Här är toppresultaten vid de tre senaste högskoleproven. - Expressen

objective is to find a stabilizing controller which minimizesthe H ∞ -norm of the closed-loop transfer function from w toz that is given by the linear fractional transformationz = T zw w = G 11 + G 12 K(I − G 22 K) −1 G 21 w (8)Thus the H ∞ control problem is formalized as synthesizing acontroller K such that‖T zw ‖ ∞ < γ zw (9)for some value γ zw > 0.V. CONTROL DESIGNThe plant G from the general control problem in Figure 3is selected as the EDFA state space model in (7). It followsthat the average inversion level is taken as the internal state,x ⇒ N 2 (t), the disturbance signal is selected as the inputchannel powers w ⇒ p IN , the measurement signal v ⇒ y Gand the control signal u ⇒ p p . However the original plantG is modified so that the performance variable is taken asthe average inversion z ⇒ N 2 (t), as opposed to the outputchannel powers.Given a pilot tone disturbance with the following properties:• Disturbance signals reside in a specified bandwidth[f min ,f max ] where f max < T ransmissionBitRate(typically on the order of Gb/s and Tb/s)• Disturbance signals have a maximum modulation indexm maxwe now provide a methodology for designing a suitable H ∞controller.Fig. 4.w pu pW wĜK ∞W yG pz py GpWeighted augmented EDFA for H ∞ controller synthesisBased on this format we propose that the following parametersbe selected accordingly:• Choose w L and w H to describe the frequency bandwidth(rad/s) where the input signal resides. ie. w L =2π · f minand w H =2π · f max• Choose α w to describe the maximum amplitude or modulationdepth of the input disturbances. ie. α w =1+m max• Choose k w < 1 to describe the amount of cross gainmodulation within the disturbances.C. Integral ControlIt is a desired performance objective to minimize the steadystate error of output signals. This can be accomplished byensuring that the average total gain y G is maintained. Inorder to achieve this, integral control should be imposed onthe resulting H ∞ controller. But, in order to meet the H ∞assumptions, the selected weighting function W y must beproper, in the form:A. Plant AugmentationIn order to meet the H ∞ assumptions in [15], the originalEDFA model must augmented. Specifically the EDFA modelin (7) must be augmented to ensure that D 12 has full columnrank. In order to meet this assumption, ẑ is introduced:[ ] [ ]C1 x + Dẑ = =11 w + D 12 u(10)z˜zαuwhere α is a design parameter to be selected. This gives theaugmented plant G ⇒ Ĝ with modified state space matrices[ ] [ ]DˆD1111 = , DˆD120 12 =(11)αThe selection of α represents a control penalty, limiting thecontrol signal the designer is able to provide to the EDFA.Therefore, the selection of α should reflect the EDFA’s pumpcharacteristics, most notably the pump response time andmaximum deliverable pump power.B. Disturbance WeightingThe selection of the disturbance weighting is strictly basedon characteristics of the input disturbances of the system.Specifically, using the known transfer function for a secondorder bandpass filter the disturbance weighting W w is used todescribe the frequency content of pilot tone disturbances:W w =k w · w H · s(s + w L )(s + w H ) + α w (12)15W y = β ys + k y(13)s + α yBased on the above form, we select the following parameters:• Choose k y =1as the controller gain• We select the bandwidth in which integral control shouldbe imposed. That is, we choose β y