Hi : ) today I’ll focus on the most important
thing, which designing is highly recommended. After my last accident we decided
that until we start flying again, we have to prepare some automatic stability
control based on PID controller.
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| Fig. 1 - Schematic of PID Controller |
The main idea of using controller is to
compensate position by using a feedback. At first, we have to achieve some
state. We starting from initial position, and our goal is to obtain assuming
position. After every step we have to measure somehow our state and compensate
it. Main idea is shown in above picture.
PID controller contains of few independent parts. It comes from age of
analog devices and names of particular parts are cooperated with some setups of
operation amplifiers. The simplest controller consist of proportional (P) part
only. In more complicated circuits it can be found also other parts, like
integral (I) or derivative (D). Depending
of using one or few parts in same time, the controller could be called: P (only
proportional), PI (proportional + integral) or PID (proportional + integral +
derivative). In above picture are
presented some equations which are describing each part. There are also
some mixed controllers (ID or PD only) but I don’t want to describe it. There is some differences between using all of
mentioned parts:
-
- Proportional:
is the easiest feedback control to
implement, and simple proportional control is probably the most common
kind of control loop.
A proportional controller is
just the error
signal multiplied by a constant and fed out to the drive. Below picture
shows the differences between setting some gains. For small gains (kp = 1) the
motor goes to the correct target, but it does so quite slowly. Increasing the
gain (kp = 2) speeds up the response
to a point.
Beyond that point (kp = 5, kp = 10) the motor starts out faster, but it overshoots the target. In the end
the system doesn’t settle out any quicker than it would have with lower gain,
but there is
more overshoot. If we kept increasing the gain
we would eventually reach a point where the system
just oscillated around the target and never settled out-the system would be
unstable.
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| Fig 2. Impulse response of proportional part |
-
- Integral:
Integral control is used to add long-term precision to a control loop. It is
almost always used in conjunction with proportional control. Integral control
by itself usually decreases stability,
or destroys it
altogether. Next figure shows the
motor and gear with pure integral control
(pGain = 0). The
system doesn’t settle. Like
the precision actuator
with proportional control, the
motor and gear system
with integral control alone
will oscillate with bigger
and bigger swings
until something hits a limit. Another figure shows
the motor and gear
with proportional and integral
(PI) control. The position takes
longer to settle out
than the system
with pure proportional control,
but it will not settle to the wrong spot.
 |
| Fig 3. Impulse response of pure integral part |
 |
| Fig 4. Impulse response of proportional + integral parts together |
-
In
general, if you can’t stabilize a plant with proportional control, you can’t
stabilize it with PI control. We know that proportional control deals with the
present behavior of
the plant, and that
integral control deals
with the past behavior of the plant. If we had
some element that
predicts the plant
behavior then this might
be used to
stabilize the plant. A differentiator will do the trick. The differential
term itself is the
last value of
the position minus the current
value of the position. This gives you a rough estimate of the velocity
(delta position/sample time), which predicts where the position will be in
a while. With differential control
you can stabilize the precision actuator system. Below figure shows the
response of the
precision actuator system with
proportional and derivative (PD) control. The next figure shows the
heating system with PID control. You can see the
performance improvement to be had
by using full PID control with this plant.
 |
| Fig 5. Impulse response of proportional + derivative parts together |
 |
| Fig 6. Impulse response of proportional + integral + derivative parts all together |
When PID is implemented, an
important issue is to set gains of every part in that way, to achieve proper
characteristic. The most common cases are presented below:
 |
| Fig 7. Different impulse responses |
Above figure illustrates the two
mutual excluding wishes presented above, and the compromise. The figure shows
the response in the process output variable due to a step change of the set point.
(The responses are with three different controller gains in a simulated control
system.) It is useful to have this compromise in mind when you perform
controller tuning.
Depending on every information
presented earlier, we perform a small application to tune PID, which we want to
apply in quadrocopter. For now it is a very simplified version, and it will
extended in nearest future. Below we are presenting how it behaves:
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| Fig 8. Our PID simulation and overshot parameters |
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| Fig 9. Our PID and long-time stabilizing signal (a lot of oscillations) |
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| Fig 10. Our PID and quite fast stabilizing of signal |
As you can see, tuning of PID controller is not
a simplest issue. That’s all for today. Have a nice weekend :)
References:
1. http://blog.nikmartin.com/2012/11/process-control-for-dummies-like-me.html
2. http://home.hit.no/~hansha/documents/control/theory/tuning_pid_controller.pdf
3. http://nicisdigital.wordpress.com/2011/06/27/proportional-integral-derivative-pid-controller/
4. http://igor.chudov.com/manuals/Servo-Tuning/PID-without-a-PhD.pdf
References:
1. http://blog.nikmartin.com/2012/11/process-control-for-dummies-like-me.html
2. http://home.hit.no/~hansha/documents/control/theory/tuning_pid_controller.pdf
3. http://nicisdigital.wordpress.com/2011/06/27/proportional-integral-derivative-pid-controller/
4. http://igor.chudov.com/manuals/Servo-Tuning/PID-without-a-PhD.pdf
