AS-74.3199 Wireless Automation

Implementing PIDPLUS for Halvari system

The goal is to keep the ball and the cart in the middle.

Can be controlled by a manual joystick or a computer.

The position of the ball (angle φ) and of the cart (coordinate y) are measured and their derivatives calculated. Thus our system has four states from which a control variable (force F) can be derived.

In optimal control force is calculated as a linear combination of the states

Halvari – a ball balancing system

4321 KKyKyKF

Halvari – a ball balancing system

Halvari can be described with two nonlinear differential equations

FrRmmMy )sincos)(()( 2

coscossin)cossinsin)(()( 22 FryMrmrgrRmr

r

rRJ

The constants used in the equations

kg 4Mkg 7.0m

23 kgm10175.0 J

m 5.0Rm 2.0r

sm 81.9 2g

Mass of the cart without the ball

Mass of the ball

Ball’s moment of inertia

Radius of the arch

Radius of rotation

Acceleration of gravity

Halvari – a ball balancing system

The linearized form is derived by assuming that the angle φ is small. Thus we have the following assumptions:

FMmrJmMrR

mr

MmrJmMrR

mMmgr

)))((()))(((

)(2

2

2

2

FMmrJmM

Jmr

MmrJmM

grmy

2

2

2

22

)()(

sin

0sin 2 1cos

02

The linearized equations are

Traditional PID PID controller consists of three terms: proportional,

integral and derivative.

The input of a PID controller is an error signal which is the difference between a reference signal and the measured output of the process:

dt

tdeTde

TteKtu

t

di

p

)()(

1)()(

0

)()()( tytyte r P

I

D

processΣ Σ+

-

yyr e u

PIDPlus

In wireless automation loss of measurement and control data is common

Traditional PID doesn’t handle packet loss that well and gives a poor dynamic response

The solution is to replace PID’s integral part with a filter that takes packet loss into account PIDPlus

PIDPlus holds on to the last filter output until a new measurement is received and after that it calculates a new filter output from the last controller output and time elapsed since the last communication

PIDPlus integral

The filter eguation:

)1)(( 111resetT

T

NNNN eFOFF

outputfilter NF

outputfilter last the1 NF

output controllerlast the1 NO

packages receivedbetween timeT

I

PTreset

Source:Addressing Control Applications Using Wireless Devices,Emerson Global Users Exchange

PIDPlus derivative

The derivative term is described by the equation

Because the reference signal is 0 thereis a connection

A filter has to be added to the derivativeterm. The filter equation is

packagesbetween timeT

termderivative controllerDO

constant efilter timfT

(angle φ or place y)

T

eeKO NN

DD

1 errorcurrent Neerrorlast 1 Ne

outputfilter , Nfe

outputfilter last 1, Nfe

variablecontrolled Ny

NN ye

11,,

NN

fNf

f

fNf ee

TT

Te

TT

Te

• N2 receives packet and uses external pins to relay information to N3• Range: [0,4095] Physical values: [0,2.5 V]• Measurements range: [1300, 1900] [0.79,1.15 V]

N1Joystic

kCH 12.

N2

• Joystick measurement and Joystick ON/OFF

N3DAC1

ON/OFF

• N3 calculates control speed and direction and sends packet to N4

CH 18.

N4

• N4 assigns external pins for speed and direction• Signals are amplified from 0-2.5 V to 0-5.0 V.

SYSTEMDIR

SPEED

Halvari: Joystick Control

• N3.5 uses PIDplus algorithm to determine control• Sends control command to N4 and waits for ACK packet• If measurements from N3 not received, old control

values used

• N3 receives cart and ball position measurements from the system• If control switch is activated, PIDplus control used• Measurements are sent to N3.5

BALL POS.

CART POS.CH 16.

N4

• N4 assigns external pins for speed and direction• Signals are amplified from 0-2.5 V to 0-5.0 V.

SYSTEMDIR

SPEED

N3.5

CH 18.

ACKCH 18.

N3

Halvari: PIDplus controllerHalvari: PIDplus controller

Some simulations

PIDPlus without any packet loss.

red = angle φblue = place ygreen = y’turquoise = φ’

Some simulationsPIDPlus with somepacket loss.Sampling packet loss probability = 20%andActuator packet loss probability = 20%

red = angle φblue = place ygreen = y’turquoise = φ’

The longest timebetween updates is about 3 times thesampling time.

Some simulationsPIDPlus with a lot ofpacket loss.Sampling packet loss probability = 40%andActuator packet loss probability = 40%

red = angle φblue = place ygreen = y’turquoise = φ’

Time between updatescan be even 9 times thesampling time.

References

Addressing Control Applications Using Wireless Devices, Emerson Global Users Exchange, PowerPoint-show

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