- Source: Popov criterion
In nonlinear control and stability theory, the Popov criterion is a stability criterion discovered by Vasile M. Popov for the absolute stability of a class of nonlinear systems whose nonlinearity must satisfy an open-sector condition. While the circle criterion can be applied to nonlinear time-varying systems, the Popov criterion is applicable only to autonomous (that is, time invariant) systems.
System description
The sub-class of Lur'e systems studied by Popov is described by:
x
˙
=
A
x
+
b
u
ξ
˙
=
u
y
=
c
x
+
d
ξ
{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+bu\\{\dot {\xi }}&=u\\y&=cx+d\xi \end{aligned}}}
u
=
−
φ
(
y
)
{\displaystyle {\begin{matrix}u=-\varphi (y)\end{matrix}}}
where x ∈ Rn, ξ,u,y are scalars, and A,b,c and d have commensurate dimensions. The nonlinear element Φ: R → R is a time-invariant nonlinearity belonging to open sector (0, ∞), that is, Φ(0) = 0 and yΦ(y) > 0 for all y not equal to 0.
Note that the system studied by Popov has a pole at the origin and there is no direct pass-through from input to output, and the transfer function from u to y is given by
H
(
s
)
=
d
s
+
c
(
s
I
−
A
)
−
1
b
{\displaystyle H(s)={\frac {d}{s}}+c(sI-A)^{-1}b}
Criterion
Consider the system described above and suppose
A is Hurwitz
(A,b) is controllable
(A,c) is observable
d > 0 and
Φ ∈ (0,∞)
then the system is globally asymptotically stable if there exists a number r > 0 such that
inf
ω
∈
R
Re
[
(
1
+
j
ω
r
)
H
(
j
ω
)
]
>
0.
{\textstyle \inf _{\omega \,\in \,\mathbb {R} }\operatorname {Re} \left[(1+j\omega r)H(j\omega )\right]>0.}
See also
Circle criterion
References
Haddad, Wassim M.; Chellaboina, VijaySekhar (2011). Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach. Princeton University Press. ISBN 9781400841042.