Specifications

covar
16-43
p =
30.3167
You can compare this output of covar to simulation results.
randn('seed',0)
w = sqrt(5)
∗randn(1,1000); % 1000 samples
% Simulate response to w with LSIM:
y = lsim(sys,w);
% Compute covariance of y values
psim = sum(y .∗ y)/length(w);
This yields
psim =
32.6269
The two covariance values p and psim do not agree perfectly due to the finite
simulation horizon.
Algorithm Transfer functions and zero-pole-gain models are first converted to state space
with
ss.
For continuous-time state-space models
is obtained by solving the Lyapunov equation
The output response covariance is finite only when and then
.
In discrete time, the state covariance solves the discrete Lyapunov equation
and is given by
x
·
Ax Bw+=
yCxDw+=
Q
AQ QA
T
BWB
T
++ 0=
PD0=
PCQC
T
=
AQA
T
Q– BWB
T
+ 0=
PPCQC
T
DWD
T
+=