do not do homework until it is assigned. the …

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EE 5323 Homeworks Spring 2017

Updated: Saturday, April 08, 2017 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. Some homework assignments refer to the textbooks: Slotine and Li, etc. For full credit, show all work. Some problems require hand calculations. In those cases, do not use MATLAB except to

check your answers.

It is OK to talk about the homework beforehand. BUT, once you start writing the answers, MAKE SURE YOU WORK ALONE. The purpose of the Homework is to evaluate you individually, not to evaluate a team. Cheating on the homework will be severely punished. The next page must be signed and turned in at the front of ALL homeworks submitted in this course.

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EE 5323 Nonlinear Control Systems

Homework Pledge of Honor

On all homeworks in this class - YOU MUST WORK ALONE.

Any cheating or collusion will be severely punished.

It is very easy to compare your software code and determine if you worked together It does not matter if you change the variable names.

Please sign this form and include it as the first page of all of your submitted homeworks.

.......................................................………………………………………………………………........

Typed Name: ____________________________________________

Pledge of honor:

"On my honor I have neither given nor received aid on this homework.”

e-Signature: ________________________________________

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EE 5323 Homework 1 State Variable Systems, Computer Simulation 1. Simulate the van der Pol oscillator 0')1(" 2 yyyy using MATLAB for various ICs.

Plot y(t) vs. t and also the phase plane plot y'(t) vs. y(t). Use y(0)=0.1, y'(0)= 0.1 a. For = 0.04. b. For= 0.85.

2. Do MATLAB simulation of the Lorenz Attractor chaotic system. Run for 150 sec. with all

initial states equal to 0.4. Plot states versus time, and also make 3-D plot of x1, x2, x3 using PLOT3(x1,x2,x3).

2133

31212

211 )(

xxbxx

xxxrxx

xxx

use = 10, r= 28, b= 8/3. 3. Consider the Voltera predator-prey system

1 1 1 2

2 2 1 2

x x x x

x x x x

.

Simulate the system using MATLAB for various initial conditions. Take ICs spaced in a uniform mesh in the box x1=[-2,2], x2=[-2,2]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5].

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EE 5323 Homework 2 Nonlinear Systems and Equilibrium 1. Consider the Voltera predator-prey system

1 1 1 2

2 2 1 2

x x x x

x x x x

.

Find the equilibrium points and their nature. 2. Duffing’s equation is interesting in that it exhibits bifurcation, or dependence of stability

properties and number of equilibrium points on a parameter. The undamped Duffing equation is

3 0x x x a. Find the equilibrium points. Show that for 0 there is only one e.p. b. For 0 there are 3 eps. Linearize the system and study the nature of these 3 e.p.s c. Simulate the Duffing oscillator and make time plot and phase plane plot. Do for

a. 1 b. 0.1 c. 1

3. Consider the system

2

2

(1 )

(1 )

x y x y

y x y x

.

Simulate the system using MATLAB for various initial conditions for the two cases:

a. Take ICs spaced in a uniform mesh in the box x1=[-10,10], x2=[-10,10]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-15,15]x[-15,15].

b. Take ICs spaced in a uniform mesh in the box x1=[-3,3], x2=[-3,3]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5].

4. The system of equations

21 1 1 2 1

22 2 1 2 2

x ax bx x cx

x dx ex x fx

describes the growth of two competing species that prey on each other. The constants are positive parameters. Pick a=c=d=f=2, b=e=3. Simulate the system using MATLAB for various initial conditions. Take ICs spaced in a uniform mesh in the box x1=[-2,2], x2=[-2,2]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5].

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EE 5323 Homework 3 Vector Fields, Flows, First Integrals 1. Consider the undamped oscillator

0x x a. Write position-velocity state space form ( )X f X . b. Plot the trajectories ( ), ( )x t x t vs. time. Use initial conditions of (0) 0.1, ( ) 0x x t

c. Plot the vector field 1 1 2

2 1 2

( , )( )

( , )

f x xf X

f x x

in the phase plane 1 2( , ) ( , )x x x x . Plot for

points spaced in a uniform mesh in the box x1=[-10,10], x2=[-10,10]. d. Plot the system trajectories (flows or orbits) in the phase plane. Take ICs spaced in a

uniform mesh in the box x1=[-10,10], x2=[-10,10]. e. Derive the First Integral of Motion 1 2( , )F x x as done in class. Plot the FIM as a 3-D

surface over the phase plane on the x1=[-10,10], x2=[-10,10]. 2. Repeat for the unstable system

0x x

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EE 5323 Homework 4 E.P.s, Lyapunov Stability Analysis 1. Equilibrium points and linearization System is

1 2 1 2

2 1 1 2

( 1)

( 1)

x x x x

x x x x

a. Find all equilibrium points b. Find Jacobian c. Find the nature of all e.p.s

2. Use Lyapunov to examine the stability of these systems. Simulate time histories from many uniformly spaced ICs to verify your results.

a. 2

1 1 2 1

22 1 2 2

x x x x

x x x x

b. 2

1 2 1 1

2 1

( 2)x x x x

x x

3. Use Lyapunov to show that the system

)3(

)3(2

22

1222

12

22

211

2211

xxxxxx

xxxxxx

is locally asymptotically stable. Find the Region of Asymptotic Stability. Simulate the system from many uniformly spaced ICs. 3. Use Lyapunov to show that the system

2 2

1 2 1 1 2

2 22 1 2 1 2

( 1)

( 1)

x x x x x

x x x x x

is UUB. Simulate the system from many uniformly spaced ICs.

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EE 5323 Homework 5 Lyapunov’s Method 1. UUB of system with disturbance.

Consider the system on S&L p. 66 with a disturbance d ( ) 0x c x d

Assume that 2( )xc x ax with 0a a known positive constant

a. Assume that d is unknown but is bounded by d D with D a known positive constant.

Prove that the system is UUB and find the bound on x(t). b. Assume that d is unknown but is bounded by d D x with D a known positive constant.

Prove that the system is UUB and find the bound on x(t).

2. UUB

Use Lyapunov to show that the system

)3(

)3(2

22

1222

12

22

211

2211

xxxxxx

xxxxxx

is uniformly ultimately bounded UUB. That is, show that the Lyapunov derivative is NEGATIVE OUTSIDE A BOUNDED REGION. Find the radius of the bounded region outside which V <0. Any states outside this region are attracted towards the origin. 3. Use Lyapunov Equation to check the stability of the linear systems

a. 0 1

6 5x Ax x

b. 4 2

1 1x Ax x

c. 0 1

4 0x Ax x

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EE 5323 Homework 6 Lyapunov Controls Design, Feedback Linearization 1. On Slotine & Li p. 71 we used the Lyapunov function

4 2 21 2

1( ) ( 2 10)

2V x x x

and we obtained Lyapunov derivative 10 6 4 2 21 2 1 2( ) (4 12 )( 2 10)V x x x x x

Plot ( )V x and ( )V x using MATLAB. Pick a region for the domains that reveals a nice plot. 2. A system is given by

1 2 1

2 1 2

sgn( )x x x

x x x u

Select Lyapunov function candidate 2 211 22( ) ( )V x x x

Use Lyapunov to design a controller u(x) to make system SISL.

3. For each of these systems, Use Lyapunov to Design feedback control u(x) to make the system i. SISL, and then ii. AS

a. 2

1 1 2 1

22 1 2

x x x x

x x x u

b. 2

1 1 2

22 1 2 2

x x x u

x x x x

c. 2

1 1 2 1

2

x x x x

x u

4. Multi-input Control. Use Lyapunov to design controls 1 2,u u to make this system

i. SISL, and then ii. AS

21 1 2 1

3 72 1 2 2

x x x u

x x x u

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EE 5323 Homework 7

I/O Feedback Linearization 1. A system is given by

1 2

42 1 2

sin

cos

x x

x x x u

with output 1( ) (t)y t x

a. Design a FB linearization controller to make the output follow a desired trajectory ( )dy t

That is, find ( )u t b. Discuss the internal dynamics. Are they a problem?

2. NMP System A system is given by

4

2 5( 1) 0

y zy u

z y z z

a. Take output ( )y t and find the FB linearization controller ( )u t to follow the prescribed

trajectory ( )dy t .

b. Find the internal dynamics. Set ( ) 0y t to get the zero dynamics. Are the ZD stable? Does the FB linearization controller work?

3. Effect of Output Choice in i/o FB Linearization It is desired to stabilize a system given by

2

212

1121 sin

xxx

uxxxx

a. Select the output as 1xy and use FB lin. design to select the control u(t) to follow the

desired trajectory ( )dy t . Check the internal dynamics. Set y=0 to get the zero dynamics.

Is the system minimum phase? b. Select the new output 2xy . Find the FB lin. controller u(t). Does this work? What

about the internal dynamics?

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DO NOT WORK BEYOND THIS PAGE The following hwks have not yet been assigned and may change.

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EE 5323 Homework 3 Chaos, Phase Plane 1. A system that exhibits chaos is the logistic function

)1()1(1 kkkk xxx

However, chaos only occurs for certain values of k . Rather than try all values of k , we can

sweep through the k values using

kk 1

for fixed less than but close to 1. These two equations form a dynamical system. Perform a MATLAB simulation to reproduce this plot of xk vs k, which was taken for 9995.0

and initial value of 0 3 . Interpret the plot with

some discussion in terms of bifurcation theory. Plot also k . Show your MATLAB code.

It is indeed interesting that the logistic function appears in economic systems and military supply systems. 2. Slotine and Li p. 39 problem 2.2. Simulate and plot phase plane trajectories for various ICs. Do not do the problem requested in the book. 3. Slotine and Li p. 35 Example 2.7b. Show that this system has an unstable limit cycle.

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EE 5323 Homework 4 Lyapunov Stability Analysis 4. Slotine and Li p. 97 problem 3.1. 5. Use Lyapunov to show that the system

)3(

)3(2

22

1222

12

22

211

2211

xxxxxx

xxxxxx

is locally asymptotically stable. Find the Region of Asymptotic Stability 6. Use Lyapunov to examine the stability of these systems. Simulate time histories from many

uniformly spaced ICs to verify your results.

c. 2

1 1 2 1

22 1 2 2

x x x x

x x x x

d. 1 2 1 1

2 1 1 2

sin

sin

x x x x

x x x x

e. 2

1 2 1 1

2 1

( 2)x x x x

x x


2 2

1 2 1 1 2

2 22 1 2 1 2

( 1)

( 1)

x x x x x

x x x x x

has a stable limit cycle. Simulate the system from many uniformly spaced ICs.

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c. Assume that d is unknown but is bounded by d D with D a known positive constant.

Prove that the system is UUB and find the bound on x(t). d. Assume that d is unknown but is bounded by d D x with D a known positive constant.


2. UUB


)3(

)3(2

22

1222

12

22

211

2211

xxxxxx

xxxxxx

is uniformly ultimately bounded UUB. That is, show that the Lyapunov derivative is NEGATIVE OUTSIDE A BOUNDED REGION. Find the radius of the bounded region outside which V <0. Any states outside this region are attracted towards the origin. 3. Use Lyapunov Equation to check the stability of the linear systems

d. xx

56

10

e. 4 2

1 1x Ax x

f. 0 1

4 0x Ax x

2. Barbalat’s Lemma and LaSalle extension

a. Use quadratic Lyapunov Function to show this system is locally AS

2

1 2 1 1

2 1 2

( 2)x x x x

x x x

Find a ball within which 0V . This region is contained in the region of attraction.

b. Use quadratic Lyapunov Function to show this system is locally SISL

2

1 2 1 1

2 1

( 2)x x x x

x x

Find a region within which 0V .

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c. Use Barbalat’s Lemma to verify that 0V . Check uniform continuity of the Lyapunov

derivative of the system in part b.

d. Use LaSalle’s extension to verify that the system in part b is actually AS. Find the equilibrium point.

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EE 5323 Homework 6 Lyapunov Controls Design, I/O Feedback Linearization 1. A system is given by

1 2 1

2 1 2

sgn( )x x x

x x x u

Select Lyapunov function candidate 2 211 22( ) ( )V x x x

Use Lyapunov to design a controller u(t) to make system SISL. 2. A system is given by

uxxx

xx

24

12

21

cos

sin





2

212

1121 sin

xxx

uxxxx

c. Select the output as 1xy and use FB lin. design to select the control u(t) to follow the


Is the system minimum phase? d. Select the new output 2xy . Find the FB lin. controller u(t). Does this work? What


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EE 5323 Homework 7 Feedback Linearization, backstepping 1. I/O feedback linearization. Slotine and Li Problem 6.3. System is

1 2

42 1 2

1

sin

cos

x x

x x x u

y x

Do i/o feedback linearization to make output track desired trajectory ( )dy t .

2. Backstepping. The system is

1 2

42 1 2 1

sin

cos

x x u

x x x x

Do backstepping to stabilize this system. Select the desired value 1dx to yield the first step

dynamics of 2 2 0x x .

Compare this to Problem 1, which uses i/o FB linearization. 3. Backstepping. Slotine and Li Problem 6.11. ‘Globally stabilize’ means backstepping. System is

4 5

2 5( 1) 0

y zy u

z y z z

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The following homeworks are out of date. Do not do homeworks until they are assigned. They may change.

18

Old hwk 1 2014

8. Obtain the linear model of the system described by

2( 1) 0y y y y around the equilibrium point. 9. The system of equations

21 1 1 2 1

22 2 1 2 2

x ax bx x cx

x dx ex x fx

describes the growth of two competing species that prey on each other. The constants are positive parameters and it is assumed that the two states are positive. Determine the linear model of the system around the equilibrium point (0,0).

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EE 5323 Homework 2 Nonlinear Systems and Equilibrium 10. Consider the Voltera predator-prey system

1 1 1 2

2 2 1 2

x x x x

x x x x

.

a. Find the equilibrium points and their nature. b. Simulate the system using MATLAB for various initial conditions. Take ICs spaced in a

uniform mesh in the box x1=[-2,2], x2=[-2,2]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5].

11. Consider the system

2

2

(1 )

(1 )

x y x y

y x y x

.

Simulate the system using MATLAB for various initial conditions for the two cases:

c. Take ICs spaced in a uniform mesh in the box x1=[-10,10], x2=[-10,10]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-15,15]x[-15,15].

d. Take ICs spaced in a uniform mesh in the box x1=[-3,3], x2=[-3,3]. Make one phase plane plot with all the trajectories on it. Plot phase plane on square [-5,5]x[-5,5].


21 1 1 2 1

22 2 1 2 2

x ax bx x cx

x dx ex x fx

describes the growth of two competing species that prey on each other. The constants are positive parameters. Simulate the system using MATLAB for various initial conditions. Make one phase plane plot with all the trajectories on it. 13. Duffing’s equation is interesting in that it exhibits bifurcation, or dependence of stability

properties and number of equilibrium points on a parameter. The undamped Duffing equation is

3 0x x x d. Find the equilibrium points. Show that for 0 there is only one e.p. e. For 0 there are 3 e.p.s Linearize the system and study the nature of these 3 e.p.s f. Simulate the Duffing oscillator for 1 . Make time plot and phase plane plot.

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EE 5323 Homework 3 Chaos, Phase Plane 1. A system that exhibits chaos is the logistic function

)1()1(1 kkkk xxx



kk 1

for fixed less than but close to 1. These two equations form a dynamical system. Perform a MATLAB simulation to reproduce this plot of xk vs k, which was taken for 9995.0

and initial value of 0 3 . Interpret the plot with

some discussion in terms of bifurcation theory. Plot also k . Show your MATLAB code.

It is indeed interesting that the logistic function appears in economic systems and military supply systems. 2. For Slotine & Li Example 2.2 on P. 20- a. Find equilibrium points b. Linearize the system about each equilibrium point. Find poles in each case. c. Simulate the system to find the Region of Attraction. 3. Slotine and Li p. 39 problem 2.2. Simulate and plot phase plane trajectories for various ICs. Do not do the problem requested in the book. 4. Slotine and Li p. 35 Example 2.7b. Show that this system has an unstable limit cycle.

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EE 5323 Homework 4 Lyapunov Stability Analysis 14. Slotine and Li p. 97 problem 3.1. 15. Use Lyapunov to show that the system

)3(

)3(2

22

1222

12

22

211

2211

xxxxxx

xxxxxx

is locally asymptotically stable. Find the Region of Asymptotic Stability 16. Use Lyapunov to examine the stability of these systems. Simulate time histories from many

uniformly spaced ICs to verify your results.

f. 2

1 1 2 1

22 1 2 2

x x x x

x x x x

g. 1 2 1 1

2 1 1 2

sin

sin

x x x x

x x x x

h. 2

1 2 1 1

2 1

( 2)x x x x

x x


2 2

1 2 1 1 2

2 22 1 2 1 2

( 1)

( 1)

x x x x x

x x x x x

has a stable limit cycle. Simulate the system from many uniformly spaced ICs.

22




e. Assume that d is unknown but is bounded by d D with D a known positive constant.

Prove that the system is UUB and find the bound on x(t). f. Assume that d is unknown but is bounded by d D x with D a known positive constant.


2. UUB


)3(

)3(2

22

1222

12

22

211

2211

xxxxxx

xxxxxx

is uniformly ultimately bounded UUB. That is, show that the Lyapunov derivative is NEGATIVE OUTSIDE A BOUNDED REGION. Find the radius of the bounded region outside which V <0. Any states outside this region are attracted towards the origin.

3. Lyapunov Theorem for Control Design. A system is given by

uxxx

uxxx

23

22

111

a. Use Lyapunov Linearization Method to show that the open-loop system with u(t)= 0 is unstable about the origin.

b. Select the nonlinear feedback control input 22xu . Find the closed-loop system. Use

a Lyapunov extension to show that the nonlinear closed-loop system is UUB.

That is, select the quadratic Lyapunov function and find V along the closed-loop system trajectories. Then show that V is negative outside a region (i.e. if x is large enough).

If you cannot solve for x such that ( )V x is negative, then plot ( )V x using MATLAB and draw conclusions about stability.

c. Discuss the stability. When is the Lyapunov derivative negative? Can you use a LaSalle Extension to show AS?

4. Use Lyapunov Equation on p. 81 to check the stability of the linear systems

g. xx

56

10

h. 4 2

1 1x Ax x

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EE 5323 Homework 6 Redo Exam 1 and turn in as Hwk 6 1. Lyapunov Functions- limit cycle, UUB Use Lyapunov functions to examine the stability of the following systems. Be clear and show all steps.

a. Pick a suitable complicated Lyapunov function to study the stability of the limit cycle for the system

3 2 4

1 2 1 1 2

2 42 1 2 1 2

6 ( 3 3)

( 3 3)

x x x x x

x x x x x

b. Use the standard simple quadratic Lyapunov function to show that this system is UUB.

Describe the region outside which the Lyapunov derivative is negative.

3 2 4

1 2 1 1 2

2 2 42 1 2 2 1 2

6 ( 3 3)

6 ( 3 3)

x x x x x

x x x x x x

2. Barbalat’s Lemma and LaSalle extension

e. Use quadratic Lyapunov Function to show this system is locally AS

2

1 2 1 1

2 1 2

( 2)x x x x

x x x

Find a ball within which 0V . This region is contained in the region of attraction.

f. Use quadratic Lyapunov Function to show this system is locally SISL

2

1 2 1 1

2 1

( 2)x x x x

x x

Find a region within which 0V .

g. Use Barbalat’s Lemma to verify that 0V . Check uniform continuity of the Lyapunov derivative of the system in part b.

h. Use LaSalle’s extension to verify that the system in part b is actually AS. Find the

equilibrium point. 3. Equilibrium points and linearization

System is 1 2 1 2

2 1 1 2

( 1)

( 1)

x x x x

x x x x

d. Find all equilibrium points

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e. Find Jacobian f. Find the nature of all e.p.s

4. Lyapunov Equation, AS, SISL

a. Use Lyapunov equation to show this system is AS 0 1

8 6x Ax x

Use Q=I>0. Is the solution P to the Lyapunov equation unique?

b. Use Lyapunov equation to show this system is SISL 0 1

0 6x Ax x

Use 0 0

00 1

Q

. Now, you have to find ANY positive definite P that solves the Lyapunov

equation. Is the solution P unique for this case?

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EE 5323 Homework 7 i/o Feedback Linearization 1. A system is given by

uxxx

xx

24

12

21

cos

sin




2. NMP System A system is given by

0)1( 52

54

zzyz

uzyy

a. Take output ( )y t and find the FB linearization controller ( )u t to follow the prescribed

trajectory ( )dy t .

b. Find the internal dynamics. Set ( ) 0y t to get the zero dynamics. Are the ZD stable? Does the FB linearization controller work?


2

212

1121 sin

xxx

uxxxx

e. Select the out as 1xy and use FB lin. design to select the control u(t) to follow the


Is the system minimum phase? f. Select the new output 2xy . Find the FB lin. controller u(t). Does this work? What


26

EE 5323 Homework 8 Slotine and Li

Feedback Linearization, backstepping 1. I/O feedback linearization. Slotine and Li Problem 6.3. System is

1 2

42 1 2

1

sin

cos

x x

x x x u

y x

Do i/o feedback linearization to make output track desired trajectory ( )dy t .

2. Backstepping. The system is

1 2

42 1 2 1

sin

cos

x x u

x x x x

Do backstepping to stabilize this system. Select the desired value 1dx to yield the first step


Compare this to Problem 1, which uses i/o FB linearization. 3. Backstepping. Slotine and Li Problem 6.11. ‘Globally stabilize’ means backstepping. 4. I/O fb linearization. It is desired to stabilize a system given by

2

212

1121 sin

xxx

uxxxx

a. Select the out as 1xy and use FB lin. design to select the control to make output go to

zero. Is the system minimum phase? b. Select the new output 2xy . Does this work? c. Design a backstepping controller.

27

The following homeworks are out of date. Do not do homeworks until they are assigned. They may change.

28

EE 5323 Homework 1 Fall 2009

Nonlinear Systems and Equilibrium 1. Obtain the linear model of the system described by

2( 1) 0y y y y around the equilibrium point.


21 1 1 2 1

22 2 1 2 2

x ax bx x cx

x dx ex x fx

describes the growth of two competing species that prey on each other. The constants are positive parameters and it is assumed that the two states are positive. Determine the linear model of the system around the equilibrium point. Simulate the system using MATLAB for various initial conditions. Make phase plane plot. 3. Determine the equilibrium points and their nature for the predator-prey system

1 1 1 2

2 2 1 2

x x x x

x x x x

.

Simulate the system using MATLAB for various initial conditions. Make phase plane plot. 4. Determine the equilibrium points and their nature for the system

2

2

(1 )

(1 )

x y x y

y x y x

.

29


State Variable Systems, Computer Simulation, Linearization 1. Simulate the van der Pol oscillator 0')1(" 2 yyyy using MATLAB for various ICs.

Plot y(t) vs. t and also the phase plane plot y'(t) vs. y(t). Use y(0)=0.2, y'(0)= 0. c. For = 0.05. d. For= 0.9.

2. Do MATLAB simulation of the Lorenz Attractor chaotic system. Run for 150 sec. with all

initial states equal to 0.3. Plot states versus time, and also make 3-D plot of x1, x2, x3 using PLOT3(x1,x2,x3).

2133

31212

211 )(

xxbxx

xxxrxx

xxx

use = 10, r= 28, b= 8/3.

3. A system has transfer function 134

4)(

2

ss

ssH

a. Use MATLAB to make a 3-D plot of the magnitude of H(s) b. Use MATLAB to make a 3-D plot of the phase of H(s) c. Use MATLAB to draw magnitude and phase Bode plots

4. Use separation of variables to verify the formula for x(t) in Slotine & Li ex. 1.2 on p. 7. 5. Duffing’s equation is interesting in that it exhibits bifurcation, or dependence of stability properties and number of equilibrium points on a parameter. The undamped Duffing equation is 3 0x x x

g. Find the equilibrium points. Show that for 0 there is only one e.p. h. For 0 there are 3 e.p.s Linearize the system and study the nature of these 3 e.p.s i. Simulate the Duffing oscillator for 1 . Make time plot and phase plane plot.

30


Chaos, Phase Plane 1. A system that exhibits chaos is the logistic function

)1()1(1 kkkk xxx



kk 1

for fixed less than but close to 1. These two equations form a dynamical system. Perform a MATLAB simulation to reproduce this plot of xk vs k, which was taken for 9995.0 . Interpret the plot with some discussion in terms of bifurcation theory. Plot also k . Show your

MATLAB code. It is indeed interesting that the logistic function appears in economic systems and military supply systems. 2. For Slotine & Li Example 2.2 on P. 20- a. Find equilibrium points b. Linearize the system about each equilibrium point. Find poles in each case. c. Simulate the system to find the Region of Attraction. 3. Slotine and Li p. 39 problem 2.2. Simulate and plot phase plane trajectories for various ICs. Do not do the problem requested in the book. 4. Slotine and Li p. 39 problem 2.3 Simulate using MATLAB using various initial conditions. Do not do the problem requested in the book.

31

EE 5323 Homework 4 Fall 2009 Slotine and Li

Lyapunov’s Method 1. Slotine and Li p. 97 problem 3.1. 2. Use Lyapunov Equation on p. 81 to prove asymptotic stability of the system

xx

56

10


)3(

)3(2

22

1222

12

22

211

2211

xxxxxx

xxxxxx

is locally asymptotically stable. Find the Region of Asymptotic Stability 4. UUB


)3(

)3(2

22

1222

12

22

211

2211

xxxxxx

xxxxxx

is uniformly ultimately bounded UUB. That is, show that the Lyapunov derivative is NEGATIVE OUTSIDE A BOUNDED REGION. Find the radius of the bounded region outside which V <0. Any states outside this region are attracted towards the origin. 5. Lyapunov Theorem for Control Design.

A system is given by

uxxx

uxxx

23

22

111

a. Use Lyapunov Linearization Method to show that the open-loop system with u(t)= 0

is unstable about the origin.

b. Select the nonlinear feedback control input 22xu . Find the closed-loop system.

Use a Lyapunov extension to show that the closed-loop system is UUB. That is, select the quadratic Lyapunov function and find V along the closed-loop

system trajectories. Then show that V is negative outside a region (i.e. if x is large enough).

c. Discuss the stability. When is the Lyapunov derivative negative? Can you use a

LaSalle Extension to show AS?

32

EE 5323 Homework 5 Fall 2009, Slotine and Li

Lyapunov 1. S&L p. 103, Example 4.2.

a. For the 3 systems given, prove the stability claimed by verifying the 3 conditions given.

b. Integrate the state equations to find the solutions x(t) of the three systems.

2. S&L p. 105, Example 4.3. Integrate the state equation to verify the solution given. 3. S&L p. 155 problem 4.9, parts a and b. 4. Consider the nonlinear dynamics for an m-link robot manipulator,

,M q q C q q q Dq g q ,

where , mq R . M q accounts for the robot inertia. ,C q q q accounts for centrifugal and

Coriolis forces. Dq accounts for viscous damping. g q accounts for gravity forces. In addition,

we have the following properties:

i. M q is a symmetric positive definite matrix of q .

ii. 2M C is a skew symmetric matrix of q , q . iii. g q W q q , where W q is a positive definite function of q .

Show that for 0D , the map from to q is passive lossless. And when D is positive definite, the map from to q is passive dissipative.

Hint: Use the total energy 1

2TV q M q q P q as a storage function. Select an appropriate

P q , a positive definite function of q .

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EE 5323 Homework 6 Fall 2009, Slotine and Li

Feedback Linearization, backstepping 1. I/O feedback linearization. Slotine and Li Problem 6.3 2. Backstepping. The system is

1 2

42 1 2 1

sin

cos

x x u

x x x x

Do backstepping to stabilize this system. Select the desired value 10x to yield the first step


Compare this to Problem 6.3, which uses i/o FB linearization. 3. Backstepping. Slotine and Li Problem 6.11. ‘Globally stabilize’ means backstepping. 4. I/O linearization. It is desired to stabilize a system given by

2

212

1121 sin

xxx

uxxxx

d. Select the out as 1xy and use FB lin. design to select the control. Is the system minimum phase? e. Select the new output 2xy . Does this work. f. Design a backstepping controller.

5. Input-State Linearization. Slotine and Li Problem 6.7.

a. Write ( )f x , ( )g x , fad g .

b. Is the system input-state linearizable. c. Check the given 1( )z x . Does it work?

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EE 5323 Homework 7 Fall 2009, Strogatz book

Bifurcations 1. Plot with MATLABthe following vector fields as 3-D surfaces in the (x,r) plane. Also plot the bifurcation diagrams in the plane.

a. 2( , )x f x r r x

b. 2( , )x f x r r x

c. 2( , )x f x r rx x

d. 3( , )x f x r rx x

e. 3( , )x f x r rx x

f. 3 5( , )x f x r rx x x

x

r

do not do homework until it is assigned. the …

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