Simple PI controller (MATLAB)

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AS550 Aerospace Systems

Control and Estimation

Indian Institute of Technology Madras

AE11B026

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Simple PI ControllerBy C R Rakesh, AE11B026

30 Oct 2014

Problem 1: KI = 0

Let us consider the case of Vref as a constant (= 200) w.r.t time and Vi = 100

0.01 0.4(The necessary changes can be made in the GUI): It is seen that the value ofV asymptotically reaches a steady state value. The steady state value then increases with

increase in KP. Hence, for KP in the above mentioned range, the gap between the steady state

value and Vref decreases with increase in KP. A sample graph is shown below (KP = 0.09):

0.5 0.75, then the change in V is also high ( ~ ), hence V may overshootVref. But

since KI is zero, there is no smooth correction to Vref, but an abrupt jumpis observed. This

may cause enormous loads on the aircraft. A sample graph for this case is shown (KP = 0.7).

The overshoot is not smoothly brought back to Vref when KI = 0.


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KP 0.75 0.8: We see that the oscillation spread increases.

KP 0.8: In this case, the overshoots grow and the system state diverges. The overshoot istoo large that the system oscillates very strongly. Such a control should not be used.Sample

graph (KP=1):

Problem 2: Different values of KP and KI: We shall explore the addition of KI into the above 3 cases

and see if KI solves any of the above problems.

KP = 0.09, KI = 0.1: We give a low KI to start with and observe that the velocity now

overshoots Vref slightly, but the overshoot dies down because of KI. But, it should be noted

that the steady state value (reached after some finite time) is now exactly equal to Vref

(confirmed from numerical values). This compensates the disadvantage of a purely

proportional controller. It should be noted that the overshoot die-down is gradual and not

abrupt.


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As KI increases, the number of overshoots increase, but the time at which steady state value

(Vref) is reacheddecreases, i.e., the overshoot die-down is more abrupt than when KI is low.

Note in the figure below that Vmax is higher (KI = 10)

KP = 0.7, KI = 0 to 10:Earlier, we said that for KI = 0, we observed oscillations till a particulartime, and which had very high frequency, i.e., abrupt changes in state of the system. For this

case, when you increase KI, it is seen that the frequency reduces, but the time till which the

oscillations persist increase. Also, at very high values of KI, the final steady state value of

velocity is equal to Vref.


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KP = 0.78, KI from 0 to 10:For low values of KI, we observe a graph similar to above. But,as KI increases, something strange happens:


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KP = 1, KI from 0 to 10: For low values of KI itself, the system diverges as so:

The response (on increasing KI) is similar to the last graph in the previous section (when KP

= 0.78 and KI = high). The only difference is that the value of KI at which the system truly

diverges (i.e., Nan encountered numerically) reduces when KP is high.

Some additional responses: It can be seen from the GUI that this code was built for a generic case

when Vref can vary with time. In this section, we show that if the PI controller is tuned properly, the

system can achieve any reference state. The response on changing KP and KI can be played aroundwith in the GUI.

Sine variation:


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Cosine variation:

Ramp


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Square

Sawtooth

Rectangular pulse


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Triangular Pulse

Conclusion: PI controller is a versatile tool for ensuring that the system state comes close to the

required state as soon as possible. The tuning method seems to be arbitrary but the general

consensus is that a higher value of KP and KI cause the system to diverge and a very low values of

both parameters are insufficient to bring the system state to the required state. The system dynamics

too play a vital role in deciding the response of the system. But it is seen that the PI controller toohas a very vital role in the final system response, sometimesit can be used to get a required system

response regardless of the system dynamics.

That is, in the above example, the system parameters are the constants A and B; It can be shown that

mild changes in A and B can have little effect on the final system response, if KI and KP are chosen

appropriately.

This process is called the PI controller tuning, on which significant care/time has to be taken.

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Simple PI controller (MATLAB)