Fa10_midterm1_sol

Fa10_midterm1_sol - ECE 483 Midterm 1 Solution Problem 1...

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Unformatted text preview: ECE 483 Midterm 1 Solution Problem 1 (20 points) For the above simulation diagram with input u ( k ) and output y ( k ), (a) (10 pt) Identify a set of state variables and write the corresponding state-space model. Solution: Denote the output of the left and the right time delay units by x 1 ( k ) and x 2 ( k ), respectively, then the corresponding state-space model is x ( k + 1) = bracketleftbigg- 4 1- 1 0 bracketrightbigg x ( k ) + bracketleftbigg 4 1 bracketrightbigg u ( k ) y ( k ) = bracketleftbig- 1 0 bracketrightbig x ( k ) + u ( k ) . You may switch the roles of x 1 ( k ) and x 2 ( k ). Then the corresponding state-space model is x ( k + 1) = bracketleftbigg- 1 1- 4 bracketrightbigg x ( k ) + bracketleftbigg 1 4 bracketrightbigg u ( k ) y ( k ) = bracketleftbig- 1 bracketrightbig x ( k ) + u ( k ) . (b) (10 pt) Find the transfer function Y ( z ) U ( z ) . Solution: Y ( z ) U ( z ) = bracketleftbig- 1 0 bracketrightbig bracketleftbigg z + 4- 1 1 z bracketrightbigg- 1 bracketleftbigg 4 1 bracketrightbigg + 1 = bracketleftbig- 1 0 bracketrightbig 1 z 2 + 4 z + 1 bracketleftbigg z 1- 1 z + 4 bracketrightbigg bracketleftbigg 4 1 bracketrightbigg...
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Fa10_midterm1_sol - ECE 483 Midterm 1 Solution Problem 1...

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