Circle criterion

Overview

Consider a linear system subject to non-linear feedback, i.e., a nonlinear element \varphi(v, t) is present in the feedback loop. Assume that the element satisfies a sector condition [\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 [-1/\mu_1,-1/\mu_2] located on the x-axis.

General description

Consider the nonlinear system

: \dot{\mathbf{x}} = \mathbf{Ax} + \mathbf{Bw}, : \mathbf{v} = \mathbf{Cx}, : \mathbf{w} = \varphi(v, t).

Suppose that

1. \mu_1 v \le \varphi(v,t) \le \mu_2 v,\ \forall v,t
2. \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. \Re\left[(\mu_2 C(i\omega I_n-A)^{-1}B-1)(1-\mu_1C(i\omega I_n-A)^{-1}B)\right]

Then \exists c0,\delta0 such that for any solution of the system, the following relation holds:

:: |x(t)| \le ce^{-\delta t}|x(0)|,\ \forall t \ge 0.

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

References