Answer Happy

where index i in Aj, a; and hi refer to tank i and index j in k; and Yjrefer to pump i and valve j, with Aidenotes the cross section area of the tank i and hiits water level, a¡is is the cross section area of the outlet hole, k; is a constant and Y; € [0,1). Questions a. Denote the deviations from the equilibrium as Auị = Uị – uand Ahị = hi-hi, write down the equations which describe an equilibrium u, u,hi,hi,h, h3, h. b. Show that the linearized system is given by 0 1 1 AT3 0 2ks A1 0 0 22k2 dr dt A2T4 + 0 0 0 u 0 0 TE 01月 (1-12)k2 ཡྻ ས བ 0 (1-1) A4 0 0 y = 1 0 0 0 0 1 0 0 c Where Tiis a constant to be determined and --(A). -- -- U Au Au2 I Ah Ah, Ahz Ah4 Ahi Ah2 c. Show that the transfer matrix from u to y is given by G(s) = 91(s) 912(s) 921(s) 922(s) 31-1 ka 1+$T (1 - 1)k1c2 (1+871) (1+872) (1-2)ke1. (1+873)(1+671) 2622 1+81) Where Ci = Aj
d. The zeros of G(S) are given by the zeros of kikucior fa+s det G(S) (1+T3)(1+ STA) (1 - 71)(1 – 12)] 7172 II–1(1 + $T;) Discuss the values of Y and Y2 for which G(S) is minimum phase and for which G(S) is non- minimum phase e. Express the RGA associated to the transfer function matrix G(s) in terms of Y and Y2. Calculate the RGA matrix when Y1=Y2=0.625 (for minimum phase) and Y1=Y2=0.375 (for non-minimum phase). f. Given that Ti = 52, T2 = 85, k1C1 = 2.43 and k2C2 = 2.5. Design PI controllers of the form K(1 + for the transfer functions 911(s) and 922(s) such that their step response are characterized by a second order behavior with ratio damping $1 = 0.9 and Xi = 0.7 respectively for a natural frequency wo = 1 rad.s-1. g. Design decouple controllers (cross-controllers) to eliminate the interaction occurring between the inputs and outputs. < ww Reminder: Minimum phase is when all the poles and zeros are in left hand side of the s-plane and non-minimum phase is when at least one zero should be on the right side of the s-plane.