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Solutions_hw3

# Solutions_hw3 - Homework 3 ECE147A For problems that are...

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Homework 3 ECE147A October 19, 2009 For problems that are taken from the text book, I’ll just give you excerpts from the solution manual. Those pages are going to be marked with a black frame. Problem 1 In order to obtain the transfer function from v 1 := u 1 - u 1 to z 3 := x 3 - x 3 , we need to linearize the state space model around ( x , u ). As usual, we define deviation variables z := x - x and v := u - u and compute the Jacobians: (right hand side) ∂x = - 3( u 1 - x 1 ) 2 0 - 1 0 0 1 1 - 1 dr dx 3 (1) (right hand side) ∂u = 3( u 1 - x 1 ) 2 1 0 0 0 0 (2) The term dr dx 3 is a little tricky, but since we are going to insert the equilibrium x 3 , all we need to know is the slope of r (0), which from the figure is easily obtained as 2. The linearized system thus looks like this: ˙ z = - 27 0 - 1 0 0 1 1 - 1 2 | {z } =: A z + 27 1 0 0 0 0 | {z } =: B v (3) Since we are only interested in the transfer function from v 1 to z 3 , we set c T = 0 0 1 and v 2 = 0 (which is the same as disregarding the second column in the B -matrix) and obtain: H ( s ) = Z 3 ( s ) V 1 ( s ) = c T ( sI - A ) - 1 B 1 = 27 s s 3 + 25 s 2 - 52 s + 27 . (4) 5 points for the model, 1 point for the equilibrium, 7 points for the linearization if you got it all, including the definiton of deviation variables. 1

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9005 3. Consider the circuit shown in Fig. 9.58; u 1 and u 2 are voltage and current sources, respectively, and R 1 and R 2 are nonlinear resistors with the following characteristics: Resistor 1 : i 1 = G ( v 1 ) = v 3 1 Resistor 2 : v 2 = r ( i 2 ) ; where the function r is de°ned in Fig. 9.59. (a) Show that the circuit equations can be written as _ x 1 = G ( u 1 ° x 1 ) + u 2 ° x 3 _ x 2 = x 3 _ x 3 = x 1 ° x 2 ° r ( x 3 ) : Suppose we have a constant voltage source of 1 Volt at u 1 and a constant current source of 27 Amps; i.e., u o 1 = 1 , u o 2 = 27 . Find the equilibrium state x o = [ x o 1 ; x o 2 ; x o 3 ] T for the circuit. For a particular input u o , an equilibrium state of the system is de°ned to be any constant state vector whose elements satisfy the relation _ x 1 = _ x 2 = _ x 3 = 0 : Consequently, any system started in one of its equilibrium states will remain there inde°- nitely until a di/erent input is applied. (b) Due to disturbances, the initial state (capacitance, voltages, and inductor current) is slightly di/erent from the equilibrium and so are the independent sources; that is, u ( t ) = u o + °u ( t ) x ( t 0 ) = x o ( t 0 ) + °x ( t 0 ) : Do a small-signal analysis of the network about the equilibrium found in (a), displaying the equations in the form ° _ x 1 = f 11 °x 1 + f 12 °x 2 + f 13 °x 3 + g 1 °u 1 + g 2 °u 2 : (c) Draw the circuit diagram that corresponds to the linearized model. Give the values of the
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Solutions_hw3 - Homework 3 ECE147A For problems that are...

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