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Adding PID controller Chapter. #346
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Original file line number | Diff line number | Diff line change |
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#include <stdio.h> | ||
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struct pid_context { | ||
double kp; | ||
double ki; | ||
double kd; | ||
double setpoint; | ||
double last_error; | ||
double integral; | ||
double dt; // Normally you calculate the change in time. | ||
}; | ||
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struct pid_context get_pid(double setpoint, double dt, double kp, double ki, | ||
double kd) { | ||
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struct pid_context ctx = {0}; | ||
ctx.setpoint = setpoint; | ||
ctx.dt = dt; | ||
ctx.kp = kp; | ||
ctx.ki = ki; | ||
ctx.kd = kd; | ||
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return ctx; | ||
} | ||
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double pid_calculate(struct pid_context ctx, double input) { | ||
// Here you would calculate the time elapsed. | ||
double error = ctx.setpoint - input; | ||
ctx.integral += error * ctx.dt; | ||
double derivative = (error - ctx.last_error) / ctx.dt; | ||
ctx.last_error = error; | ||
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return ctx.kp * error + ctx.ki * ctx.integral + ctx.kd * derivative; | ||
} | ||
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int main() { | ||
struct pid_context ctx = get_pid(1.0, 0.01, 1.2, 1.0, 0.001); | ||
double input = 0.0; | ||
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for (int i = 0; i < 100; ++i) { | ||
input += pid_calculate(ctx, input); | ||
printf("%g\n", input); | ||
} | ||
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return 0; | ||
} |
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#Proportional-Integral-Derivative Controller | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Does there need to be a space here?
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The proportional-integral-derivative controller (PID controller) is a control loop feedback mechanism, used for continuously modulated control. | ||||||||||
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capitalize PID because the acronym follows |
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The PID controller is in three parts proportional controller, integral controller, and derivative controller. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I think this would sounds better as: |
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Before we get into how a PID controller works, we need a good example to use to explain how it work. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. This sentence sounds pretty strange. The repetition ( |
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Imagine you are making a self driving rc car that drives on a line, how wuld make it work given that the car moves with a constent speed. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. self driving -> self-driving |
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There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Maybe add a sentence here like:
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I prefer introducing it over time, to make it easier to follow. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. My argument was that it was not followable without a transitional sentence. |
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### Proportional Controller | ||||||||||
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If the car is too far to the right then you would turn left and visa versa. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. visa versa -> vice versa |
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Since there is a range of angles you can turn the wheel, you can turn with proportion to how far you are from the line. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I'm not the biggest fan of this sentence either. The Something like Even if you leave the current version: |
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This is what the proportional controller (P controller) does, which is given by, | ||||||||||
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There were just too many commas and it was confusing. Same problem with the D and I controllers below. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. It's not grammatically incorrect. What would you change it to? |
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$$ P = K_{p} e(t), $$ | ||||||||||
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Where $K_{p}$ is a constant and $e(t)$ is the current error. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. How is the error defined? Is it the distance from the line? There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. |
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The performance of the controller improves with larger $K_{p}$; | ||||||||||
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if $K_{p}$ is too high then when the error is too high, the system becomes unstable, i.e. the rc car drives in a circle. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Wouldn't it zig zag out of control? There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Yeah, this is also not clear to me. I would be expecting an instability like what we see with euler methods. Also: what if K is a function of the error? Is this common practice? |
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There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I think a note should be made here that the car's motion can be completely corrected with the Proportional controller, but it has the problem of overshooting a lot, thus additional controllers are necessary to maintain proper control of the car. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Also, this was not addressed. |
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### Derivative Controller | ||||||||||
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The P controller works well but it has the added problem of overshoting a lot. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. overshoting -> overshooting There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Noted that I believe this statement should be in the previous section. There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I prefer it here. |
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we need to dampen the oscillation, on way to solve this is to make the rc car resistant to sudden changes of error. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. we -> We. I would split the sentences instead of using the comma There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more.
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This is what the derivative controller (D controller) does, which is given by, | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. The controller is not "given" by the formula. The behavior of the controller is described by the formula. The same problem is present in the P and I parts as well. |
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$$ D = K_{d} \frac{de(t)}{dt}$$ | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. What are There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. It is a derivative. |
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Where $K_{d}$ is a constant. | ||||||||||
If $K_{d}$ is too high then the system is overdamped, i.e. the car takes too long to get back on track. | ||||||||||
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If it's too low the system is underdamped, i.e. the car oscillates around the line. | ||||||||||
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When the car is getting back on track quickly with little to no oscillations then the system is called critically damped. | ||||||||||
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### Integral Controller | ||||||||||
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I looks like we are done, we start driving but if some wind starts pushing the car then we get a constant error. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I -> It There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Maybe something like:
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We need to know if we are spending too long on one side and account for that. | ||||||||||
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The way to do that is to sum up all the errors and multiply it by a constant. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Maybe
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This is what the integral controller (I controller) does, which is given by, | ||||||||||
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$$ I = K_{i} \int_{0}^{t} e(x) dx, $$ | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. is this a different Also, is the integral going from 0 -> t but in terms of |
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Where $K_{i}$ is a constant. | ||||||||||
The peformance of the controller is better with higher $K_{i}$; but with higher $K_{i}$ it can introduce oscillations. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. Again, a side-by-side animation would do well here |
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### Proportional-Integral-Derivative Controller | ||||||||||
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The PID controller is just a sum of all there three constrollers, of the form, | ||||||||||
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$$ U = K_{p} e(t) + K_{i} \int_{0}^{t} e(x) dx + K_{d} \frac{de(t)}{dt} $$ | ||||||||||
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To use a PID controller, you need to tune it, by setting the constants, $K_{p}$, $K_{i}$, and $K_{d}$. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. tune it, -> tune it (it doesn't need the comma) |
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There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, and more. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. I'd love to see one of those method expanded later, I've tried to make a PID controlled segway-type device using an arduino once, but I could never quite get the constants right manually. |
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The uses of PID controllers are theoretically any process which has mesurable output and a known ideal output, | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. mesurable -> measurable |
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but controllers are used mainly for regulating temperature, pressure, force, flow rate, feed rate, speed, and more. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. speed, and more -> speed and more |
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## The Algorithm | ||||||||||
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Luckily the algorithm is very simple, You just need to make the PID equation discrete. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. You -> you |
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Thus, the equation looks like this, | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. this, -> this: |
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$$ U = K_{p} e(t_{j}) + \sum_{l=0}^{j} K_{i} e(t_{l}) \Delta t + K_{d} \frac{e(t_{j-1}) - e(t_{j})}{\Delta t}. $$ | ||||||||||
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In the end the code looks like this: | ||||||||||
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{% method %} | ||||||||||
{% sample lang="c" %} | ||||||||||
[import:26-34, lang:"c_cpp"](code/c/pid_controller.c) | ||||||||||
{% endmethod %} | ||||||||||
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## Example Code | ||||||||||
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This example is not calculating the time elapsed, instead it is setting a value called dt. | ||||||||||
There was a problem hiding this comment. Choose a reason for hiding this commentThe reason will be displayed to describe this comment to others. Learn more. It is not clear from the code what this example is supposed to output or do, in general. I just need a clear description of what this example code is trying to do. I am having a little trouble figuring out what the code is supposed to be doing, but I think it's a 1D analog to the car example, right? We are trying to keep the car on |
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{% method %} | ||||||||||
{% sample lang="c" %} | ||||||||||
[import, lang:"c_cpp"](code/c/pid_controller.c) | ||||||||||
{% endmethod %} | ||||||||||
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<script> | ||||||||||
MathJax.Hub.Queue(["Typeset",MathJax.Hub]); | ||||||||||
</script> |
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