## Description

1. An operator introduces a step change in the flow rate Fi to a particular process at 9:05

A.M., changing the flow from 500 to 540 l / min. The first significant change in the process

temperature T (initially at 50◦C) occurs 9:09 AM. Subsequently, response in temperature is

quite rapid, slowing down gradually until it appears to reach a steady-state value of 55.7 ◦C.

The operator notes in the logbook that there is no change after 9:34 A.M. What approximate

TF might be used to relate temperature to flow rate for this process in the absence of more

accurate information? What should the operator do next time to obtain a better estimate?

2. Consider a process whose TF is given by G(s) = (2s + 1)e

−3s

(20s + 1)(15s + 1)(4s + 1)(0.5s + 1)

(a) Simulate the step response of this system and fit an FOPTD model using Krishnaswamy

and Sundaresan’s method (of two points)

(b) From the transfer function, directly obtain the FOPTD and SOPTD approximations using

Skogestad’s half-rule method

(c) Fit an SOPTD model using the frequency-domain (magnitude and phase) least-squares

approximation method.

(d) Compare the step responses of the models obtained in parts (a)-(c) with that of the

original one. Tabulate your observations.

3. It is desired to develop an empirical model for a process. The exercise will provide insights into

the data generation and subsequent model identification.

(a) Assume that the process is G(s) = 4(−s + 1)e

−2s

(6s + 1)(8s + 1). Set up the SIMULINK diagram

for the sampled-data system consisting of a ZOH, the process and the sampler in series.

Choose Ts = 0.8 s.

(b) Design a pseudo-random binary signal (PRBS) input sequence. Use the idinput routine

for this purpose. Generate N = 2555 long sequence with B = 0.2 and amplitudes

between -2 and 2. Simulate the process using this input to the ZOH. Add measurement

noise (of variance 1.2) at the output to obtain the measurement y[k]. Partition the data

into training and test data sets.

[For the remainder of this exercise, you shall assume that no process knowledge

is available.]

(c) Estimate the non-parametric, i.e., response-based models to obtain estimates of delay,

steady-state gain and qualitative guess of the process order.

(d) Next assume an appropriate model for the system,

y

?

[k] +Xn

i=1

aiy

?

[k − i] = Xm

j=d

bju[k − j] (1)

where the values of d, m and n have to be chosen as per your analysis of impulse and

step responses (you are not allowed to use any knowledge of the process). Assuming

white-noise errors in the measurements, estimate the parameters of (1) using the oe

routine.

(e) Assess the goodness of the model estimated in (3d) for underfit using the residual analysis.

Use the resid routine for this purpose. Is the model satisfactory? If no, refine the model

structure (by increasing the output and/or input orders) until the model passes this test

satisfactorily. Subsequently, examine the errors in parameter estimates (using the present

routine) and compare the gains of this model and the one obtained in (3c).

(f) Report the final discrete-time model after cross-validation with the test data.