MATLAB ASSIGNMENT HW #7

Due Tuesday, February 9

You go stereo shopping at a disreputable stereo shop for an amplifier that has a gain of about 40 dB and a bandwidth of about 20,000 Hz. The disreputable salesman sells you an amp with the following transfer function:

1.a) Use the Matlab bode command to make a Bode plot of your amp for the frequencies w = 100 to 106 radians/sec. Print out your bode plot (both magnitude and phase) and answer the following questions:

i) What is the actual bandwith of your amp in Hz?

ii) What is the pole of your amp?

iii) Is it asymptotically stable?

iv) What is the low frequency gain of your amp? Is it about 40 db?

In order to improve your amplifier, you invoke your new found knowledge of negative feedback and modify your amplifier in the following fashion:

b) Use the Matlab bode command to make a Bode plot of your modified amp for the frequencies w = 100 to 10⁶ radians/sec. Print out your bode plot (both magnitude and phase) and answer the following questions:

i) What is the actual bandwith of your modified amp in Hz?

ii) What is the pole of your modified amp?

iii) Is it asymptotically stable?

iv) What is the low frequency gain of your modified amp? Is it about 40 db?

c) You go shopping again and buy the latest CD by the group, UB40-IBONLY39. The title cut from this CD is quite repetitive and can be modelled by the following input signal, w(t)=10cos50000t. If your amp were ideal, the sinusoidal steady-state output due to this input would be y(t)=1000cos(50000t+0° ). Use your Matlab Bode plots for both the original amp and your modified amp to answer the following questions:

i) What is the aproximate steady-state output of your original amp due to the input w(t)=10cos50000t?

ii) What is the aproximate steady-state output of your modified amp due to the input w(t)=10cos50000t?

iii) Which amp produces an steady-sate output closer to the ideal output?

d) Suppose you accidently changed the negative feedback in your modified design to POSITIVE feedback (i.e., change the minus sign to a plus sign on the feedback summer). Answer the following questions about this accidental amplifier incorporating POSITIVE feedback:

i) What is the pole of your accidental amp?

ii) Is it asymptotically stable?

iii) Can you think of other examples of accidental POSITIVE feedback causing systems to go unstable (hint: think about your high school auditorium PA system)?

You graduate from UK and take a job with a major electronics corporation which is developing a DSS satellite tracking system to compete with RCA's DSS system. You are put in the position of designing the control for the satellite system. The current satellite system has the following transfer funditon where w(t) is the desired tracking angle and y(t) is the actual tracking angle:

Your manager instructs you to design a control system which meets the following specifications:

i) System is asymptotically stable and the steady-state error is zero when w(t) is a unit step. That is as t ® ¥ then y(t) ® w(t).

ii) The output due to a step settles to within 2% of its final value within 2 seconds (this will occur if all poles are to the left of the line s =-2.

2.a) Use the Matlab step command and make a plot of the unit step response of the original satellite system versus time. Does the system meet specifications?

b) In an effort to meet specifications, you use your newfound feedback knowledge to form a feedback loop around the satellite system as follows:

where K is a gain to be determined. Use the Matlab rlocus() command to make a plot of the poles of your feedback system as you vary k from 0.1 to 100 and answer the following questions:

i) Does the feedback system go unstable? If so, where does the root locus plot cross the jw -axis?

ii) Is there a point on the root locus where all three poles are to the left of the line s =-2? If yes, then try to estimate the value of K which produces these poles by repeatedly using the Matlab rlocus() command for various ranges of K.

c) Use this value of K and find the transfer function, Y(s)/W(s). Use the Matlab step() command to plot the unit step response of this new feedback transfer function. Use this plot to answer the following questions:

i) What is the steady-state error for the feedback system?

ii) What is the 2% settling time for the feedback system?

iii) Will your manager give you a big bonus?