EE 5323 Homeworks Fall 2009 Updated: Sunday, April 21, 2013 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.

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EE 5323 Homework 1 Fall 2009 Nonlinear Systems and Equilibrium 1. Obtain the linear model of the system described by & & & y + ( y 2 1) y +y=0 around the equilibrium point. 2. The system of equations 2 & x 1 = ax1 bx1 x2 cx12 & x 2 = dx2 ex1 x2 fx2 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 & x 1 = x1 + x1 x2 . & x 2 = x2 x1 x2 Simulate the system using MATLAB for various initial conditions. Make phase plane plot. 4. Determine the equilibrium points and their nature for the system &= y (1 + x y 2 ) x . &= x(1 + y x 2 ) y

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EE 5323 Homework 2 Fall 2009 State Variable Systems, Computer Simulation, Linearization 1. Simulate the van der Pol oscillator

y"+ (y2 1) y '+ y =0

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. a. For = 0.05. b. 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 x 1, x2, x3 using PLOT3(x1,x2,x3). 1 = ( x1 x 2 ) x 2 = rx1 x 2 x1 x3 x 3 = bx3 + x1 x 2 x

use = 10, r= 28, b= 8/3. 3. A system has transfer function H ( s) =s+4 s + 4s + 132

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. Duffings 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 & & x + x + x3 = 0 a. Find the equilibrium points. Show that for > 0 there is only one e.p. b. For < 0 there are 3 e.p.s Linearize the system and study the nature of these 3 e.p.s c. Simulate the Duffing oscillator for = 1 . Make time plot and phase plane plot.

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EE 5323 Homework 3 Fall 2009 Chaos, Phase Plane 1. A system that exhibits chaos is the logistic function x k +1 = (1 + k ) x k (1 x k ) However, chaos only occurs for certain values of k . Rather than try all values of k , we can sweep through the k values using k +1 = k 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 = 0.9995 . 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. 20a. 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.

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EE 5323 Homework 4 Fall 2009 Slotine and Li Lyapunovs Method 1. Slotine and Li p. 97 problem 3.1. 2. Use Lyapunov Equation on p. 81 to prove asymptotic stability of the system0 = x 6 1 x 5

3. Use Lyapunov to show that the system 1 = x1 x 2 2 + x1 ( x1 2 + x 2 2 3) x

2 = x1 2 x 2 + x 2 ( x1 2 + x 2 2 3) x is locally asymptotically stable. Find the Region of Asymptotic Stability4. UUB Use Lyapunov to show that the system

1 = x1 x 2 2 x1 ( x1 2 + x 2 2 3) x 2 = x1 2 x 2 x 2 ( x1 2 + x 2 2 3) x 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