7-6 Analysis of Dynamic Performance for Discrete Systems

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7-6 Analysis of Dynamic Performance for Discrete
Systems
-Time response of discrete systems -Influence on
dynamic performance by sampler and
holder -Relation between CL poles and dynamic
response
1. Time response
Similar to continuous systems,in general the
analysis of dynamic performance is based on the
assumption that the system takes input of
unit-step signal.
Expanded in series,obtained is the output pulse
sequence
transitional and steady state (SS) performance

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In case the PTF cannot be sought,
but the input is known,
can still be solved from .
Ex 1Given system with ZOH, where
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2. Influence on dynamic performance by sampler
and holder
After sampling in continuous systems,the dynamic
properties will be alteredInducing the holder
will change the poles and zeros for the CL
systems. Take Ex.1 as an example
(1)Response to unit-step in the continuous system
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(2)Response with sampler only
(3)Response with both sampler and ZOH
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Specifications in time domain
Cont. sys Disc.
sys(Samp.) Disc. sys(Samp-Z)
Conclusions (1)With sampler only peak-time and
setting time are decreased,but with increase of
overshoot. Normally the system stability may be
deteriorated after sampling. For large time-delay
systems, error sampling may increase system
stability on the contrary.
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(2)ZOH and sampler will compromise both stability
and response speed. In addition to sampling,
phase lag of ZOH reduces system stability, too.
3. Relation between CL poles and transitional
response
Irrespective of all poles located inside the unit
circle, the distribution of the poles in the
circle may influence transitional response
greatly.
Let CL PTF be
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Assume no repeated CL poles?If the system is
stable, then
Let response to the unit step
Take Partial fraction expansion
The first terms inverse transform is the stable
component ,and the followed terms
inverse transform are the dynamic components of
.
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Observe the dynamic component of
(1)CL poles on the real axis
(2)CL Conjugate complex poles in Z plane.
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CL real pole distribution and its corresponding
dynamic response patterns.
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CL complex pole distribution and its
corresponding dynamic response patterns
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Dynamic response is a oscillated
blow-up pulse sequence.
Dynamic response is a oscillated
pulse sequence.
Dynamic response is a
oscillated convergent pulse sequence.
Origin in Z plane
in S plane
For cont. sys., is called stability degree.
A system that its CL poles are located at origin
in Z plane is called the discrete system with
infinite stability.
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7-7 Digital Compensation for Discrete Systems
Design approach for linear discrete
systems (1)Analog design (2)Discrete
design(direct digitization)
Analog design approach
Control system is analyzed and controller is
designed as a continuous system, then the
controller is digitized.
Direct digitization design approach
Control system is analyzed as a discrete system,
find the system PTF, and the controller is
designed using discrete system theory.
A typical design of digital control
systemminimum beat design.
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PTF of Digital Controller
CL PTF
Error PTF
Controller PTF
or
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Basic thinking for controller designdetermine CL
PTF or error PTF ,
then controller
Minimum beat (MB) design
MB systemexcited by typical input,the system is
able to end the transitional process with finite
number of bears (or within finite of T), and
has no SS error at sampling instance. Such a
system is called MB system.
Commonly used typical inputs
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General form of typical input
Design guide-line for MB system(1)G(z) dose not
have time-delay,has no poles on and outside the
unit circle. (2)Select to make the
system with the typical input be able to ,after
minimum number of T (or beats),reach the state
that its output sequence has zero SS error at
each sampling instance, i.e., that the complete
tracking can be realized. In so doing, the
controller is thus designed.
Controller design
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Normally, to simplify the controller, take lower
order, let
After compensation, all CL poles will be located
on the origin, i.e., the compensated system is
with infinite stability.
(1)Input of unit-step
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Conclusionafter one beat (T),the system is able
to completely track the unit step input signal.
(2)Input of unit ramp
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Conclusionafter two beats (2T),the system is
able to completely track the unit ramp input
signal.
(3)Input of unit acceleration
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Digital controller
Conclusionafter three beats (3T),the system is
able to completely track the unit acceleration
input signal.
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The setting time of MB system is determined by
the form of CL PTF, and is irrelevant with type
of input signals.
Ex.with unit ramp input
(1)with unit step
(2)with unit ramp
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(3)with unit acceleration
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Conclusions (1)Fastness
the MB system designed by unit ramp input, its
transitional process takes two beats. (2)Precisene
ss for unit step and unit ramp input,
after two beats, SS error disappears. But, for
unit acceleration input, the SS error remain
for sampling instance. (3)Transition
good for unit ramp inputnot-so-good for unit
step input,its overshoot reaches
100 (4)Smoothness after into steady
state,wave normally exists between sampling
points.
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ExLet
Design with unit ramp input ,
find PTF of controller .
Controller