Box-Jenkins Model Definitions

When A(z) equals 1, the general-linear polynomial model reduces to the Box-Jenkins model.

This model provides a complete model of a system because this model represents disturbance properties separately from system dynamics. This model is useful when you have disturbances that enter late in the process, such as measurement noise on the output.

Use the SI Estimate BJ Model VI to estimate Box-Jenkins models. The identification method of the Box-Jenkins model is the prediction error method, which is the same as that of the ARMAX model.

The following equation shows the form of the Box-Jenkins model:

where

• y(k) represents the system outputs
• u(k) represents the system inputs
• n is the system delay
• e(k) is the system disturbance

B(z), C(z), D(z), and F(z) are polynomial with respect to the backward shift operator z^-1 and defined by the following equations:

The following figure depicts the signal flow of a Box-Jenkins model:

where

• u is the system inputs
• y is the system outputs
• e is the system disturbance
• v and ω are the auxiliary variables

SISO

The following are the time domain equations for the Box-Jenkins SISO model:

$w\left(k\right)+f_{1}w\left(k-1\right)+...+f_{k_f}w\left(k-k_f\right)=b_{0}u\left(k-n\right)+b_{1}u\left(k-n-1\right)+...+b_{k_b-1}u\left(k-n-k_b+1\right)$

$v\left(k\right)+d_{1}v\left(k-1\right)+...+d_{k_d}v\left(k-k_d\right)=e\left(k\right)+c_{1}e\left(k-1\right)+...+c_{k_c}e\left(k-k_c\right)$

$y\left(k\right)=w\left(k\right)+v\left(k\right)$

where

• k_f is the F order
• k_b is the B order
• k_c is the C order
• k_d is the D order
• u(k) is the system input
• n is the system delay
• e(k) is the system disturbance
• w is the auxiliary variable