The meaning of process gain [1st order model]

First-order model: (tao)*(dx/dt) + x = (Kp)*u

dx/dt = derivative of state variable, x
u = input
tao = time-constant of the system
Kp = process gain

The above equation is used to represent the process or system which you considering. It can describe the behaviour of the system, so by using the model you can predict what will happen to your process in case of un- or expected incident. The parameters such, tao and Kp are very important. For time constant (tao), it describes how fast of the process dynamic to reach its new steady state after perturbed by the input as i described in the article 'The meaning of time-constant'. Now in this article, i will describe the meaning of process gain (Kp).

Time constant (tao) describes how fast to reach the new steady-state value, while process gain (Kp) describes the value of the new steady-state that process output can be changed with respect to process input. The following example, the process model has same time-constant (tao = 1) with different process gain (Kp = 1, 1.5, 2). You can see from the output profile, time to reach new steady-state is very close for all cases but new steady values of the output are different in each case.

Before i will describe how to obtain the values of process gain based gradient concept, I will show you the solution of the 1st order equation in time domain. The solution is as

following: x(t) = (Kp)*u(t)*[1-exp(-t/tao)]. This example time constant and input change are set to be 1 then the equation will be reduced to x(tao) = 0.623*(Kp). Then if Kp = 1 and you change input value from 0 to 1, the process output (x) will be reach new steady at 0.623. Thus at time = 1 (at time-constant), process output will reach 62.3% of total change (new steady-state).

Process gain	New-steady	Output	Time	Time-constant estimate
1 1.5 2	1 1.5 2	0.6247 0.93701 1.2494	1.98 1.98 1.98	0.98 0.98 0.98

Estimate of time-constant from the output data is close to real time-constant of the process, the truncation error is occurred because of the software (MATLAB) precisiton. This example just used to illustrate the concept of process gain. If process gain is large then small change of input has large effect to the output.

Process gain of the process describes sensitivity of process output [or new steady-state] with respect to input change.