# 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

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