Input Tracking

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Input Tracking

• Chapter 10

2
Learning Objectives Student will be able to

• Put a transfer function in Bode form
• Determine steady state errors from system type
• Verify the entries in the steady-state error
  table found in FE Reference Manual
• Determine the system type of the controller
  required to produce required steady state error
  of closed loop system

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Output matches Input
Objective of Control Engineering
Stability
Output tracks input
Input tracking
In spite of
Disturbances
Noise
Input tracking
4
Time Out Before Starting
What does 1/s represent?
Signal? Hardware?
1/s2?
What are the 4 common inputs?

• Step
• Ramp
• Parabolic
• Sinusoid

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For Input tracking, L0, N0

• H(s)1, GR(s)1
• C(s)G1(s)G2(s)G(s)
• See simplified BD in FE Handbook and at right

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Intuitive Input Tracking
For Y(s) R(s),
For Y(s) R(s),
How?
How?
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Intuitive Input Tracking

• Focus attention on common inputs
• Focus attention on E(s) S(s)R(s)
• For the error to be small either
• the input is small, or
• the transfer function S(s) is small

• G(s) needs to be VERY large where R(s) is large.
• Where are step, ramp, and parabolic inputs large?
• Where must G(s) be large?
• What must G(s) contain?

G(s) must contain an integrator!!!
Its denominator must have an un-canceled factor
of s in it.
- What mathematical operation must G(s) perform
(among others)?
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Input Tracking
KB is the Bode Constant
T is the system Type
Bode Form
n(s) and d(s) are polynomials
Example Put the given TF in Bode Form and
identify KB, T, n(s) and d(s). Be sure that
n(0)d(0)1.
Homework 1 and 2 is now assigned. (see pg. 10-3)
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Input Tracking
We will now explore how the Bode Form of G(s)
provides insight into the input tracking problem.
Compute S(s) assuming G(s) is in Bode Form.
Assume feedback system is STABLE.

• We will consider various inputs and various
  system types.
• Step inputs (T0, T1, T2)
• Ramp inputs (T0, T1, T2)
• Parabolic inputs (T0, T1, T2)

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Input Tracking, Step Inputs
The value of A depends on T and KB.
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Input Tracking, Step Inputs
T0
T1
T2
Compare to Table in FE Reference Manual
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Input Tracking, Ramp Inputs
The values of A and B depend on T and KB.
If B is not zero, then the error has a ramp
component and the error explodes, regardless of
what value A has.
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Input Tracking, Ramp Inputs
T0
Infinite error no matter what value for A
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Input Tracking, Ramp Inputs
T1
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Input Tracking, Ramp Inputs
T2
ess 0
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Input Tracking, Parabolic Inputs
The values of A, B and C depend on T and KB.
If B or C are not zero, then the error has a
parabolic component and the error explodes,
regardless of the value A.
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Input Tracking, Parabolic Inputs
T0
The error has a parabolic term hence it explodes
and goes to infinity
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Input Tracking, Parabolic Inputs
T1
The error has a ramp term hence it explodes and
goes to infinity
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Input Tracking, Parabolic Inputs
T2
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A little Design
Suppose the plant has transfer function

• Determine the system type of the controller to
  guarantee 0 steady state error due to a
• Step input?
• Ramp input?
• Parabolic input?

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Quiz/Homework

• Un-graded homework
• Be able to reproduce the mathematical
  verification of the of the entries in the
  steady-state error table found in the FE
  reference Manual.
• Quiz problem
• Be able to verify any element in the steady-state
  error table found in the FE reference manual
  using the techniques just described.