Correlation analysis refers to methods
that estimate the impulse response of a linear model, without specific
assumptions about model orders.
The impulse response, g, is the system's
output when the input is an impulse signal. The output response to
a general input, u(t), is obtained
as the convolution with the impulse response. In continuous time:
$$y(t)={\displaystyle {\int}_{-\infty}^{t}g\left(\tau \right)u\left(t-\tau \right)}d\tau $$
In discrete-time:
$$y\left(t\right)={\displaystyle \sum _{k=1}^{\infty}g\left(k\right)u\left(t-k\right)}$$
The values of g(k) are
the discrete time impulse response coefficients.
You can estimate the values from observed input-output data
in several different ways. impulseest
estimates
the first n coefficients using the least-squares
method to obtain a finite impulse response (FIR) model of order n.
Several important options are associated with the estimate:
Prewhitening —
The input can be pre-whitened by applying an input-whitening filter
of order PW
to the data. This minimizes the effect
of the neglected tail (k > n
) of the impulse
response.
A filter of order PW
is applied
such that it whitens the input signal u
:
1/A = A(u)e
, where A
is
a polynomial and e
is white noise.
The inputs and outputs are filtered using the filter:
uf = Au
, yf = Ay
The filtered signals uf
and yf
are
used for estimation.
You can specify prewhitening using the PW
name-value
pair argument of impulseestOptions
.
Regularization —
The least-squares estimate can be regularized. This means that a prior
estimate of the decay and mutual correlation among g(k)
is
formed and used to merge with the information about g
from
the observed data. This gives an estimate with less variance, at the
price of some bias. You can choose one of the several kernels to encode
the prior estimate.
This option is essential because, often, the model order n
can
be quite large. In cases where there is no regularization, n
can
be automatically decreased to secure a reasonable variance.
You can specify the regularizing kernel using the RegulKernel
Name-Value
pair argument of impulseestOptions
.
Autoregressive Parameters —
The basic underlying FIR model can be complemented by NA
autoregressive
parameters, making it an ARX model.
$$y\left(t\right)={\displaystyle \sum _{k=1}^{n}g\left(k\right)u\left(t-k\right)}-{\displaystyle \sum _{k=1}^{NA}{a}_{k}y\left(t-k\right)}$$
This gives both better results for small n
and
allows unbiased estimates when data are generated in closed loop. impulseest
uses NA
= 5 for t>0 and NA
= 0 (no autoregressive component) for t<0.
Noncausal effects —
Response for negative lags. It may happen that the data has been generated
partly by output feedback:
$$u(t)={\displaystyle \sum _{k=0}^{\infty}h(k)y\left(t-k\right)}+r\left(t\right)$$
where h(k) is the impulse
response of the regulator and r is a setpoint or
disturbance term. The existence and character of such feedback h can
be estimated in the same way as g, simply by trading
places between y and u in the
estimation call. Using impulseest
with an indication
of negative delays, $$\text{mi}=\text{impulseest}(data,nk,nb),\text{}nk0$$,
returns a model mi
with an impulse response
$$\left[h(-nk),h(-nk-1),\mathrm{...},h(0),g(1),g(2),\mathrm{...},g(nb+nk)\right]$$
aligned so that it corresponds to lags $$\left[nk,nk+1,\mathrm{..},0,1,2,\mathrm{...},nb+nk\right]$$.
This is achieved because the input delay (InputDelay
)
of model mi
is nk
.
For a multi-input multi-output system, the impulse response g(k)
is an ny-by-nu matrix, where ny is
the number of outputs and nu is the number of inputs.
The i–j element of the
matrix g(k) describes the behavior
of the ith output after an impulse in the jth
input.