DP12.12 A benchmark problem consists of the mass–

spring system shown in Figure DP12.12, which rep

resents a flexible structure. Let m m 1 2 = = 1 and 0.5 ≤ ≤k 2.0 2[ ] 9 . It is possible to measure x t 1 ( ) and

x t 2 ( ) and use a controller prior to u t( ). Obtain the system description, choose a control structure, and de

sign a robust system. Determine the response of the system to a unit step disturbance. Assume that the output x t 2 ( ) is the variable to be controlled.

FIGURE DP12.12 Two-mass cart system. u(t) x(1) X(1) m m2 Input d(t) Disturbance