From: Pete Seiler <guiness@vehicle.ME.Berkeley.EDU>
Subject: Re: meaning hankelSV
Date: Sat, 20 May 2000 19:47:30 -0700
Newsgroups: sci.math
To: vincent lampaert <vincent.lampaert@mech.kuleuven.ac.be>
Summary: [missing]

> Is there someone who can tell me the physical meaning of the hankel
> singular values of a mechanical system? Do they have a similar meaning
> as the eigenvalues (measure for the resoncance frequency) and
> eigenvector(measure for the eigenmodes) of the A matrix of the state
> space description of the mechanical system?

The hankel singular values (hsv) measure the gain of the system from past
inputs to future outputs.  

Consider the linear time-invariant system:
        dx/dt = Ax+Bu
        y = Cx+Du   
The largest hankel singular value of this system is equal to:
	max ||Py||    subject to u in L_2^- and ||u||=1
where P projects L_2 onto L_2^+.  This notation is tough, but it says the
following: Assume that the system was initially at rest, x(-oo)=0.  
Choose u(t) defined on (-oo,0] in any way such that it has norm 
||u||=sqrt[int_{-oo}^{0} u(t)^2 dt] = 1.  This input will cause the state
of the system to have some value x0 at t=0.  You can think of x0 as the
effect of the past inputs on the system.  Next, set u(t)=0 for all t>0 and
observe the output, y(t) for all t>0.  The notation above says as long as
you choose u(t) for t<=0  with ||u||=1 (||u||<1 is fine too) the norm of
the output for t>0:  ||Py||=sqrt[int_{0}^{+oo} y(t)^2 dt]
will always be less than the maximum hankel singular value.
In other words, the maximum hsv is the gain from past inputs to future
outputs.

The hsv are especially useful for model truncation:
The controllability and observability grammians are respectively given by:
	P = integral_{0}^{+oo} exp(At)B B* exp(A*t) dt
	Q = integral_{0}^{+oo} exp(A*t)C* C exp(At) dt
(where * denotes hermitian transpose)
Assuming the system is stable (i.e. all eigenvalues of A are in the left
half of the complex plane), the Hankel singular values are given by the
eigenvalues of PQ.  It is possible to do a state transformation on the
linear system to get a balanced realization,
i.e. P=Q=diag(sigma_1,...,sigma_n) where sigma_1 through sigma_n are the
hsv.  You can then show that if sigma_1>>sigma_n then the last state (in
the transformed coordinates) is much more difficult to observe and
control than the first state.  From an input-output point-of view, this
last state is not important and can be neglected.  In short, you can to
model reduction by doing a balanced realization, chopping off states which
have small hsv, and then transforming back to your original state
coordinates.

If you are interested in more on this stuff, check out the first 10 pages
or so of the paper "All optimal Hankel norm approximations of linear
multivariable systems and their L^oo error bounds" by Keith Glover 
(International Journal of Control, p 1115-1193, 1984). This paper assumes
you have a strong knowledge of linear systems and control.

Pete