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Input-Output Linear Methods

Every identified linear input-output model is returned according to the following structure:

\[ y_k = G(z)u_k + H(z)e_k \]

where \(G(z)\) and \(H(z)\) are transfer function matrices of polynomials in \(z\), which is the forward shift operator.

The underlying generalized model structure is:

\[ A(z)y_k=\frac{B(z)}{F(z)}u_k+\frac{C(z)}{D(z)}e_k \]

which can be visualized as block diagram:

in-out-model

For each structure, the corresponding orders of the various blocks have to be set, since a specific parameter vector (\(\Theta\)) and a corresponding regressor matrix (\(\phi\), in which the various regressor vectors \(\varphi_k\) are stacked) have to be defined Ljung, 1999.`

Here's a summary of all the available model structures and polynomials used:

Model structure \(G(z)\) \(H(z)\)
FIR (Finite Impulse Response) \(B(z)\) 1
ARX (AutoRegressive with eXogenous inputs) \(A^{-1}(z)B(z)\) \(A^{-1}(z)\)
ARMAX (AutoRegressive with Moving Average eXogenous inputs) \(A^{-1}(z)B(z)\) \(A^{-1}(z)C(z)\)
ARMA (AutoRegressive with Moving Average) \(A^{-1}(z)\) \(A^{-1}(z)C(z)\)
ARARX \(A^{-1}(z)B(z)\) \(A^{-1}(z)D^{-1}(z)\)
ARARMAX \(A^{-1}(z)B(z)\) \(A^{-1}(z)D^{-1}(z)C(z)\)
OE (Output Error) \(F^{-1}(z)B(z)\) \(1\)
BJ (Box-Jenkins) \(F^{-1}(z)B(z)\) \(D^{-1}(z)C(z)\)
GEN (Generalized) \(A^{-1}(z)F^{-1}(z)B(z)\) \(A^{-1}(z)D^{-1}(z)C(z)\)

For example, in the case of a SISO (single-input single-output) ARMAX model, we have:

\[ \Theta=[ a_1 \: a_2 \: ... \: a_{n_a} \: b_1 \: ... \: b_{n_b} \: c_1 \: ... \: c_{n_c}]^T \\ \varphi_k= [ -y_{k-1} \: \dots \: -y_{k-n_a}\: u_{k-1-\theta} \: \dots \: u_{k-n_b-\theta} \: \epsilon_{k-1} \: \dots \: \epsilon_{k-n_c} ]^T \]

where \(\epsilon_k = y_k - \hat{y}_k\) is the prediction error, being \(\hat{y}_k\) the output model, and \(\theta\) is the time-delay.

Following naming convention for orders is used throughout SIPPY:

  • \(n_a\) - orders of \(A(z)\): the order of \(n_y\) outputs.
  • \(n_b\) - orders of \(B(z)\): the orders of \(n_y \times n_u\) input-output relations. Note that if an input does not influence an output, set that order equal to \(0\);
  • \(n_c\) - orders of \(C(z)\): the order of \(n_y\) error models (numerator in \(H(z)\)).
  • \(n_d\) - orders of \(D(z)\): the order of \(n_y\) error models (denominator in \(H(z)\)).
  • \(n_f\) - orders of \(F(z)\): the order of \(n_y\) undisturbed output model.
  • \(\theta\): the \(n_y \times n_u\) input-output time-delays.

Here an example for an ARMAX model for a MIMO system (4 inputs \(\times\) 3 outputs) is illustrated. Denoting \(z^{-1}\) as the backward shift operator, a MISO approach is employed, that is, for each of the three outputs the identified structure is:

\[ \begin{split} (1+a_1 z^{-1}+...+a_{n_a} z^{-n_a}) y_k & = (b_{1,1} z^{-(1+\theta_1)}+...+b_{1,n_{b1}} z^{-(n_{b1}+\theta_1)}) u^{(1)}_k + \\ & +(b_{2,1} z^{-(1+\theta_2)}+...+b_{2,n_{b2}} z^{-(n_{b2}+\theta_2)}) u^{(2)}_k \\ & + (b_{3,1} z^{-(1+\theta_3)}+...+b_{3,n_{b3}} z^{-(n_{b3}+\theta_3)}) u^{(3)}_k+ \\ & + (b_{4,1} z^{-(1+\theta_4)}+...+b_{4,n_{b4}} z^{-(n_{b4}+\theta_4)}) u^{(4)}_k+ \\ & + ... + (1+c_{1} z^{-1}+...+c_{n_{c}} z^{-n_{c}}) e_k \end{split} \]
Output 1 Output 2 Output 3
Input 1 \(g_{11}=\frac{4 z^3 + 3.3 z^2}{z^5 - 0.3 z^4 - 0.25 z^3 - 0.021 z^2}\) \(g_{21}=\frac{-85 z^2 - 57.5 z - 27.7}{z^4 - 0.4 z^3}\) \(g_{31}=\frac{0.2 z^3}{z^4 - 0.1 z^3 - 0.3 z^2}\)
Input 2 \(g_{12}=\frac{10 z^2}{z^5 - 0.3 z^4 - 0.25 z^3 - 0.021 z^2}\) \(g_{22}=\frac{71 z + 12.3}{z^4 - 0.4 z^3}\) \(g_{32}=\frac{0.821 z^2 + 0.432 z}{z^4 - 0.1 z^3 - 0.3 z^2}\)
Input 3 \(g_{13}=\frac{7 z^2 + 5.5 z + 2.2}{z^5 - 0.3 z^4 - 0.25 z^3 - 0.021 z^2}\) \(g_{23}=\frac{-0.1 z^3}{z^4 - 0.4 z^3}\) \(g_{33}=\frac{0.1 z^3}{z^4 - 0.1 z^3 - 0.3 z^2}\)
Input 4 \(g_{14}=\frac{-0.9 z^3 - 0.11 z^2}{z^5 - 0.3 z^4 - 0.25 z^3 - 0.021 z^2}\) \(g_{24}=\frac{0.994 z^3}{z^4 - 0.4 z^3}\) \(g_{34}=\frac{0.891 z + 0.223}{z^4 - 0.1 z^3 - 0.3 z^2}\)
Error model \(h_{1}=\frac{z^5 + 0.85 z^4 + 0.32 z^3}{z^5 - 0.3 z^4 - 0.25 z^3 - 0.021 z^2}\) \(h_{2}=\frac{z^4 + 0.4 z^3 + 0.05 z^2}{z^4 - 0.4 z^3}\) \(h_{3}=\frac{z^4 + 0.7 z^3 + 0.485 z^2 + 0.22 z}{z^4 - 0.1 z^3 - 0.3 z^2}\) 
n_a = [3, 1, 2]
n_b = [[2, 1, 3, 2], [3, 2, 1, 1], [1, 2, 1, 2]]
t_h = [[1, 2, 2, 1], [1, 2, 0, 0], [0, 1, 0, 2]]