In nonlinear control

Nonlinear control

Nonlinear control is the area of control engineering specifically involved with systems that are nonlinear, time-variant, or both. Many well-established analysis and design techniques exist for LTI systems ; however, one or both of the controller and the system under control in a general control...

, the circle criterion is an important theorem in showing the stability of nonlinear time-varying systems. It can be viewed as a generalization of the Nyquist stability criterion

Nyquist stability criterion

When designing a feedback control system, it is generally necessary to determine whether the closed-loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that overcompensates -- if the car drifts in one direction, the control system...

 for LTI 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 (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 located on the x axis.

General description

Consider the nonlinear system

Suppose that

1. 
2. is stable
3. 

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.

External links

• Sufficient Conditions for Dynamical Output Feedback Stabilization via the Circle Criterion
• Popov and Circle Criterion (Cam UK)
• Popov and Circle Criterion (ETH)