Relationship between roots of the characteristic equation and stability

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Relationship between roots of the characteristic
equation and stability

• Stability Analysis

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Premise
Consider the general solution of a differential
equation by Laplace Transforms where R is a
transform of the response and C is a single term
of the solution and D is the remainder of the
solution.
The denominator of C forms a root or pair of
roots of the denominator of R.
Inverting C allows us to find a single term of
the solution of R
Example First order system with sine input
The possible roots of the characteristic equation
and the component of the solution they produce
are shown in the following table Any of the
roots of the characteristic equation can result
in an unstable root.
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Table of Laplace Transforms
Factor
Solution
Root
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Location of roots
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Conclusions
Conclusions
Denominator terms that have roots with positive
real parts result in instable solutions.
A single instable root in the characteristic
equation renders the whole solution instable.
(assumes there is no cancellation)
Instability Unbounded output for a bounded input
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Closed loop example
Consider the following closed loop system where
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Characteristic equation
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Method of direct substitution

• Variation of a parameter in a system can cause
  the system to become instable.
• That point at the verge of instability is
  characterized by a sinusoidal response.
• Generally, any further increase (or sometimes
  decrease) in the parameter will cause exponential
  increasing response.
• That point at the verge of instability can be
  determined by letting s in the characteristic
  equation equal iw and solving for the parameter.
  This would be the value of the parameter that
  would result in a sinusoidal root.

Example
Let tp1 3, tm 2, tp2 1, K2 1
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Finding the Characteristic Equation
The characteristic equation is
Letting s iw and solving for Kc and w
or
For the imaginary part
only
makes sense
and for the real part
and for the only real solution,
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Solving for the ultimate Gain
The system oscillates at a frequency of
When the controller gain is
This value of controller gain is know as ultimate
gain
A gain lower that this value will normally result
in stable response. See table 7-1.1 for
recommended gain settings.

• Field tuning using ultimate gain
• Turn OFF integral and derivative action.
• Increase proportional gain until a continuous
  sinusoidal oscillation occurs
• The gain is the ultimate gain. (the period is
  the ultimate period)
• Use table 7-1.1 to set the controller

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Example
Consider the following first order system
The characteristic equation is
Determine the response of the system if input is
a sinusoidal
Which may be transformed to
the system response, R(s), is then