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:
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.
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:
The input can be pre-whitened by applying an input-whitening filter
PW to the data. This minimizes the effect
of the neglected tail (
k > n) of the impulse
A filter of order
PW is applied
such that it whitens the input signal
1/A = A(u)e, where
a polynomial and
e is white noise.
The inputs and outputs are filtered using the filter:
uf = Au,
yf = Ay
The filtered signals
used for estimation.
You can specify prewhitening using the
pair argument of
The least-squares estimate can be regularized. This means that a prior
estimate of the decay and mutual correlation among
formed and used to merge with the information about
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
be quite large. In cases where there is no regularization,
be automatically decreased to secure a reasonable variance.
You can specify the regularizing kernel using the
Name-Value pair argument of
Autoregressive Parameters —
The basic underlying FIR model can be complemented by
parameters, making it an ARX model.
This gives both better results for small
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:
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, ,
returns a model
mi with an impulse response
aligned so that it corresponds to lags .
This is achieved because the input delay (
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.