Discretetime Systems and zTransform

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Discrete-time Systems and z-Transform

• Digital Control Theory
• Lecture 2

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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

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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.

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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?
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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.
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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
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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.
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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?
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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
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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
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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.

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z- and s-transform pairs
Ref Control Systems Engineering
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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

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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.
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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,
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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
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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

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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.
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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
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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
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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)
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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
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Example - partial-fraction expansion for complex
poles (contd)
Thus
According to
we get
So y(k) can be written as