The normalized, scaled equations of a cart as drawn in Fig. 5.66 of mass mc holding an

Chapter 5, Problem 5.37

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The normalized, scaled equations of a cart as drawn in Fig. 5.66 of mass mc holding an inverted uniform pendulum of mass mp and length l with no friction are where is a mass ratio bounded by 0 < < 0.75. Time is measured in terms of = o t where . The cart motion y is measured in units of pendulum length as and the input is force normalized by the system weight . These equations can be used to compute the transfer functions In this problem you are to design a control for the system by first closing a loop around the pendulum, Eq. (5.96), and then, with this loop closed, closing a second loop around the cart plus pendulum, Eq. (5.97). For this problem, let the mass ratio be mc = 5mp . (a) Draw a block diagram for the system with V input and both Y and as outputs. (b) Design a lead compensation for the loop to cancel the pole at s = 1 and place the two remaining poles at 4 j4. The new control is U(s), where the force is V (s) = U(s) + D(s)(s). Draw the root locus of the angle loop. (c) Compute the transfer function of the new plant from U to Y with D(s) in place. (d) Design a controller Dc(s) for the cart position with the pendulum loop closed. Draw the root locus with respect to the gain of Dc(s). (e) Use MATLAB to plot the control, cart position, and pendulum position for a unit step change in cart position. Figure 5.66 Figure of cart pendulum for 5.37