## What Is a Process Model?

The structure of a process model is a simple continuous-time transfer function that describes linear system dynamics in terms of one or more of the following elements:

• Static gain Kp.

• One or more time constants Tpk. For complex poles, the time constant is called ${T}_{\omega }$—equal to the inverse of the natural frequency—and the damping coefficient is $\zeta$ (`zeta`).

• Process zero Tz.

• Possible time delay Td before the system output responds to the input (dead time).

• Possible enforced integration.

Process models are popular for describing system dynamics in many industries and apply to various production environments. The advantages of these models are that they are simple, support transport delay estimation, and the model coefficients have an easy interpretation as poles and zeros.

You can create different model structures by varying the number of poles, adding an integrator, or adding or removing a time delay or a zero. You can specify a first-, second-, or third-order model, and the poles can be real or complex (underdamped modes). For more information, see Process Model Structure Specification.

For example, the following model structure is a first-order continuous-time process model, where K is the static gain, Tp1 is a time constant, and Td is the input-to-output delay:

`$G\left(s\right)=\frac{{K}_{p}}{1+s{T}_{p1}}{e}^{-s{T}_{d}}$`

Such that, $Y\left(s\right)=G\left(s\right)U\left(s\right)+E\left(s\right)$, where Y(s), U(s), and E(s) represent the Laplace transforms of the output, input, and output error, respectively. The output error, e(t), is white Gaussian noise with variance λ. You can account for colored noise at the output by adding a disturbance model, H(s), such that $Y\left(s\right)=G\left(s\right)U\left(s\right)+H\left(s\right)E\left(s\right)$. For more information, see the `NoiseTF` property of `idproc`.

A multi-input multi-output (MIMO) process model contains a SISO process model corresponding to each input-output pair in the system. For example, for a two-input, two-output process model:

`$\begin{array}{l}{Y}_{1}\left(s\right)={G}_{11}\left(s\right){U}_{1}\left(s\right)+{G}_{12}\left(s\right){U}_{2}\left(s\right)+{E}_{1}\left(s\right)\\ {Y}_{2}\left(s\right)={G}_{21}\left(s\right){U}_{1}\left(s\right)+{G}_{22}\left(s\right){U}_{2}\left(s\right)+{E}_{2}\left(s\right)\end{array}$`

Where, Gij(s) is the SISO process model between the ith output and the jth input. E1(s) and E2(s) are the Laplace transforms of the two output errors.

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