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EE 141 Final

03/18/10Duration: 3 hours

The final is closed book and closed lecture notes. No calculators.

You can use a single page of handwritten notes.

Please carefully justify all your answers.

Problem 1: A simple model describing the concentration of insulin in patients with diabetesis given by:

x1 = 0.4x1 + 0.2x2 + 0.02x3 + ux2 = 0.2x1 0.4x2x3 = 0.2x1 0.02x3

where x1 describes the concentration of insulin in the plasma, x2 the concentration of insulinin the liver, and x3 the concentration of insulin in the body tissues. The variable u represents

the control input and describes the amount of insulin injected by the patients.

1. Derive the transfer function for this system by taking the concentration of insulin in theplasma as the output.

2. Is this system stable?

3. If we model an insulin injection as a step input of magnitude 0.5, what is the steadystate concentration of insulin in the plasma upon an injection?

4. The transfer function derived in question 1 can also be written in the following form:

G(s) =(s + 0.4)(s + 0.02)

(s + 0.6035)(s + 0.2102)(s + 0.006306)

We now seek to approximate this transfer function by eliminating the zeros and retainingtwo of the three poles. The possible approximations are:

G1(s) =k

(s + 0.2102)(s + 0.006306)

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G2(s) =k

(s + 0.6035)(s + 0.006306)

G3(s) = k

(s + 0.6035)(s + 0.2102)

where k and k are two positive gains. Which step response in Figure 1 corresponds toG1 and which corresponds to G2, and to G3? Carefully justify your answer.

0 100 200 300 400 500 600 700 800 9000

100

200

300

400

500

600

700

800Step Response

Time (sec)

Amplitude

0 5 10 15 20 25 300

1

2

3

4

5

6

7

8Step Response

Time (sec)

Amplitude

0 100 200 300 400 500 600 700 800 9000

50

100

150

200

250

300Step Response

Time (sec)

Amplitude

Figure 1: Step responses for Problem 1.

5. We will now use G2 to design a controller. To simplify the computations we will movethe poles slightly, take k = 1, and let our new transfer function be:

H(s) =

1

(s + 0.6)(s + 0.006)

Design a controller, placed in the feedback path, so that:

(a) the settling time is no more than one second;

(b) the rise time is no more than 0.9 seconds.

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Problem 2 Consider the transfer function:

G(s) = 54s + 4

2s2 + 4s + 10

1. Sketch the root locus for G.

2. Based on your root locus, design a proportional controller to reduce the settling time asmuch as possible.

3. Based on your root locus, what is the gain margin? Carefully justify your answer.

4. Verify your answer to the previous question by drawing the magnitude and phase plotsfor G. Recall that

5 2.2.

5. Design a compensator so that the magnitude plot decays at 40 dB/dec for all the fre-quencies greater than 100 rad/s and making the DC gain is 20 dB.

6. Can the closed-loop system track step inputs?

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