## 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

*K*._{p}One or more time constants

*T*. For complex poles, the time constant is called $${T}_{\omega}$$—equal to the inverse of the natural frequency—and the damping coefficient is $$\zeta $$ (_{pk}`zeta`

).Process zero

*T*._{z}Possible time delay

*T*before the system output responds to the input (_{d}*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, *T _{p1}* is a
time constant, and

*T*is the input-to-output delay:

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

Such that, $$Y(s)=G(s)U(s)+E(s)$$, 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 $$Y(s)=G(s)U(s)+H(s)E(s)$$. 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:

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

Where, *G _{ij}(s)* is the SISO process
model between the

*i*

^{th}output and the

*j*

^{th}input.

*E*and

_{1}(s)*E*are the Laplace transforms of the two output errors.

_{2}(s)