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 output when the input is an impulse signal. The output response to a general input, u(t), is the convolution with the impulse response. In continuous time:
In discrete 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.
impulseest estimates the first
n coefficients using the
least-squares method to obtain a finite impulse response (FIR) model
of order n.
impulseest provides several important options for the estimation:
Regularization — Regularize the least-squares estimate. With regularization, the algorithm forms an estimate of the prior decay and mutual correlation among g(k), and then merges this prior estimate with the current information about g from the observed data. This approach results in an estimate that has less variance but also some bias. You can choose one of several kernels to encode the prior estimate.
This option is essential because the model order
n can often be quite
large. In cases without regularization,
n can be automatically decreased
to secure a reasonable variance.
Specify the regularizing kernel using the
name-value argument of
Prewhitening — Prewhiten the input by applying an input-whitening
filter of order
PW to the data. Use prewhitening when you are performing
unregularized estimation. Using a prewhitening filter minimizes the effect of the neglected
n—of the impulse response. To achieve
prewhitening, the algorithm:
Defines a filter
A of order
PW that whitens
the input signal
1/A = A(u)e, where
A is a polynomial and
e is white noise.
Filters the inputs and outputs with
uf = Au,
yf = Ay
Uses the filtered signals
Specify prewhitening using the
PW name-value pair argument of
Autoregressive Parameters — Complement the basic underlying FIR model by NA autoregressive parameters, making it an ARX model.
This option both gives better results for small n values and allows
unbiased estimates when data are generated in closed loop.
sets NA to
5 when t > 0 and sets NA to
0 (no autoregressive
component) when t < 0.
Noncausal effects — Include response to negative lags. Use this option if the estimation data includes output feedback:
h(k) is the impulse response of the regulator and
r is a setpoint or disturbance term. The algorithm handles the
existence and character of such feedback h, and estimates
h in the same way as g by simply trading places
between y and u in the estimation call. Using
impulseest with an indication of negative delays,
nk < 0, returns a model
mi with an impulse response
that has an alignment that corresponds to the
lags . The algorithm achieves this alignment because the input delay
InputDelay) of model
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.