Digital Control Systems The zTransform The z Transform

Digital Control Systems The z-Transform

The z Transform Definition of z-Transform The z transform method is an operational method that is very powerful when working with discrete time systems. In considering the z-transform of a time function x(t), we consider only the sampled values of x(t), that is: x(0), x(T), x(2 T), x(3 T)…. The z-Transform of a time function x(t), where t is nonnegative, or of a sequence of values x(k. T), where k takes zero or positive integers and T is sampling period is defined by the following equation: For a sequence of numbers x(k), the z-transform is defined by In the one sided z transform we assume x(t)=0 for t<0 or x(k)=0 for k<0. Note that z is a complex variable

The z Transform • In the two sided z transform, the time function x(t) is assumed to be nonzero values for k<0 we assume x(t)=0 for t<0 or x(k)=0 for k<0. Note that z is a complex variable

Z Transforms of Elementary Functions •

Z Transforms of Elementary Functions Decaying Exponential Function Damped Cosine Wave

Z Transforms of Elementary Functions Unit Ramp Function

Z Transforms of Elementary Functions Polynomial Function ak

Table of Z Transforms

Z-Transform Examples *

Z Transform Examples * *

Properties of the z-Transform Linearity Time Shift Multiplication by an Exponential Sequence Differentiation in z-Domain

Properties of the z-Transform Time Reversal Initial Value Theorem Convolution

Properties of the z-Transform Example:

Properties of the z-Transform Example Find the Z-transform of the causal sine sequence.

Properties of the z-Transform

Inverse z-Transform The inverse Z-transform can yield the corresponding time sequence f(kt) uniquely. However, it says nothing about f(t). There might be numerous f(t) for a given f(k. T). Other than referring to z transform tables, four methods for obtaining the inverse z transform are available: • Direct Division Method • Computational Method • Partial Fraction Expansion Method • Inversion Integral Method

Inverse z-Transform Direct Division Method Express X(z) in powers of z− 1

Inverse z-Transform Computational Method-Difference Equation Approach Example:

Inverse z-Transform Computational Method-Difference Equation Approach Example:

Inverse z-Transform Computational Method-Difference Equation Approach Example(cntd. ): Finding difference equation from TF. By substituting k=-2 into eqn. , we find y(0)=0 By substituting k=-1 into eqn. , we find y(1)=0. 4673 By substituting k=0 into eqn. , we find y(2)=0. 3769 . ………………. ………

Inverse z-Transform Computational Method-Matlab Approach Example:

Inverse z-Transform Computational Method-Matlab Approach Example (cntd. ):

Inverse z-Transform Partial Fraction Expansion

Inverse z-Transform Example:

z-Transform Method for Solving Difference Equation Time shift Property of Z-transform

z-Transform Method for Solving Difference Equation Example:

z-Transform Method for Solving Difference Equation Example: