Circle criterion

Consider a linear system subject to non-linear feedback, i.e. a non linear element ${\displaystyle \varphi (v,t)}$ is present in the feedback loop. Assume that the element satisfies a sector condition ${\displaystyle [\mu _{1},\mu _{2}]}$, and (to keep things simple) that the open loop system is stable. Then the closed loop system is globally asymptotically stable if the Nyquist locus does not penetrate the circle having as diameter the segment ${\displaystyle [-1/\mu _{1},-1/\mu _{2}]}$ located on the x-axis.

Consider the nonlinear system

${\displaystyle {\dot {\mathbf {x} }}=\mathbf {Ax} +\mathbf {Bw} ,}$

${\displaystyle \mathbf {v} =\mathbf {Cx} ,}$

${\displaystyle \mathbf {w} =\varphi (v,t).}$

Suppose that

1. ${\displaystyle \mu _{1}v\leq \varphi (v,t)\leq \mu _{2}v,\ \forall v,t}$
2. ${\displaystyle \det(i\omega I_{n}-A)\neq 0,\ \forall \omega \in R^{-1}{\text{ and }}\exists \mu _{0}\in [\mu _{1},\mu _{2}]\,:\,A+\mu _{0}BC}$ is stable
3. ${\displaystyle \Re \left[(\mu _{2}C(i\omega I_{n}-A)^{-1}B-1)(1-\mu _{1}C(i\omega I_{n}-A)^{-1}B)\right]<0\ \forall \omega \in R^{-1}.}$

Then ${\displaystyle \exists c>0,\delta >0}$ such that for any solution of the system the following relation holds:

${\displaystyle |x(t)|\leq ce^{-\delta t}|x(0)|,\ \forall t\geq 0.}$

Condition 3 is also known as the frequency condition. Condition 1 the sector condition.

• Sufficient Conditions for Dynamical Output Feedback Stabilization via the Circle Criterion
• Popov and Circle Criterion (Cam UK)
• Stability analysis using the circle criterion in Mathematica

• Haddad, Wassim M.; Chellaboina, VijaySekhar (2011). Nonlinear Dynamical Systems and Control: a Lyapunov-Based Approach. Princeton University Press. ISBN 9781400841042.