Stability in electronic circuits
dx
 f x , t 
dt
Definition 1
Consider a solution x0(t) generated by an initial condition x0(0) and a neighboring solution
x(t). The solution x0(t) is stable if for any ε > 0 there exists a δ > 0 such that if
x0  x 0 0  
then
x t   x 0 t   
for all
t0
If x0(t) is not stable, it is said to be unstable.
Definition 2
x0 is said to be asymptotically stable if it is stable, and if
lim x t   x 0 t   0
t 
x0t
2
x1t
2
x2t
0
t
Fig. 1
1
Autonomous equation
dx
 f x 
dt
 
x0- the equilibrium point, f x 0  0
x  x0  y
dy
 f y  x0
dt


dy
 gy
dt

g y   f y  x0

 
g 0   f x 0  0
0 is the equilibrium point of
dy
 gy
dt
dx
 f x , t 
dt
x  x1  xn 
T
x  x12    xn2
Stability of the circuit as a whole (global stability)
Definition 3
A circuit is said to be completely stable if for any solution x(t)
lim xt   0
t 
Definition 4
A circuit is said to be Lagrange stable if all solutions remain bounded as t →  ; that is,
given any initial condition x0  , there exists an M (a function of x0  ), such that xt   M
for all t  0 . If a circuit is not Lagrange stable, it will be called unstable.
Local stability, Lyapunov’s first method
dx
 f x 
dt
x  x0  y
2
dy
 f x0  y
dt


  
f x0  y  f x0 
dy
 Ay
dt

df 0
x y  h.o.t.
dx
 
y  0 - the equilibrium point
A
df 0
x
dx
 
Theorem 1
If all the eigenvalues of the matrix A have negative real parts, then the origin is asymptotically
stable. If at last one eigenvalue has positive real part, the origin is unstable.
Example 1
dx
 f x 
dt
i = f(v)
i1
iC
I2
i
C
v
vC
Is
I1
0
Fig. 2
C
Fig. 3
dvC
 I s  f vC 
dt
dvC
1
   f vC   I s 
dt
C
dvC
1
  i1
dt
C
1.
Is  0 ,
v
dvC
  f vC 
dt
v0  0 ,
3
A
1 df
C dvC
0
vC  0
2.
i1=( f (vc)-Is)
I s  I1
Ab  
1 df
vC  b   0
C dvC
Aa  
1 df
vC  a   0
C dvC
0 b
a
v=vc
Fig. 4
C
dv C
 i1
dt
vC  a  
dv C
0
dt
dv C
0
dt
  0 : i1  0
  0 : i1  0
vC  a
unstable
3.
i1=( f (vC)-Is )
Is
Aa  0
Ab  0
Ac  0
I1  I s  I 2
b
0
c
a
vC
Fig. 5
The direct method of Lyapunov: local stability
Definition 1
Consider a function V(x) = V (x1, ..., xn), defined in some neighborhood R of the origin. If
(i)
V(x) is continuously differentiable in R,
(ii) V(0) = 0,
4
(iii) V(x) > 0, x ≠ 0, x  R,
then V(x) is said to be positive definite in R
Definition 2
A function V(t, x), defined in some neighborhood R of the origin is positive define in R if
V(t, x) has continuous derivatives in R for t  0,
(i)
(ii) V(t, 0) = 0, t  0
(iii) there exists a scalar function W(x), positive definite, such that
W(x) ≤ V(t, x), x  R, t  0
Example
2 x

 3 x22 e  t is positive for all x ≠ 0 and for each fixed t, but does not fit the definion of a
positive definite function
2
1
Definition 3
Consider a neighborhood R of the origin and a function V(t, x), positive definite in R. V(t, x)
dx
is said to be a Lyapunov function for an equation
 f t , x  if
dt
dV
0
for all xR and t  0
dt
where
dV V V dx V V




f t , x 
dt
t x dt
t x
Theorem
If there exists a neighborhood R of the origin over which a Lyapunov function can be defined,
then the origin is stable
Example
L=1H
iL
iC
vC
R1 
1

2
C=1F
1
R2  
3
Fig. 6
5
dvC
 3vC  iL
dt
di L
1
 vC  iL
dt
2
 3vC  iL  0
1
 vC  iL  0
2
iL0   0
vC0   0 ,
v 
x  C
 iL 
T
1
1
1
1
V x   C vC2  L iL2  vC2  iL2
2
2
2
2
 dvC 
  3vC  iL 
 dt 
dv dV dx
 2 1 2





 vC iL 
  vC iL  v  1 i    3vC  iL 
d
i
dt dx d t
2 
 C
L

 L

2 
 dt 
x  0:
dV
0
dt
vC0   0 ,
6
iL0   0
is stable