Slide# 1

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Control of Structural Vibrations

Lecture #7_4

H2 - H Control Algorithms

Instructor:

Andrei M. Reinhorn P.Eng. D.Sc.

Professor of Structural Engineering

Slide# 2

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.

Slide# 3

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:

dete2

2)(

ete max)(

deT

teT

TTRMS

2

2

1lim)(

Slide# 5

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.

• = eigenvalues (ATA)

•

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

2max A

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 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.

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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 FormulationDisturbance Regulated Output

Controller

FeedbackControl Action

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 -synthesis toolbox of Matlab

Plant