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dsc_hw_5_w09 - R = 0.25 Ω L = 0.01 H C = 0.002 F and...

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Homework Set #5: MAE 3600 – Dynamic Systems and Control, Winter 2009 Due date: Friday, March 20, 2009 Problem 1 . Recall the MEMS tuning-fork gyroscope model from Problem 3, HW#1. First, rename all position variables as z 0 , z 1 and z 2 (for example, don’t use x 1 , since this symbol is reserved for the first state variable). Derive a complete state-space representation (SSR) for this system, where the two relative displacements are the two output (measured) variables; i.e., y 1 = z 1 z 0 and y 2 = z 2 z 0 . The two inputs are the two applied forces, f 1 and f 2 . Show your steps. Problem 2 . Recall the circuit with a single inductor and single capacitor (Problem 1, HW#2). Derive a complete state-space representation (SSR) for this system, where the single input is the source voltage e in ( t ) and the single output is voltage across the resistor R . Show your steps. Problem 3 . Simulate the RLC circuit in Problem 2 using Simulink and the following numerical values:
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Unformatted text preview: R = 0.25 Ω , L = 0.01 H, C = 0.002 F, and input voltage e in ( t ) = 0.5sin10 π t V. There is no current in the circuit, or charge in the capacitor at time t = 0. Plot the output (voltage across resistor R ) and input e in ( t ) [on the same figure] for a simulation time of 0.5 sec. Hand in a printout of your Simulink model. Problem 4 . Given the system matrix --= 2 30 1 A a) Determine the system’s eigenvalues “by hand” (you can check with Matlab). b) Based on knowledge of the eigenvalues, describe the system’s “natural” response to arbitrary initial conditions and zero input. c) Verify your description in part b by simulating the system’s response to an initial state vector x (0) = [ 0 2 ] T . There is no system input. You may use Matlab (LSIM.m) or Simulink. Hand in a plot of state x 1 ( t ), and hand in a printout of either your Mfile or Simulink model....
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