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In order to find the optimal compensator K, for a given K it must first be established for which input the largest value of the 2 norm is achieved sup‖el S(w)w(u )y(w)l-dwl (3.31) 00 It can be shown, that this integral assumes its largest value if v is a sinusoid, modified in such a way that the 2 norm exists. This could for example be a signal equal to a const ant times a sinusoid for a long time interval and zero otherwise. The const ant must be chosen such that the signal has unit norm. The angular frequency of the sinusoid has to be the value for which S(u)w(w)l assumes its maximal value. Then sup‖ell=sup|s(ju)W(j)=‖sW‖ (3.32) Hence. the value of the error assumed for any v E v simply equals the Hoo norm of ompensator is the solution to the following minimization problem P2=SWHx=是s|s()W( (3.33) In other words, the Hoo optimal control minimizes the Hoo norm of the sensitivity function s weighted by w In comp arison, the H2 optimal compensator minimizes the mean value over all frequencies of the square of S(ju)w(u )12, whereas the Hoo optimal compensator minimizes the maximal S(u)w(w) By scaling wow) it is possible to formulate the Hoo nominal performance condition in the form Sgu)w gw)< 1 (3.35) The advant ages by applying the Hoo norm for specification of the performance requirements above the先2 norm are: The designer can bound the peak value of S(u) directly by chosing the input weight w Gw) appropriately Also using the input weight www), the designer can specify the desired bandwidth of the sensitivity function S(w), defined as the value of w where S(a) st ays beyond (3dB) Moreover, the Hoo norm facilit ates a tool for sp ecifying robust performance as we shall see t section 3. 7 Robust performance If a compensator is designed only based on requirement s for nominal performance and robust st ability, g might cont ain a model which is close to instability for the closed loop sy stem This is likely to give a very poor performance. To ensure the compensator to work well for all models in g, robust performance should is required for the models in g. ( " , ! ( " " , & ' & ,! = " & ' ! ( " ' ( " ' & & " ' < -! 7! & # " " C <! 6 " & ' ( & " " &