CSCI1600: Embedded and
Real Time Software
Lecture 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/PIDControlOfATankL
evel
 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