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Discrete-time Systems and z-Transform

- Digital Control Theory
- Lecture 2

Outline

- Discrete-time systems
- The z-transform and examples
- The inverse z-transform and examples
- Power series method
- Partial-fraction expansion method
- Real poles
- Complex poles

Discrete-time systems

- A discrete-time system is a dynamic system which

evolves discretely in time, thus a discrete-time

system has signals that can change values only at

discrete instants at time. - The terms, sampled-data systems, discrete-time

systems and digital control systems have been

used loosely (or even interchangeably) in the

control literature. In a broad sense, they refer

to systems in which some form of digital or

sampled signal takes place. - The discrete-time systems we are studying are LTI

(linear time-invariant). A linear system

satisfies the property of superposition and

homogeneity. Time-invariance means that system

parameters are constant with respect to time.

Discrete-time systems (contd)

- Digital computers are often used in control

systems as digital compensators or digital

controllers.

A digital control system is shown in the figure.

Assume that all the numbers enter or leave the

computer at the same fixed period, T, known as

the sampling period. Hence, the reference input

in the figure is a sequence of sample values

r(kT). The variables r(kT), m(kT), and u(kT) are

discrete signals in contrast to m(t) and y(t),

which are continuous functions of time. A

question arises naturally how do we describe the

operation of a discrete-time system

mathematically?

Discrete-time systems (contd)

- The describing equation of an LTI discrete-time

system is a difference equation of the form

where x(k) is the output sequence and e(k)

input sequence, which can be generated from

corresponding time functions by sampling every T

seconds. So x(k) and e(k) are understood to be

x(kT) and e(kT), respectively, but T is often

dropped for simplicity.

The z-transform

For LTI continuous-time systems, the Laplace

transform is used in system analysis and design.

Accordingly, z-transform is utilised in the

analysis of LTI discrete-time systems. For a

number sequence e(k), k0,1,2, , which may

represent the sampled time function of e(t) with

the sampling period T omitted for convenience,

the z-transform is defined by

The inverse z-transform is written as

The z-transform pair is then

The variable z is related to the Laplace

transform variable s by

z-transform examples

Example 1. Given that e(k)1 for all k0, find

E(z). By definition, E(z) is

The following power series is very useful in

expressing E(z) in closed form

xlt1 is the region of convergence.

Therefore, the closed form of E(z) is obtained as

Obviously, e(k) may be generated by sampling a

unit step function.

z-transform examples (contd)

Example 2. Given that e(k)e-akT, find

E(z). E(z) can be written in power series form as

In this example, e(k) may be generated by

sampling the exponential function e(t)e-at.

What is the Laplace transform of e-at?

z-transform examples (contd)

Example 3. Find the z-transform of a sampled

unit ramp. A sampled unit ramp can be written as

f(kT)kT. By the definition of the z-transform

In order to find a closed form of F(z), we

multiply the above equation by z,

and

Thus

Properties of the z-transform

Sequence z-transform Name

Linearity theorem

Real translation

Complex translation

Convolution

Initial value theorem

Final value theorem

In the table, u(k) is the discrete unit step

function

Properties of the z-transform (contd)

- In the property of real translation,
- shifting right is e(k-n)
- shifting left is e(kn)
- Example 1. Find Zr(k) when r(k) is shifted one

place to the right. - Zr(k-1) z-1R(z)
- Example 2. Find Zr(k) when r(k) is shifted n

places to the left. - Since z-transform is defined only for k 0, this

extra term represents the term that is lost after

the shift to the left.

z- and s-transform pairs

Ref Control Systems Engineering

The inverse z-transform

- Generally, to find a sampled time waveform,

inverse z-transform is to be performed. Since

the z-transform comes from sampled waveforms, the

inverse z-transform will only yield the values of

the time function at the sampling instants. - For example, the z-transform of unit step

function is z/(z-1), but the inverse z-transform

of z/(z-1) can be any time function which has a

value of unity at t0, T, 2T, - We present two methods for finding the inverse

z-transform - Power series method
- Partial-fraction expansion method

Power series method

When E(z) is expressed as the ratio of two

polynomials in z, the power series method

involves dividing the denominator of E(z) into

the numerator such that a power series of the

form

is obtained . From the definition of the

z-transform, the values of e(k) are simply the

coefficients in the power series. Although this

method does not yield a closed form expression,

it can be used for plotting.

Example - power series method

Example. Find the values of e(k) for E(z) given

by the expression

Using long division, we obtain

and therefore e(0)23, e(1)36, e(2)-2,

e(3)-107,

Partial-fraction expansion method

In a manner similar to that employed with the

Laplace transform, E(z) can be expanded in

partial fractions and then the z-transform table

can be used to determine the inverse z-transform.

In the table, d(k) is called discrete unit

impulse function

Partial-fraction expansion method (contd)

Because a factor of z appears in the numerator of

the z-transforms given in the z-transform table,

the partial-fraction expansion should be

performed on E(z)/z, and finally multiply the

result by z to replace the z in the

numerator. In the partial-fraction expansion of

E(z), the following cases can occur

- E(z) has only real poles
- E(z) has complex conjugate poles

Partial-fraction expansion method (contd)

Before presenting examples, consider the function

We can expand E(z) in power series

By definition of the z-transform, the inverse

z-transform of E(z) above is ak. The z-transform

pair

is most commonly encountered in the

partial-fraction expansion method.

Example - partial-fraction expansion for real

poles

Example. Find the inverse z-transform of the

following function

Begin by dividing E(z) by z and perform a

partial-fraction expansion

Next, multiply through by z

Using the important result , we

get

Partial-fraction expansion for complex poles

When E(z) contains complex poles, we can apply

the same partial-fraction expansion procedure.

Alternatively, a different approach can be used

to find the inverse z-transform. Recall Eulers

relation, given by

Applying Eulers formula to the following

function yields

Partial-fraction expansion for complex poles

(contd)

The z-transform of

is given by

where indicates complex conjugate. Hence, the

following relationship holds

Thus, when Y(z) is written in terms of k1 and p1,

we can solve for y(k)

Example - partial-fraction expansion for complex

poles

Example. Find the inverse z-transform of the

following function

We rewrite Y(z) as

Dividing both sides by z, we have

From the partial-fraction expansion, we obtain k1

Example - partial-fraction expansion for complex

poles (contd)

Thus

According to

we get

So y(k) can be written as

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