EE 5323 Homeworks

Fall 2009

Updated: Thursday, November 05, 2009

  • 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.

EE 5323 Homework 1

Fall 2009

Nonlinear Systems and Equilibrium

  1. Obtain the linear model of the system described by

around the equilibrium point.

  1. The system of equations

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.

  1. Determine the equilibrium points and their nature for the predator-prey system

.

Simulate the system using MATLAB for various initial conditions. Make phase plane plot.

  1. Determine the equilibrium points and their nature for the system

.

EE 5323 Homework 2

Fall 2009

State Variable Systems, Computer Simulation, Linearization

  1. Simulate the van der Pol oscillator 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.
  1. For = 0.05.
  2. For= 0.9.
  1. 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).

use = 10, r= 28, b= 8/3.

3. A system has transfer function

  1. Use MATLAB to make a 3-D plot of the magnitude of H(s)
  2. Use MATLAB to make a 3-D plot of the phase of H(s)
  3. 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

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

EE 5323 Homework 3

Fall 2009

Chaos, Phase Plane

1. A system that exhibits chaos is the logistic function

However, chaos only occurs for certain values of . Rather than try all values of , we can sweep through the values using

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 . Interpret the plot with some discussion in terms of bifurcation theory. Plot also . Show your MATLAB code.

It is indeed interesting that the logistic function appears in economic systems and military supply systems.

2. For Slotine & LiExample 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.

EE 5323 Homework 4

Fall 2009 Slotine and Li

Lyapunov’s Method

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

3. Use Lyapunov to show that the system

is locally asymptotically stable. Find the Region of Asymptotic Stability

4. UUB

Use Lyapunov to show that the system

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 <0. Any states outside this region are attracted towards the origin.

5. Lyapunov Theorem for Control Design.

A system is given by

  1. 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 . 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 along the closed-loop system trajectories. Then show that 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?

EE 5323 Homework 5

Fall 2009, Slotine and Li

Lyapunov

  1. S&L p. 103, Example 4.2.
  1. For the 3 systems given, prove the stability claimed by verifying the 3 conditions given.
  1. Integrate the state equations to find the solutions x(t) of the three systems.
  1. S&L p. 105, Example 4.3. Integrate the state equation to verify the solution given.
  1. S&L p. 155 problem 4.9, parts a and b.

4. Consider the nonlinear dynamics for an m-link robot manipulator,

,

where . accounts for the robot inertia. accounts for centrifugal and Coriolis forces. accounts for viscous damping. accounts for gravity forces. In addition, we have the following properties:

  1. is a symmetric positive definite matrix of .
  2. is a skew symmetric matrix of ,.
  3. , where is a positive definite function of .

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

Hint: Use the total energy as a storage function. Select an appropriate , a positive definite function of .

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

Do backstepping to stabilize this system. Select the desired value to yield the first step dynamics of .

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

  1. Select the out as and use FB lin. design to select the control. Is the system

minimum phase?

  1. Select the new output . Does this work.
  1. Design a backstepping controller.

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

  1. Write ,,.
  2. Is the system input-state linearizable.
  3. Check the given. Does it work?

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.

1