1. Slide# 1 212 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

2. 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.

3. Slide# 3 Measures of “Size” - Norms  Properties of Norms:  Vector Norms:

4. Slide# 4 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:

5. 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:  v i = pre-images of the principal semi-axes.  = eigenvalues (A T A) Unit Sphere Mapped to an Ellipsoid – Singular values, , are the lengths of the principal semi-axes. or Singular Value Decomposition (SVD)

6. Slide# 6 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.

7. Slide# 7 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

8. Slide# 8 Differences between H 2 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 H 2 norm does not.  Example:

9. Slide# 9 Problem Formulation Disturbance Regulated Output Controller Feedback Control Action 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  -synthesis toolbox of Matlab Plant