hw2-fa10 - CALIFORNIA INSTITUTE OF TECHNOLOGY Control and...

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CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 101 D. MacMynowski Fall 2010 Problem Set #2 Issued: 4 Oct 10 Due: 11 Oct 10 Note: In the upper left hand corner of the second page of your homework set, please put the number of hours that you spent on this homework set (including reading). 1. Choose any one of the following systems, locate the equilibrium points for the system and indicate whether each is asymptotically stable, stable (but not asymptotically stable) or unstable. To determine stability, you can either use a phase portrait (if appropriate), analyze the linearization or simulate the system using multiple nearby initial conditions to determine how the state evolves. (a) Nonlinear spring mass. Consider a nonlinear spring mass system with dynamics m ¨ q = - k ( q - aq 3 ) - c ˙ q, where m = 1000 kg is the mass, k = 250 kg/s 2 is the nominal spring constant, a = 0 . 01 represents the nonlinear “softening” coefficient of the spring and c = 100 kg/s is the damping coefficient. Note that this is very similar to the spring mass system we have studied in Section 2.2, except for the nonlinearity. (b) Genetic toggle switch. Consider the dynamics of two repressors connected together in a cycle. It can be shown (Exercise 2.9) that the normalized dynamics of the system can
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