H 2 and H Control
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Control of Structural Vibrations
Lecture #7_4
H
2
- 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
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 e ( t )
2
e
2 d
– - norm e ( t )
max
– Power or RMS Norm
T lim
a semi-norm.
e ( t )
RMS
1
2 T
T
T e
2
d
Signal Norm: A signal norm consists of two parts:
This is only
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
v i
= pre-images of the principal semi-axes.
Mapped to an Ellipsoid – Singular values, s , are the lengths of the principal semi-axes.
or
•
• s = eigenvalues ( A T A ) s max
A
2
Singular Value
Decomposition (SVD)
Slide# 5
H
2
Norm of a Transfer
Function
The H
2 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 z i
( t ) is the response to a unit impulse applied to state variable i .
Thus the H
2 norm, can be interpreted as:
Also, the H
2 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 H
2
Norms and H
We can write the Frobenius Norm in terms of Singular Values as
This shows that:
The H
norm satisfies the multiplicative property while the H
2 norm does not.
Example:
,
Slide# 8
Problem Formulation
Plant
Disturbance Regulated Output
Control Action Feedback
Controller
Problem: To find the gain matrix K that minimizes the H
2 or H
norm of H zd
.
This can be done for example using functions from the m -synthesis toolbox of
Matlab
Slide# 9
