Advanced Embedded Systems Design

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) BAE 3023

Elements of a feedback control system • Review elements and variables BAE 3023

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. BAE 3023

Digital form of a classic feedback controlled system • If sampling rate is fast and holds are employed, this system approaches the analog system BAE 3023

Time Domain PID Controller Equation One of the conventional models used to express a PID controller is: Where: BAE 3023

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. BAE 3023

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: BAE 3023

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: BAE 3023

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. BAE 3023

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. BAE 3023

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. BAE 3023

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

Assignment • Complete CAN message demo • Turn in course portfolio by 5:00 PM Wednesday Dec. 8th BAE 3023

Advanced Embedded Systems Design