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Advanced Embedded Systems Design

Lecture 14 Implementation of a PID controllerBAE 5030 - 003

Fall 2004Instructor: Marvin Stone

Biosystems and Agricultural EngineeringOklahoma State University

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Goals for Class Today

• Questions over reading / homework (CAN Implementation)

• Zigbee and 802.14.5 – (Kyle)

• PID implementation (Stone)

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Elements of a feedback control system

• Review elements and variables

Gc G2

G3

Error

Manipulated

VariableD

ControlledVariable

out+

+

G1

out

-

+ out

Load

in

Setpoint

set

outout_measured

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)1()1( 23

2

23

1

GGG

GG

GGG

G

c

cset

c

inout

)( _measuredoutsetcGD

21 DGGinout 3_ Goutmeasuredout

Output (out) is readily calculated as a function of:Load (in) andSetpoint (set)

Manipulation is a simple function of the controller TF and error.

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Digital form of a classic feedback controlled system

• If sampling rate is fast and holds are employed, this system approaches the analog system

Gc G2

G3

Error

ManipulatedVariableD* D

ControlledVariable

out+

+

G1

out

-

+ out

Load

in

Setpoint

set

out

out_measured

out_measured*

Computer based controller

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One of the conventional models used to express a PID controller is:

dt

deedteKM d

t

tic

0

1

rateDerivitive

rateReset

signalErrore

gainControllerK

onManipulatiM

d

i

c

Time Domain PID Controller Equation

Where:

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Derivitive Form of a PID Controller

A convenient way to implement this equation in a controller is as the derivative of manipulation known as the velocity form of the equation as shown below:

2

2

dt

ede

dt

deK

dt

dMd

ic

In a practical system this equation will work well and does not require

any steady-state references, but eliminating the i and d term

completely results in:

dt

deK

dt

dMc deKdM cor,

This equation has no positional reference and error accumulation is a problem. Use velocity form only for PI or PID modes.

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Conversion of the DE to a Difference Equation

To begin the conversion of the PID equation to a difference

equation, the equation is multiplied by dt.

dt

ded

edtdeKdM d

ic

Note that since M is a differential and ess is zero, this equation

conveniently applies to the absolute variables as well as the deviation variables.

For small t, the equation can be approximated as:

t

tKm d

ic

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Representation with Discrete Time Variables

Each of the differences () can be expressed as discrete values of

each of the variables ( m and ) at the times 0, 1, and 2 as shown below:

t0 t2t1

M

1 2

0

M 0 M 1M 2

The equation can be simplified with the assumption that t is constant:

t

tKm d

ic

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Discrete form of PID controller

Replacement of the differences () with the discrete

variables ( m and ) results in:

12

21212 )(

t

tKmm d

ic

0112

21212 )(

t

tKmm d

ic

012

21212 2)(

t

tKmm d

ic

Note that is assumed to be a constant. If t varies, the equations should be derived with that in mind.

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Discrete form of PID controller

This equation can be solved for the current manipulation, m2, in terms

of values known at time t2: m1 , e2 , e1, and e0.

The other parameters in the equation are constants.

01212

211

ttt

tKmm ddd

ic

Where C1,C2, and C3 are constants and the current manipulation is

expressed in terms of known values, the current and past errors.

03122112 cccKmm c or,

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Translation of PID Equation into Algorithm

This equation may be translated directly into a computer language, for example:

m2 = m1 + k*(C1*e2 – C2*e1 + C3*e0);

Within a computer program, the current erroris calculated from the current measurement ofthe controlled variable and the setpoint, for example:

e2 = T_setpoint – T_measured;

The current manipulation m2 is then computed using the previous controller equation, and finally, at the end of the time step, each of the variables is shifted forward for the next calculation.

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Translation of PID Equation into Algorithm

For example in C, the code might look like:

measure_and_manipulate() //Call once per delta T begin T_measured = measure_T(); //Get the measured temperature e2 = T_setpoint – T_measured //Calculate the current error m2 = m1 + k*(C1*e2 – C2*e1 + C3*e0); //Calculate the manipulation set_manupilation(m2); //Output the manipulation e0 = e1; //Shift the error and manipulation e1 = e2; //forward one time step m1= m2; end;

Note that the time step t is controlled by the time required to execute

the loop. C1,C2 and C3 are all functions of t. The equation will probably be executed as floats! (Or very special care must be taken with scaling.

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Assignment

• Complete CAN message demo• Turn in course portfolio by 5:00 PM Wednesday Dec.

8th