10/28/2012 Due: 2 Weeks Homework 6 1. Consider the model of the distillation column with profiles of concentration and temperature: Extractant(glycol)FA(Water and isoproponal mixture)Vapor sidestreamSUHeating StreamBottom product(glycol)T1T2Concentration Temperaturez2z1 waterproponalg lycolT1T2 A LTI model of this process is described by the following block diagram H(s)K1K2∆s∆u∆z1∆z2 where the plant is given by H(s) = 3.04s-278.2s(s+6.02)(s+30.3)0.052s-206.6s(s+6.02)(s+30.3) and the proportional controller is K1 = 5 and K2 = -10. a) Apply the multivariable Nyquist theorem to this system, examining the return difference matrix I+KH, K = diag{K1,K2}, and determine stability. Turn in the multivariable Nyquist plot. This is a plot of the [ ]det I KH+, indicate the number of encirclements. b) Plot the singular values of the return difference matrix and stability robustness matrix versus frequency. Compute the singular value gain and phase margins for this system. This is a plot of [ ]ILσ+ and 1ILσ−+ versus frequency. Plot these using a log scale for frequency and magnitude in dB.10/28/2012 Due: 2 Weeks Partial answer: Using Matlab I calculated ( )min 0.3883I Ljσωω+= and ( )1min 0.4526IL jσωω−+=. c) Turn in your Matlab code used to compute the answers above. 2. Consider the missile problem presented in class. A block diagram for the system is The state space model the for the missile is given as 10001 0zZZA VZ VZMMqq q qαδαδαδαα αδδ         =+=+                   where: Za = -1.3046 (1/s); Zd = -0.2142(1/s); Ma = 47.7109 (1/s2); Md = -104.8346 (1/s2);V = 886.78 (fps); The autopilot (the controller) is a classical PI controller where ( )( )( )( )qqazAz qK saK saK s Ksss++= = where Ka=-0.0015; Kq=-0.32; az=2.0; aq=6.0; a) Build state space models for the 1) controller, 2) the closed loop system, 3) the loop transfer function L at the plant input. b) Simulate the closed loop system to a unit step response. Compute the rise time and settling time for the acceleration command response. c) Evaluate this design in the frequency domain. 1) Plot a Nyquist plot for L determined in part a) with axis scaling limited to +/- 3. 2) Plot a Bode plot for L determined in part a). 3) Plot ( )ILσ+ for L determined in part a). 4) Plot ( )1ILσ−+ for L determined in part a). 5) Compute classical stability margins from part 1) and 2) and singular value stability margins from part 3) and 4). 3. a) Repeat problem 2) and include in the missile dynamics a second order actuator10/28/2012 Due: 2 Weeks 2222nc nnssωδδ ζω ω=++ where wn = 150 (rps) and z = 0.6. b) What happens to the stability and performance of the system if the actuator natural frequency is reduced to wn = 60 (rps)? What is the lowest actuator natural frequency (using these same gains) that produces a closed loop stable