• Source: 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 (LTI) systems.


      Overview


      Consider a linear system subject to non-linear feedback, i.e., a nonlinear element



      φ
      (
      v
      ,
      t
      )


      {\displaystyle \varphi (v,t)}

      is present in the feedback loop. Assume that the element satisfies a sector condition



      [

      μ

      1


      ,

      μ

      2


      ]


      {\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



      [
      −
      1

      /


      μ

      1


      ,
      −
      1

      /


      μ

      2


      ]


      {\displaystyle [-1/\mu _{1},-1/\mu _{2}]}

      located on the x-axis.


      General description


      Consider the nonlinear system








      x

      ˙



      =

      A
      x

      +

      B
      w

      ,


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






      v

      =

      C
      x

      ,


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






      w

      =
      φ
      (
      v
      ,
      t
      )
      .


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


      Suppose that





      μ

      1


      v
      ≤
      φ
      (
      v
      ,
      t
      )
      ≤

      μ

      2


      v
      ,

      ∀
      v
      ,
      t


      {\displaystyle \mu _{1}v\leq \varphi (v,t)\leq \mu _{2}v,\ \forall v,t}





      det
      (
      i
      ω

      I

      n


      −
      A
      )
      ≠
      0
      ,

      ∀
      ω
      ∈

      R

      −
      1



      and

      ∃

      μ

      0


      ∈
      [

      μ

      1


      ,

      μ

      2


      ]

      :

      A
      +

      μ

      0


      B
      C


      {\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




      ℜ

      [

      (

      μ

      2


      C
      (
      i
      ω

      I

      n


      −
      A

      )

      −
      1


      B
      −
      1
      )
      (
      1
      −

      μ

      1


      C
      (
      i
      ω

      I

      n


      −
      A

      )

      −
      1


      B
      )

      ]

      <
      0

      ∀
      ω
      ∈

      R

      −
      1


      .


      {\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



      ∃
      c
      >
      0
      ,
      δ
      >
      0


      {\displaystyle \exists c>0,\delta >0}

      such that for any solution of the system, the following relation holds:





      |

      x
      (
      t
      )

      |

      ≤
      c

      e

      −
      δ
      t



      |

      x
      (
      0
      )

      |

      ,

      ∀
      t
      ≥
      0.


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


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


      External links


      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


      References


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

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