Digital Control Systems

The z-Transform

The z Transform

Definition of z-TransformThe 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(2T),x(3T)….

The z-Transform of a time function x(t), where t is nonnegative, or of a sequence of values x(kT), 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

Definition of z-TransformNote that, when dealing with a time sequence x(kT) obtained by sampling a time signal x(t), the z transform X(z) involves T explicitly. However for a number sequence x(k), the z transform X(z) does not involve T explicitly.

The z transform of x(t), where <t< or of x(k), where k takes integer values (k=0,) is defined by

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

By definition

Step Function

Impulse Function

Z Transforms of Elementary Functions

Decaying Exponential Function

Damped Cosine Wave

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

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Z Transform Examples

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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

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

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(kT).

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.):

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

. ………………………………. ………

Finding difference equation from TF.

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: