State Space Modelling and Control

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State Space Modelling and Control

• Morten Kristiansen
• Bjørn Langeland

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Lektion 5
3
Regulator and servo

• Regulator
• Servo

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Design of a servo

• Steady state error
• Design
• Reference magnification
• Error integration

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Todays agenda

• Today's lesson
• Observers
• Observability
• Observer with regulator

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Observers

• State feedback (closed loop) is used to determine
  the control input u(t).
• Full state feedback is not always possible
  because the states are
• insufficient for measuring
• impaired by error or noisy from measuring signals
• large expenses for measuring
• An observer is built with a model of the real
  plant/system

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Simulation of model and observer
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Respons model og observer
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Selecting Ke

• Feedback through the observer gain matrix Ke
  serves as a correction signal to the model of the
  plant.
• If significant unknowns are involved, Ke should
  be relatively large.
• If output signal are contaminated significantly
  by disturbances and measurement noises, Ke should
  be relatively small.
• General rule observer poles should be two to
  five times faster than controller poles.
• If observer poles are placed to the right of the
  controller poles in the left half plane, then the
  system response will be dominated by the observer
  poles.

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Eigenvalues for observer
Eig_obs 4eig_reg
Eig_obs eig_reg
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Observability

• Given a system and a model
• Model
• A system is fully observable if it from a known
  output y(t) and a known input u(t), can determine
  an arbitrary state. (e.g. the start condition
  x(t0))
• Requirement for observability
• Required full rank and invert able

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Observer with regulator
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Respons model, observer and regulator
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Exercises

• Assignment