Process Control Paper on the Approximation of the FOPTD Model

Given :

Approximation of FOPTD Model

(a)Two Points Method

[See Appendix-A for MATLAB Code]

Figure (1): Process reaction curve using Matlab

At 28.4%, t1 = 2.48s
At 63.2%, t2 = 4.18s

T = 1.5(t2 – t1) = 1.5(4.18-2.48)=2.55s
L = 0.5(3t1-t2)=0.5(3*2.48-4.18)=1.63s

Gain K=1

Approximated TF is : Gs=1e-1.63s2.55s+1

Verification of Two Point Method in Time domain and Frequency domain

[See Appendix-A for MATLAB Code]

Figure (2): Verification of Two Point Method in Time domain

Figure (3): Verification of Two Point Method in Frequency domain

(b) Log Method

For this method, we need samples of the output which can be done using Matlab, the following function takes samples every 1 second starting from 1 to 20, and the results are stored in arrays x and y.

[See Appendix-B for MATLAB Code]

Figure (4): Log method curve

From the graph, we can get cross-axis value. Thus:
K=1, L/T=0.9081, L=1.89
T=1.89/0.9081=2.08

Approximated TF is : Gs=1e-1.89s2.08s+1

Verification of Log method in Time domain and Frequency domain

[See Appendix-B for MATLAB Code]

Figure (5): Verification of Log Method in Time domain

Figure (6): Verification of Log Method in Frequency domain

(c) Area Method

We sample the output with a time interval of 0.5s from time 0 to 20, 41 samples totally. And calculate the area between every adjacent sample as shown in Figure 7.

[See Appendix-C for MATLAB Code]

Figure (7): Log method curve

Using Matlab, Area A0 and A1 are determined using cftool Code.

A0 = 3.9698

For A1, the area is taken between the time 0 and the time when the output reaches 63.2% of the final value. So the time is from 0 to 4.17s. The area from time 0 to 4s can be calculated using the same method as when we calculate A0. Totally 4/0.5+1 = 9 samples

A1 = 0.9787

According to the Formulas:

T=eA1/K=0.9787e=2.660
L= (A0 /K)-T=3.9698-2.660=1.3098

And hence, the resultant model is Gs=1e-1.3098s2.66s+1