CSCI1600: Embedded and Real Time SoftwareLecture 12: Modeling V: Control Systems and Feedback

Steven Reiss, Fall 2015

Control Systems Desired output value: target value

Actual output value: measured value

Actuator input: controls the plant’s behavior

Error: desired - actual

Control Variables

The actuator input can be binary or continuous Amount of heat, turn, gas, …

Turn left/right, turn on heat, accelerate

The outputs (and error) can be a vector or a scalar Optimize for a single factor (speed, temperature, …)

Optimize for multiple factors (temp + humidity, …)

On-Off Control

Suppose we do the simple thing for a heater If actual temp < target then turn on heater, else off

What is going to happen to the temperature Overshoot

Time to heat up (undershoot)

Oscillation

Smarter On-Off Control

A little more sophisticated temp < target – delta1 : HEAT ON

temp >= target – delta2 : HEAT OFF

temp > target + delta3 : COOL ON

temp <= target + delta4 : COOL OFF

What’s going to happen here What is it is very cold (hot) outside

Proportional Control

Suppose we have control over the actuator Can give it a range of values (low/high, continuous, …)

Acceleration in a car, heater with low/high flame (emergency mode), variable speed fan

What would we want to do in that case

Proportional Control

Make the actuator input proportional to the error Large error -> large input (accelerate fast)

Small error -> small input (accelerate slow)

No error -> do nothing

Assume doing nothing drives system the other way

Or that there is a corresponding input on the other side

Actuator = Kp * Error

Problem: What should Kp be

Should be > 0

Actual value depends on the system

How could you determine the value? Modeling

Mathematics

Experimentation

Is This Sufficient

Will it eliminate overshoot, oscillation, slow rise time Depends on the actual system

If the system is not perfectly linear or the actuator is not immediate, then probably not

We can do better

Proportional-Derivative Control

A and B are two situations leading to point T

What should the output be for each?

Proportional-Derivative Control

Want to take the rate of change into account Fast rate – slow down the response

Slow rate – speed up the response

Actuator = Kp * error - Kd * deriv deriv = the derivative of the error

deriv = change in error over time

deriv = change in error from last time to this

Choosing Kp and Kd

Now we have two parameters to determine How could you do this

Generally Kd is > Kp Note the Kd is subtracted, but stated as positive

Is This Sufficient

Steady state error How could this occur

Determining Steady State Error

Look at the sum of the error In the past

Not necessarily full past

Or constrain in bounds

This is the integral of the error How might you compute this

Computing Integral of Error

Approximate with sum integ = integ + error;

if (integ > MAX) integ = MAX;

else if (integ < MIN) integ = MIN

Actuator = Kp*error – Kd*deriv + Ki*integ

Ki now needs to be chosen Typically much smaller than Kp

Issues in Controllers

Actual input might have a limit range/set of values Set the actuator to the nearest value

Off/on based on threshold

Sampling rate affects the computation Might want to average the derivative

Computations are typically non-integer

Understanding PID

http://demonstrations.wolfram.com/PIDControlOfATankLevel

http://sites.google.com/site/fpgaandco/pid

PID Tuning

Set Ki=0, Kd=0, Kp=1

Increase Kp until the actual oscillates with a constant amplitude Let U = this Kp

Let P = oscillation period (in seconds)

Set Kp = U/1.7, Ki = (Kp*2), Kd = (Kp*P)/8

PID Tuning

In general requires a bit of sophistication Control theory

Control system design

Control engineers

For More Information

Wikipedia : PID

http://www.embedded.com/design/embedded/4211211/PID-without-a-PhD

Homework

Design a SIMON game https://www.youtube.com/watch?v=4YhVyt4q5HI

What are the tasks

What types of models are appropriate

Develop appropriate models (of at least one task)

Be prepared to show and explain models for the tasks

Be prepared to hand in the models