1. CSCI1600: Embedded and Real Time Software
Lecture 14: Modeling V: Control Systems and Feedback
Steven Reiss, Fall 2016

2. Control Systems Desired output value: target value
Actual output value: measured value
Actuator input: controls the plant’s behavior
Error: desired - actual
Controller changes the actuator input to attempt to make the actual output value match the desired output value.
Lets look at this from

3. 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, …)

4. 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

5. 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

6. 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

7. 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

8. Demo

9. Problem: What should Kp be
Should be > 0
Actual value depends on the system
How could you determine the value?
Modeling
Mathematics
Experimentation

10. 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

11. Proportional-Derivative Control
A and B are two situations leading to point T
What should the output be for each?

12. 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

13. 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
Try different values on the demo

14. Is This Sufficient
Steady state error
How could this occur

15. 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

16. 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
Again play with demo

17. 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

18. 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

19. PID Tuning Requires sophistication Control theory
Control system design
Control engineers

20. For More Information Wikipedia : PID
PID-without-a-PhD

21. Homework
LAB
Design a SIMON game
Read the lab and do the pre-lab work