7. (a) Using your favorite continuous-time modeling software (such as LabVIEW Simulink, or Ptolemy II), construct a model of the helicopter control system shown in Figure 2.4. Choose some reasonable parameters and plot the actual angular velocity as a function of time, assuming that the desired angular veloc- ity is zero, (t) 0, and that the top-rotor torque is non-zero, Tt(t) but) Give your plot for several values of K and discuss how the behavior varies with K. (b) Modify the model of part (a) to replace the Controller of Figure 2.4 (the sim- ple scale-by-K actor) with the alternative controller shown in Figure 2.6. This 40 Lee &Seshia, Introduction to Embedded Systems 2. CONTINUOUS DYNAMICS Controller Figure 2.6: A PI controller for the helicopter. alternative controller is called a proportional-integrator (PI) controller. It has two parameter Ki and K2. Experiment with the values of these parame- ters, give some plots of the behavior with the same inputs as in part (a), and discuss the behavior of this controller in contrast to the one of part (a) Helicopter Controller2 Scale2 Integrator2 Controller4 Scale3 Integrator3 ControllerS 1/K Original Control System Scale Integrator Controller3 1/K Figure 2.4: (a) Helicopter model with separately controlled torques for the top and tail rotors. (b) Transformation to an equivalent model (assuming K >0). (c) Further transformation to an equivalent model that we can use to understand the behavior of the controller 36 Lee & Seshia, Introduction to Embedded Systems