212 Ketter Hall, North Campus, Buffalo, NY 14260 civil.buffalo

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 H2 - H Control Algorithms Instructor: Andrei M. Reinhorn P.Eng. D.Sc. Professor of Structural Engineering

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. Measures of “Size” - Norms • Properties of Norms: • Vector Norms:

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. 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)

6. 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.

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. 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:

9. 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