.7. HOMEWORK 39 3.7 Homework Make sure that you can perform each of the chapter objectives defined in Section 3.1. In addition, turn in the problems below. Problems 1-3 are to be done by everyone individually. Problem 4 is to be done by each group. As before, only the group secretary needs to turn in group problems. Please write your name clearly on the top of each page, and for group problems include the group name at the top of each page 1. (10 Points) To support the results of group task 3.3.2, use quantitative analysis to show why the equation (3.1) converges to the fixed point 0.7391. That is, show that this fixed point is stable and attractive. Hint: the model is nonlinear, and for f(z)co(), the derivative is f,(z) =-sin(x). Also, your analysis should make use of equation (3.3) 2. (10 Points) Turn in your answers to Tasks 3.3.3 3. (10 Points) Turn in your answers to Tasks 3.4.1 and 3.4.2 4. Group Problem: Car Speed Model - Exploring Stability Properties The purpose of this problem is for you to use simulations (that is, to plot orbits) to determine stability properties for the fixed points cal- culated in group task 3.4.4. Where possible you will also use mathe- matical analysis to justify your stability conclusions. In addition, you will examine the settling time for the orbits computed, which is a commonly used mesure of performance in the dynamics and control community The model of the car's speed is .7. HOMEWORK 39 3.7 Homework Make sure that you can perform each of the chapter objectives defined in Section 3.1. In addition, turn in the problems below. Problems 1-3 are to be done by everyone individually. Problem 4 is to be done by each group. As before, only the group secretary needs to turn in group problems. Please write your name clearly on the top of each page, and for group problems include the group name at the top of each page 1. (10 Points) To support the results of group task 3.3.2, use quantitative analysis to show why the equation (3.1) converges to the fixed point 0.7391. That is, show that this fixed point is stable and attractive. Hint: the model is nonlinear, and for f(z)co(), the derivative is f,(z) =-sin(x). Also, your analysis should make use of equation (3.3) 2. (10 Points) Turn in your answers to Tasks 3.3.3 3. (10 Points) Turn in your answers to Tasks 3.4.1 and 3.4.2 4. Group Problem: Car Speed Model - Exploring Stability Properties The purpose of this problem is for you to use simulations (that is, to plot orbits) to determine stability properties for the fixed points cal- culated in group task 3.4.4. Where possible you will also use mathe- matical analysis to justify your stability conclusions. In addition, you will examine the settling time for the orbits computed, which is a commonly used mesure of performance in the dynamics and control community The model of the car's speed is
