6.302 Feedback Systems

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science

Fall Term 2002 Problem Set 4

6.302 Feedback Systems

Issued : September 30, 2002 Due : Monday, October 7, 2002

Problem 1: More Root Locus Practice

For the following loop transmissions, sketch the root locus for positive and negative K, locate any points of interest (centroids, points of entry and exit, etc.), and specify whether the closed loop system will ever be stable.

1.

L(s)

=

K(s2 + 1) (s + 1)

(14)

2.

L(s)

=

K(s + 1)(s + 2) s3(s + 10)2(s + 100)2

(15)

3.

L(s)

=

Ks(s + 1.5) (s + 4)(s + 1)(8s - 1)(8s - 7)

(16)

Problem 2: Strange Root Locus

You are sitting in the final exam for 6.302, and upon turning to the second page, you are coPnSffrroangterdepwlaictehmtehnetbs lock diagram illustrated in Figure 1.

R

G1(s)

G2(s)

C

Figure 1: High-Stress System

where

G1(s)

=

0.9(s+

8 9

)

(s+

8 10

)

G2(s)

=

K (s+0.8)(s2+0.6s+64.09)

(17)

In order to maximize your suffering, the exam asks you to answer the following questions:

1. Sketch (by hand) the step response of G1(s). What is this type of behavior called, and why is the practice of avoiding this type of behavior recommended?

2. Sketch the root locus of L(s) = G1(s)G2(s).

3.

Use

MATLAB

to draw

the

step

response

of

C R

(s)

with

K

= 1.

Provide an

explanation

for the atypical characteristics of this plot.

Problem 3: Nyquist Mania

One of the most attractive features of the Nyquist criterion is that it provides a convenient

method for predicting what values of K will fall in a certain region of the complex plane. In

most circumstances, we are only interested in knowing when the closed loop poles leave the

left half plane. Rather than restrict ourselves to this mundane application of such a powerful

tool, consider the following problem. In designing a feedback system, your specifications

require

that

the

system

have

a

damping

ratio

less

than

cos-1(

4

).

Use Figures 2 and 3 to

meet this specification.

PSfrag replacements

j

r

45 45

PSfrag replacements

G(s)

R

K s+1

C

1

s

H (s)

Figure 3: Block Diagram

Figure 2: Nyquist Contour

1. Draw a Nyquist plot for the loop transmission L(s) = G(s)H(s) and the Nyquist contour as given above. Determine the values of K (both positive and negative) for which the poles of the closed loop system lie outside the shaded region. For these corresponding values of K, determine how many poles lie in the shaded region from the encirclement information in the diagram.

2. Plot the response of c(t) for r(t) = t, i.e. a unit ramp and the value of K being such that the poles of the closed loop system lie just on the border of the shaded region. Assume that K > 0. What is the value of tp for this particular c(t)?

Computer Project 4: Root Locus

This computer project should be completed using Octave, MATLAB or similar software. You may find it helpful to save your work as it may be useful for future projects. Please hand in clearly labelled printouts.

Compensation techniques using the Root Locus Rules

1. You were recently hired as a consultant to a small liberal arts college up the creek. They are somewhat distraught by the fact that they are using feedback and the system cannot be made stable. After a brief look you discover that the system is, in fact, a double integrator. You decide that the best way to compensate the

system is in a unity feedback loop with a series compensator. That is, the transfer function of the forward path is

L(s)

=

K

Gc(s) s2

,

where Gc is the compensating block.

(a) Assume Gc = 1. Is it possible to stabilize the system by varying K alone? Produce a root locus plot to demonstrate your answer.

(b) You decide to use a lead compensator. This is a compensator with a low frequency zero and a pole at a higher frequency. Being well aware of the Root Locus rules, you know that the pole must be placed at a high enough frequency that it will not affect the poles and zeros near the origin. Write down a transfer function for the compensator in the form

Gc(s)

=

s + 1 s + 1

where 10 and produce a root-locus plot for the compensated system. Over what range of K is the system stable?

2. Given your success with the double integrator, you decide to attempt to stabilise a triple integrator in a similar manner. Your forward path is

L(s)

=

K

Gc(s) s3

,

and you decide to compensate using two lead compensators

Gc(s)

=

(s + 2)2 (s + 100)2 .

(a) Produce a root-locus plot for positive and negative gains of the uncompensated and compensated system. Find the range of K over which the compensated system is stable.

(b) Using K = 2.5 ? 105 plot the closed-loop step response.

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