process dynamics and control 3rd edition chapter 12 problem 3e solution - chegg.com

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Chegg Guided Solutions for Process Dynamics and Control 3rd EditionChapter 12 Problem 3E

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Step 2 of 11Write the expressions for controller settings derived from Equation (1).

(2)

(3)

Consider the process transfer function.

(4)

Compare Equations (1) and (4).

Provide feedback (0)

Step 3 of 11(a)

Determination of the PI controller settings for the design using direct synthesis method follows:

Substitute 2 for , 0.2 for , 1 for , 0.2 for in Equation (2).

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Substitute 1 for in Equation (3).

Therefore, the PI controller settings and are and .


Step 4 of 11(b)

Determination of the PI controller settings for the design using direct synthesis method follows:

Substitute 2 for , 0.2 for , 1 for , 1.0 for in Equation (2).

Substitute 1 for in Equation (3).

Therefore, the PI controller settings and are , and .


Step 5 of 11(c)

Consider the design relation for integral of Time Absolute Error (ITAE) of disturbance input as follows:

(5)

Where,

for the proportional mode

for the integral mode.



Refer Table 12.3 for the values of A and B.

Substitute for in Equation (5).

(6)


(7)

For P mode, substitute 2 for , 0.2 for , 1 for , 0.859 for A, and for B, in Equation (6).

For I mode, substitute 2 for , 0.2 for , 1 for , 0.674 for A, and for B, in Equation (7).

The PI controller settings are and and .


Step 6 of 11(d)

The design relation for integral of Time Absolute Error (ITAE) of set point input is,

(8)

Where,

for the proportional mode

for the integral mode.



Refer to Table 12.3 for the values of A and B.


(9)


(10)

For P mode, substitute 2 for , 0.2 for , 1 for , 0.586 for A, and for B, in Equation (6).

For I mode, substitute 2 for , 0.2 for , 1 for , for A, and for B, in Equation (7)

The PI controller settings are , and .


Step 7 of 11(e)

A conservative controller setting typically has small values of and large values of

The direct synthesis method in part (b) has , and

The controller using direct synthesis method in part (b) gives more conservative settings.

The ITAE method in part (c) has , and .

The controller using ITAE method in part (c) gives less conservative settings.



Thus, controller in the gives more conservative settings.

The controller in gives less conservative settings.


Step 8 of 11(f)

Steps to create the SIMULINK model:

Take the required blocks from the library browser and connect all the blocks as in Figure 1.

Double click the step response block and set step time value as 1, initial value as 1 and final value as 0.

Double click the transfer function block (Transfer Fcn) and set numerator coefficients as [2] and setdenominator coefficients as [1 1].

Double click the PID controller block and change the controller to PI type. Set the PI controller to idealform. Set the value as 0.42 and value as 1 for part (b) controller. Set the value as 2.07 and value as0.49 for part(c) controller.

Double click the transport delay block and set the delay value of input as 0.2

Double click the simout block and change the parameter to array.

Run the SIMULINK model using F5 key.

Consider the Figure 1.

Figure 1




Step 9 of 11Type the following in the MATLAB command window and press enter.

plot(tout,simout,tout,simout1)

legend('controller b)','controller c)');


Step 10 of 11The comparison of closed-loop step response to a unit step disturbance for controller b) and controller c) isshown in Figure 2.

Figure 2


Step 11 of 11Therefore, the simulation of closed-loop step response to a unit step disturbance is executed and the graphis drawn.




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process dynamics and control 3rd edition chapter 12 problem 3e solution - chegg.com

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