# A concentration control loop in an industrial chemical process can be represented by a second order dynamics with a long transmission delay. The two time constants of the second order dynamics are T1 = 4 min. and T2 = 6 min. respectively. The steady state gain is 5 and dead time is 12 min. The loop is to be controlled to achieve a desired dynamics of first order with time constant Td = 2 min. and zero steady-state offset. Produce solutions to the following questions.

Coursework Specification and Corresponding Marks

A concentration control loop in an industrial chemical process can be represented by a second order dynamics with a long transmission delay. The two time constants of the second order dynamics are T1 = 4 min. and T2 = 6 min. respectively. The steady state gain is 5 and dead time is 12 min. The loop is to be controlled to achieve a desired dynamics of first order with time constant Td = 2 min. and zero steady-state offset. Produce solutions to the following questions.

(i)    Identify the plant transfer function and the desired closed-loop system transfer function. Then, Design a feedback control system, with the controller designed using Direct Synthesis Method, where the time delay is approximated using the first-order Taylor expansion. Implement the controller with a standard industrial PID controller.
(ii)    Design a feedback control system with a PID controller designed using the closed-loop Ziegler-Nichols empirical method. The Simulink model used in the experiment to obtain the sustained output curve should be displayed. The sustained output curve should also be displayed. The controller design must be presented.

(iii)    Design a Smith Predictor control system for the given plant with the controller designed using the Direct Synthesis Method.  A block diagram must be shown to demonstrate the control system structure. The design details are also required.

(iv)    Simulate the three control systems designed using the three methods with a unity step input for 100 minutes. The simulation must use one Simulink model with three sub-models, each for a control system. Display the Simulink model and sub-models, and present the simulation parameter setting. Display the three outputs and the set-point in one figure for comparison. Display also the three corresponding control variables in another figure. Compare the three output curves and comment on the performances of the three control systems.
(v)    Write a report with content, a brief introduction, description of the process to be controlled, presentation of the control system design with each method, simulation model and sub-models, control systems outputs and corresponding control variables, and finally the comparison, comments and conclusion.

The assignment should be presented logically, clearly and completely. The assignment should be written using Microsoft Word. Mathematical expressions must be presented using Microsoft Equation. All figures must have title, labels and legends.

References
Course Material Book
Author:    Seborg, Edgar and Mellichamp, Publishing Year:     2011, Title: Process dynamics and control, Edition:    3rd, Publisher:     McGraw-Hill, ISBN: 978-0-470-64610-6
Author: Ogata, K., Publishing Year:1997, Title: Modern Control Engineering, Edition:3rd, Publisher:     Prentice Hall International, ISBN: 0-13-261389-1
Guide to Performance Criteria (The Module Leader is advised to delete sections not applicable to the coursework set and if necessary modify the criteria accordingly)
70% and above:
Your work must be of outstanding quality and fully meet the requirements of the coursework specification and learning outcomes stated. You must show independent thinking and apply this to your work showing originality and consideration of key issues. There must be evidence of wider reading on the subject. Key words which may describe a coursework at this level include: appraises, compares, concludes, contrasts, criticizes, critiques, defends, discriminates, evaluates, explains, interprets, justifies, relates, supports.

Chapter 2 Processes with Time Delay

2.1 Stability of Time Delay Systems

A feedback control system structure is shown in Fig.1. When time delays are involved in the closed-loop system, the system stability is influenced and the relative stability will be reduced.

Fig.1 feedback control system structure

We study the stability of closed-loop system when time delay is involved. Generally, the characteristic equation (CE) of the closed loop system is   1+Gc(s)Gp(s) = 0. The roots of the CE are the poles of the closed-loop system. If all poles have negative real parts, the system is stable. If some poles have zero real parts, the system is critical stable. If any pole has positive real part, system is unstable.

Method 1
As TF of time delay is exponential function and when it is included in a CE, the CE is not easy to be solved. Therefore, use Pade approximation of time delay and then use Routh stability criterion.

Example 1

A feedback control system has a process TF

and a P controller with Gc(s) = K. The actuator and sensor TF are first order with small time constants and unity gains. Determine range of K using the Routh table so that the system is stable.

Solution

As the process model is first order, the system would be stable for any value of K if it has no time delay. However, time delay will greatly reduce the relative stability of the closed-loop system.

The closed-loop TF is

The CE is

Using first-order Pade approximation,  , we have approximated CE,

or

or

The Routh table is

s2    s1    s0
s2    5    1+2K    0
s1    6-2K    0    0
s0    1+2K    0

System stable requests that the elements in the first column are greater than zero, or
6-2K > 0
1+2K > 0
Solve K from the above inequalities,
K < 3
K > -1/2

Consider the controller gain is always positive, we have finally,
0 < K < 3

The stable range obtained is not accurate as an approximation is used.

Method 2
Next, a method is introduced which can evaluate accurately the closed-loop system stability when time delay is involved.
Step 1: Obtain sinusoidal transfer function  by replacing s in  with .
Step 2: Using frequency analysis method, determine the critical value of frequency  and gain  by solving the CE:  , or
(1)

As  is a complex number, (1) is equivalent to the following two equations,
(3)
(4)
Step 3: From (4) solve  , then substitute  into (3) to solve  . Thus,             K <  the system will be stable.

Example 2

For the same system in example 2.2, determine exact range of K so that the closed-loop system is stable.

Solution
As                    Gc(s) = K,

So,
Step1:

Step 2:
(5)

(6)

Step 3:
Equation (6) is difficult to solve analytically, we here use trial and error method.
1    0.8    0.9    0.895

-3.3734    -2.9258    -3.1521    -3.1405

So, it is obtained that,

Sub  into (5)

Then,

This implies that the stable range of K is

0 < K < 2.2927

When the second order Pade approximation is used, the range of K is (calculation is omitted),
0 < K < 2.32

These results give us a general idea of how accurate these approximations are. Simulation of the closed-loop system using Simulink is shown below.

Fig.2b Responses with K=2(green), 2.29(blue), 2.5(red)

2.2 Controller Design for Time Delay Systems

Direct Synthesis Method

Direct synthesis method can be used to design a controller TF, Gc(s), with which the closed-loop TF, Gcl(s) of the system will be equal to the given desired closed-loop TF, Gr(s). See Fig.2.4, the TF from set-point to output can be derived as
(7)
Using Gr(s) to replace Gcl(s) in (17), then Gc(s) can be solved as

(8)
Now, the question is how to select a desired TF to satisfy performance requirements. Usually a first order TF is used as the desired TF with time constant to be chosen to satisfy the requirement for setting time:
(9)

As many industrial processes can be represented by first-order plus time delay or second order plus time delay models, the design to these models are demonstrated below.

Example 3

Design feedback controllers for the following processes using the direct synthesis method,

(a)     ,        (b)

Solution
Directly using formula (8) with the desired TF given in (9), we have for (a)

Compare with the standard PI controller
(10)
It is confirmed that the controller is a PI controller with  .

For (b), we have

Compare with the standard PID controller

(11)

It is confirmed that the controller is a PID controller with

.

Processes with Time Delay

If the process has a time delay  , then a reasonable choice for the desired closed-loop TF is
(12)

and   must be selected so that  , because the controller variable cannot response to a set point change in less than   time units due to the process time delay  .

Example 4

Design feedback controllers for the following processes using the direct synthesis method,

(a)     ,    (b)
Solution

Select   in (12) and directly using formula (8) with the desired TF given in (12), we have for (a)

The first-order Tayler approximation,  , is used for the time delay. We have

Comparing with (2.20) we know that this is a PI controller with

(13)
For (b),

It is a PID controller with
(14)

It can be seen in (13) and (14) that Kc is reduced by   appearing in denominator. This is because the system with time delay is easy to oscillate, and therefore the controller gain is designed smaller.

2.3 Discrete-Time PID Control

Analogue PID control has been learned in level 5 module. Here we introduce discrete-time PID control. Continuous-time PID controller is a function of error signal,
(15)

To get the digital version of the PID controller, the following approximations are used.

Substitute these to (15),
(16)
(16) is called Position form.

The position form is not easy to use due to the accumulation term of the error. Another is the Velocity form. Write (16) for sampling instant, k-1, as below,

Subtracting it from (2.26) yields,

(17)

And the control variable can be obtained by updating its last time value.

(18)

Equation (17) can be of the following form which is easier to implement (see example 5(d))

(19)

Example 5

A second order plus time delay process with TF of

(a)    Design an analogue PID controller
(b)    Simulate the control system using Simulink
(c)    Fine-tune the controller to improve the performance
(d)    Convert the controller to a discrete PID controller
(e)    Simulate the discrete control system
(f)    Fine-tune the discrete controller to improve the performance

Solution

(a)    Design the analogue PID controller using the Ziegler-Nichols closed-loop method. Construct a Simulink model of the unit feedback closed-loop system with P control of gain Kc=1 as shown in Fig.2.6a. Tune gain till Kc=1.055 to achieve sustained oscillation in output as shown in Fig.2.6b.

Note: use block: To Workspace to send the variables to Matlab workspace, here need to choose “Save Format: Array” for the block.

Thus, we obtain from tuning and response curve,

Fig.3a Simulink model of closed-loop control system

Fig.3b Closed-loop response: sustained oscillation

Using the following table

Controllers    Kc
P    0.5Kcr
0
PI    0.45Kcr    Pcr/1.2    0
PID    0.6Kcr    0.5Pcr    0.125Pcr

We have

So the analogue PID controller TF is

(b)    Simulate the Control system as Fig.2.7a and run it with a unit step input to get output in Fig.2.7b

Fig.4a Continuous PID control system

Fig.4b Continuous PID control performance

(c)    It can be seen that about 30% overshoot exists in the output. A fine tuning of reducing Kc to 0.45 results in the performance in Fig.2.7c with reduced overshoot of 5%.

Fig.4c Continuous PID control performance with fine tuning, Kc=0.45

(d)    The discrete PID controller is obtained using equation (2.28), where sampling period  ,

The controller is

(e)    Set up a discrete Simulink model

Fig.5a Simulink model of discrete PID control system

This is a continuous-time and discrete-time mixed simulation, as the process is continuous while the controller is discrete. Remember here for every block need to set the sampling time (delt T=0.1sec).

Run the Simulink model to obtain the performance as below.

Fig.5b output of discrete PID control system

The output is not as smooth as the analogue PID output. To improve this the smaller sample time can be used and more accurate approximation is used to obtain discrete PID controller. Fine tuning can also reduce the overshoot. Kc is reduced to 0.45 the output is as below.

Fig.5c fine-tuned discrete PID output with Kc = 0.45

Tutorial Questions:

1.    A system TF has a first-order lag with time constant 7 minutes, a static gain K and a time delay of 3 minutes. Determine the maximum K for unity feedback system stability by (i) Routh stability criterion with the first order Pade approximation for the time delay; (ii) the Bode stability criterion.   (answer for (ii):  )

2. A chemical process has inlet and outlet flows of the same flow rate controlled by a flow control as shown in Fig.Q2. The inlet flow compound includes chemicals A and the concentration is measured as c1(t), while the concentration in the tank is stirred to be even and is c(t). The concentration of the liquid flowed into the tank, ci(t), is   minutes delay from the measured concentration c1(t) . The flow rate q = 3 litre/min and the volume of the tank V = 18 litres. Determine the critical gain K for the unity feedback control system stable when the proportional controller is used (using the two methods).

Fig.Q2 process system for concentration

3.    When the time delay in Question 1 is approximated using the first-order Taylor expansion, determine the critical gain K such that the unity feedback control system is stable when the proportional controller is used. Then compare the K with that obtained in Q1 when the first order Pade approximation is used.
(answer: (i) Kcr = 2.33 for first order Talor, (ii) Kcr = 5.67 for first order Pade, true Kcr = 4.33)

4.    Simulate the closed-loop system response to a unity step set point for Question 3. Tune K to obtain a continuous cycling output, and then compare the tuned K with calculated K in Question 3 and Question 1.

5.    Three TFs are as follows,

Design controllers for the three processes above using the Direct Synthesis method. Required closed-loop is for a time constant of ¼ of dominant process time constant and zero steady-state error. In each case, how can the controllers be implemented? For G3 use a first order Pade approximation.
(answer: (i) Gc(s)=8(1+1/4s), (ii) Gc(s)=6/5(1+1/3s+5/3*s), (iii) Gc(s)=(12s+1)/3s*(1.5s+1)/(4.5s+6))

6.    For a process with the TF  , (i) Design a PID control system using the Ziegler-Nichols closed-loop method to set the  . (ii) Then, simulate the system to track a unit step set-point and plot the output. (iii) Convert the obtained analogue PID controller to a discrete one. (iv) Finally, simulate the control system with the discrete PID controller and plot the output.

7.    In Question 6, tune the analogue and discrete PID controller parameters to improve the system performance, and plot the outputs before and after the tuning for comparison for both continuous and discrete time cases. 