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problem10

# problem10 - EE221A Linear System Theory Problem Set 10...

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EE221A Linear System Theory Problem Set 10 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2011 Issued 12/2; Due 12/9 Problem 1: Feedback control design by eigenvalue placement. Consider the dynamic system: d 4 θ dt 4 + α 1 d 3 θ dt 3 + α 2 d 2 θ dt 2 + α 3 dt + α 4 θ = u where u represents an input force, α i are real scalars. Assuming that d 3 θ dt 3 , d 2 θ dt 2 , dt , and θ can all be measured, design a state feedback control scheme which places the closed-loop eigenvalues at s 1 = - 1, s 2 = - 1, s 3 = - 1 + j 1, s 4 = - 1 - j 1. Problem 2: Controllability of Jordan Forms. Given the Jordan Canonical Form of Problem Set 7: A = - 3 1 0 0 0 0 0 0 - 3 1 0 0 0 0 0 0 - 3 0 0 0 0 0 0 0 - 4 1 0 0 0 0 0 0 - 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Suppose this matrix A were the dynamic matrix of a system to be controlled. What is the minimum number of inputs needed for the system to be controllable? Problem 3: Observer design. Figure 1 shows a velocity observation system where x 1 is the velocity to be x

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