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Problem 7.39PP
(a) Using the capacitor voltage and inductor current as state-variables, write state and output 
equations for the system.
This problem is intended to give you more insight into controllability and observability. Consider
the circuit In Fig., with an Input voltage source u(t) and an output current y(t).
(b) Find the conditions relating R^. R2, C. and L that render the system uncontrollable. Find s 
similar set of conditions that result In an unobservable system.
(c) Interpret the conditions found In part (b) physically in terms of the time constants of the 
system.
system.
(d) Find the transfer function of the system. Show that there is a pole-zero cancellation for the 
conditions derived in part (b) (that is, when the system is uncontrollable or unobservable). 
Figure Electric circuit
Step-by-step solution
step 1 of 10
y « )
Step 2 of 10
From the above circuit,
y ( ‘ ) = ic + h
where ic = C ^
” dt
= C Jfi
in ly in g KVL to the circuit.
Writing the above equation in terms o f state variables.
-CX^ - X i + LX^ + = 0
The second loop equation is:
- t t + Xj + io ^ i = 0
= ~Xj + tt
ia = — +—
s , s,
(1)
(2)
(3)
(■4)
(5)
Step 3 of 10
Equating Equations (2) & (5),
■ '. . Xi = ----- Xi + ------ u
 ̂ R^C ^
(6)
Step 4 of 10
From Equations (4) & (6),
f l 1 1 „ 1
0 )
step 5 of 10
From Equations (1) & (5),
;y = ^ + — +x^
From equations (6),(7) & (8)
■ -1 o ' 1
R^C R^C
1 - A U J 1
L R^L ^
a -
■[-i ']H 5 }
(8)
(9)
Step 6 of 10
RG =
1 0 1
R^C R^C
1_ 1 -A 1
L R^L . A.
f e ) ’
3^- 1 I ^
■ Jt,
^ - W 'A C [ A c J
-1
B, B}LC
Step 7 of 10
For the system to be uncontrollable,
R^-\-¥R^R^LC 1
---------
B ^-\+ B iB ^L C = L
| f i ^ ( i + ^ Z c ) 7 l + I l
Step 8 of 10
1 ^ 1 -1 ' r
c) F 'H ’ ' = ~R^C R^L
0 -R^ 1 _
L a . a 1'
7* =
R^C R^L
A
- L + B f i { B i - \ )
B^LC
-A
H ’ ’ F ’ ’ / f ]
__1_ j;iC (fii-l)-£
1 -B ,
^ _ jj[C (i^-l)-£
Bi B^LC
B^B^LC-B jC + B f;+ L
For the sjrstem to be unobservable,
B iB ^LC -B ^C -^B iC + LC = 0 
B fH ^ B i-B ^ L - i) = LC
Step 9 of 10
det
T .F :
s l - F -O 
H J
d e t[f i/-F ] 
DENQMINATQR:
___l_
~ R^C 
^ 1 -1 
R,L
S + - L
R^C
I z A
RiL
0
S+R2
= s^ + —̂ s+J^s + - ^
R^C ^ R^C
R^C R^C
NU1£ERATQR:
Step 10 of 10
1
S +-----R̂C ° B f l
det
1 -R ,
R^L
s . B , - 1
1
• k \A
=
{ ie .cs+ i'l 
" 1 AC )
£ ± 5 l± 1 ] + 
A J
1 ] + A [ l A + i ± ^ l 
R,] r,c [ r,l j
1~
B,L J
« ^ C + l+ s C + J ^ C + C ̂ l - ^ i+ £ s + j ! , £
B^C ^ L C
^ si!i‘Z.C + + s j^ £ C + j! i^ Z .C + jj^ £ C + l- i i i+ £ s + /^ Z .
B lL C