Chapter 5 Transient and Steady State Response

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Chapter 5 Transient and Steady State Response

• I will study and get ready and someday my chance
  will come
• Abraham Lincoln

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Transient vs Steady-State

• The output of any differential equation can be
  broken up into two parts,
• a transient part (which decays to zero as t goes
  to infinity) and
• a steady-state part (which does not decay to zero
  as t goes to infinity).

Either part might be zero in any particular case.
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Prototype systems
1st Order system
2nd order system
Agenda transfer function response to test
signals impulse step ramp parabolic sinu
soidal
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1st order system
Impulse response Step response Ramp
response Relationship between impulse, step and
ramp Relationship between impulse, step and ramp
responses
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1st Order system
Prototype parameter Time constant
Relate problem specific parameter to prototype
parameter.
Parameters problem specific constants. Numbers
that do not change with time, but do change from
problem to problem.
We learn that the time constant defines a problem
specific time scale that is more convenient than
the arbitrary time scale of seconds, minutes,
hours, days, etc, or fractions thereof.
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Transient vs Steady state
Consider the impulse, step, ramp responses
computed earlier. Identify the steady state and
the transient parts.
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1st order system
Consider the impulse, step, ramp responses
computed earlier. Identify the steady state and
the transient parts.
Impulse response Step response Ramp
response Relationship between impulse, step and
ramp Relationship between impulse, step and ramp
responses
Compare steady-state part to input function,
transient part to TF.
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2nd order system

• Over damped
• (two real distinct roots two 1st order systems
  with real poles)
• Critically damped
• (a single pole of multiplicity two, highly
  unlikely, requires exact matching)
• Underdamped
• (complex conjugate pair of poles, oscillatory
  behavior, most common)
• step response

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2nd Order System
Prototype parameters undamped natural
frequency, damping ratio
Relating problem specific parameters to prototype
parameters
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Transient vs Steady state
Consider the step, responses computed earlier.
Identify the steady state and the transient parts.
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2nd order system

• Over damped
• (two real distinct roots two 1st order systems
  with real poles)
• Critically damped
• (a single pole of multiplicity two, highly
  unlikely, requires exact matching)
• Underdamped
• (complex conjugate pair of poles, oscillatory
  behavior, most common)
• step response

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Use of Prototypes
Too many examples to cover them all We cover
important prototypes We develop intuition on the
prototypes We cover how to convert specific
examples to prototypes We transfer our insight,
based on the study of the prototypes to the
specific situations.
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Transient-Response Spedifications

• Delay time, td The time required for the
  response to reach half the final value the very
  first time.
• Rise time, tr the time required for the response
  to rise from
• 10 to 90 (common for overdamped and 1st order
  systems)
• 5 to 95
• or 0 to 100 (common for underdamped systems)
• of its final value
• Peak time, tp
• Maximum (percent) overshoot, Mp
• Settling time, ts

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Derived relations for 2nd Order Systems
See book for details. (Pg. 232)
Allowable Mp determines damping ratio. Settling
time then determines undamped natural frequency.
Theory is used to derive relationships between
design specifications and prototype parameters.
Which are related to problem parameters.
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Chapter 5 Homework
B problems 1, 2, 3, 7, do one of the
following 9, 10, 11, 27 15, R-H problems,
23, 24, 25, 26, 28, 30, 31, 32
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Example Figure 5-5
Choose physical parameters to achieve a Rise
time of .5 seconds Maximum overshoot of 10 2
settling time of 1.3 seconds
Relate physical parameters to prototype
parameters. Use prototype relationships. Three
requirements, two parameters.
See also example 5-2, pg 236-237
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Higher order system

• PFEs have linear denominators.
• each term with a real pole has a time constant
• each complex conjugate pair of poles has a
  damping ratio and an undamped natural frequency.
• Read section 5-4

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What block diagram?
Rational functions are ratios of polynomials.
PFE for step input and only distinct real
poles. PFE for step input complex roots. PFE for
step input and repeated real or complex roots.
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Poles of C(s) come from a) TF or b) input
function Real Poles in LHP produce decaying
exponentials. Complex Poles in LHP produce
decaying sinusoids. Simple pole at origin produce
step functions. Simple complex poles on imaginary
axis produce sinusoids. Multiple poles on
imaginary axis produce unbounded terms. Any poles
in RHP produce unbounded terms.
TF of Stable systems have poles only in LHP. CL
poles that are located far from the imaginary
axis have real parts that are large
negative. These poles decay to zero very
fast. Dominant poles produce the terms that
dominate the response. Close to imaginary axis,
with large residues.
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Rouths Stability Criterion
How do we determine stability without finding all
poles? Actual poles provide more info than is
needed. All we need to know if any poles are in
LHP. Rouths stability criterion (Section 5-7).
What values of K produce a stable system?
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Proportional Control of plant w/o integrator
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Integral control of plant w/o integrator
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Proportional control of plant w integrator
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PI Control of plant w disturbance
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Integral control of Plant w disturbance
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Derivative Control Action
Read at bottom of pg 285-286.
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Proportional control of system with inertial load
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PD control of a system with inertial load
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PD control of 2nd order systems
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Steady State Errors
Bode Form
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Steady State Errors-cont.
Bode Form
Now compute ess for N 0, 1, 2 and R(s) 1/s,
1/s2, 1/s3
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Steady State Errors-cont.
Now compute ess for N 0, 1, 2 and R(s) 1/s,
1/s2, 1/s3
Type 1 (N) systems. Type 2 (N) systems. Type 3
(N) systems. Steady state error table, pg. 293,
FE Reference manual, notice differences.
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Read Chapter 6