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212 Ketter Hall, North Campus, Buffalo, NY 14260 www.civil.buffalo.edu Fax: 716 645 3733 Tel: 716 645 2114 x 2400 Control of Structural Vibrations Lecture #7_4 H 2 - H  Control Algorithms Instructor: Andrei M. Reinhorn P.Eng. D.Sc. Professor of Structural Engineering.

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212 ketter hall north campus buffalo ny 14260 civil buffalo

212 Ketter Hall, North Campus, Buffalo, NY 14260

www.civil.buffalo.edu

Fax: 716 645 3733 Tel: 716 645 2114 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


Frequency domain methods

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.


Measures of size norms

Measures of “Size” - Norms

  • Properties of Norms:

  • Vector Norms:


Measures of size norms1

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:


Singular values

Singular Values

Unit Sphere

Mapped to an Ellipsoid – Singular values, s, are the lengths of the principal semi-axes.

  • 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)


H 2 norm of a transfer function

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 Parseval’s 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.


H norm of a transfer function

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


Differences between h 2 and h norms

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:


Problem formulation

Problem Formulation

Regulated Output

Disturbance

Control Action

Feedback

Controller

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


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