No model detection for a input / output vectors in a MIMO Syetem

How can I determine if a pair of input/output vectors (y and u) have no model for a MIMO transfer function? y and u are i-th and j-th output and input vectors, respectively. I am trying to know if there is not a model for them, in order to put a zero in that position, otherwise a proper system identification method would be used to determine the existing model. Attention: cross correlation analysis between y and u is not enough to determine this. I will appreciate any suggestion.

Answer 1

[Rajiv Singh]

What do you mean by "no model for a MIMO transfer function"? If you are trying to analyze if j:th output is affected by i:th input or not, you can do some sort of regression analysis (partial least squares, ARX modeling). For each output, you could try a order determination exercise (see ARXSTRUC, SELSTRUC). The optimal orders shown by SELSTRUC would indicate the weights assigned to each input signals towards determination of the output. Note that it may not be easy to isolate the effect of one input signal on a particular output or determine such effects uniquely. However, Principal Component analysis or Partial Least Squares techniques would let you determine a minimal set of inputs that appear to be strongly correlated with the output.

[Osmel Reyes]

Dear Rajiv, that´s exactly what I meant. I´m trying to construct a function to detect when the i-th input and j-th output have no relation or model in the transfer function, in order to put or to suggest putting a zero in (i-th, j-th) position on the TF. I am not trying to make an input-output selection to exclude inputs, but to detect the influence of an input in some output. Do you think tha it could be done with PLS or PAC?

I´ve been working with the Shell Benchmark (Cott, 1995). This is a 2X2 plant, that has no model between the output 2 and the input 1. I´ve tried to detect this with techniques such as cross correlation and Single-Input Effectiveness (SIE) (Cao;Rossiter, 1996), but the results have been no good enough. For example, in SIE, the smallest (etaI)^2(j) coefficient indeed corresponds to output 2; input 1, but its value is in the same order of magnitude of other combinations coefficients with model. Something similar occurs for cross correlation analysis. I´ve been working with input / output vectors but in closed-loop, because in realistic application of this function will be unlikely that the control loop could be opened.

Thanks for your comments.

Answer 2

[Osmel Reyes]

Dear Rajiv, that´s exactly what I meant. I´m trying to construct a function to detect when the i-th input and j-th output have no relation or model in the transfer function, in order to put or to suggest putting a zero in (i-th, j-th) position on the TF. I am not trying to make an input-output selection to exclude inputs, but to detect the influence of an input in some output. Do you think tha it could be done with PLS or PAC?

I´ve been working with the Shell Benchmark (Cott, 1995). This is a 2X2 plant, that has no model between the output 2 and the input 1. I´ve tried to detect this with techniques such as cross correlation and Single-Input Effectiveness (SIE) (Cao;Rossiter, 1996), but the results have been no good enough. For example, in SIE, the smallest (etaI)^2(j) coefficient indeed corresponds to output 2; input 1, but its value is in the same order of magnitude of other combinations coefficients with model. Something similar occurs for cross correlation analysis. I´ve been working with input / output vectors but in closed-loop, because in realistic application of this function will be unlikely that the control loop could be opened. Thanks for your comments.