Performance Specifications

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Performance Specifications

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? synopsis

• ? Tracking Systems
• ? Forced Response
• ? Power-of-Time Error Performance
• ? Performance Indices and Optimal System
• ? System Sensitivity
• ? Time Domain Design

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? Analyzing Tracking Systems
? Tracking Systems
Control system that creates an output which
tracks the input to
some level of tolerance.
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Ex)
- Step input
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? The analysis and design of tracking systems
can be separated into two parts
1. The characteristic roots (poles)
2. Tracking of the reference input
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? Natural Response, Relative Stability, and
Damping
- The relative stability the distance into the
left half of the complex plane
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- A pair of complex conjugafe characteristic
roots
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? Forced Response
? Steady State Error
Zero initial conditions
? The poles of The poles of T(s)
The forced part of the error signal
gt Perfect tracking
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? Initial and Find Values
- Initial value
Ex)
? If there is m impulse in y(t) at t0, then y(0)
will be infinite
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- The final value (steady state value)
Ex)
Multiple poles
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?
RHP
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? Initial and Final Value Theorems to
Representative Laplace Transform terms.
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? Steady State Errors to Power-of-time Inputs

Error signal is
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Ex)
Its error to the standard ramp input
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? The forced component of the error can do only
one of three things
1. The forced error can be zero
Output equals the power-of-time reference input
2. The forced error can be a constant
Output and reference input differ by a constant
3. The forced error can involve a nonzero term
proportional to t or a higher power of t
The error grows without bound
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? These three situaticns are easily
distinguished, without calculating e(t), by
applying the final-value theorem to E(s)
The final value of e(t) is zero
situation 1
The final value of e(t) is a finite nonzero
constant situation 2
E(s) has more than a single pole at s0,
the fimal value of e(t) approaches infinity
situation 3
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? Power-of-time Error Performance
? System type number
the order of the pole of T(s) at s0
- the type number is 0
i) Step input
Constant
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ii) Ramp input
? Higher power-of-t input give infinite steady
state error
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- the type number is 1 has one
factor of s in the numerator
i) Step input
ii) Ramp input
? Higher power-of-t input, the error of such a
system is infinite, since has a repeated s0
denominator root
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- the type number is 2 has one
factor of in the numerator
i) Step input , ramp input zero steady
state error
ii) Parabolic input
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? Unity Feedback Systems
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i) step input
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Can be made aibitraily small in this case by
choosing a sufficiently large amplifier gain K
ii) Ramp input
? ramp, paratolic , or higher power-of-t input

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i) step input
ii) ramp input
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? Unity Feedback Error Coefficients
Steady state error coefficients of unity feedback
system
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i) Step input ( i 1 )
Steady state error to input
ii) Ramp input ( i 2 )
Steady state error to input
iii) Higher power-of-t input ( i 2, 3, 4 ... )
Steady state error to input
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? Steady State Error Coefficients of Unity
Feedback System
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Ex)
Type 0 system
Steady state error to input
type 1
? When an integrator is added
Steady state error to input
Steady state error to input
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? Performance Indices and Optimal Systems
A commonly used performance index
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Ex)
A conmonly used performance index is the integral
of the square of the error to a step input
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Desired to choose the parameter K to give minimum
Integral square error to a step input
The error to a step input
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For unit step input
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Figure Integral square error performance
measure
for a certain second order system with adjustable
damping ratio
? Minimal mean square error to a step input for
the system occurs for
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- Other useful performance indices
Figure other performance measure
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- Hurwitz determinant method
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Ex)
Using Table
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? System Sensitivity
? Calculating the Effects of changes in Parameters
Unit step input
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Unit step input
Unit step input
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? Sensitivity Functions
Ex)
The sensitivity of T to changer in G
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The sensitivity of T to changer in H
The sensitivity of T(s) to changer in a
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Ex)
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For the nominal value of a2
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For the nominal value of b1
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? Sensitivity to Disturbance signals
For a unit step disturbance input
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For a unit step disturbance input
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Ex)
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The system is stable
A unit step disturbance
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? Consider the open-loop system
The same transfer function relating Y(s) and R(s)
as does the feedback system
A unit step disturbance
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? Time Domain Design
? Ziegler-Nichols compensation ( type 0
)
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1. A proportional compensator is applied so that
2. The compensator is defined by
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Ex)
i) The fist step
The Routh-Hurwitz table is formed
Row to zero
The complex conjugate roots
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Normal Ziegler-Nichols
P compensator
PI compensator
PID compensator
Quarter-Wave Ziegler Nichols
PID compensator
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? Chien-Hrones-Reswick Compensation ( type
1 )
1.
2.
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Ex)
PID compensator
Overdamped
20 overshoot