ABARAJITHAN . G

• Quadcopter

• Flight Control

• Robots

• Self Driving Cars

• Air conditioner

When we need to automatically control something…

• We are a feedback control system!!

• Try walking / writing with your eyes closed!

Feedback Control

Observe the Effect

Make changes

Air conditioner

• Our room is at 30°C.

• We need to :

Cool down it to 22°C

Within a shortest time.

Maintain the temperature at 22°C against external effects.

On-Off control

• Can’t control the power: only ON - OFF

• Disadvantage:

oTemperature oscillations

oUnstable System

Switch On A/C

Switch Off A/C

Temperature < 22°C

Temperature > 22°C

PID Control System

Remember the

PAST

(Integral)

Consider the

PRESENT

(Proportional)

Predict the

FUTURE

(Derivative)

And adjust power accordingly…

ERROR

• Air condition in a room

• Error will be changing over time.

Error8°C

Process Variable30°C

Set Point Value22°C

= -

Proportional ControlConsider the present

Concept : Reduce power gradually

𝑃𝑜𝑤𝑒𝑟 = 𝐾𝑝(𝐸𝑟𝑟𝑜𝑟)

Where 𝐾𝑝 is proportional gain

(we need to tune)

Reduce power of A/C gradually, until temperature = 22°C

• Steady state error may occur in pure proportional control.

• We don't have this in reality.

Systems have momentum –The room has heat capacity

Add small overshoot

Proportional control…

Integral ControlRemember the past!

Concept : If past has high errors, increase the power

Integral : Sum of errors over time

𝑃𝑜𝑤𝑒𝑟 = 𝐾𝑝 𝐸𝑟𝑟𝑜𝑟 + 𝐾𝑖(𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑜𝑓 𝑝𝑎𝑠𝑡 𝑒𝑟𝑟𝑜𝑟𝑠)

Where 𝐾𝑖 is integral gain (we need to tune)

If temperature doesn’t settle for a long time, apply more power

Proportional-Integral control…

• Removes steady state error

• Tends to introduce overshoot!

• Increases relaxation time

• 𝐾𝑖 should be very small to prevent overshoot

Derivative ControlPredict the future!

Concept : If room cools slowly, increase the power

𝑃𝑜𝑤𝑒𝑟 = 𝐾𝑝 𝐸𝑟𝑟𝑜𝑟 + 𝐾𝑖 𝐼𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑜𝑓 𝑝𝑎𝑠𝑡 𝑒𝑟𝑟𝑜𝑟𝑠 + 𝐾𝑑 𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑒𝑟𝑟𝑜𝑟

Where 𝐾𝑑 is derivative gain (we need to tune)

eg: Empty room cools fast Use low powerWater filled room cools slowly Use high power

PID Equation

Power = 𝐾𝑝 Error 𝐾𝑖

Sum of past

errors over

time

𝐾𝑑

How fast

error

changes

+ +

𝑃𝑜𝑤𝑒𝑟(𝑡) = 𝐾𝑝 ∙ 𝑒𝑟𝑟𝑜𝑟(𝑡) + 𝐾𝑖 ∙ 𝑠𝑡𝑎𝑟𝑡

𝑛𝑜𝑤

𝑒𝑟𝑟𝑜𝑟(𝑡) ∙ 𝑑𝑡 + 𝐾𝑑 ∙𝑑(𝑒𝑟𝑟𝑜𝑟(𝑡))

𝑑(𝑡)

Math : Second Order Ordinary Differential Equation

𝐾𝑝 , 𝐾𝑖 , 𝐾𝑑 are constants to be determined by careful tuning

• 𝐾𝑝 , 𝐾𝑖 , 𝐾𝑑 are tuned to have

• Minimum relaxation time

• Minimum steady state error

• Minimum oscillations / vibrations

PID Equation

Power = 𝐾𝑝 Error 𝐾𝑖

Sum of past

errors over

time

𝐾𝑑

How fast

error

changes

+ +

Arduino Code

No need of advanced math!

• To self balance

• 3 axes – Adjust angle (3 PID equations, 9 constants to be tuned)

• Error - Measured by a gyroscope module, Power - Given to the motors

Quadcopters, Aircrafts

THANK YOU

Presentation by:

ABARAJITHAN . G