mor.norm

Main interface for LTI dynamical system norms computation.

Syntax

N = mor.norm(H)
N = mor.norm(H,ntype)
N = mor.norm(H,ntype,freqBand)
N = mor.norm(H,ntype,freqBand,opt)

Description

This routine enables to compute different norms for dynamical systems.

If H is a state-space model, ntype can be:
- 2 or 'H2': compute the (frequency-limited) H2 norm of the system.
- 'L2': compute the (frequency-limited) L2 norm of the system. Unlike the H2-norm, this one is finite even for unstable systems.
Note that results might become inaccurate if the model has some poles very close to the imaginary axis.
If H is a couple of frequencies and associated transfer matrices {w, H}, ntype can be:
- inf: maximum of the maximum absolute value of the transfer function at each sample point.
- 2: averaged sum of squared Frobenius norm at each sample point.
- 'L2': trapezoidal approximation of the L2-norm.
- 'Linf': maximum of the 2-norm of the samples.
Differences between these norms will appear for MIMO data.

Input arguments

sys	Dynamical system or input-output data. More specifically `sys` can either be a state-space ODE (DAE) realization described as `sys = ss(A,B,C,D)` (or `sys = dss(A,B,C,D,E)`) a matrix set describing an ODE (or DAE) described as `sys = {A,B,C,D}` (or `sys = {A,B,C,D,E}`) a set of N pulsations of an nu inputs, ny outputs responses described `sys = {w_i,H_i}`
ntype	Type of the norm to be computed (string, see in the above description part).
freqBand	Frequency band on which the norm must be considered (2x1 vector with non-negative values).
opt	Structure of options.

Output arguments

N	Norm value (positive real scalar).

Examples

Norms of state-space model

Example 1: H2 and frequency-limited H2 norms. This example shows how to use the MOR Toolbox to compute the H2 and frequency-limited H2 norms of a state-space model. The latter is actually computed for several frequency intervals to illustrate that it tends towards the H2-norm.

H        = ss(zpk([],[-0.1+3*1i,-0.1-3*1i,-0.05+10*1i,-0.05-10*1i,-0.01+20*1i,-0.01-20*1i],1));
h2norm   = norm(H);       % Matlab routine
h2norm2  = mor.norm(H);   % MOR routine
% Computing the frequency-limited h2-norm for various intervals
W        = logspace(0,1.4,100);
h2wnorm  = zeros(20,1);
for i = 1:length(W)
  interval    = [0, W(i)];
  h2wnorm(i)  = mor.norm(H, 2, interval);
end
% Illustration of the frequency-limited h2-norm
G        = bode(H,W);
figure
subplot(211)
semilogx(W, abs(squeeze(G)),'b')
set(gca(),'Xlim',[W(1),W(end)])
subplot(212)
semilogx([W(1),W(end)],h2norm * [1, 1], 'b')
hold on
semilogx([W(1),W(end)],h2norm2 * [1, 1], 'r--')
semilogx(W, h2wnorm, 'm')
set(gca(),'Xlim',[W(1),W(end)])
legend('h2-norm (Matlab)', 'h2-norm (MOR)','Frequency limited h2-norm','Location','SouthEast')

Example 2: L2-norm of unstable systems. This example shows how to compute the L2-norm for an unstable system and what it actually represents in terms of norms of the stable and unstable parts of the system.

% System composed by a stable and an unstable part
stable_part = ss(tf(1,[1, 2]));
unstab_part = ss(tf(1,[1, -1]));
H           = stable_part + unstab_part;
% The H2-norm is infinite
mor.norm(H)
% The L2-norm is finite
mor.norm(H, 'L2')
% and it is equal to the sqrt of the squared sum of the H2-norm of the
% stable part and the H2-norm of the mirror of the unstable part
mirrored    = ss(tf(1,[1, 1]));
sqrt(mor.norm(stable_part)^2 + mor.norm(mirrored)^2)

Norm of input-output data

This example gives an overview of the norms for input-output data and how some of them are related to the norm of the underlying system.

rng(3)
H     = rss(20,2,3,1);
H.D   = 0;
fprintf('h2-norm of the model: %.3f\n',mor.norm(H))
fprintf('hinf-norm of the model: %.3f\n',norm(H,inf))
% Sampling the transfer function
w = logspace(-2,2,100);
h = freqresp(H,w);
% Pointwise norms of the data
fprintf('Mean square Frobenius norm of the data: %.3f\n',mor.norm({w,h},2))
fprintf('Pointwise 2-norm of the data: %.3f\n',mor.norm({w,h},inf))
% System related norms of the data
fprintf('Approximation of the L2-norm from the data: %.3f\n',mor.norm({w, h}, 'l2'))     % Should be 'close' to the h2 norm
fprintf('Approximation of the Linf-norm from the data: %.3f\n',mor.norm({w, h}, 'linf'))   % Should be 'close' to the hinf norm