Correlation Analysis Algorithm
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
ncan often be quite large. In cases without regularization,
ncan be automatically decreased to secure a reasonable variance.
Specify the regularizing kernel using the
RegularizationKernelname-value argument of
Prewhitening — Prewhiten the input by applying an input-whitening filter of order
PWto the data. Use prewhitening when you are performing unregularized estimation. Using a prewhitening filter minimizes the effect of the neglected tail—
n—of the impulse response. To achieve prewhitening, the algorithm:
Defines a filter
PWthat whitens the input signal
1/A = A(u)e, where
Ais a polynomial and
eis white noise.
Filters the inputs and outputs with
uf = Au,
yf = Ay
Uses the filtered signals
Specify prewhitening using the
PWname-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.
impulseestsets NA to
5when 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:
where 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
impulseestwith an indication of negative delays,
mi = impulseest(data,nk,nb), where
nk< 0, returns a model
miwith 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.