1 0 0 ' ■ 0 0 ■
A = W 0 0 2a> 
0 0 0 1 . B = 1 0 
0 0
0 -2o» 0 0 0 1
« _ r I 0 0 0 1 
[ o 0 1 Q J*
-[S]-
The inputs u ̂ and u2 are the radial and tangential thrusts, the state-variables x1 and x3 are the 
radial and angular deviations from the reference (circular) orbit, and the outputs y1 and y2 are 
the radial and angular measurements, respectively.
The inputs u ̂ and u2 are the radial and tangential thrusts, the state-variables x^ and x3 are the 
radial and angular deviations from the reference (circular) orbit, and the outputs y1 and y2 are 
the radial and angular measurements, respectively.
(a) Show that the system is controllable using both control inputs.
(b) Show that the system is controllable using only a single input. Which one is it?
(c) Show that the system is observable using both measurements.
(d) Show that the system is observable using only one measurement. Which one is it?
Step-by-step solution
step 1 of 9
(a)
Write the state description matrix.
A s
B s
0 0 0
3 ^ fro m equations (1) and (2).
AB«
A B s
0
3(»*
0
0
1
0
0
1 0 0 
0 0 
0 0 1 
0 0
(7)
-2 ® 0
Substitute equations (2) and (7) in equation (6).
■(8)
0 0 1 0
1 0 0 lea
0 0 0 1
0 1 -lea 0
From equation (8), the rank of matrix Is [ ^ . So this system Is controllable using both control 
inputs.
Step 2 of 9
(b)
Write the BiValueatzero upvalue. 
O'
...... (9)B,
Modify equation (6).
C ; = [ b , AB, ... A - ’ B , ] (10)
Calculate A B jfrom equations (1) and (9).
A B ,*
A B ,*
0 1 0 0
0 0 2d>
0 0 0 1
0 - 2 ^ 0 0 
1
0
0
-2a)
■ (11)
Calculate A^B,from equations (1) and (11).
A *B ,s
A *B ,s
0 1 0 0 ~ r ~
3B,=
0
0
...... (13)
2a>‘
Substitute equations (9), (11), (12) and (13) in equation (10).
.(14)
0 1 0
1 0 - o ’ 0
0 0 -2a> 0
0 - 2 « 0 2®’
Step 4 of 9
Writethe B^ value at zero u,value.
0
...... (15)B ,s
Modify equation (6).
C i= [ B , A B j ... A - % ] (16)
Calculate A B , from equations (1) and (15).
A B ,*
A B ,-
1
0
0 0
-2 B ,=
A > B ,=
Substitute equations (15), (17), (18) and (19) in equation (16).
0 1 0 0 2fl)
3®’ 0 0 2fl) 0
0 0 0 1 0
0 -2fl) 0 0 -4fl)*
0
-2fl)*
^ f l ) *
(19)
0
0 0 2d) 0
0 2fl) 0 - la / ...
0 0 - t o * ...
1 0 -4fl)^ 0 ...
(20)
From equations (14) and (20), the rank of matrix in C, matrix is [^and rank of matrix in Q 
matrix Is. So this system is controllable only in |B,|value at zero u,value.
Step 5 of 9
(c)
Write the general formula for observability matrix O- 
Write the observable condition from description matrices. 
C 
CA
(21)
Step 6 of 9
Calculate CA^rom equations (1) and (3).
CA
CA
0 0 
0
fl 0 0
' [ o 0 1
fo 1 0 0] 
' [ o 0 0 i j
■ 0 1 0 o '
3d)* 0 0 2d)
0 0 0 1
0 -2d) 0 0
(22)
Substitute equations (3) and (22) in equation (21).
1 0 0 O'
O s
0 0 1 0 
0 1 0 0 
0 0 0 1
(23)
Therefore, from equation (23), the rank of matrix is [ ^ . So this system is observable using both 
measurements.
Step 7 of 9
(d)
Writethe C, value from equation (3).
C ,= [l 0 0 0 ] ...... (24)
Modify equation (21).
 ̂ C,
o ,=
C,A
C,A'
C,A*
(25)
Calculate C.Affom equations (1) and (24).
■ 0 1 0 0 '
3®’ 0 0 2d)
0 0 0 1
0 -2d) 0 0
C ,A -[1 0 0 0]
C ,A -[0 1 0 0 ] ...... (26)
Calculate C,A*from equations (1) and (26).
C,A’ =[0 I 0 0]
C ,A '= [3 o ’ 0 0 2 =
■ 0 0 0
0 I 0 0
3d)* 0 0 2d) ...... (29)
0 -d)* 0 0
Step 8 of 9
Write the C, value from equation (3).
C, = [0 0 I 0 ] ...... (30)
Modify equation (21).
c,
C,A
0 ,= C,A’ ...... (31)
C.A’
C ,A -[0 0 I 0]
Calculate C ,A from equations (1) and (30).
0 1 0 0 '
3o’ 0 0 2o
0 0 0 1
0 -2 o 0 0
C,A = [0 0 0 1 ]...... (32)
Calculate C ,A ’ from equations (1) and (32).
0 1 0 0 ‘
C,A’ =[0 0 0 1]
3o’ 0 0 2o
0 0 0 1 
0 -2 o 0 0
C,A’ =[0 -2 o 0 0 ] ...... (33)
Step 9 of 9
Calculate C,A*from equations (1) and (33).
C,A’ =[0 -2 o 0 0]
C,A’ = [-6 o * 0 0 -4 o ’ ] ...... (34)
Substitute equations (30), (32), (33) and (34) in equation (31).
■ 0 1 0 0'
3d)̂ 0 0 2d)
0 0 0 1
0 -2d) 0 0
■ 0 0 1 0
0 0 0 1
0 -2d) 0 0
-6®* 0 0 -t®’
.(35)o , =
Therefore, from equations (29) and (35), the rank of matrix in matrix is [^and rank of matrix 
in 0,matt1xis. So this system is observable only in j^ jva lu e .