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Question about the Proof of bounded Real Lemma

  1. Apr 9, 2013 #1
    My question is: it is possible to proof the bounded real lemma for $H_\infty$ performance with the following procedure?

    The $H_\infty$ performance is defined as:
    \begin{align}
    \parallel G_{ru}\parallel_{\infty} & <\gamma
    \end{align}
    where $G_{ru}$ represent the transfer from the input $u$ to the output $y$. It is commonly mentioned that this performance is achieved with the following criteria performance

    \begin{align}
    J_{ru}=\int_{0}^{\infty}y^{T}(t)y(t)d\tau & \leq & \gamma^{2}\int_{0}^{\infty}u^{T}(t)u(t)d\tau\
    \end{align}

    Then, I suppose that the proof of the lemma can done by deriving a Lyapunov equation through $J_{ru}$
    \begin{align}
    V(x)=X^TPX
    \end{align}

    such that

    \begin{align}
    J_{ru}=\int_{0}^{\infty}\left(y^{T}(t)y(t)- \gamma^{2}u^{T}(t)u(t)+\frac{d\dot{V}(x(t))}{dt}\right)d\tau-V(x)<0
    \end{align}

    Then by deriving the Lyapunov equation the typical representation of bounded real-lemma can be achieved. My question is about if this proof is correct?. Before hands thanks for your answer.
     
  2. jcsd
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