Final Exam for System Analysis - Fall 2002 | 332 501 | Exams Electrical and Electronics Engineering

Rutgers University

The State University Of New Jersey

College of Engineering

Department of Electrical and Computer Engineering

332:501 Systems Analysis Fall 2002

FINAL EXAMINATION

You have three hours to answer the following questions with point values as shown (total 150). You

are allowed two sides of handwritten notes on an



sheet of paper. Think through each problem

BEFORE you begin to write and don’t get stuck on one problem. Move on if you are stumped. YOU

MUST SHOW ALL WORK. ANSWERS GIVEN WITHOUT WORK RECEIVE NO CREDIT.

GOOD LUCK!

1. (100 points) A Smorgasbord of Systems Analysis Facts:

Here are a set of questions which test your general knowledge of the course material. In

many ways, they constitute the basic knowledge you MUST have. Keep your answers short

and to the point.

(a) (10 points) Suppose a set of



linearly independent vectors



span some space



Let







. Is there another different set of coefficients



such that





 

Why/why not?

(b) (10 points) A system

!#" $

operates on elements

of some metric space to produce

'!"(%)$

. If

is linear, what properties must it satisfy? Let

!" *,+.-0/1$2



*,+3-0/546

where

#798

is a constant. Is this system linear?

:*;4=<>-1*?8

. Is this equation linear? Is it time invariant? What is

*,+.-0/

-A@98

with

*,+B8C/ED

(d) (10 points) Describe the fundamental idea behind linearization of a nonlinear system.

Describe the method by which we perform linearization. Be general and precisely

mathematical starting from the differential equation

FG+.IHKJHL-0/

(e) (10 points) Carefully define the concept of “system state” for a system which evolves

with time.

(f) (10 points) The characteristic equation of an LTI system is

MNO4MQPO4MRSD

. Is this

system stable or unstable? How about

RTM NRUM PRUM2RVD

(g) (10 points) What is a Lyapunov function for a system? Find a Lyapunov function for

the system

W0XY

W1Z

4[<\W1Y

W1Z

4]*^_8

(h) (10 points) What is BIBO stability? If a general system is globally asymptotically

stable, must it be BIBO stable as well? NOTE: not just LTI.

(i) (10 points) For a continuous time system, what is the definition of complete controlla-

bility? How do we determine in general whether a linear system is completely control-

lable? What is the controllability condition for a linear time invariant system.

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Rutgers University The State University Of New Jersey College of Engineering Department of Electrical and Computer Engineering

332:501 Systems Analysis Fall 2002 FINAL EXAMINATION

You have three hours to answer the following questions with point values as shown (total 150). You

are allowed two sides of handwritten notes on an

sheet of paper. Think through each problem

BEFORE you begin to write and don’t get stuck on one problem. Move on if you are stumped. YOU

MUST SHOW ALL WORK. ANSWERS GIVEN WITHOUT WORK RECEIVE NO CREDIT.

GOOD LUCK!

(100 points) A Smorgasbord of Systems Analysis Facts: Here are a set of questions which test your general knowledge of the course material. In many ways, they constitute the basic knowledge you MUST have. Keep your answers short and to the point.

(a) (10 points) Suppose a set of linearly independent vectors span some space .

Let . Is there another different set of coefficients such that ?

Why/why not?

(b) (10 points) A system !#" $ operates on elements % of some metric space to produce

'!"(%)$. If! is linear, what properties must it satisfy? Let !" ,+.-0/1$2 ,+3-0/546

where #7 98 is a constant. Is this system linear?

(c) (10 points) Let ;4=<>-1?8:. Is this equation linear? Is it time invariant? What is *,+.-0/ ,

- A@98 with *,+B8C/ED?

(d) (10 points) Describe the fundamental idea behind linearization of a nonlinear system. Describe the method by which we perform linearization. Be general and precisely

mathematical starting from the differential equation FG+. IHKJHL-0/:.

(e) (10 points) Carefully define the concept of “system state” for a system which evolves with time.

(f) (10 points) The characteristic equation of an LTI system is MNO4MQPO4MRSD. Is this

system stable or unstable? How about RTM N RUM P RUM2RVD?

(g) (10 points) What is a Lyapunov function for a system? Find a Lyapunov function for

the system W0XY

W1Z X

[<\W1Y

W1Z

]*^_.

(h) (10 points) What is BIBO stability? If a general system is globally asymptotically stable, must it be BIBO stable as well? NOTE: not just LTI. (i) (10 points) For a continuous time system, what is the definition of complete controlla- bility? How do we determine in general whether a linear system is completely control- lable? What is the controllability condition for a linear time invariant system.

(j) (10 points) Repeat the previous question for observability.

(40 points) Swaying Bridges:

Once when driving into Rutgers from New York City, I got stuck on the George Washington bridge in a traffic jam. Nothing moved for about an hour. During that hour I noticed just how much a bridge deck jostles up and down in the wind. This problem is inspired by that experience. The abstraction of my situation is depicted in FIGURE 1

(a) (10 points) Derive a differential equation which describes the motion along the vertical

axis ( axis) of the mass at the center of the bridge. Assume the springs themselves are

massless. Assume the rest length of the springs is < and that when the mass in in

position ;_8 , there is no tension in the springs.

(b) (20 points) Is this system linear or nonlinear?

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GOOD LUCK!

(a) (10 points) Suppose a set of linearly independent vectors span some space .

Let . Is there another different set of coefficients such that ?

(b) (10 points) A system !#" $ operates on elements % of some metric space to produce

'!"(%)$. If! is linear, what properties must it satisfy? Let !" ,+.-0/1$2 ,+3-0/546

where #7 98 is a constant. Is this system linear?

(c) (10 points) Let ;4=<>-1?8:. Is this equation linear? Is it time invariant? What is *,+.-0/ ,

- A@98 with *,+B8C/ED?

mathematical starting from the differential equation FG+. IHKJHL-0/:.

(f) (10 points) The characteristic equation of an LTI system is MNO4MQPO4MRSD. Is this

system stable or unstable? How about RTM N RUM P RUM2RVD?

the system W0XY

W1Z X

[<\W1Y

W1Z

]*^_.

axis ( axis) of the mass at the center of the bridge. Assume the springs themselves are

massless. Assume the rest length of the springs is < and that when the mass in in

position ;_8 , there is no tension in the springs.

(c) (10 points) If we want a rest displacement of R D with a mass of 1000kg, what should

the spring constant be? ]ED .

M

K

F(t)

K

2L

Mg

Final Exam for System Analysis - Fall 2002 | 332 501, Exams of Electrical and Electronics Engineering

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Partial preview of the text

Download Final Exam for System Analysis - Fall 2002 | 332 501 and more Exams Electrical and Electronics Engineering in PDF only on Docsity!

GOOD LUCK!

(a) (10 points) Suppose a set of linearly independent vectors   span some space .

Let    . Is there another different set of coefficients  such that   ?

(b) (10 points) A system !#" $ operates on elements % of some metric space to produce

'!"(%)$. If! is linear, what properties must it satisfy? Let !" *,+.-0/1$2  *,+3-0/546

where #7 98 is a constant. Is this system linear?

(c) (10 points) Let ;4=<>-1?8:. Is this equation linear? Is it time invariant? What is *,+.-0/ ,

- A@98 with *,+B8C/ED?

mathematical starting from the differential equation FG+. IHKJHL-0/:.

(f) (10 points) The characteristic equation of an LTI system is MNO4MQPO4MRSD. Is this

system stable or unstable? How about RTM N RUM P RUM2RVD?

the system W0XY

W1Z X

[<\W1Y

W1Z

]*^_.

axis ( axis) of the mass at the center of the bridge. Assume the springs themselves are

massless. Assume the rest length of the springs is < and that when the mass in in

position ;_8 , there is no tension in the springs.

(c) (10 points) If we want a rest displacement of R D with a mass of 1000kg, what should

the spring constant  be? ]ED  .

M

K

F(t)

K

2L

Mg

(a) (10 points) Suppose a set of linearly independent vectors span some space .

Let . Is there another different set of coefficients such that ?

'!"(%)$. If! is linear, what properties must it satisfy? Let !" ,+.-0/1$2 ,+3-0/546

where #7 98 is a constant. Is this system linear?

(c) (10 points) Let ;4=<>-1?8:. Is this equation linear? Is it time invariant? What is *,+.-0/ ,

- A@98 with *,+B8C/ED?

mathematical starting from the differential equation FG+. IHKJHL-0/:.

(f) (10 points) The characteristic equation of an LTI system is MNO4MQPO4MRSD. Is this

the system W0XY

]*^_.

position ;_8 , there is no tension in the springs.

(c) (10 points) If we want a rest displacement of R D with a mass of 1000kg, what should

the spring constant be? ]ED .