CSE 416 DIGITAL CONTROL Lecture 02 Z Transform

CSE 416: DIGITAL CONTROL Lecture 02: Z Transform Dr. Ahmed Mahmoud, --/10/2020

What Are Difference Equations? • An equation of the change in a variable over discrete time. • The main mathematical tool for modeling discrete systems. • Differential equation in continuous, difference equation in discrete. • Denoted by subscripts or as a function of sample times or indices. • Same classification as differential equations: Orders, linear/nonlinear, autonomous/ nonautonomous. --/10/2020 CSE 416: DIGITAL CONTROL 2

Fibonacci Series --/10/2020 CSE 416: DIGITAL CONTROL 3

Bank Interest Rate --/10/2020 CSE 416: DIGITAL CONTROL 4

Why Difference Equations? (1) --/10/2020 CSE 416: DIGITAL CONTROL 5

Why Difference Equations? (2) --/10/2020 CSE 416: DIGITAL CONTROL 6

Z Transform --/10/2020 CSE 416: DIGITAL CONTROL 7

Z Transform Example (1) --/10/2020 CSE 416: DIGITAL CONTROL 8

Z Transform Example (2) --/10/2020 CSE 416: DIGITAL CONTROL 9

Z Transform Properties: Addition • The z-transform of a sum of number sequences is equal to the sum of the ztransforms of the number sequences: --/10/2020 CSE 416: DIGITAL CONTROL 10

Z Transform Properties: Multiplication by a Constant • The z-transform of a number sequence multiplied by a constant is equal to the constant multiplied by the z-transform of the number sequence: • The previous two properties form the linearity property. --/10/2020 CSE 416: DIGITAL CONTROL 11

Z Transform Properties: Linearity --/10/2020 CSE 416: DIGITAL CONTROL 12

Z Transform Properties: Unit Sample Function --/10/2020 CSE 416: DIGITAL CONTROL 13

Z Transform Properties: Translation --/10/2020 CSE 416: DIGITAL CONTROL 14

Z Transform: Initial and Final Value Theorems • Initial Value: • Final Value: --/10/2020 CSE 416: DIGITAL CONTROL 15

Z Transform: Initial and Final Value Theorems --/10/2020 CSE 416: DIGITAL CONTROL 16

Digital Control Systems --/10/2020 CSE 416: DIGITAL CONTROL 17

Solving Difference Equations (1) --/10/2020 CSE 416: DIGITAL CONTROL 18

Solving Difference Equations (2) --/10/2020 CSE 416: DIGITAL CONTROL 19

Solving Difference Equations (3) --/10/2020 CSE 416: DIGITAL CONTROL 20

Inverse Z Transform: Power Series --/10/2020 CSE 416: DIGITAL CONTROL 21

Inverse Z Transform: Partial Fractions --/10/2020 CSE 416: DIGITAL CONTROL 22

Simulation Diagram --/10/2020 CSE 416: DIGITAL CONTROL 23

Simulation Diagram --/10/2020 CSE 416: DIGITAL CONTROL 24