Transcript H 2 and H  Control

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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# 1
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# 2
Measures of “Size” - Norms
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Properties of Norms:

Vector Norms:
Slide# 3
Measures of “Size” - Norms

Matrix Norms:
– Matrix Norm Induced by Vector Norm:
– Frobenius Norm:

Temporal Norms: Norm over time or frequency.
– 2-norm

– Power or RMS Norm
a semi-norm.

2
 e  d
 max  e  

e(t ) 2 
–  - norm e(t )


e(t )
RMS
 1
 lim 
T 
 2T

T
T

e  d 

2
This is only
Signal Norm: A signal norm consists of two parts:
Slide# 4
Singular Values

The action of a matrix on a vector can be viewed as a combination of rotation and
scaling, as shown below:
Unit Sphere

Mapped to an Ellipsoid – Singular
values, s, are the lengths of the
principal semi-axes.
vi = pre-images of the principal semi-axes.
or
•
s = eigenvalues (ATA)
•
s max  A 2
Singular Value
Decomposition (SVD)
Slide# 5
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.
Slide# 6
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
Slide# 7
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:
,
Slide# 8
Problem Formulation
Disturbance
Plant
Control Action
Regulated Output
Feedback
Controller
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
Slide# 9