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(a) [2 marks] List the conditions placed on a controller Q(s), designed using affine param- to control a stable plant G.(s) with the control structure shown in Figure 1 eterisation, A(s) U(s) Y(s) Plant Gos) Figure 1: Block Diagram: Affine Parameterisation in terms of Q (b) [4 marks] Show that the nominal loop shown in Figure is internally stable for any stable nominal model G,(s) il Q(s) C(s)-1-Qis)G, (s) where Q(s) is any stable proper transfer function. R(s) U(s) Y(s) C(s) G₂(s) Figure 2: Block Diagram: Affine Parameterisation in terms of C (c) [3 marks] A typical requirement in the design of a controller is that S.w)| be small over a selected frequency bandwidth w, so as to reject output disturbances. Discuss the tradeoffs that have to be considered with respect to the other sensitivity functions in the loop if the bandwidth w, is large. (d) [6 marks] Using affine parametrisation, calculate the parallel PI controller parameter val ues K₂ (proportional gain) and K, (integral gain) in terms of the parameters of the nominal model, K Go(s)- TS + 1 and of the desired bandwidth ww.