Time Response Characteristics

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Time Response Characteristics

• Digital Control Theory
• Lecture 7

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Outline

• System time response
• Characteristic equation
• Mapping the s-plane into the z-plane
• Mathematical expression of s- and z-plane pole
  mapping
• Steady-state accuracy

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System time response
The time response of discrete-time systems is
introduced via examples.
Consider the system shown in the figure. We want
to find the unit step response of the system.
The system output can be expressed as
This comes from C(s).
where G(z) is
Refer to Lecture 5 as to how to derive A(z) from
a general function A(s).
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System time response (contd)
Thus G(z) is
The unit step response at the sampling instants
is
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System time response (contd)
To compare the output response of the
sampled-data system and its analog counterpart, we
remove the sampler and ZOH. The analog
closed-loop TF is
Thus the analog system unit step response is
given by
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System time response (contd)
We have found the unit step response of the
system at sampling instants. Suppose we want to
calculate the
response at all instants of time for the
sampled-data system.
Since
the continuous output of the system is
In this expression,
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System time response (contd)
The continuous output of the sampled-data system
is
time delay
open-loop step response
Laplace transform delay property
and hence
In summary, the continuous output response of the
sampled-data system in the figure is the
superposition of a number of delayed step
responses of the open-loop system.
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System time response (contd)
The response of the sampled-data system
and the response of the analog system
are both plotted in the figure.
From the previous analysis, the steps that
appear in the response curve of the sampled-data
system is a result of the sampler and ZOH.
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System time response (contd)
The final value or steady-state gain (i.e., dc
gain) of the unit step response of the system is
For the system shown in the figure, if the input
to the sampler is constant, the output of ZOH is
also constant and equal to the sampler input.
Thus the sampler/ZOH has no effect and may be
removed when calculating the steady-state gain,
resulting in a continuous system.
which agrees with the previous result.
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dc gain calculation
We summarise two points about the calculation of
the dc gain (i.e., the steady-state gain of a
sampled-data system with a constant input
applied).

• For a stable system with a constant input, the
  system output approaches a constant value as time
  increases. The dc gain may be calculated by
  evaluating the closed-loop transfer function
  (CLTF) with z1.
• The same value of dc gain is obtained by
  evaluating the CLTF of the analog system (with
  sampler/ZOH removed) with s0.

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Characteristic equation
Consider the closed-loop sampled-data system in
the figure.
Using partial-fraction expansion, we can write
C(z) as
CR(z) contains the terms which originate in the
poles of R(z).
natural response terms
These terms determine the characteristic of the
system natural response. The pis originate in
the roots of the characteristic equation
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Characteristic equation (contd)
The system characteristic equation is
The roots of the characteristic equation are the
poles of CLTF.
If a TF cannot be written, the characteristic
equation is the part of the denominator of C(z)
that is independent of input function. For
example, for a certain system, the output C(z)
The characteristic equation should be
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Mapping the s-plane into the z-plane
For continuous-time systems, the time response
characteristics are closely related to
closed-loop pole locations. This is also true
for discrete-time systems. Since zeTs, for
different pole locations in the s-plane, we want
to find out the corresponding locations in the
z-plane.
Along the j?-axis,
Hence, poles located on the unit circle in the
z-plane are equivalent to poles on the imaginary
axis in the s-plane. For ??s/2, ?Tp, thus the
j?-axis between j?s/2 and j?s/2 maps into the
unit circle in the z-plane. In fact, any portion
of the j?-axis of length ?s maps into the unit
circle in the z-plane.
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Mapping the s-plane into the z-plane (contd)

• Poles located on j?-axis in the s-plane are
  equivalent to poles on the unit circle in the
  z-plane.

• The left-half-plane portion of the primary strip
  maps into the interior of the unit circle.
• The right-half-plane portion of the primary strip
  maps into the exterior of the unit circle.

Since the stable region of the s-plane is the
left half plane, the stable region of the z-plane
is the interior of the unit circle.
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Mapping the s-plane into the z-plane (contd)

• Constant damping loci (i.e., straight lines with
  s constant) in the s-plane map into circles in
  the z-plane.

• Constant frequency loci (i.e., straight lines
  with ? constant) in the s-plane map into rays in
  the z-plane.

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Mapping the s-plane into the z-plane (contd)
The correspondence of some s-plane and z-plane
pole locations is illustrated in the figure.
z
s
The s-plane poles of the form ssj? result in a
transient response of the form Aestcos(?tF).
When sampling occurs, these s-plane poles result
in z-plane poles at
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Transient response characteristics of z-plane
pole locations

• What form of transient response do poles at
  produce?
• In terms of magnitude, is there any difference in
  transient response resulted from poles inside the
  unit circle and outside the unit circle?

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Mathematical expression of s- and z-plane pole
mapping
Consider a 2nd-order s-plane TF in standard form
which has poles
The equivalent z-plane poles occur at
Hence
Solving the above equations for ? and ?n yields
The time constant, t, of the poles is given by
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Mathematical expression of s- and z-plane pole
mapping (contd)
TF pole locations in the s-plane transform into
z-plane poles as
In order for the sampling to have negligible
effect, T must be much less than the time
constant t, which means (T/t)ltlt1, thus placing
the z-plane poles in the vicinity of z1. For
the complex pole, an additional requirement is
that several samples be taken per cycle of the
sinusoid, or that ?Tlt1.
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Steady-state accuracy
An important characteristic of a control system
is its ability to track certain inputs with a
minimum of error. Consider the unity feedback
sampled-data system in the figure.
The TF of the system is
where G(z)ZG(s).
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Steady-state error
From the final value theorem, the steady-state
error