2102332 Linear Control System 1 Semester 2/2010 Homework #4 Assign: Wed December 29, 2010. Due: Fri January 7, 2011. Name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ID: . . . . . . . . . . . . . . . . . Sec: . . . . . . . . . . . . . . . . . . . . . . . Collaborator: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ID: . . . . . . . . . . . . . . . . . Sec: . . . . . . . . . . . . . . . . . . . . . . .

Instruction: Always remember that the primary goal is to learn the course material. You are encouraged to collaborate on the homework, however, you must write up your homework yourself. You should not copy somebody else’s homework or lend your homework to others. If you choose to collaborate, you should be able to recreate all of the steps involved in solving a problem yourself, and should do so in your writeup. Please list the names of your collaborators on the first page of each homework. If any student breaks the homework regulation in the first time, his/her score for that homework will be zero. If you violate the regulation more than once, your entire homeworks and project will be zero. Turn in your work in class on Friday or put in the homework box located in front of EE 409 by 4 pm. 1. For a position tracking system shown in Figure 1, the control configuration can be viewed as proportional plus derivative (PD) control where the control parameters consist of Kp and Kt . Design Kp and Kt to meet the following performance specifications. • The steady state error due to a unit ramp input is less than 0.3. • The maximum overshoot of the unit step response is less than 4.3% • The settling time of the unit step response is less than 3 seconds. Td (t) r(t)

e(t)

-e - 1 + 6 −

- e - Kp +6 −

+ 1 -? e+ s(s + 2)

y(t)

-

Kt s 

Figure 1: Block diagram of the position tracking system.

2. Suppose a reaction curve of an open-loop system is shown Figure 2. Determine the transfer function of the open-loop system that fits the reaction curve. Design P, PI, and PID controllers using Ziegler-Nichols tuning rule and plot the output responses of the feedback system. Briefly discuss the output performance in terms of rise time, settling time, % maximum overshoot, and stead-state error for each controller. 3. Consider a unity feedback system with PID controller. An engineer has applied the PID tuning techniques learned from control engineering course to obtain parameters for P, PI, and PID controllers. Afterward, he performs closed-loop experiment due to a unit step input The control input (U1 , U2 , U3 ) and process output (Y1 , Y2 , Y3 ) of control systems are depicted in Figure 3. Unfortunately, the controller information for each response is missing. Match the control input and process output to each controller. Briefly explain your answer. response control input process output

P controller

PI controller

PID controller

Step Response 0.8

0.7

0.6

Amplitude

0.5

0.4

0.3

0.2

0.1

0

0

0.2

0.4

0.6

0.8

1 Time (sec)

1.2

1.4

1.6

1.8

2

Figure 2: Reaction curve of an open-loop system.

2.5

1.4 U1

Process Output

Control Input

Y1

1.2

2 1.5 1 0.5

1 0.8 0.6 0.4 0.2

0 0

5

10

15

0

20

2.5

5

10

15

20

1.4 U2

Y2

1.2 Process Output

2 Control Input

0

1.5 1 0.5

1 0.8 0.6 0.4 0.2

0 0

5

10

15

0

20

0

5

10

15

20

1.4

2.5 U3

Y3

1.2 Process Output

Control Input

2 1.5 1

1 0.8 0.6 0.4

0.5

0.2 0 0

5

10

Time (sec)

15

20

0

0

5

10

15

Time (sec)

Figure 3: Time response due to a unit step input.

20