Generally, you can describe a discrete system by using the general-linear polynomial model. This model provides flexibility for both system dynamics and stochastic dynamics.

Use the SI Estimate General Linear Model VI to estimate general-linear polynomial models. The following equation describes this model:

yk=z-nGz-1, θuk+Hz-1, θek

where

  • u(k) is the input of the system
  • y(k) is the output of the system
  • e(k) is the disturbance of the system which usually is zero-mean white noise
  • G(z–1, θ) is the transfer function of the deterministic part of the system
  • H(z–1, θ) is the transfer function of the stochastic part of the system

The deterministic transfer function specifies the relationship between the output and the input signal. The stochastic transfer function specifies how the random disturbance affects the output signal. Often the deterministic and stochastic parts of a system are referred to as system dynamics and stochastic dynamics, respectively.

The term z–1 is the backward shift operator, which is defined by the following equations:

z-1xk=xk-1
z-2xk=xk-2

...

z-nxk=xk-n

z–n defines the number of delay samples between the input and the output.

G(z–1, θ)u(k) and H(z–1, θ)e(k) are rational polynomials as defined by the following equations:

Gz-1, θ=Bz, θAz, θFz, θ
Hz-1, θ=Cz, θAz, θDz, θ

The vector θ is the set of model parameters. The following equations do not display θ to make the equations easier to read.

The following equation shows the form of the general-linear model:

Azyk=BzFzuk-n+CzDzek

where

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

A(z), 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:

Az=1+a1z-1+...+akaz-ka
Bz=b0+b1z-1+...+bkbz-kb-1
Cz=1+c1z-1+...+ckcz-kc
Dz=1+d1z-1+...+dkdz-kd
Fz=1+f1z-1+...+fkfz-kf

The following figure depicts the signal flow of a general-linear model:

Figure 188. Signal Flow of a General Linear Model

where

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

Setting one or more of A(z), C(z), D(z), and F(z) to 1 can create simpler models such as autoregressive with exogenous terms (ARX), autoregressive-moving average with exogenous terms (ARMAX), output-error, and Box-Jenkins models, which you commonly use in real-world applications.

SISO

The following are the time domain equations for the general-linear SISO model:

w k + f 1 w k - 1 + . . . + f k f w k - k f = b 0 u k - n + b 1 u k - n - 1 + . . . + b k b - 1 u k - n - k b + 1
v k + d 1 v k - 1 + . . . + d k d v k - k d = e k + c 1 e k - 1 + . . . + c k c e k - k c
y k + a 1 y k - 1 + . . . + a k a y k - k a = w k + v k

where

  • kf is the F order
  • kb is the B order
  • kc is the C order
  • kd is the D order
  • ka is the A order
  • u(k) represents the system inputs
  • n is the system delay
  • e(k) is the system disturbance
  • w is the auxiliary variable