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CATHOLIC UNIVERSITY OF SANTA MARIASCHOOL OF PHYSICS AND FORMAL SCIENCES PROFESSIONAL PROGRAM OF ELECTRONICS ENGINEERINGAutomatic Control III

HOMEWORK

STUDENT:Portugal Vargas, Stephanie Patricia

TEACHER:Eng. Lucy Angela Delgado Barra

2012

CONTROL AUTOMTICO IIILABORATORY-HOMEWORK

I. ACTIVITIES

Using MatLab, Simulink and Sisotools:1. Design a PI controller for de plant

Can you get a closed-loop settling time less than 50 sec and M>20o?

As we can see in the following image this system has a stable loop and even if we do have the phase margin requested, the time response is really slow as shown in the step response figure.

If we place a PI controller, we have:

After having added the PI controller we can now have a faster response and analyzing the bode diagrams we can be sure that our controller is fulfilling the requests we were given.

2. Design a PD controller for de plant

Can you get a closed-loop settling time less than 1.5 sec and overshoot less than 10%?

System response without the PD controller

Adding the PD controller we have a new response and a new bode diagram as shown below and as well see, we are now designing around the parameters given and we add a real zero at -2.73 and adjust the gain just by the limit to make sure we have the response requested.

3. Use the plant (SISOTOOL)

This is a second order system. Our goal is to speed up the closed-loop response so that the setting time is less than 1 second, produce a position error of 0.1, and percent overshoot less than 10%.

At first glance, given the step response, it seems impossible to move the gain value of the system to warrantee getting the values were looking for, we can move the pair of complex roots to their limit and still we couldnt reach the values requested as it will be proven here:

a) Proportional (P) ControlWith a proportional control, looks the step response as the gain increases. What do you see about this? And how is the error? Is the system unstable to any value of Kp?

As the gain increases we can observe that oscillations begin to increase as well as it will be shown in the pictures below.At first the error is quite big and as we change the value of gain our response begins to show too many oscillations that ultimately tend to the value of 1.This system is stable for any value of K, however it has lots of oscillations before reaching the stability we look for.

Gain value of 1 Gain value of 10

Gain value of 50 Gain value of 100 Gain value of 1000

b) Proportional + Derivative (PD) Control Edit your compensator and add a real zero. Note that in this PD design that you can select where you place this real zero along the real axis. Take a moment to explore what happens to the root locus, the step response, and the control effort as you move the zero.

As we move the zero along the real axis we can observe that at first the complex roots that the system has, become simple roots. As we keep moving the zero further from the origin, the roots become complex again and the step response shows more stability and the time of response becomes faster, however the error is still considerably big.

Now move the zero between 1 and 4. Find a configuration with a position error less than 0.5. Save the step response and control effort figure and the controller that produced it.

The zero is placed at 3.94 and it gives us an error of 0.093, in the following images we can observe the step response as well as the bode diagrams.

Next move the zero between 7 and 9. What happens to the root locus? Are we likely to get a faster response of the closedloop system with this design than the previous one? Specify a controller with a zero in this range that produces a settling time of 0.1 seconds or less. (Dont forget that k