Equações de Estado do Circuito

Prévia do material em texto

Problem 7.37PP
Consider the electric circuit shown in Fig.
(a) Write the internal (state) equations for the circuit. The input u(t) is a current, and the output y 
is a voltage. Let x1 = iL and x2 = vc.
(b) What condition(s) on R, L, and C will guarantee that the system is controllable?
(c) What condition(s) on R, L, and C will guarantee that the system is observable? 
Figure Electric circuit
Step-by-step solution
step ̂ of 7 ^
I
Step 2 of 7
As there are two energy storage elements, at least two state variables are required to
describe the system
Let
^ i= h
X2=V(j
at node,
CdvcU =tr + -------
‘ dt 
u = X i+C i^
:.X2 = — + —
^ C C
(1)
Step 3 of 7
Applying RVL,
+ V c+ icR = 0
where
ic = “ -*s
Writing the above eqi»tioa In terms of state variables.
+/? [a - i ^ ] = 0
-x ^ R -L x i +X2 +R u - XiR = 0
. -2 i? _^1
‘ L ' L ' L
and y = ic^
= u R - i iR 
= u R ~ XiR
(?)
(3)
Step 4 of 7
FromEqu(l), (2)& (3)
Xi - 22 r R '
L L LXj — - 1 ^ UJ 1
s . — 0
. c . c . (4)
Step 5 of 7
F =
-2R r
L L a =
'R '
L
=1 0
L c J
1
.c .
H = [-R 0] j = R
FO-.
-2R r R'
L L L
0 1
. c C.
2/? r
L
_ R
. LC
T = [ a j i o ] ]
Step 6 of 7
i
1
LC LC 
As for the system to be completely controllable,
-R ^ 1
I?C ~ C LC
-R-
IV VC
;e0
1 2 ^ ^ 
K — K + A ^ 0
r C
. -R ^C -L-¥ 2 R ^C ^
- i > 0
step 7 of 7
H ’ =
■-2^ -V
L c
1
. L
0
2R^'
L
-R
L .
F
2R}
L
L
d e t r = — > 0 ; R }> \
L