Homework+11

# Homework+11 - " ˙ x 1 ˙ x 2 = A" x 1 x 2 with...

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ME 132 Homework # 11 Issued: November 09, 2009 Due: November 13, 2009 1 Output feedback stabilization Consider a linear time-invariant plant with state-space realization: ˙ x 1 ˙ x 2 ˙ x 3 y = 0 1 0 0 0 0 1 0 2 - 3 - 4 - 1 1 0 0 0 x 1 x 2 x 3 u (a) Find the transfer function associated with this plant. (b) Suppose we control the plant using the output feedback law u = - ky + r . Find a state-space realization for the closed loop system. (c) Is it possible to stabilize the plant using this output feedback controller? If not, explain your answer. If so, what are the possible values of the controller gain k that will stabilize the plant? 2 Pole-zero diagrams Consider a linear time-invariant system H ( s ). Shown below are several pole-zero diagrams for H ( s ) together with several possible unit-step response plots. Pair each pole-zero diagram (A-E) with the most appropriate step response (AA-EE). Explain your reasoning. AA BB CC DD B C D E EE -1 0 1 2 -1 0 1 2 -1 0 1 2 -1 0 1 2 -1 0 1 2 A 1

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3 Phase portraits Consider the second order linear time-invariant system
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Unformatted text preview: " ˙ x 1 ˙ x 2 # = A " x 1 x 2 # with initial condition x 1 (0) = . 5 ,x 2 (0) = 0 . 5 In each of the cases below, simulate the system for 5 seconds, and turn in plots of x 1 ( t ) versus x 2 ( t ). These are called phase portraits. On your plots, mark the initial condition by a hollow circle (a) A = " 1-1 0 # (b) A = "-2-1 1-2 # (c) A = " 0 1 1 0 # (d) A = " 0 0 0 1 # (e) A = "-1 0 1 # (f) A = " 1-1 # (g) A = "-1 1 # (h) A = " 2-1 1 2 # 4 Controllable canonical form Consider the system ˙ x ( t ) = Ax ( t ) + Bu ( t ) where A = 6 13 7-2-4-2-2-4-3 , B = -3 1 1 (a) Transform this system into controllable canonical form. (b) Design a state feedback law u = Lx + r that places the closed loop eigenvalues at-1 ,-1 ,-1. 2...
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Homework+11 - " ˙ x 1 ˙ x 2 = A" x 1 x 2 with...

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