If you need to design an optimized control loop for a hardware
control problem， consider trying a proportional integral derivative
（PID） controller. In this article， Robert explains that PID
regulations are simple to code， wire， and tune to find the results
you‘re looking for.
Welcome back to the Darker Side. The vast majority of real-world
systems are based around feedback loops， which are used to
manipulate the inputs to a system to obtain a desired effect on its
outputs in a controlled way. Your DVD player uses a feedback loop
to drive its spinning motor for a precise rotation speed. Your
mobile phone has a feedback loop to adjust its transmit power to
the required level. And， of course，your car has plenty of feedback
loops（not only in the cruise control module）。Systems incorporating
feedback work even if the relationship between the desired value
（e.g.， the spinning speed of a motor） and the controlled one （in
that case， the current applied to the motor） is not
straightforward. Delay effects， inertia， and nonlinearities make
life more interesting， and so do external conditions. For example，a
cruise control device needs to apply more torque on the motor if
you are climbing a hill. It should keep overshoots as small as
possible to avoid getting a fine after you get to the top of a
hill.
On the theoretical side， these problems have been well-known for
years and are covered by the Control System theory. The theory
explains how to design an optimized control loop for a given
problem， at least if the problem is well formalized. The
theory-which started with the works of James Clerk Maxwell in 1868，
followed by a lot of inspired mathematicians，such as Alexander
Lyapunov and Harry Nyquist， to name a few-is not simple. The first
pages of control system books usually start with mathematical
notions like pole placement，Z-transforms， and sampling
theorems，which nonspecialists may find difficult to deal with， even
if good books have adopted a more engineer-oriented approach （e.g.，
Tim Wescott‘s Applied Control Theory for Embedded Systems，Newnes，
2006）。 Should you give up？No. Fortunately， some classical control
algorithms are applicable to a lot of problems. More importantly，
for design guys like us， they are easy to implement in firmware or
hardware. The most usual is the ubiquitous proportional integral
derivative （PID） control， which has one interesting characteristic：
it is tuned by only three or four parameters. As you will see， the
parameters can be determined by empirical methods， without even
knowing the exact behavior of the controlled process. Don‘t get me
wrong. I am not saying that a PID will solve any control problem，
especially if the system has multiple inputs and outputs. But you
may want to give it a try before digging into more complex
solutions. In any case， it's a must-have for every engineer. So，
let‘s go with PID！
A BASIC CASE
The most basic example of a feedback loop is a temperature
controller used to drive a heater or cooler to get a precise
temperature （or temperature profile） on a temperature sensor on a
device. Regular readers may remember my article about a home-made
reflow oven controller based on a firmware PID loop （"Easy
Reflow：Build an SMT Reflow Oven Controller，"Circuit Cellar 168，
2004）。This month， I will take an even simpler example. Twenty years
ago， I bought a Weller VR20 soldering iron，which had a built-in
resistive temperature sensor. At that time， I couldn‘t afford the
corresponding power supply，so of course I built my own （see Photo
1）。 It was easy. I used a 24-V transformer， a potentiometer to tune
the temperature， a digital voltmeter （Remember the old
CA3161/CA3162 three-digit voltmeter chipset？）， and a crude "control
system." I used an LM311 comparator to drive an output TRIAC on and
off， whether the measured temperature was above or below the
threshold. For control system specialists，this is the most basic
form of a "bang-bang control" algorithm-and it is so for obvious
reasons. （Just imagine yourself driving your car with only two
settings： throttle fully open or brakes fully engaged.） It was
working，but the temperature regulation was oscillating around 3°C
above and below the preset value. That corresponded to an 18-mV
oscillation on the sensor measurement （see Photo 2）。
Why？ Just because the heater is not in direct contact with the
sensor and because the assembly does not weigh 0 g. The heat takes
some time to go from the heater to the tip， so if you wait for the
sensor to reach the target temperature before switching the power
off， it is already too late. The heat will continue to flow from
the heater to the tip and you will get a significant overshoot，
which ultimately gives oscillation around the threshold. It is easy
to model this behavior （see Figure 1）。 I have even coded a Scilab
simulation of this system （see Figure 2）。 Close to the behavior of
my old iron controller， isn‘t it？
点击查看Figure 1Just as a reminder： Scilab is an open-source Matlab-like tool with
great simulation toolboxes. The Scilab simulation sources I coded
for this article are posted on the Circuit Cellar FTP site. Don‘t
hesitate to read them because I've commented them heavily.
As I will show you at the end of this article， I improved this iron
controller a couple of years after its assembly and got a
drastically improved regulation. On the hardware side， I do not
recommend that you duplicate this design because it is based on
obsolete technology. But it is a perfect example to introduce PID
controls.
PROPORTIONAL？How do you improve the thermal regulation？Using a full on/off drive
on the heater is simple， but it is not the best solution. Why
should you use 80 W of heating power if you are close to the
target？It should be reduced more and more as you approach. The
simplest way to do so would be to calculate the heating power as
proportional to the distance to the target. This is where the "P"
of PID comes from. Rather than just comparing the measured value to
a threshold and switching the output on and off， a proportional
controller manages it more smoothly. The algorithm calculates the
error， which is the target value minus the measurement，multiplies
it by a given gain（usually denoted Kp）， and uses the result of this
calculation to drive the output after limiting it to reasonable
values. Refer to the pseudocode algorithm in Figure 3a.
You may criticize this approach as a linear power supply stage
would be needed to implement such a proportional control， with its
power inefficiency and added complexity， and you would be right.
But for a thermal controller，nothing forbids you from using the
"Pheater" value to directly drive a high-speed PWM output rather
than a DC power generator. The heating power will be proportional
to the mean PWM value，which is the desired command. What
improvement could you get with such a proportional control system？
I did the simulation in Figure 2. The results show that the
temperature oscillation is reduced from 12° down to 7°C， at least
under the hypothesis of the simulation （see Figure 4）。
The optimal value of Kp must be determined for each application
because it is dependent on the system parameters and your
preferences. Figure 5 illustrates the system‘s behavior with
different Kp values. My experience tells me that it is to start
with low Kp values and increase it up to a point where oscillations
and ringing starts to be a little too high.
DERIVATIVE HELPS！A proportional controller uses only the current measurement to
determine the output value. It doesn‘t have memory or forecasting
to improve the regulation. When you press the brake as you park
your car in your garage， you don't apply only a pressure
proportional to the distance between your car and the back wall.
You also use other information， such as the speed at which your car
is approaching the wall， and this makes sense. As you may remember
from your youth， speed is the derivative of the distance over time.
A proportional- derivative control （PD） adds the derivative of the
error to the calculation with another gain noted as Kd（see Figure
3b）。
For discrete time systems （e.g.，microcontroller-based）， the
derivative of the error can be approximated as the current error
minus the previous one， divided by the duration of the calculation
time step. The result is the algorithm in Figure 3c.
How does it work？ If the error is increasing quickly， then the
d（Error）/dt term is high. So， for the same absolute error， the
command applied on the output is higher. This enables you to return
to the target quicker. On the contrary， if the error is quickly
reducing， then the d（Error）/dt term is negative. This reduces the
power applied on the output，which reduces risks of overshoots. For
simple systems， like the one I simulated， a PD algorithm provides
an impressive improvement，even if the situation is not so easy in
real life （see Figure 6）。
How do you tune the Kd coefficient？As illustrated in Figure 7， the
system is more and more damped when Kd is higher. If Kd is null，
then we are back to a simple proportional control. An increased Kd
reduces oscillations and overshoots but also increases the time
needed to reach a new setpoint. If you increase Kd even more， the
system starts to be soft and often too soft. This is why I
previously said that the Kp proportional parameter should usually
be set a little higher than the value which gave the first
oscillations because Kd will help reduce them. So， in a
nutshell，with a PD regulation， you can first set Kd and Kp to 0，
increase Kp until small oscillations appear on the output，and then
increase Kd until there is enough damping but not more. The
resulting parameter set is often close to optimal for common
systems.
INTEGRAL， FOR FULL PIDTake another look at the PD control simulation in Figure 6 and you
will see that the system reaches a steady state， which is not
exactly the target value. On this simulation， the long-term sensor
value is 347°C， 3°lower than the preset 350°C target. This is
because the system is dissipative，meaning that some energy flows
from the iron to the ambient air. At equilibrium， the heat loss to
the ambient air is exactly equal to the 3°C error multiplied by Kd.
The system is stable but will never reach its target setpoint： the
error stays constant so its derivative is null and Kd is useless.
"PID" appears in this article‘s title. I already covered "PD"
feedback loops， so we need to add the "I" in order to avoid such
long-term errors. Not only do you need to take into account the
error and its derivative over time， but also its integral over
time. If there is a constant error， this integral will be higher
and higher over time. If you add it to the command through another
Ki gain， then the equilibrium state will be forced to be exactly at
the setpoint value（see Figure 3d）。 The result， for timesampled
systems， is the algorithm in Figure 3e.
This works and it is the final form of the PID control algorithm
（see Figure 8）。However， please take care. The integral term must be
manipulated with caution. Contrary to the Kd and Kp gains （at least
with reasonable values）， an improper Ki gain can make the system
unstable. Moreover， the effect of Ki is usually opposed to Kd： a
higher Ki gives higher oscillations and longer stabilization time.
Practically speaking， it is best to always start with Ki = 0. If it
is mandatory，increase Ki a little after determining the optimal Kp
and Kd parameters，just to a value providing a good， longterm
convergence. Then， you will probably need to retune Kd and Kp to
readapt the short-term behavior， and then Ki again， and more.
Another good way to reduce the risk of instability is to limit the
maximum and minimum value of the integral term to a given range
with a new parameter MaxIntegral. This is another value to
determine by experimentation； but globally， you end up with only
four numbers to optimize， which is far easier than going through
the full control system theory.
It is time to go back to my dear 1987 soldering iron regulator.
What did I do in 1989 to reduce the temperature oscillations？ I
simply added a differential term. I didn‘t even change the output
stage， which is still a TRIAC driven by a comparator in full on/off
mode， but I no longer compared the measured value to the preset
value. I compared the measured value plus Kd times d（measured
value）/dt to the preset value. Think about it twice. This is
exactly the same as the PD algorithm as long as the preset value is
constant. A PD control loop can be 100% analog （see Figure 9）。 What
were the actual improvements？ Compare the oscillogram in Photo 3
with the initial one（see Photo 2）。 The addition of a Kd parameter
reduced the temperature oscillation from 6° down to 1.3°C. That's
not bad with just a couple of op-amps more. Photo 3 shows the
derivative term in action. The output is no longer fully in phase
with the sensor. It starts to increase as soon as the sensor
temperature starts to reach its maximum， even if the actual
temperature is still above target. This is anticipation.
点击查看Figure 9WRAPPING UPTo be honest， I no longer use the iron controller on a daily basis
because I have a newer one. However，I still use it from time to
time， even if it is not lead-free compatible. Anyway，I hope I have
demonstrated that PID regulations are simple to code， or even to
wire with a couple of opamps. Moreover， they are easy to tune at
least for simple systems. Using only the proportional term may
already give good results. The derivative term could be added to
provide damping （or shaping） of the response. The addition of an
integral term ensures that there will be no systematic errors. If
the average error is not zero， the integral term will increase over
time， reducing the error. However， the integral term is a little
more difficult to manage than the Kp and Kd terms because it can
make the system unstable. Handle it with care，or add limits on the
integral term allowed values. Don‘t forget that these coefficients
can be negative too.
Lastly， note that playing with the Kp， Ki， and Kd coefficients is
easy，and looking at their effects on the real-life controlled
system is fun. Consider experimenting with a small controlled
system rather than exercise yourself directly on your nearby
nuclear plant. Anyway， I hope that PID is no longer on the darker
side for you！
Robert Lacoste lives near Paris， France. He has 18 years of
experience working on embedded systems， analog designs，and wireless
telecommunications. He has won prizes in more than 15 international
design contests. In 2003， Robert started a consulting company，
ALCIOM，to share his passion for innovative mixed-signal designs.
You can reach him at rlacoste@alciom.com. Don‘t forget to write
"Darker Side" in the subject line to bypass his spam filters.
PROJECT FILESTo download code, go to
ftp://ftp.circuitcellar.com/pub/Circuit_Cellar/2008/221.
RESOURCEST. Wescott, Applied Control Theoryfor Embedded Systems,
Newnes,Burlington, MA, 2006.
Wikipedia, "Control Theory,"
http://en.wikipedia.org/wiki/Control_theory.
SOURCE
WaveRunner 6050A Oscilloscope
LeCroy Corp.
www.lecroy.com
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