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Unformatted text preview: V ( x ) = 1 2 x 2 , compute the range of the parameter α for which this Lyapunov function shows that the system is stable. (c) Now repeat this procedure using linearization: Instead of the non-linear dynamics ˙ x = F ( x ) , linearize F ( x ) around the equilibrium point, x e = 0, to obtain: F l ( x ) = F (0) + F (0) x. Consider the linearized dynamics ˙ x = F l ( x ) , and ±nd the range of the parameter α for which the system is neutrally stable, and the range over which ths system is asymptotically stable, with this linear feedback policy. 3. Exercise 4.4 from the book: Lyapunov functions and stability. 1 4. Exercise 4.10 from the book: Eigenvalue placement as a function of the control parameters (root locus). 5. Exercise 4.14 from the book. 6. (Optional) Read the section on bifurcation, and do exercise 4.6 from the book. 7. (Optional) Exercise 4.7 from the book. 2...
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- Fall '08