In nonlinear control and stability theory, the circle criterion is a stability criterion for nonlinear time-varying systems. It can be viewed as a generalization of the Nyquist stability criterion for linear time-invariant (LTI) systems.
Consider a linear system subject to non-linear feedback, i.e., a nonlinear element
\varphi(v,t)
[\mu1,\mu2]
[-1/\mu1,-1/\mu2]
Consider the nonlinear system
x |
=Ax+Bw,
v=Cx,
w=\varphi(v,t).
Suppose that
\mu1v\le\varphi(v,t)\le\mu2v, \forallv,t
\det(i\omegaIn-A) ≠ 0, \forall\omega\inR-1and\exists\mu0\in[\mu1,\mu2]:A+\mu0BC
\Re\left[(\mu2C(i\omega
-1 | |
I | |
n-A) |
B-1)(1-\mu1C(i\omega
-1 | |
I | |
n-A) |
B)\right]<0 \forall\omega\inR-1.
Then
\existsc>0,\delta>0
|x(t)|\lece-\delta|x(0)|, \forallt\ge0.
Condition 3 is also known as the frequency condition. Condition 1 is the sector condition.