Consider a unity feedback system where the open-loop transfer function is given by kg(s)h(-, (sa ros)- and k(s+z) is contributed by a pd controller. assume that the parameter α is determined mathematically to be 1 1) sketch the root locus for z = α = 1 . determine the range of k for which the system is stable. 2) if the actual parameter α is 1 .5 instead, sketch qualitatively the "actual" root locus 3) explain why one should not cancel an unstable open-loop pole (ie, choose z =-a ) in designing a controller

A3-mm-thick panel of aluminum alloy (k 177 w/m-k, c 875 j/kg-k and ? = 2770 kg/m) is finished on both sides with an epoxy coating that must be cured at or above t,-150°c for at least 5 min. the production line for the curing operation involves two steps: (1) heating in a large oven with air at ts,0-175°c and a convection coefficient of h, 40 w/m2. k, and (2) cooling in a large chamber with air at 25°c and a con- vection coefficient of he 10 w/m2.k. the heating portion of the process is conducted over a time interval te which exceeds the ime required to reach 150°c by 5 min (h = r + 300 s). the coating has an emissivity of ? = 0.8, and the temperatures of the oven and chamber walls are 175 and 25°c, respectively. if the panel is placed in the oven at an initial temperature of 25°c and removed from the chamber at a safe-to-touch tempera ture of 37°c, what is the total elapsed time for the two-step curing operation?