Friday, December 25, 2015

More PID ramblings

First of all, Merry Christmas!

Since I cannot seem to turn my brain off on this topic I have been semi-immersed in for a number of weeks, I wanted to jot down a few thoughts on the topic of PID controllers.

My goals in using a PID controller with a mechanical fader that is driven by DC motor are simple, to wit:

  1. Instantaneous snapping to a new position once loaded into the controller.
  2. Positional accuracy and repeatability when the same position is loaded.
  3. Smooth movement of the fader control under PID control when set point changes are small and frequent.
  4. Tolerant of "stair-stepped" set point changes arriving at non-regular or varying intervals while still maintaining smooth movement.
So far, I have achieved the first two goals.  To date, I have been using a positional PID in an attempt to meet all these requirements.  A positional PID is one that takes a positional set point and a current position as input and produces the necessary output to drive the fader to its new position.  It takes care of acceleration and deceleration as necessary to arrive quickly at the new position.

However, when trying to achieve smooth movement that is quite slow, I find that a positional PID is insufficient for the task for the following reasons:
  1. A PID tuned to aggressively snap to a new position is a poor experience when trying to achieve slow but smooth changes to the position.  The fader tends to be very jerky and/or noisy.  The effect looks like a long-term crack user trying to be cool when the authorities arrive.  I have attempted to use multiple tuning settings where I reduce the gains of the P, I and D terms, but while it improves things, it is still unacceptable.
  2. I find that it takes approximately 30% of the motor speed range just to get the fader to move if it is already stopped.  Once it overcomes inertia and friction it will suddenly lunge forward and likely overshoot the desired set point if it is close at hand.
  3. Once the fader is in motion, it takes a lot less energy from the motor to keep it in motion, but there is still a fairly significant dead band in the motor speed range where the fader will stop again if the PID output drops too low.  Anytime the PID calculation must reverse direction in order to slow motion, it must necessarily pass through this dead band on both sides of zero.
What is needed for smooth motion out of a PID is to have the PID control the velocity of the motor, not just to move it to a new position.  I have attempted to simulate this using a positional PID by limiting the range of the output of the PID to something lower than the maximum speed of the motor when it is desired to be non-aggressive about positioning.  However, what I find to be true is that while it helps the overall effect, the results are not sufficiently predictable or repeatable from one fader to the next.  They all have their own friction profiles and while they are carefully manufactured, the motor movement transfer function from one fader to the next varies considerably.

So, to provide smooth movement for non-aggressive position changes, what is needed is a velocity PID that will control the speed of the motor according to a motion profile.  I will be implementing a trapezoidal motion profile where the motor will accelerate to a maximum speed, hold that speed and then decelerate to a stop at the desired position.

So, I think I need the following:
  1. ​Desired velocity - I will calculate desired velocity based off the current positional error.  When the desired position changes, we start moving in that direction according to a motion profile.
  2. Current velocity - I currently calculate this as the derivative term of positional PID.  The first derivative of position is velocity.
  3. A second PID to control motor velocity when being non-aggressive about positioning that will control motor speed according to a motion profile in a tight control loop.

We can visualize position, acceleration and motion profiles as follows:


To determine the desired velocity, I will calculate the positional PID error.  This is simply the difference between the current position and the desired position.  The further I have to go, the faster I will go up to a maximum limit.  As the fader moves, the error changes (approaches zero) as the fader approaches its target.  The fader loop will calculate the desired velocity as the current position changes towards the target.  As friction changes, voltage changes and other external disturbances affect the actual velocity, the PID will calculate the necessary correction in velocity to stay on the motion profile.

More to come...

Sunday, December 20, 2015

Christmas Math

Just for fun...



I wish you all the best in this holiday season.

Saturday, November 28, 2015

PID Controllers

Wow...  Where to begin.  I have had the pleasure of trying to get my head around implementing a fairly simple controller for a linear fader control that is also motor controlled.

This is an Alps fader which is basically a linear 10K potentiometer that has a small DC motor mounted on the back.  It has a 100mm travel length and the fader control is touch sensitive.  Mouser and other suppliers have them, though I cannot recommend hobbyists buying them through these channels as they are quite expensive.  I have seen them on eBay and other auction sites at significant savings.


The motor is mounted along the axis of the linear fader making a very low profile solution.  The motor drives a toothed belt to move the fader to any position desired remotely.  The motor is a simple DC motor that can be driven by up to 10 VDC.


So, one could envision simply putting a voltage across the pot and connecting the wiper to an Arduino analogue input.  Moving the pot would report a digital number from 0-1023 at any given instant in time representing the voltage across the pot at the wiper and therefore, the relative position of the fader control.

 However, if we wanted to simply tell the fader to position itself to any of its 1024 positions it is capable of, we would basically need to turn the motor on and drive it in the correct direction while reading the pot position until it attains the correct value.  This would be an appropriate application for a PID controller.

To control a DC motor, it is most convenient to use an H-Bridge circuit.  As can be seen by the circuit diagram below, turning on Q1 and Q4 will drive the motor in one direction whilst turning on Q3 and Q2 will drive the motor in the opposite direction.  As an addition feature turning on Q1 and Q2 or Q3 and Q4 will short out the power supply and allow for stress testing your circuit protection implementation...


In reality, there are nice single chip H-Bridge solutions that not only allow for controlling the motor direction, but also allow for PWM control of motor speed.  Most suppliers of robotics components will have typical examples of these controllers as well as places like Adafruit and Sparkfun.

In my next posting, I will get into the details of the hookup and programming of the PID controller for this fader.  More to come...

Saturday, October 17, 2015

Fun with PIDs

PID?  What the heck is a PID?

According to Wikipedia, PID is a Proportional-Integral-Derivative controller (PID controller) which is any kind of control loop feedback mechanism.  These controllers find common usage in industrial control systems.

What a PID controller does is continuously calculate an error value as the difference between the measured value of something that is changing and the desired value.

Perhaps a simple example is in order.  Let's say you want to fill a pan of water at a specific temperature from the tap.  You have two sources of water, cold and hot.  You will need to control the mix of hot and cold water to obtain the desired temperature which you are going to measure with your finger.  So you start with a guess of the hot and cold water settings, measure the result and adjust the hot and cold flow until satisfied.  Making a change that is too big may result in overshooting or undershooting the desired temperature.

Lets call the water temperature sensed by your finger the "process variable" or PV.  The temperature that is desired, let's call the "set point" or SP.  The amount of change of the water flow mix of hot and cold, let's call the "control variable" or CV.  And lastly, the difference between the measured temperature and the desired temperature is the "error" or e.

The most obvious way to change the water flow is in proportion to the current error.  The bigger the error, the bigger the change.  A slightly more complex methodology might want consider the rate of change of the error adding more water flow control depending on how fast the error is approaching zero (a derivative action).  Another refinement might be to consider some historical accumulated error information to detect whether the temperature is settling out too low or too high and make the appropriate corrections (an integral action).  Another way to perform this integral action would be change the current water tap position in steps proportional the current error.

When making changes that result in overshoot or undershoot of the desired set point, continuing to do so will result in oscillations around the desired set point that either grow or decay with time.  If the oscillations decay, the system is stable.  If they grow, the system is unstable.  If they remain at a constant amplitude, the system is marginally stable.  The desired goal is a gradual convergence to the desired set point, so the controller may try to dampen future oscillations by tempering its adjustments by a process called reducing the loop gain.

When the controller starts from a stable state with no error (PV = SP), any changes made by the controller will be in response to changes in other inputs to the process that affect it.  These changes are known as disturbances.  An example of a disturbance in our simple system described above would be a change in the tap water temperature caused by an external event such as the water heater deciding to turn on its heating element.

In theory, a PID controller can be used to control any process which has a measurable output (PV), a known ideal for that output (SP) and an input to the process (CV) that will affect the PV.  So, as you can see PID controllers find uses in any problem domain that needs to regulate temperature, force, speed, pressure, flow rate, weight, position and just about any other variable for which a measurement exists.

So, now just a little bit about the theory of PID controllers.  The name as we have seen above comes from its three terms (Proportional, Integral and Derivative) which are summed to calculate the output of the controller.

The clasasic formula for the algorithm (from Wikipedia) is:


where:
   Kp - Proportional gain.  These first three are algorithm tuning parameters
   Ki  - Integral gain
   Kd - Derivative gain

   e   - Error = SP - PV

   t   - Time or instantaneous time (the present)
   T  - Variable of integration (takes values from time 0 to present time t)

What is desired is a controller that will smoothly make adjustments to the desired set point with minimal over-shoot and under-shoot.  Directly implementing this formula in code would lead to some simple solutions that would most likely suffer from a number of short-comings such as the following:

  • Sample Time - The formula requires recalculation at a regular interval since both integration and derivatives are a function of time.
  • Derivative Kick - Any time the set point (SP) is changed, the error is changed (SP-PV) and the derivative of this change is infinity.  In practice however, since the change in time is never zero, it ends up being a very big number which when fed into the calculation results is an undesirable spike in the output.
  • On the fly Tuning Changes - Changes in the three tuning parameters affect the integration of the error value over time.  Any change in Ki will be multiplied by the entire error sum that has been accumulating when we really only want it to affect the result going forward.
  • Reset Wind-up - Most controllers have some limit on operational ranges.  For example a water valve can only be set in the range of completely closed to completely open.  If the PID controller does not know about these ranges, it may calculate an output value that is out of range.  Over time when it tries to continue to add more water flow beyond maximum, only to have it clamped by the physical size of the pipe, the algorithm will continue to ask for more and more beyond the limit.  The result is that the output gets "wound up" way beyond the maximum limit.  Where the problem reveals itself is when the set point (SP) is dropped, the algorithm needs to wind back down again to the maximum before it will even affect the output.  This results in what looks like a lag in response of the controller to the new set point.
  • Switching the PID Controller on/off - This occurs when you validly decide that regardless of what the PID controller is doing, you want to override its decision for a period of time.  Now, when you stop overriding it's decision, you get a sudden, huge change in the output.  The controller keeps trying to adjust the output to get the desired result, but it doesn't see any change, so it adjusts the output a little more and so on.  It would be like externally overriding a volume control of a stereo system while trying to adjust the volume internally.  No change internally has any effect.  Then you switch off the external override and suddenly you have a huge volume change.
  • Initialization - While it is useful to be able to turn off the PID and set your own override, when you turn it back on, the PID jumps back to the last output value it had set resulting in another spike in the output.  These transitions need to be seamless.
  • Changing Direction - The PID algorithm may be used to drive a system that is either "direct acting" (an increase in the output causes an increase in the input) or "reverse acting" (an increase in the output causes a decrease in the input).  A refrigeration system for example is a reverse acting system as an increase in the cooling results in a decrease in temperature.  Changing the sign of Kp, Ki and Kd allows proper control of a reverse acting systems.
Rather than detail the solutions for all of these potential issues, let me refer you to a most excellent blog by Brett Beauregard detailing the improvements one-by-one starting with the most basic code example.  It is a great read and not one I could not improve upon.

There is a great PID implementation for the Arduino also available at the Arduino Playground which will be the subject of my next posting.

Sunday, October 11, 2015

Postage Stamp Micro-controller

I have updated my ATMega328 controller board with a new layout to fix the previously posted deficits and have sent the gerbers off to OshPark for production.  Hopefully I will have the set back in a week or so.  After validation, I will make the design public at OshPark so anyone interested can purchase the boards.  This is a one inch square PCB with an ATMega328 processor, full set of IO pins and programmable via an ISP connector.


See my previous postings for the schematic and further details.  This was originally a project done for purposes of learning how to use KiCAD.

Sunday, September 20, 2015

Arduino Satellite/Sun Tracking

I have put together the recent bits and pieces of test code that I have been playing with to control my AZ/EL servo camera mount and an Arduino port of the old Plan13 code, originally written in Basic back in 1983 by James Miller G3RUH.  You can read about his efforts over on the AMSAT web site here.  Jim gives a great treatment of the math involved and example code in Basic.


A port of the code described on the page above has been made to the Arduino and I leveraged this code from the QRPTracker project web site.  With some simple changes, I have pulled together a simple satellite tracker to drive my AZ/EL camera mount.  It should be simple to scale this up to drive an actual satellite antenna array.


Here you can see the project pointing to the point in the sky where the ISS would be located at a point in time today.  I took the AMSAT Keplerian elements and processed the six passes today where the satellite was above the horizon at my location.  The code currently processes the entire 24 hour period in a day, and tracks in real time each of the passes.

The implementation of Plan13 is not particularly accurate due to the limited precision available in single precision floating point on the Arduino, but with the broad bandwidth of most amateur 144 and 440 MHz antennas, it is of little consequence even if the accuracy was only to the degree level.  In fact most low earth orbital satellites can be worked with a fixed elevation on a standard yagi, but the project has been interesting.  If there is interest, I will post my code, though most of it was gleaned from the efforts of others.

Beyond the code I have previously published, I have changed the mapping of the azimuth servo to be clockwise from north to match a magnetic compass.  I limit the calculated elevation values to those angles physically possible with my AZ/El mount.  I prevent movement of the servos if the satellite is currently below the horizon.

I would like to put a real-time clock in the circuit and have it calculate and pre-position the antenna to the point where the satellite rises, add a simple display, add the ability to load Keplerian elements from a wifi connection to the internet and provide for calibration of the servos.

On an Arduino Uno, I am able to load the entire two line elements (TLE) set from here into flash.  I have written a routine that will print out the start of every visible pass for my location for every satellite in the list (89 of them).  I suspect that it will be unusual to have the entire set in flash at any given time, but it can be done and still have room for the code.  Alternatively a storage card could be used to store the data.

On the mighty Arduino Uno, this table fills much of the flash and calculating the position of all 89 satellites for every minute of a 24 hour period takes quite a while.  A better approach might be for the process that grabs the KEPS also pre-calculate the start of pass and just generate a table along with the KEPs table.  This table could then drive a higher level process  to allow the selection a which satellites you want to track today.

The code calculates the effects of doppler and provides a tuning frequency for both the uplink and downlink if the frequencies are provided prior to the calculation.  For now I am passing in zeros, but a table of satellite frequencies and a bit of glue code to drive tuning a transceiver could fully automate tracking and tuning tasks.

This has been lots of fun and a precursor to building a LEO satellite station, but for this particular hobby AZ/EL mount, my next task will be to calculate the position of the sun and use this mount to keep a small solar panel array pointed at the sun for battery charging.  For that particular task, I am thinking of trying out a couple of approaches.  The first would be to calculate the position of sun relative to me as an observer.  The second would be to use an analogue approach where the output of the solar panel is used as feedback to drive a correction to the AZ/EL servos to keep the panel at maximum output regardless of the position of the sun.  I suppose in theory, I could just set the device out in the sun and let it figure out where to point to maximize output.  Who knows, it might decide to point at a white wall rather than at the sun when currently behind tree cover for example.  Should be fun!