Feedback and Control Higherorder Systems

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Feedback and ControlHigher-order Systems

• Thao Doan
• University of Virginia

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Outline

• Introduction
• What are higher-order systems?
• Dominant pole analysis
• HO Systems with Real Poles
• Initial condition response
• Impulse response
• Step response
• Zeros influence on systems response
• HO Systems with Complex Poles
• Impulse response
• Step response
• Example
• Conclusion

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What are high-order systems?

• N-order system
• The current output is determined by the systems
  inputs and outputs of the last n sample time.
• Difference equation
• Transfer function
• Higher-order system ngt2, 1ltmltn

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Example

• IBM Lotus Domino Server with sensor delay

Reference RIS
Max User
Actual RIS
Measured RIS
Controller
Server
Sensor
U(z)
Q(z)
M(z)
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Review poles in first-order analysis

• Life is still simple
• One pole
• Real

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Poles in higher-order analysis

• Multiple poles
• Multiple zeros
• Possibly complex poles
• Approximate high-order systems behavior with
  first-order systems.
• Dominant pole analysis

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Review SASO

• Poles affect settling time and maximum overshoot

Steady state error
Controlled variable
Overshoot
Reference
Steady State
Transient State
Settling time
Time
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Dominant pole analysis

• Which pole is dominant?
• Has largest magnitude
• Positive or negative
• Should be twice as large as the others
• Not every system has dominant pole

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Examples

• Find dominant pole of the following systems

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Dominant pole analysis
Settling time time until the output y(ks) is
within 2 its largest magnitude
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Dominant pole approximation

• p dominant pole
• If p is real, Gs approximation

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Initial condition response

• Second-order systems

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Initial condition response

• u(k) 0
• Second-order model approximation
• Settling time

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Initial condition response

• p2 0.4
• ? ks 4
• p2 0.8
• ? ks 18

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Impulse response

• y(0) y(1) 0
• u(k) 1,0,,0 with kgt2
• Second-order model approximation
• Settling time

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Impulse response
p2 0.4 ? ks 4 p2 0.8 ? ks 18
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Impulse response

• Example IBM Lotus Domino Server

Reference RIS
Max User
Actual RIS
Measured RIS
Controller
Server
Filter
U(z)
Q(z)
M(z)
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Step response

• y(0) y(1) 0
• u(k) 1
• Second-order model approximation
• Settling time

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Step response

• Example IBM Lotus Domino Server

Reference RIS
Max User
Actual RIS
Measured RIS
Controller
Server
Filter
U(z)
Q(z)
M(z)
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Zeros influence on system response

• Second order system
• q1 p1 pole-zero cancellation
• q1gt1 non-minimum-phase systems

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Matlab Examples

• Show effects of different zeros and poles
• Zero q 0.4, poles p1 0.4, p2 0.1
• q 0.2, p1 0.4, p2 0.1
• q 2, p1 0.4, p2 0.1

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Complex poles

• Second-order model
• Transfer function
• If
• complex poles

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Impulse response

• u(k) impulse signal ? U(z) 1
• Time domain equation
• ?
• ?

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Step response

• u(k) step signal ?
• Time domain equation
• ?
• ?

Max overshoot
?
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Matlab Example

• Apache HTTP Server with a filter and controller

Reference input
KeepAlive
CPU
Controller K(z)
Server G(z)
Filter H(z)
U(z)
Y(z)
W(z)
E(z)
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Matlab Example
IBM Server with a filter and controller
KeepAlive
CPU
Reference input
Controller K(z)
Server G(z)
Filter H(z)
U(z)
Y(z)
W(z)
E(z)
A -80 p1,2 0.77 /- 0.15j
-4/log0.78 16
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A -20 p10.91, p2 0.63
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Conclusion

• Response of a higher-order system with a real
  dominant pole is approximated by the first-order
  system with the same pole.
• Complex poles cause oscillation in the systems
  transient response.
• Zeros can influence systems behavior.