control postlab

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Sabanci University

Faculty of Engineering and Natural Sciences

Fall2014

ME 303

CONTROL SYSTEM

DESIGN

LAB REPORT # 2

(Group Number 4& Section B)

Lead-Lag Controller Design

Submitted by

Burak Kadayıfçı -15426

Alper Gül-14027

Alp Güçlü - 12866

on

05.12.2014

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INTRODUCTION

The main purposes of this lab are to design a lead-lag compensator and to see the Real-

Time effects of them on DC motors in lab environment. When the appropriate lead-lag

compensator design has impact on the system, desired system response and acceptable steady

state error will be observed. It is also important to understand properties of lead and lag

controller separately in terms of pole placement approach and error status.

In this experiment, these background information should be taken into consideration in order to

have a full benefits from it:

Error = Reference – Actuator =>e (t) = theta (ref) – theta (act)

Lead Controller = K(s+z1) / (s+p1)

Lag Controller = (s+z2) / (s+p2)

Lead/Lag Controller = K(s+z1)(s+z2) / (s+p1)(s+p2)

PROCEDURE

In this experiment, there are several steps that is should be completed. Firstly, we needed

to configure and design our matlab and dspace environment for convenience of the lead/lag

controller setup. For instance, we defined two values that implies pole and pole-zero with A and

C sign respectively. Thus, if we take the value of pole 10 and pole-zero 8, our pole location and

zero location will be 10 and 2. Secondly, we need to design a lead controller using appropriate

K, z1, and p1 values. When the desired response observed, it is time to add lag controller to

design lead/lag. Thirdly, need to design lag controller using appropriate z2 and p2 values. After

the observation of what we desired from lead/lag controller, it is only left to saving our

experiment, layout, files and datas.

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RESULTS

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In this part of the experiment we selected different values for lead gain. According to graphs and

the experimental date we have found the most proper gain seemed to be 79. When we selected

gain as 79 settling time and rising time decreased dramatically. The system had overshoot which

is less than gain=50 and more than gain=100. Gain=79 gave the maximum and rms value of

error less than other solutions. The maximum values of control inputs seemed to be same with

all gain values but the smallest rms value was given with the gain=50.

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In this part of the experiment theoretically we add lag compensation to our existing lead

compensator. From previous part of the experiment we selected the gain as 79 as a best option.

After that we started to give small values to zeros and poles of the compensators. When we

selected pole 0.01 and zero 0.1 rising time is less than other solutions. However the smallest

settling time value was given by pole= 0.5 and zero= 1. When we compared the overshoot

percentages the minimum value was zero which is given by the values of pole= 0.01 and zero=

0.1. The maximum values of errors were the same for each pole and zero option. However the

rms values of errors are almost same for pole=0.5 zero=1 and pole=1.5 and zero=0.5 but for

pole=0.01 and zero=0.1 the error value was relatively high. Maximum values of control input

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were all same for these corresponding values. However rms values of control input reached its

maximum value when lead-lag compensator’s pole= 0.01 and zero= 0.1. As a result the best

option was the compensator pole=0.5 and zero=1 considering small settling time and good

steady state error.

Compensator Rise time Settling

time

Over

shoot

erms emax urms umax

Lead/Gain=50 5.6553e+03 7.9285e+03 5.4130e-

04

5.2663 15 0.4674 0.9000

Lead/Gain=79 2.5711e+03 4.8317e+03 4.0606e-

04

3.1463 6.7086

0.5303

0.9000

Lead/Gain=100 5.6559e+03 7.5933e+03 0 7.8664 15 0.7372 0.9000

Lead-

lag/A=0.01,C=0.1

1.7968e+04 2.2997e+04 0 16.2933 15

0.9000

0.9000

Lead-

lag/A=0.5,C=1

5.6595e+03 1.0849e+04 13.0468 6.3998 15 0.5711 0.9000

Lead-

lag/A=1.5,C=0.5

5.6582e+03 7.5791e+03 0.0891 6.1239 15 0.5711 0.9000

DISCUSSION

Burak Kadayıfçı:

We started to the experiment as using only lead compensator. Our aim was to see

effect of lead compensator to a closed loop system. We set different values for lead gain such

as 50, 79 and 100. We saw that 79 is our best option considering their settling time and rising

time. The system was very stable with gain=79 resulting small settling time and rise time.

However there was still steady state error. Since lead compensator improves steady state error,

we should add lag compensator to lead compensator to eliminate steady state error. When we

designed lead-lag compensator we selected different values for pole and zero to see system

response. As a result of different values the best option was pole=0.5 and zero=1 considering

the smallest settling time considerable steady state error. Using lead-lag compensator in the

experiment gave the best result that desired poles with good steady state error and system

response.

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Alper Gül:

Our first challenge was designing lead compensator and to see the effects of it on the

system response. We used several values to understand working principle of lead compensator

and we concluded that the best result and desired loops and locus points have come when the

gain is equal to 79. Then we have just realized that, we had reasonable system response in terms

of settling and rise time, however our steady state error has considerably increased then at that

point we were told that we should add the lag compensator to eliminate this error. This is where

we understood the value of lag compensator in the system response. Again, we tried several

values for lag compensator. It is told that, 10 times difference between pole and zero of lag

controller is fine for taking appropriate response; however we took our best result when pole

and zero are 0.5 and 1 respectively. Furthermore, we tried different values for zero and pole

values that is not included in lab data for both lead and lag compensator to understand location

of the zero and pole with respect to each other and it was concluded that in lead, zero is closer

to the origin, in lag pole is closer to the origin. When we applied lead/lag, we combined the

effects of two different compensators to catch good steady state error and response time

Alp Güçlü:

First of all we used lead compensator. In the experiment we used a closed-loop system

and we set 3 different gains respectively 79, 50 then 100. After we gave 79 for the gain value of

the system we saw the value of the 79 was better than the other two of them. It has better

settling time and the rising time. However our steady state error did not change. For decreasing

the steady state error, we changed the system we add a lag compensator and we create lead-lag

compensator. We add some pole and zero values to decrease steady state error, we add

respectively 0.01, -0.5 and -1.5 for the poles and we add respectively 0.1, 1 and 0.5 for the zero

value. At the values for the pole -0.5 and for the zero 1, we got better results and we got less

steady state error. Also when we gave both pole and zero positive and pole was closer to the

origin we got infinite position in other words, we could not control the system and position

increased infinitely.

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CONCLUSION

As a result, we understand that lead compensator changes the root locus of the system

and pulls the roots to desired points. In lead control, zero is closer to the origin than the pole. On

the other hand, in lag compensator, it does not change the locus of the system; it helps us to have

good steady state error without overflow problems. Pole is often placed very closed to zero

(0.01) and zero is placed near pole (0.1). We understood at least ten times difference is

appropriate for lag compensator. Using both lead/lag compensators in our experiment gives us

best result that desired poles with good steady state error and system response.

APPENDIX

% %% Gain 79 figure(1) subplot(3,1,1) plot(pos.X.Data, pos.Y.Data) title('Lead Gain=79'); ylabel('Position'); subplot(3,1,2) plot(lead_error.X.Data, lead_error.Y.Data) ylabel('Error'); subplot(3,1,3) plot(input_lead.X.Data, input_lead.Y.Data) ylabel('Control Input');

% %% Gain 50 figure(2) subplot(3,1,1) plot(pos_lead50.X.Data, pos_lead50.Y.Data) title('Lead Gain=50'); ylabel('Position'); subplot(3,1,2) plot(error_lead50.X.Data, error_lead50.Y.Data) ylabel('Error'); subplot(3,1,3) plot(lead_input50.X.Data, lead_input50.Y.Data) ylabel('Control Input');

% %% Gain 100 figure(3) subplot(3,1,1) plot(pos_lead100.X.Data, pos_lead100.Y.Data) title('Lead Gain=100'); ylabel('Position'); subplot(3,1,2) plot(error_lead100.X.Data, error_lead100.Y.Data) ylabel('Error'); subplot(3,1,3) plot(input_lead100.X.Data, input_lead100.Y.Data) ylabel('Control Input');

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% %% LeadLag 0.01/0.1 figure(4) subplot(3,1,1) plot(leadlag_pos_0_01_0_1.X.Data, leadlag_pos_0_01_0_1.Y.Data) title('LeadLag Gain=79 Pole=0.01 Zero=0.1'); ylabel('Position'); subplot(3,1,2) plot(leadlag_error_0_01_0_1.X.Data, leadlag_error_0_01_0_1.Y.Data) ylabel('Error'); subplot(3,1,3) plot(leadlag_input_0_01_0_1.X.Data, leadlag_input_0_01_0_1.Y.Data) ylabel('Control Input');

% %% LeadLag -1.5/0.5 figure(5) subplot(3,1,1) plot(leadlag_pos__1_5_0_5.X.Data, leadlag_pos__1_5_0_5.Y.Data) title('LeadLag Gain=79 Pole=-1.5 Pole-Zero=0.5'); ylabel('Position'); subplot(3,1,2) plot(leadlag_error__1_5_0_5.X.Data, leadlag_error__1_5_0_5.Y.Data) ylabel('Error'); subplot(3,1,3) plot(leadlag_input__1_5_0_5.X.Data, leadlag_input__1_5_0_5.Y.Data) ylabel('Control Input');

% %% LeadLag -0.5/1 figure(6) subplot(3,1,1) plot(leadlag_pos__0_5_1.X.Data, leadlag_pos__0_5_1.Y.Data) title('LeadLag Gain=79 Pole=-0.5 Pole-Zero=1'); ylabel('Position'); subplot(3,1,2) plot(leadlag_err__0_5_1.X.Data, leadlag_err__0_5_1.Y.Data) ylabel('Error'); subplot(3,1,3) plot(leadlag_inp__0_5_1.X.Data, leadlag_inp__0_5_1.Y.Data) ylabel('Control Input');