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### Advanced Embedded Systems Design

Lecture 14 Implementation of a PID controller

BAE 5030 - 003

Fall 2004

Instructor: Marvin Stone

Biosystems and Agricultural Engineering

Oklahoma State University

BAE 3023

Goals for Class Today

- Questions over reading / homework (CAN Implementation)
- Zigbee and 802.14.5 – (Kyle)
- PID implementation (Stone)

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Output (qout) is readily calculated as a function of:

Load (qin) and

Setpoint (qset)

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

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Time Domain PID Controller Equation

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

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:

In a practical system this equation will work well and does not require any steady-state references, but eliminating the ti and td term completely results in:

or,

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.

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 Dt, the equation can be approximated as:

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Representation with Discrete Time Variables

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

Each of the differences (D) can be expressed as discrete values of each of the variables (m and e ) at the times 0, 1, and 2 as shown below:

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Discrete form of PID controller

Replacement of the differences (D) with the discrete

variables ( m ande ) results in:

Note that Dt is assumed to be a constant. If Dt 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.

or,

Where C1,C2, and C3 are constants and the current manipulation is expressed in terms of known values, the current and past errors.

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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 error

is calculated from the current measurement of

the 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 Dt is controlled by the time required to execute the loop. C1,C2 and C3 are all functions of Dt. The equation will probably be executed as floats! (Or very special care must be taken with scaling.

BAE 3023

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