# 1.8: Industrial Process Models

Industrial processes comprise procedures involving exchange of chemical, electrical or mechanical energy to aid in the manufacturing of industrial products. Industrial process models are mathematical models used to describe those processes.

An industrial process model, in its simplified form, can be represented by a first-order ODE accompanied by a dead-time, i.e., there is a finite time delay between the application of input and the appearance of the process output.

Let $$\tau$$ represent the time constant associated with an industrial process, $$\tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[3]/span[2]/span, line 1, column 1 $$ represent the dead time, and $$K$$ represent the process dc gain; then, the simplified industrial process dynamics are represented by the following delay-differential equation: $\tau \frac{{\rm d}y(t)}{{\rm d}t} +y(t)=Ku(t-t_{d} ).$ Application of the Laplace transform produces the following first-order-plus-dead-time (FOPDT) model of an industrial process: $G(s)=\frac{Ke^{-\tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[3]/span[5]/span, line 1, column 1  s} }{\tau s+1} .$
The process parameters $$\{ K,\; \tau ,\; \tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[3]/span[6]/span, line 1, column 1  \}$$
, respectively, denote the process gain, the process time constant, and the process dead time. These parameters can be identified from the process response to a unit-step input.

We note that the process model involves a transendental function, $$e^{-\tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[4]/span[1]/span, line 1, column 1  s}$$. For analysis and controller design purposes, this term is replaced by a suitable rational approximation. Typical approximations obtained from Taylor series expansion of the transendental term are given as: $e^{-\tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[4]/span[2]/span[1], line 1, column 1  s} \approx 1-\tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[4]/span[2]/span[2], line 1, column 1  s, e^{-\tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[4]/span[2]/span[3], line 1, column 1  s} =\frac{1}{e^{\tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[4]/span[2]/span[4], line 1, column 1  s} } \approx \frac{1}{1+\tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[4]/span[2]/span[5], line 1, column 1  s} , e^{-{\tau }_ds}=\frac{e^{-{\tau }_ds/2}}{e^ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[4]/span[2]/span[6], line 1, column 1  \approx \frac{1-{\tau }_ds/2}{1+{\tau }_ds/2}.$
The last expression above is called the first-order Pade’ approximation and is popular in idustrial process control applications.

Example 1.13: An industrial process model

The process parameters of a stirred-tank bioreactor are given as: $$\{ K,\; \tau ,\; \tau _ ParseError: invalid DekiScript (click for details) Callstack: at (Courses/University_of_Arkansas_Little_Rock/Introduction_to_Control_Systems_(Iqbal)/01:_Mathematical_Models_of_Physical_Systems/1.08:_Industrial_Process_Models), /content/body/p[6]/span[1]/span, line 1, column 1  \} =\left\{20,0.5,1\right\}$$. The transfer function model of the process is formed as: $$G(s)=\frac{20e^{-s} }{0.5s+1}$$.

By using a first-order Pade’ approximation, a rational transfer function model of the industrial process with delay is obtained as: $$G(s)=\frac{20\left(1-0.5s\right)}{\left(0.5s+1\right)^{2} }$$.

Further, a model of this kind is called non-minimum phase due to the extra phase added by the right-half-plane zero in the numerator (see Chapter 2 for details).