Circle criterion


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 systems.

Overview

Consider a linear system subject to non-linear feedback, i.e. a non linear element is present in the feedback loop. Assume that the element satisfies a sector condition, and 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 located on the x-axis.

General description

Consider the nonlinear system
Suppose that
  2. is stable
Then such that for any solution of the system the following relation holds:
Condition 3 is also known as the frequency condition. Condition 1 the sector condition.