# WUSTL ESE 543 - ESE 543 Homework 7 Freq Domain (3 pages)

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## ESE 543 Homework 7 Freq Domain

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- Pages:
- 3
- School:
- Washington University in St. Louis
- Course:
- Ese 543 - Control Systems Design by State Space Methods

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10 28 2012 Due 2 Weeks Homework 6 1 Consider the model of the distillation column with profiles of concentration and temperature Extractant proponal glycol FA Water and isoproponal mixture Vapor sidestream T1 T2 T1 S U z1 Heating Stream water z2 Bottom product T2 glycol Concentration Temperature glycol A LTI model of this process is described by the following block diagram K1 K2 z 1 s u H s z 2 where the plant is given by H s 3 04 s 278 2 s s 6 02 s 30 3 0 052 s 206 6 s 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 I L and I L 1 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 I L j 0 3883 and min I L 1 0 4526 j 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 Z 1 Z Az VZ 0 VZ M q q 1 q 0 0 q M 0 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 K Az s K q s aq K a s az Kq s s s 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 I L for L determined in part a 4 Plot I L 1 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 actuator 10 28 2012 Due 2 Weeks n2 c s 2 2 n s n2 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 response

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