212 Ketter Hall, North Campus, Buffalo, NY 14260

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• 212 Ketter Hall, North Campus, Buffalo, NY 14260
  
• www.civil.buffalo.edu
• Fax [phone] Tel [phone] x 2400
• Control of Structural Vibrations
• Lecture 7_4
• H2 - H? Control Algorithms
• Instructor
• Andrei M. Reinhorn P.Eng. D.Sc.
• Professor of Structural Engineering

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Frequency Domain Methods

• The Structural Model is often available in the
  frequency domain, for example, modal testing
  yields transfer functions which are in the
  frequency domain.
• Input is often specified in the frequency domain,
  for example, stochastic input such as seismic
  excitation is given in terms of Power Spectral
  Density.
• Frequency domain control algorithms allow more
  rational determination of weighting functions,
  for example, frequency domain weighting functions
  can be used to roll-off control action at high
  frequencies where noise dominates and to control
  different aspects of performance in different
  frequency ranges.
• Enable use of acceleration feedback.
• Involve shaping the size of the transfer
  function.

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Measures of Size - Norms

• Properties of Norms
• Vector Norms

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Measures of Size - Norms

• Matrix Norms
• Matrix Norm Induced by Vector Norm
• Frobenius Norm
• Temporal Norms Norm over time or frequency.
• 2-norm
• ? - norm
• Power or RMS Norm This is only a semi-norm.
• Signal Norm A signal norm consists of two parts

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Singular Values

• The action of a matrix on a vector can be viewed
  as a combination of rotation and scaling, as
  shown below
• vi pre-images of the principal semi-axes.
• s eigenvalues (ATA)

or
Singular Value Decomposition (SVD)
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H2 Norm of a Transfer Function

• The H2 norm of a transfer function is defined
  using
• 2-norm over frequency
• Frobenius norm spatially
• It is given by
• By Parsevals theorem, this is can be written in
  time domain as,
• where zi(t) is the response to a unit impulse
  applied to state variable i.
• Thus the H2 norm, can be interpreted as
• Also, the H2 norm can be interpreted as the RMS
  response of the system to a unit intensity white
  noise excitation.

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H? Norm of a Transfer Function

• The H? norm of a transfer function is defined
  using
• ? - norm over frequency
• Induced 2-norm (maximum singular value) spatially
• It is given by
• The H? norm has also several time domain
  interpretations. For example that
• H? control is convenient for representing model
  uncertainties and is therefore becoming popular
  in robust control applications

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Differences between H2 and H? Norms

• We can write the Frobenius Norm in terms of
  Singular Values as
• This shows that
• The H? norm satisfies the multiplicative
  property , while the H2 norm does not.
• Example

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Problem Formulation
Plant
Problem To find the gain matrix K that minimizes
the H2 or H? norm of Hzd. This can be done for
example using functions from the m-synthesis
toolbox of Matlab