## Description

Problem 1

(a) Find e(0), e(1), and e(10) for

E(z) = 0.1

z(z − 0.9)

using the inversion formula.

(b) Check the value of e(0) using the initial-value property.

(c) Check the values calculated in part (a) using partial fractions.

(d) Find e(k) for k = 0, 1, 2, 3, 4 if Z[e(k)] is given by

E(z) = 1.98z

(z

2 − 0.9z + 0.9)(z − 0.8)(z

2 − 1.2z + 0.27)

(e) A continuous time function e(t), when sampled at a rate of 10 Hz (T = 0.1s), has the following ztransform E(z) = 2z

z−0.8

. Find function e(t).

(f) Repeat part (e) for E(z) = 2z

z+0.8

.

(g) From parts (e) and (f), what is the effect on the inverse z-transform of changing the sign on a real pole?

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EECE 5610 (Professor Milad Siami ): Homework #3 Problem 2

Problem 2

Consider the system described by

x(k + 1) =

0 1

0 3

x(k) +

1

1

u(k),

y(k) =

−2 1

x(k).

(a) Find the transfer function Y (z)/U(z).

(b) Using any similarity transformation, find a different state model for this system.

(c) Find the transfer function of the system from the transformed state equations.

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EECE 5610 (Professor Milad Siami ): Homework #3 Problem 3

Problem 3

Given the MATLAB program

that solves the difference equation of a digital controller.

(a) Find the transfer function of the controller from input e(.) to output m(.).

(b) Find the z-transform of the controller input {e(k)}∞

k=0.

(c) Use the results of parts (a) and (b) to find the inverse z-transform of the controller output.

(d) Run the program to check the results of part (c). Please attach your MATLAB code/result (from the

command window) to your report.

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EECE 5610 (Professor Milad Siami ): Homework #3 Problem 4

Problem 4

Problem 3.7-7 of the textbook (page 113).

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