From 37b00682f798dc099b685f89985d27fc1c97dd30 Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Wed, 8 Aug 2018 17:56:35 +0100 Subject: [PATCH 01/13] Improving the discriptions of a PID controller. --- SUMMARY.md | 1 + .../pid_controller/code/c/pid_controller.c | 46 +++++++++++ contents/pid_controller/pid_controller.md | 79 +++++++++++++++++++ 3 files changed, 126 insertions(+) create mode 100644 contents/pid_controller/code/c/pid_controller.c create mode 100644 contents/pid_controller/pid_controller.md diff --git a/SUMMARY.md b/SUMMARY.md index 569142807..291cbb21c 100644 --- a/SUMMARY.md +++ b/SUMMARY.md @@ -33,6 +33,7 @@ * [Verlet Integration](contents/verlet_integration/verlet_integration.md) * [Quantum Systems](contents/quantum_systems/quantum_systems.md) * [Split-Operator Method](contents/split-operator_method/split-operator_method.md) +* [PID Controller](contents/pid_controller/pid_controller.md) * [Data Compression](contents/data_compression/data_compression.md) * [Huffman Encoding](contents/huffman_encoding/huffman_encoding.md) * [Quantum Information](contents/quantum_information/quantum_information.md) diff --git a/contents/pid_controller/code/c/pid_controller.c b/contents/pid_controller/code/c/pid_controller.c new file mode 100644 index 000000000..52bb102d9 --- /dev/null +++ b/contents/pid_controller/code/c/pid_controller.c @@ -0,0 +1,46 @@ +#include + +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. +}; + +struct pid_context get_pid(double setpoint, double dt, double kp, double ki, + double kd) { + + struct pid_context ctx = {0}; + ctx.setpoint = setpoint; + ctx.dt = dt; + ctx.kp = kp; + ctx.ki = ki; + ctx.kd = kd; + + return ctx; +} + +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; + + return ctx.kp * error + ctx.ki * ctx.integral + ctx.kd * derivative; +} + +int main() { + struct pid_context ctx = get_pid(1.0, 0.01, 1.2, 1.0, 0.001); + double input = 0.0; + + for (int i = 0; i < 100; ++i) { + input += pid_calculate(ctx, input); + printf("%g\n", input); + } + + return 0; +} diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md new file mode 100644 index 000000000..de4c3e25d --- /dev/null +++ b/contents/pid_controller/pid_controller.md @@ -0,0 +1,79 @@ +#Proportional-Integral-Derivative Controller + +You are a helmsmen keeping a vessel on course in various conditions, how do you do that. +A common way is to steer based on current course error, past error and the current rate of change. +Now, the proportional-integral-derivative controller (PID controller) works like the helmsmen on a ship. + +The PID controller is a control loop feedback mechanism, used for continuously modulated control. +The PID controller is in three parts proportional controller, integral controller, and derivative controller. + +Before we get into how a PID controller works, we need a good example to use to explain how it work. +Imagine you are making a self driving rc car that drives on a line. + +If the car is too far to the right then you would turn left and visa versa. +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. +This is what the proportional controller (P controller) does, which is given by, + +$$ P = K_{p} e(t), $$ + +Where $K_{p}$ is a constant and $e(t)$ is the current error. +The performance of the controller improves with larger $K_{p}$; +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. + +The P controller works well but it has the added problem of overshoting a lot. +we need to dampen the oscillation, on way to solve this is to make the rc car resistant to sudden changes of error. +This is what the derivative controller (D controller) does, which is given by, + +$$ D = K_{d} \frac{de(t)}{dt}$$ + +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. +If it's too low the system is underdamped, i.e. the car oscillates around the line. +When the car is getting back on track quickly with little to no oscillations then the system is called critically damped. + +I looks like we are done, we start driving but then some wind starts pushing the car and we have a constant error. +We need to know if we are spending too long on one side and account for that. +The way to do that is to sum up all the errors and multiply it by a constant. +This is what the integral controller (I controller) does, which is given by, + +$$ I = K_{i} \int_{0}^{t} e(x) dx, $$ + +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. + +The PID constroller is just a sum of all there three constrollers, of the form, + +$$ U = K_{p} e(t) + K_{i} \int_{0}^{t} e(x) dx + K_{d} \frac{de(t)}{dt} $$ + +To use a PID controller, you need to tune it, by setting the constants, $K_{p}$, $K_{i}$, and $K_{d}$. +There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, and more. + +The uses of PID controllers are thoreticaly any process which has mesurable output and a known ideal output, +but controllers are used mainly for regulating temperature, pressure, force, flow rate, feed rate, speed, and more. + +## The Algorithm + +Luckily the algorithm is very simple, You just need to make the PID equation decrete. +Thus, the equation looks like this, + +$$ U = K_{p} e(t_{j}) + \sum_{l=0}^{j} K_{i} e(t_{l}) \Delta t + K_{d} \frac{e(t_{j-1}) - e(t_{j})}{\Delta t}. $$ + +In the end the code looks like this: + +{% method %} +{% sample lang="c" %} +[import:26-34, lang:"c_cpp"](code/c/pid_controller.c) +{% endmethod %} + +## Example Code + +This example is not calculating the time elapsed, instead it is setting a value called dt. + +{% method %} +{% sample lang="c" %} +[import, lang:"c_cpp"](code/c/pid_controller.c) +{% endmethod %} + + From 874abf528094fb237b0734690f0c2254573d8a7d Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Thu, 9 Aug 2018 10:26:53 +0100 Subject: [PATCH 02/13] fixing spelling and chaning the intro. --- contents/pid_controller/pid_controller.md | 24 +++++++++++++---------- 1 file changed, 14 insertions(+), 10 deletions(-) diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index de4c3e25d..362b3073c 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -1,14 +1,12 @@ #Proportional-Integral-Derivative Controller -You are a helmsmen keeping a vessel on course in various conditions, how do you do that. -A common way is to steer based on current course error, past error and the current rate of change. -Now, the proportional-integral-derivative controller (PID controller) works like the helmsmen on a ship. - -The PID controller is a control loop feedback mechanism, used for continuously modulated control. +The proportional-integral-derivative controller (PID controller) is a control loop feedback mechanism, used for continuously modulated control. The PID controller is in three parts proportional controller, integral controller, and derivative controller. Before we get into how a PID controller works, we need a good example to use to explain how it work. -Imagine you are making a self driving rc car that drives on a line. +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. + +### Proportional Controller If the car is too far to the right then you would turn left and visa versa. 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. @@ -20,6 +18,8 @@ Where $K_{p}$ is a constant and $e(t)$ is the current error. The performance of the controller improves with larger $K_{p}$; 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. +### Derivative Controller + The P controller works well but it has the added problem of overshoting a lot. we need to dampen the oscillation, on way to solve this is to make the rc car resistant to sudden changes of error. This is what the derivative controller (D controller) does, which is given by, @@ -31,7 +31,9 @@ If $K_{d}$ is too high then the system is overdamped, i.e. the car takes too lon If it's too low the system is underdamped, i.e. the car oscillates around the line. When the car is getting back on track quickly with little to no oscillations then the system is called critically damped. -I looks like we are done, we start driving but then some wind starts pushing the car and we have a constant error. +### Integral Controller + +I looks like we are done, we start driving but if some wind starts pushing the car then we get a constant error. We need to know if we are spending too long on one side and account for that. The way to do that is to sum up all the errors and multiply it by a constant. This is what the integral controller (I controller) does, which is given by, @@ -41,19 +43,21 @@ $$ I = K_{i} \int_{0}^{t} e(x) dx, $$ 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. -The PID constroller is just a sum of all there three constrollers, of the form, +### Proportional-Integral-Derivative Controller + +The PID controller is just a sum of all there three constrollers, of the form, $$ U = K_{p} e(t) + K_{i} \int_{0}^{t} e(x) dx + K_{d} \frac{de(t)}{dt} $$ To use a PID controller, you need to tune it, by setting the constants, $K_{p}$, $K_{i}$, and $K_{d}$. There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, and more. -The uses of PID controllers are thoreticaly any process which has mesurable output and a known ideal output, +The uses of PID controllers are theoretically any process which has mesurable output and a known ideal output, but controllers are used mainly for regulating temperature, pressure, force, flow rate, feed rate, speed, and more. ## The Algorithm -Luckily the algorithm is very simple, You just need to make the PID equation decrete. +Luckily the algorithm is very simple, You just need to make the PID equation discrete. Thus, the equation looks like this, $$ U = K_{p} e(t_{j}) + \sum_{l=0}^{j} K_{i} e(t_{l}) \Delta t + K_{d} \frac{e(t_{j-1}) - e(t_{j})}{\Delta t}. $$ From 17b72263937b46930bf0c9c775a691ffefd213a5 Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Thu, 9 Aug 2018 12:43:37 +0100 Subject: [PATCH 03/13] updaing text of pid_controller.md --- contents/pid_controller/pid_controller.md | 20 ++++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index 362b3073c..b2c234873 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -1,15 +1,15 @@ #Proportional-Integral-Derivative Controller The proportional-integral-derivative controller (PID controller) is a control loop feedback mechanism, used for continuously modulated control. -The PID controller is in three parts proportional controller, integral controller, and derivative controller. +The PID controller is comprised of three parts: proportional controller, integral controller, and derivative controller. -Before we get into how a PID controller works, we need a good example to use to explain how it work. -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. +Before we get into how a PID controller works, we need a good example to explain things. +Imagine you are making a self-driving RC car that drives on a line, how would make it work given that the car moves with a constant speed. ### Proportional Controller -If the car is too far to the right then you would turn left and visa versa. -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. +If the car is too far to the right then you would turn left and vice versa. +But there are a range of angles you can turn the wheel, so you can turn proportional to how far you are from the line. This is what the proportional controller (P controller) does, which is given by, $$ P = K_{p} e(t), $$ @@ -49,16 +49,16 @@ The PID controller is just a sum of all there three constrollers, of the form, $$ U = K_{p} e(t) + K_{i} \int_{0}^{t} e(x) dx + K_{d} \frac{de(t)}{dt} $$ -To use a PID controller, you need to tune it, by setting the constants, $K_{p}$, $K_{i}$, and $K_{d}$. +To use a PID controller, you need to tune it by setting the constants, $K_{p}$, $K_{i}$, and $K_{d}$. There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, and more. -The uses of PID controllers are theoretically any process which has mesurable output and a known ideal output, -but controllers are used mainly for regulating temperature, pressure, force, flow rate, feed rate, speed, and more. +The uses of PID controllers are theoretically any process which has measurable output and a known ideal output, +but controllers are used mainly for regulating temperature, pressure, force, flow rate, feed rate, speed and more. ## The Algorithm -Luckily the algorithm is very simple, You just need to make the PID equation discrete. -Thus, the equation looks like this, +Luckily the algorithm is very simple, you just need to make the PID equation discrete. +Thus, the equation looks like this: $$ U = K_{p} e(t_{j}) + \sum_{l=0}^{j} K_{i} e(t_{l}) \Delta t + K_{d} \frac{e(t_{j-1}) - e(t_{j})}{\Delta t}. $$ From aaa8c404ae081f21f6874c852edc568b41d57340 Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Thu, 9 Aug 2018 17:23:07 +0100 Subject: [PATCH 04/13] Adding information unstable controllers --- contents/pid_controller/pid_controller.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index b2c234873..fcd27ce16 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -50,7 +50,8 @@ The PID controller is just a sum of all there three constrollers, of the form, $$ U = K_{p} e(t) + K_{i} \int_{0}^{t} e(x) dx + K_{d} \frac{de(t)}{dt} $$ To use a PID controller, you need to tune it by setting the constants, $K_{p}$, $K_{i}$, and $K_{d}$. -There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, and more. +If you choose the parameters for your PID controller incorrectly, the output will be unstable, i.e., the output diverges. +There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, Cohen–Coon, and Åström-Hägglund. The uses of PID controllers are theoretically any process which has measurable output and a known ideal output, but controllers are used mainly for regulating temperature, pressure, force, flow rate, feed rate, speed and more. From 7a1ad44d0e120c0dc6d4cf951cb267ad23504b4b Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Mon, 13 Aug 2018 10:23:37 +0100 Subject: [PATCH 05/13] pid_controller.md update --- contents/pid_controller/pid_controller.md | 32 ++++++++++++----------- 1 file changed, 17 insertions(+), 15 deletions(-) diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index fcd27ce16..9877d0d30 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -1,6 +1,6 @@ -#Proportional-Integral-Derivative Controller +# Proportional-Integral-Derivative Controller -The proportional-integral-derivative controller (PID controller) is a control loop feedback mechanism, used for continuously modulated control. +The Proportional-Integral-Derivative controller (PID controller) is a control loop feedback mechanism, used for continuously modulated control. The PID controller is comprised of three parts: proportional controller, integral controller, and derivative controller. Before we get into how a PID controller works, we need a good example to explain things. @@ -8,21 +8,22 @@ Imagine you are making a self-driving RC car that drives on a line, how would ma ### Proportional Controller -If the car is too far to the right then you would turn left and vice versa. -But there are a range of angles you can turn the wheel, so you can turn proportional to how far you are from the line. -This is what the proportional controller (P controller) does, which is given by, +If the car is too far to the right then you should turn left and vice versa. +Since there are a range of angles you can turn the wheel, so you should turn proportional to the distance from the line. +This is what the proportional controller (P controller) does, which is described by, $$ P = K_{p} e(t), $$ -Where $K_{p}$ is a constant and $e(t)$ is the current error. +Where $K_{p}$ is a constant and $e(t)$ is the current distance from the line, which is called the error. The performance of the controller improves with larger $K_{p}$; -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. +if $K_{p}$ is too high then when the error is too high, the system becomes unstable. +In this case, the car would turn in circles, since there is a maximum angle the wheel can turn, else it would zig zag around the line. ### Derivative Controller -The P controller works well but it has the added problem of overshoting a lot. -we need to dampen the oscillation, on way to solve this is to make the rc car resistant to sudden changes of error. -This is what the derivative controller (D controller) does, which is given by, +The P controller works well but it has the added problem of overshooting a lot, we need to dampen these oscillations. +One way to solve this is to make the rc car resistant to sudden changes of error. +This is what the derivative controller (D controller) does, which is described by, $$ D = K_{d} \frac{de(t)}{dt}$$ @@ -33,19 +34,20 @@ When the car is getting back on track quickly with little to no oscillations the ### Integral Controller -I looks like we are done, we start driving but if some wind starts pushing the car then we get a constant error. +The Proportional and Derivative controllers are robust enough to get the on course. +We start driving, but then some wind starts pushing the car which introduces a constant error. We need to know if we are spending too long on one side and account for that. The way to do that is to sum up all the errors and multiply it by a constant. -This is what the integral controller (I controller) does, which is given by, +This is what the integral controller (I controller) does, which is described by, -$$ I = K_{i} \int_{0}^{t} e(x) dx, $$ +$$ I = K_{i} \int_{0}^{t} e(\uptau) d\uptau, $$ 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. ### Proportional-Integral-Derivative Controller -The PID controller is just a sum of all there three constrollers, of the form, +The PID controller is just a sum of all three controllers and is of the form, $$ U = K_{p} e(t) + K_{i} \int_{0}^{t} e(x) dx + K_{d} \frac{de(t)}{dt} $$ @@ -53,7 +55,7 @@ To use a PID controller, you need to tune it by setting the constants, $K_{p}$, If you choose the parameters for your PID controller incorrectly, the output will be unstable, i.e., the output diverges. There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, Cohen–Coon, and Åström-Hägglund. -The uses of PID controllers are theoretically any process which has measurable output and a known ideal output, +Theoretically, PID controllers can be used for any process with a measurable output and a known ideal output, but controllers are used mainly for regulating temperature, pressure, force, flow rate, feed rate, speed and more. ## The Algorithm From 4c0aa459be13a2b4dc8608adfadbeb82ad0e99e7 Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Tue, 11 Sep 2018 11:21:32 +0100 Subject: [PATCH 06/13] some changes to pid_controller.md --- contents/pid_controller/pid_controller.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index 9877d0d30..9fdb455c2 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -1,4 +1,5 @@ # Proportional-Integral-Derivative Controller +Written by Gathros The Proportional-Integral-Derivative controller (PID controller) is a control loop feedback mechanism, used for continuously modulated control. The PID controller is comprised of three parts: proportional controller, integral controller, and derivative controller. @@ -9,7 +10,7 @@ Imagine you are making a self-driving RC car that drives on a line, how would ma ### Proportional Controller If the car is too far to the right then you should turn left and vice versa. -Since there are a range of angles you can turn the wheel, so you should turn proportional to the distance from the line. +Since there are a range of angles you can turn the wheel, you should turn proportional to the distance from the line. This is what the proportional controller (P controller) does, which is described by, $$ P = K_{p} e(t), $$ @@ -17,7 +18,7 @@ $$ P = K_{p} e(t), $$ Where $K_{p}$ is a constant and $e(t)$ is the current distance from the line, which is called the error. The performance of the controller improves with larger $K_{p}$; if $K_{p}$ is too high then when the error is too high, the system becomes unstable. -In this case, the car would turn in circles, since there is a maximum angle the wheel can turn, else it would zig zag around the line. +In this example, the car would turn in circles, since there is a maximum angle the wheel can turn, else it would zig zag around the line. ### Derivative Controller @@ -30,14 +31,12 @@ $$ D = K_{d} \frac{de(t)}{dt}$$ 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. If it's too low the system is underdamped, i.e. the car oscillates around the line. -When the car is getting back on track quickly with little to no oscillations then the system is called critically damped. +When the car returns to the track and there is little to no oscillations, the system is critically damped. ### Integral Controller -The Proportional and Derivative controllers are robust enough to get the on course. -We start driving, but then some wind starts pushing the car which introduces a constant error. -We need to know if we are spending too long on one side and account for that. -The way to do that is to sum up all the errors and multiply it by a constant. +The Proportional and Derivative controllers are robust enough to keep on course, but what if some wind starts pushing the car and introducing a constant error? +Well, we would need to know if we are spending too long on one side and account for it, we can figure it out by summing up all the errors and multiply it by a constant. This is what the integral controller (I controller) does, which is described by, $$ I = K_{i} \int_{0}^{t} e(\uptau) d\uptau, $$ @@ -74,7 +73,8 @@ In the end the code looks like this: ## Example Code -This example is not calculating the time elapsed, instead it is setting a value called dt. +The example code is of a 1-dimensional RC car that is trying to change from the first lane to the second lane, where the numbers represent the center of the lane. +This example is we can't calculate the time elapsed, instead we are setting a value called dt for time elapsed. {% method %} {% sample lang="c" %} From 6bdd7adec08b47a0ba65b0008aaed548636f36d6 Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Tue, 11 Sep 2018 13:24:31 +0100 Subject: [PATCH 07/13] more changes to pid_controller.md --- contents/pid_controller/pid_controller.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index 9fdb455c2..e5435e566 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -5,7 +5,7 @@ The Proportional-Integral-Derivative controller (PID controller) is a control lo The PID controller is comprised of three parts: proportional controller, integral controller, and derivative controller. Before we get into how a PID controller works, we need a good example to explain things. -Imagine you are making a self-driving RC car that drives on a line, how would make it work given that the car moves with a constant speed. +Imagine you are making a self-driving RC car that drives on a line, how would make the car stay on track given that it moves with a constant speed. ### Proportional Controller From 28c3f5ceeb212b7479c693cf01f5a72eb8fad758 Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Wed, 12 Sep 2018 10:53:50 +0100 Subject: [PATCH 08/13] more updates to pid_controller.md --- contents/pid_controller/pid_controller.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index e5435e566..91d02c7dc 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -5,11 +5,11 @@ The Proportional-Integral-Derivative controller (PID controller) is a control lo The PID controller is comprised of three parts: proportional controller, integral controller, and derivative controller. Before we get into how a PID controller works, we need a good example to explain things. -Imagine you are making a self-driving RC car that drives on a line, how would make the car stay on track given that it moves with a constant speed. +Imagine you are making a self-driving RC car that drives on a line, how could we keep the car on track if it moves with a constant speed? ### Proportional Controller -If the car is too far to the right then you should turn left and vice versa. +Imagine our RC car is moving too far to the right, in this case it makes sense to turn left. Since there are a range of angles you can turn the wheel, you should turn proportional to the distance from the line. This is what the proportional controller (P controller) does, which is described by, @@ -36,7 +36,7 @@ When the car returns to the track and there is little to no oscillations, the sy ### Integral Controller The Proportional and Derivative controllers are robust enough to keep on course, but what if some wind starts pushing the car and introducing a constant error? -Well, we would need to know if we are spending too long on one side and account for it, we can figure it out by summing up all the errors and multiply it by a constant. +Well, we would need to know if we are spending too long on one side to account for it, and we can figure it out by summing up all the errors and multiply it by a constant. This is what the integral controller (I controller) does, which is described by, $$ I = K_{i} \int_{0}^{t} e(\uptau) d\uptau, $$ @@ -74,7 +74,7 @@ In the end the code looks like this: ## Example Code The example code is of a 1-dimensional RC car that is trying to change from the first lane to the second lane, where the numbers represent the center of the lane. -This example is we can't calculate the time elapsed, instead we are setting a value called dt for time elapsed. +In this example, we can't calculate the time elapsed, so we are instead setting a value called dt for time elapsed. {% method %} {% sample lang="c" %} From 182f4bfb76dd23295f9fb1d208cb52d2169d6f05 Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Wed, 26 Dec 2018 10:26:50 +0000 Subject: [PATCH 09/13] Small chapter changes to pid_controller.md --- contents/pid_controller/pid_controller.md | 30 +++++++++++++---------- 1 file changed, 17 insertions(+), 13 deletions(-) diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index 91d02c7dc..647d59f5b 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -1,17 +1,16 @@ # Proportional-Integral-Derivative Controller -Written by Gathros The Proportional-Integral-Derivative controller (PID controller) is a control loop feedback mechanism, used for continuously modulated control. -The PID controller is comprised of three parts: proportional controller, integral controller, and derivative controller. +The PID controller has three components: proportional controller, integral controller, and derivative controller. Before we get into how a PID controller works, we need a good example to explain things. -Imagine you are making a self-driving RC car that drives on a line, how could we keep the car on track if it moves with a constant speed? +For the following sections, imagine you are designing a self driving RC car that tries to remain on a line as it is moving with a constant speed. How would you keep it on course. ### Proportional Controller -Imagine our RC car is moving too far to the right, in this case it makes sense to turn left. +Imagine our RC car is too far to the right of the line, in this case it makes sense to turn left. Since there are a range of angles you can turn the wheel, you should turn proportional to the distance from the line. -This is what the proportional controller (P controller) does, which is described by, +This is what the proportional controller (P controller) does, which is described by $$ P = K_{p} e(t), $$ @@ -22,9 +21,9 @@ In this example, the car would turn in circles, since there is a maximum angle t ### Derivative Controller -The P controller works well but it has the added problem of overshooting a lot, we need to dampen these oscillations. +The P controller works well but it has the added problem of overshooting, we need to dampen this motion. One way to solve this is to make the rc car resistant to sudden changes of error. -This is what the derivative controller (D controller) does, which is described by, +This is what the derivative controller (D controller) does, which is described by $$ D = K_{d} \frac{de(t)}{dt}$$ @@ -35,9 +34,9 @@ When the car returns to the track and there is little to no oscillations, the sy ### Integral Controller -The Proportional and Derivative controllers are robust enough to keep on course, but what if some wind starts pushing the car and introducing a constant error? -Well, we would need to know if we are spending too long on one side to account for it, and we can figure it out by summing up all the errors and multiply it by a constant. -This is what the integral controller (I controller) does, which is described by, +The Proportional and Derivative controllers are robust enough to keep on course, but what if some wind starts pushing the car constantly of track? +Well, we would need to know if we are spending too long on one side to account for it, and we can figure it out by summing up all the displacements, usually refered to as errors, and multiply it by a constant. +This is what the integral controller (I controller) does, which is described by $$ I = K_{i} \int_{0}^{t} e(\uptau) d\uptau, $$ @@ -46,7 +45,7 @@ The peformance of the controller is better with higher $K_{i}$; but with higher ### Proportional-Integral-Derivative Controller -The PID controller is just a sum of all three controllers and is of the form, +The PID controller is just a sum of all three controllers and is of the form $$ U = K_{p} e(t) + K_{i} \int_{0}^{t} e(x) dx + K_{d} \frac{de(t)}{dt} $$ @@ -68,7 +67,7 @@ In the end the code looks like this: {% method %} {% sample lang="c" %} -[import:26-34, lang:"c_cpp"](code/c/pid_controller.c) +[import:26-34, lang:"c"](code/c/pid_controller.c) {% endmethod %} ## Example Code @@ -78,9 +77,14 @@ In this example, we can't calculate the time elapsed, so we are instead setting {% method %} {% sample lang="c" %} -[import, lang:"c_cpp"](code/c/pid_controller.c) +[import, lang:"c"](code/c/pid_controller.c) {% endmethod %} +## License +The text of this of this chapter is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode) with attribution to Gathros. +The code examples are licensed under the MIT license (found in LICENSE.md). + +[

](https://creativecommons.org/licenses/by-sa/4.0/) From 6a8b5e1813b6ad0b6b7346331c977580e2c4b03a Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Sun, 30 Dec 2018 10:54:34 +0000 Subject: [PATCH 10/13] Adding wikipedia to bib file --- literature.bib | 5 +++++ 1 file changed, 5 insertions(+) diff --git a/literature.bib b/literature.bib index 0e35fc1e1..85f673620 100644 --- a/literature.bib +++ b/literature.bib @@ -31,3 +31,8 @@ @article{gs1972 publisher={Elsevier} } +@online{wikipid, + title={PID controller - Wikipedia}, + url={https://en.wikipedia.org/wiki/PID_controller}, + urldate={2018-12-30} +} From a0608f369c72d9f85af6e242012d7a7366311e5f Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Sun, 30 Dec 2018 10:56:16 +0000 Subject: [PATCH 11/13] Adding animations --- contents/pid_controller/res/circling.mp4 | Bin 0 -> 36787 bytes contents/pid_controller/res/damping.mp4 | Bin 0 -> 179060 bytes 2 files changed, 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z>E=NG7mwuW?(y0I;IrI4-2X8kV4WCqFAENUg3CYK{Idb-5a>2=Sonkgt;QdS%_U>k7q4@LZ&{zC?f;Cir&VESLx!-Mqz<$vh^!_I%z z?!T)4AG?3W|NrYa07>!Rl@GAsJo~4-{O_{+4?F*_?Eil~e*bfv_@Cn%_<_m7qEMMYoiDiJ?*eQ`y$6mXa)1oX3qfW8qyU!&0BR0E8h|Wd0w16X zfAM&#fDE1~fhP+}0Av6R0?+{P3IKRp3;?h`a84!I0q8COJpf=`AprgW!~p;kKoA!I z@VQqE01KcG_yYpZj)3V7hz$Te0B!(GfAe4&yq)nc`M*UU?8pDXKCUJR1UC%SX8-{A z0r=emG8>@A0DxG4r#BGbepvy~1%WUj02vCvH$YYc0K^ZZ4zvMZTj>A*#|hkS9{^B~ z57>tQ^I&^_*Foe0JUISfJ+O@m0BwKqBz6Fu0RXlu3;-M_u>X7j*a7?_KHwMuJV_Qn z;{k{Q0PBGD!Q;dW06gzu{+oh%Ag2Gf`h?p+J7)ku9RBzL*gyvR@c{tXS3{uh;AtIr zOo80PTLJP7oDc2*i~yJcvI0O|0RXxn;8Gpp0ibSw@iag_K%oP?CjfBXnF98}{)2gN zdmy$D;M{ow02m*ZR6qvy1F{4Ft_SiNJT4$*0N~sM_Y)i&aLxg-UlDXF# O9)4C%c2*8{_WuQTWS{&1 literal 0 HcmV?d00001 From 45e66ef493f0ef5a9d0afa4443c3d9ede5fed832 Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Sun, 30 Dec 2018 10:57:13 +0000 Subject: [PATCH 12/13] Adding animations and citations --- contents/pid_controller/pid_controller.md | 34 +++++++++++++++++++++-- 1 file changed, 31 insertions(+), 3 deletions(-) diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index 647d59f5b..c1cf7804f 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -19,6 +19,13 @@ The performance of the controller improves with larger $K_{p}$; if $K_{p}$ is too high then when the error is too high, the system becomes unstable. In this example, the car would turn in circles, since there is a maximum angle the wheel can turn, else it would zig zag around the line. +

+ ### Derivative Controller The P controller works well but it has the added problem of overshooting, we need to dampen this motion. @@ -32,6 +39,13 @@ If $K_{d}$ is too high then the system is overdamped, i.e. the car takes too lon If it's too low the system is underdamped, i.e. the car oscillates around the line. When the car returns to the track and there is little to no oscillations, the system is critically damped. +
+ +
+ ### Integral Controller The Proportional and Derivative controllers are robust enough to keep on course, but what if some wind starts pushing the car constantly of track? @@ -51,11 +65,15 @@ $$ U = K_{p} e(t) + K_{i} \int_{0}^{t} e(x) dx + K_{d} \frac{de(t)}{dt} $$ To use a PID controller, you need to tune it by setting the constants, $K_{p}$, $K_{i}$, and $K_{d}$. If you choose the parameters for your PID controller incorrectly, the output will be unstable, i.e., the output diverges. -There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, Cohen–Coon, and Åström-Hägglund. +There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, Cohen–Coon, and Åström-Hägglund.{{ "wikipid" | cite }} Theoretically, PID controllers can be used for any process with a measurable output and a known ideal output, but controllers are used mainly for regulating temperature, pressure, force, flow rate, feed rate, speed and more. +### Bibliography + +{% references %} {% endreferences %} + ## The Algorithm Luckily the algorithm is very simple, you just need to make the PID equation discrete. @@ -84,7 +102,17 @@ In this example, we can't calculate the time elapsed, so we are instead setting MathJax.Hub.Queue(["Typeset",MathJax.Hub]); ## License -The text of this of this chapter is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode) with attribution to Gathros. -The code examples are licensed under the MIT license (found in LICENSE.md). + +##### Code Examples + +The code examples are licensed under the MIT license (found in [LICENSE.md](https://github.com/algorithm-archivists/algorithm-archive/blob/master/LICENSE.md)). + +##### Text + +The text of this chapter was written by Gathros and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). + +##### Images/Graphics +- The animation "[damping](res/damping.mp4)" was created by Gathros and is licenced under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The animation "[circling](res/circling.mp4)" was created by Gathros and is licenced under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). [

](https://creativecommons.org/licenses/by-sa/4.0/) From 2d2be13d5fcebdb74d71d149cf39d8ac046e1fec Mon Sep 17 00:00:00 2001 From: Gathros <6323830+Gathros@users.noreply.github.com> Date: Fri, 7 Aug 2020 12:40:44 +0100 Subject: [PATCH 13/13] Adding small fixes and another video --- contents/pid_controller/pid_controller.md | 55 ++++++++++++-------- contents/pid_controller/res/oscillation.mp4 | Bin 0 -> 35822 bytes 2 files changed, 33 insertions(+), 22 deletions(-) create mode 100644 contents/pid_controller/res/oscillation.mp4 diff --git a/contents/pid_controller/pid_controller.md b/contents/pid_controller/pid_controller.md index c1cf7804f..a95286115 100644 --- a/contents/pid_controller/pid_controller.md +++ b/contents/pid_controller/pid_controller.md @@ -4,20 +4,22 @@ The Proportional-Integral-Derivative controller (PID controller) is a control lo The PID controller has three components: proportional controller, integral controller, and derivative controller. Before we get into how a PID controller works, we need a good example to explain things. -For the following sections, imagine you are designing a self driving RC car that tries to remain on a line as it is moving with a constant speed. How would you keep it on course. +For the following sections, imagine you are designing a self driving RC car that tries to remain on a line as it is moving with a constant speed. How would you keep it on course? ### Proportional Controller Imagine our RC car is too far to the right of the line, in this case it makes sense to turn left. -Since there are a range of angles you can turn the wheel, you should turn proportional to the distance from the line. +Since there are a range of angles you could turn the wheel by, it is unclear what strategy would work best to return the RC car to the line; however, it is clear that if the angle chosen is proportional to the distance from the line, the car will always be moving towards it. This is what the proportional controller (P controller) does, which is described by $$ P = K_{p} e(t), $$ -Where $K_{p}$ is a constant and $e(t)$ is the current distance from the line, which is called the error. -The performance of the controller improves with larger $K_{p}$; -if $K_{p}$ is too high then when the error is too high, the system becomes unstable. -In this example, the car would turn in circles, since there is a maximum angle the wheel can turn, else it would zig zag around the line. +Where $$K_{p}$$ is an arbitrary constant and $$e(t)$$ is the current distance from the line, which is called the error. +The performance of the controller improves with larger $$K_{p}$$; +however, if $$K_{p}$$ and $$e(t)$$ are too high then the system becomes unstable. +If the proportional controller is used alone, the car will not move appropriately. + +For example, if the car is too far out, the angle will be above $$2\pi$$, which means that the car cannot quite return to the line and will instead move in circles, as shown below:
+On the other hand, if the car is sufficiently close to the line, it will oscillate back and forth, but never rest on the line, itself, as shown below: + +
+ +
+ ### Derivative Controller The P controller works well but it has the added problem of overshooting, we need to dampen this motion. -One way to solve this is to make the rc car resistant to sudden changes of error. +One way to do this is to make the RC car resistant to sudden changes of error. This is what the derivative controller (D controller) does, which is described by $$ D = K_{d} \frac{de(t)}{dt}$$ -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. -If it's too low the system is underdamped, i.e. the car oscillates around the line. +Where $$K_{d}$$ is a constant and $$\frac{de(t)}{dt}$$ is the derivative of the error function $$e(t)$$. +If $$K_{d}$$ is too high then the system is overdamped, i.e. the car takes too long to get back on track. +If it's too low, the system is underdamped, i.e. the car oscillates around the line. When the car returns to the track and there is little to no oscillations, the system is critically damped.
@@ -48,14 +59,14 @@ Your browser does not support the video tag. ### Integral Controller -The Proportional and Derivative controllers are robust enough to keep on course, but what if some wind starts pushing the car constantly of track? -Well, we would need to know if we are spending too long on one side to account for it, and we can figure it out by summing up all the displacements, usually refered to as errors, and multiply it by a constant. +The Proportional and Derivative controllers are robust enough to keep the RC car on course, but what if some wind or other external force starts pushing the car constantly off track? +Well, we would need to know if we are spending too long on one side to account for this, and we can figure that out by taking the sum of all the displacements, usually referred to as errors, and multiplying it by a constant. This is what the integral controller (I controller) does, which is described by -$$ I = K_{i} \int_{0}^{t} e(\uptau) d\uptau, $$ +$$ I = K_{i} \int_{0}^{t} e(\uptau) d\tau, $$ -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. +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. ### Proportional-Integral-Derivative Controller @@ -63,18 +74,14 @@ The PID controller is just a sum of all three controllers and is of the form $$ U = K_{p} e(t) + K_{i} \int_{0}^{t} e(x) dx + K_{d} \frac{de(t)}{dt} $$ -To use a PID controller, you need to tune it by setting the constants, $K_{p}$, $K_{i}$, and $K_{d}$. +To use a PID controller, you need to tune it by setting the constants, $$K_{p}$$, $$K_{i}$$, and $$K_{d}$$. If you choose the parameters for your PID controller incorrectly, the output will be unstable, i.e., the output diverges. -There are multiple methods of tuning like, manual tuning, Ziegler–Nichols, Tyreus Luyben, Cohen–Coon, and Åström-Hägglund.{{ "wikipid" | cite }} +There are multiple methods of tuning like manual tuning, Ziegler–Nichols, Tyreus Luyben, Cohen–Coon, and Åström-Hägglund.{{ "wikipid" | cite }} Theoretically, PID controllers can be used for any process with a measurable output and a known ideal output, but controllers are used mainly for regulating temperature, pressure, force, flow rate, feed rate, speed and more. -### Bibliography - -{% references %} {% endreferences %} - -## The Algorithm +## Putting it all together Luckily the algorithm is very simple, you just need to make the PID equation discrete. Thus, the equation looks like this: @@ -98,6 +105,10 @@ In this example, we can't calculate the time elapsed, so we are instead setting [import, lang:"c"](code/c/pid_controller.c) {% endmethod %} +### Bibliography + +{% references %} {% endreferences %} + diff --git a/contents/pid_controller/res/oscillation.mp4 b/contents/pid_controller/res/oscillation.mp4 new file mode 100644 index 0000000000000000000000000000000000000000..d4620953199dc6c4375982293ed9359f87f934e8 GIT binary patch literal 35822 zcmeFYWpo`&vL?E>n38UC}Ajl|W?+7zVBKRZ}R99;fPVrp$~ z?Dks`YX>)T7dvAR5vbq9&fUe>+tAd}-pSYvbZrXSBR3agYX^`B5TT3lZyF01V|#N~ z&~BL+I(dWI)@FiiptiA@vD5Emm>8N^8@v9V#M;ID&%!*-tu3wGOh99fPUa4VmX1!K z(Z4#KK&-ar-XPh(7w}ih(B9esMC)p5?qF`}?k32=@>@w4 z0RVvi>knMIX3LXN;sb5(pDx$}KztA?0Bjym6hsF=B7vv_ZUZ_1b^ulpl7I31pEkgj z+ka7hw*V-l(&kf1^~!!eQ8#i(6JMW@Cl@ns8P!_%r76)J0-SP^1=A7*HhoB2BhlWW zTp`KulC9zMLo)&ZGHV~RIk_2%3#F}d>l@;Kwc!8&ETDdH08k6GP;j775TK*$f6xaw zfVJ`v{^GI`^X4?iB8jX|i5m34ZP8>&<@uA~C=6e#<>{WNK){>_f|2m@95U^m5Q6&G 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