Due Friday, December 5

1. Consider the pendulum from Homework #30 shown below.

a) Go back and write the nonlinear model found in HW#30, in the form of . Find the nonlinear controllability matrix, . Is the original nonlinear system controllable?

b) Find a diffeomorphism, , such that z is in nonlinear controllable canonical form.

c) Find a nonlinear feedback control, such that the system in the z coordinates is linear and has eigenvalues of {-2, -3}

d) Compare your answer above to the control derived in HW#30 using the perturbation method. When (if ever) are these controls the same?

2a) Prove that for the system, , our nonlinear controllability matrix, evaluates to our old controllability matrix, M=[b Ab...A^n-1b]

b) Again, consider . We have already derived one expression for the similarity transformation, z=T_pv^-1x, which places the system in phase variable form (i.e., T_pv^-1 = M_pvM^-1). Use the method developed in class to derive a new expression for T_pv^-1 in terms of the last row M^-1 (that is, find our diffeomorphism, )