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Optimal control model of an articulated part that starts from station A and arrives at station B. [closed]
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I have the following optimal control problem:
Formulate a model of optimal control of a joint that starts from station A and arrives at station B and you want to know how far you should start to decelerate and you also want to know the optimal time.
- I have an articulated bus that moves in an exclusive way and part of
stop A with initial velocity equal to zero and arrives at stop B
with final velocity equal to zero and the speed limit on the road is $40 km / h$. - I am considering that the traffic lights that are in the exclusive way are in green light every time the articulated bus is near.
- The driver of the articulated bus is interested in knowing at what distance he should stop accelerating the articulated to reach stop B with zero speed. With this we are indicating that we must control is the acceleration due to the accelerator, that is, the acceleration will be my control variable.. From here I got the following question: How to find an equation of the speed that is a function of the position ?, that is, $v = v (x)$ and also how this variable could add variables such as speed in curves, if the road has slopes, among others.
- The maximum acceleration I think will be limited by the capacity of
the engine and the maximum deceleration will be limited by the
capacity of the braking system. - Finally, How can I formulate a cost functional that minimizes travel time?
optimization mathematical-modeling control-theory
asked Jul 26 at 23:27
Alex Pozo
488214
closed as unclear what you're asking by Ross Millikan, Shailesh, Claude Leibovici, Taroccoesbrocco, Mostafa Ayaz Jul 29 at 19:54
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
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up vote
0
down vote
favorite
I have the following optimal control problem:
Formulate a model of optimal control of a joint that starts from station A and arrives at station B and you want to know how far you should start to decelerate and you also want to know the optimal time.
- I have an articulated bus that moves in an exclusive way and part of
stop A with initial velocity equal to zero and arrives at stop B
with final velocity equal to zero and the speed limit on the road is $40 km / h$. - I am considering that the traffic lights that are in the exclusive way are in green light every time the articulated bus is near.
- The driver of the articulated bus is interested in knowing at what distance he should stop accelerating the articulated to reach stop B with zero speed. With this we are indicating that we must control is the acceleration due to the accelerator, that is, the acceleration will be my control variable.. From here I got the following question: How to find an equation of the speed that is a function of the position ?, that is, $v = v (x)$ and also how this variable could add variables such as speed in curves, if the road has slopes, among others.
- The maximum acceleration I think will be limited by the capacity of
the engine and the maximum deceleration will be limited by the
capacity of the braking system. - Finally, How can I formulate a cost functional that minimizes travel time?
optimization mathematical-modeling control-theory
asked Jul 26 at 23:27
Alex Pozo
488214
closed as unclear what you're asking by Ross Millikan, Shailesh, Claude Leibovici, Taroccoesbrocco, Mostafa Ayaz Jul 29 at 19:54
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
You need to specify the constraints as otherwise the optimum is infinite speed until you reach the destination then stop. A typical one is a limit on acceleration, which might be different between acceleration and deceleration. Another is a limit to speed. If there are special speed limits at places along the way, for curves or hills, you need to say that.
– Ross Millikan
Jul 26 at 23:37
@amWhy I made a first sketch of this model based on the classical theory of optimal control ece.gmu.edu/~gbeale/ece_620/notes_620_minimum_time.pdf But then I realized that what really should be interesting is when the accelerator should be released in order to arrive with speed and also that the speed would be a function of distance and not of time.
– Alex Pozo
Jul 27 at 0:00
@RossMillikan Could you recommend books about this problem?
– Alex Pozo
Jul 27 at 2:08
No, I don't know any.
– Ross Millikan
Jul 27 at 3:05
1
@AlexPozo Could you specify the required dynamics? This knowledge is essential in order to formulate conveniently the problem.
– Cesareo
Jul 27 at 12:57
 |Â
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have the following optimal control problem:
Formulate a model of optimal control of a joint that starts from station A and arrives at station B and you want to know how far you should start to decelerate and you also want to know the optimal time.
- I have an articulated bus that moves in an exclusive way and part of
stop A with initial velocity equal to zero and arrives at stop B
with final velocity equal to zero and the speed limit on the road is $40 km / h$. - I am considering that the traffic lights that are in the exclusive way are in green light every time the articulated bus is near.
- The driver of the articulated bus is interested in knowing at what distance he should stop accelerating the articulated to reach stop B with zero speed. With this we are indicating that we must control is the acceleration due to the accelerator, that is, the acceleration will be my control variable.. From here I got the following question: How to find an equation of the speed that is a function of the position ?, that is, $v = v (x)$ and also how this variable could add variables such as speed in curves, if the road has slopes, among others.
- The maximum acceleration I think will be limited by the capacity of
the engine and the maximum deceleration will be limited by the
capacity of the braking system. - Finally, How can I formulate a cost functional that minimizes travel time?
optimization mathematical-modeling control-theory
asked Jul 26 at 23:27
Alex Pozo
488214
I have the following optimal control problem:
Formulate a model of optimal control of a joint that starts from station A and arrives at station B and you want to know how far you should start to decelerate and you also want to know the optimal time.
- I have an articulated bus that moves in an exclusive way and part of
stop A with initial velocity equal to zero and arrives at stop B
with final velocity equal to zero and the speed limit on the road is $40 km / h$. - I am considering that the traffic lights that are in the exclusive way are in green light every time the articulated bus is near.
- The driver of the articulated bus is interested in knowing at what distance he should stop accelerating the articulated to reach stop B with zero speed. With this we are indicating that we must control is the acceleration due to the accelerator, that is, the acceleration will be my control variable.. From here I got the following question: How to find an equation of the speed that is a function of the position ?, that is, $v = v (x)$ and also how this variable could add variables such as speed in curves, if the road has slopes, among others.
- The maximum acceleration I think will be limited by the capacity of
the engine and the maximum deceleration will be limited by the
capacity of the braking system. - Finally, How can I formulate a cost functional that minimizes travel time?
optimization mathematical-modeling control-theory
asked Jul 26 at 23:27
Alex Pozo
488214
edited Jul 26 at 23:50
asked Jul 26 at 23:27
Alex Pozo
488214
asked Jul 26 at 23:27
Alex Pozo
488214
asked Jul 26 at 23:27
Alex Pozo
488214
488214
closed as unclear what you're asking by Ross Millikan, Shailesh, Claude Leibovici, Taroccoesbrocco, Mostafa Ayaz Jul 29 at 19:54
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Ross Millikan, Shailesh, Claude Leibovici, Taroccoesbrocco, Mostafa Ayaz Jul 29 at 19:54
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
You need to specify the constraints as otherwise the optimum is infinite speed until you reach the destination then stop. A typical one is a limit on acceleration, which might be different between acceleration and deceleration. Another is a limit to speed. If there are special speed limits at places along the way, for curves or hills, you need to say that.
– Ross Millikan
Jul 26 at 23:37
@amWhy I made a first sketch of this model based on the classical theory of optimal control ece.gmu.edu/~gbeale/ece_620/notes_620_minimum_time.pdf But then I realized that what really should be interesting is when the accelerator should be released in order to arrive with speed and also that the speed would be a function of distance and not of time.
– Alex Pozo
Jul 27 at 0:00
@RossMillikan Could you recommend books about this problem?
– Alex Pozo
Jul 27 at 2:08
No, I don't know any.
– Ross Millikan
Jul 27 at 3:05
1
@AlexPozo Could you specify the required dynamics? This knowledge is essential in order to formulate conveniently the problem.
– Cesareo
Jul 27 at 12:57
 |Â
show 2 more comments
You need to specify the constraints as otherwise the optimum is infinite speed until you reach the destination then stop. A typical one is a limit on acceleration, which might be different between acceleration and deceleration. Another is a limit to speed. If there are special speed limits at places along the way, for curves or hills, you need to say that.
– Ross Millikan
Jul 26 at 23:37
@amWhy I made a first sketch of this model based on the classical theory of optimal control ece.gmu.edu/~gbeale/ece_620/notes_620_minimum_time.pdf But then I realized that what really should be interesting is when the accelerator should be released in order to arrive with speed and also that the speed would be a function of distance and not of time.
– Alex Pozo
Jul 27 at 0:00
@RossMillikan Could you recommend books about this problem?
– Alex Pozo
Jul 27 at 2:08
No, I don't know any.
– Ross Millikan
Jul 27 at 3:05
1
@AlexPozo Could you specify the required dynamics? This knowledge is essential in order to formulate conveniently the problem.
– Cesareo
Jul 27 at 12:57
You need to specify the constraints as otherwise the optimum is infinite speed until you reach the destination then stop. A typical one is a limit on acceleration, which might be different between acceleration and deceleration. Another is a limit to speed. If there are special speed limits at places along the way, for curves or hills, you need to say that.
– Ross Millikan
Jul 26 at 23:37
You need to specify the constraints as otherwise the optimum is infinite speed until you reach the destination then stop. A typical one is a limit on acceleration, which might be different between acceleration and deceleration. Another is a limit to speed. If there are special speed limits at places along the way, for curves or hills, you need to say that.
– Ross Millikan
Jul 26 at 23:37
@amWhy I made a first sketch of this model based on the classical theory of optimal control ece.gmu.edu/~gbeale/ece_620/notes_620_minimum_time.pdf But then I realized that what really should be interesting is when the accelerator should be released in order to arrive with speed and also that the speed would be a function of distance and not of time.
– Alex Pozo
Jul 27 at 0:00
@amWhy I made a first sketch of this model based on the classical theory of optimal control ece.gmu.edu/~gbeale/ece_620/notes_620_minimum_time.pdf But then I realized that what really should be interesting is when the accelerator should be released in order to arrive with speed and also that the speed would be a function of distance and not of time.
– Alex Pozo
Jul 27 at 0:00
@RossMillikan Could you recommend books about this problem?
– Alex Pozo
Jul 27 at 2:08
@RossMillikan Could you recommend books about this problem?
– Alex Pozo
Jul 27 at 2:08
No, I don't know any.
– Ross Millikan
Jul 27 at 3:05
No, I don't know any.
– Ross Millikan
Jul 27 at 3:05
1
1
@AlexPozo Could you specify the required dynamics? This knowledge is essential in order to formulate conveniently the problem.
– Cesareo
Jul 27 at 12:57
@AlexPozo Could you specify the required dynamics? This knowledge is essential in order to formulate conveniently the problem.
– Cesareo
Jul 27 at 12:57
 |Â
show 2 more comments
1 Answer
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up vote
1
down vote
The basic equations for constant acceleration with $s$ being position, $v$ being velocity, and $a$ being acceleration are
$$v=v_0+at\s=s_0+v_0t+frac 12at^2$$
with the zero subscripts representing starting values. If you have a limiting velocity $v_lim$and fixed acceleration, you reach that velocity after $frac v_lima$ at distance $frac 12at^2$. That is the distance from the end you should start to decelerate at.
answered Jul 27 at 3:05
Ross Millikan
275k21186351
I think they do not understand the concerns I have. If we are driving the bus and we want to travel in the shortest possible time, the idea would be to press the accelerator thoroughly, but we have a speed limit of $40km/h$, remember that we want to reach stop B with zero speed, that is, we want to know how far we should go stop pressing the accelerator, with this I think we should have $v = v (x)$. I think that using equations of uniformly varied linear motion is not useful for this kind of problem.
– Alex Pozo
Jul 27 at 12:30
1
Without information on other constraints, the equation for uniform motion gives the optimum. You start from zero speed, accelerate as quickly as possible to maximum speed, then at the end decelerate as quickly as possible to reach the target with zero velocity. You can compute the deceleration distance as$frac 12frac v^2a$ so you start decelerating that far from the end.
– Ross Millikan
Jul 27 at 15:37
Can I see this problem only with a variation calculation problem? Since I could get a function $T(v)=int_A^Bfracdxv$ where $v$ I would get it from the free-body diagram of the bus, considering that the mobile is on an inclined plane and there is a presence of friction.
– Alex Pozo
Jul 28 at 14:43
That is a completely different problem than you have been talking about. Earlier somebody was choosing the acceleration within certain constraints. Now you have a mechanics problem driven by gravity. Please figure out what you are asking.
– Ross Millikan
Jul 28 at 14:51
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
The basic equations for constant acceleration with $s$ being position, $v$ being velocity, and $a$ being acceleration are
$$v=v_0+at\s=s_0+v_0t+frac 12at^2$$
with the zero subscripts representing starting values. If you have a limiting velocity $v_lim$and fixed acceleration, you reach that velocity after $frac v_lima$ at distance $frac 12at^2$. That is the distance from the end you should start to decelerate at.
answered Jul 27 at 3:05
Ross Millikan
275k21186351
I think they do not understand the concerns I have. If we are driving the bus and we want to travel in the shortest possible time, the idea would be to press the accelerator thoroughly, but we have a speed limit of $40km/h$, remember that we want to reach stop B with zero speed, that is, we want to know how far we should go stop pressing the accelerator, with this I think we should have $v = v (x)$. I think that using equations of uniformly varied linear motion is not useful for this kind of problem.
– Alex Pozo
Jul 27 at 12:30
1
Without information on other constraints, the equation for uniform motion gives the optimum. You start from zero speed, accelerate as quickly as possible to maximum speed, then at the end decelerate as quickly as possible to reach the target with zero velocity. You can compute the deceleration distance as$frac 12frac v^2a$ so you start decelerating that far from the end.
– Ross Millikan
Jul 27 at 15:37
Can I see this problem only with a variation calculation problem? Since I could get a function $T(v)=int_A^Bfracdxv$ where $v$ I would get it from the free-body diagram of the bus, considering that the mobile is on an inclined plane and there is a presence of friction.
– Alex Pozo
Jul 28 at 14:43
That is a completely different problem than you have been talking about. Earlier somebody was choosing the acceleration within certain constraints. Now you have a mechanics problem driven by gravity. Please figure out what you are asking.
– Ross Millikan
Jul 28 at 14:51
add a comment |Â
up vote
1
down vote
The basic equations for constant acceleration with $s$ being position, $v$ being velocity, and $a$ being acceleration are
$$v=v_0+at\s=s_0+v_0t+frac 12at^2$$
with the zero subscripts representing starting values. If you have a limiting velocity $v_lim$and fixed acceleration, you reach that velocity after $frac v_lima$ at distance $frac 12at^2$. That is the distance from the end you should start to decelerate at.
answered Jul 27 at 3:05
Ross Millikan
275k21186351
I think they do not understand the concerns I have. If we are driving the bus and we want to travel in the shortest possible time, the idea would be to press the accelerator thoroughly, but we have a speed limit of $40km/h$, remember that we want to reach stop B with zero speed, that is, we want to know how far we should go stop pressing the accelerator, with this I think we should have $v = v (x)$. I think that using equations of uniformly varied linear motion is not useful for this kind of problem.
– Alex Pozo
Jul 27 at 12:30
1
Without information on other constraints, the equation for uniform motion gives the optimum. You start from zero speed, accelerate as quickly as possible to maximum speed, then at the end decelerate as quickly as possible to reach the target with zero velocity. You can compute the deceleration distance as$frac 12frac v^2a$ so you start decelerating that far from the end.
– Ross Millikan
Jul 27 at 15:37
Can I see this problem only with a variation calculation problem? Since I could get a function $T(v)=int_A^Bfracdxv$ where $v$ I would get it from the free-body diagram of the bus, considering that the mobile is on an inclined plane and there is a presence of friction.
– Alex Pozo
Jul 28 at 14:43
That is a completely different problem than you have been talking about. Earlier somebody was choosing the acceleration within certain constraints. Now you have a mechanics problem driven by gravity. Please figure out what you are asking.
– Ross Millikan
Jul 28 at 14:51
add a comment |Â
up vote
1
down vote
up vote
1
down vote
The basic equations for constant acceleration with $s$ being position, $v$ being velocity, and $a$ being acceleration are
$$v=v_0+at\s=s_0+v_0t+frac 12at^2$$
with the zero subscripts representing starting values. If you have a limiting velocity $v_lim$and fixed acceleration, you reach that velocity after $frac v_lima$ at distance $frac 12at^2$. That is the distance from the end you should start to decelerate at.
answered Jul 27 at 3:05
Ross Millikan
275k21186351
The basic equations for constant acceleration with $s$ being position, $v$ being velocity, and $a$ being acceleration are
$$v=v_0+at\s=s_0+v_0t+frac 12at^2$$
with the zero subscripts representing starting values. If you have a limiting velocity $v_lim$and fixed acceleration, you reach that velocity after $frac v_lima$ at distance $frac 12at^2$. That is the distance from the end you should start to decelerate at.
answered Jul 27 at 3:05
Ross Millikan
275k21186351
answered Jul 27 at 3:05
Ross Millikan
275k21186351
answered Jul 27 at 3:05
Ross Millikan
275k21186351
answered Jul 27 at 3:05
Ross Millikan
275k21186351
275k21186351
I think they do not understand the concerns I have. If we are driving the bus and we want to travel in the shortest possible time, the idea would be to press the accelerator thoroughly, but we have a speed limit of $40km/h$, remember that we want to reach stop B with zero speed, that is, we want to know how far we should go stop pressing the accelerator, with this I think we should have $v = v (x)$. I think that using equations of uniformly varied linear motion is not useful for this kind of problem.
– Alex Pozo
Jul 27 at 12:30
1
Without information on other constraints, the equation for uniform motion gives the optimum. You start from zero speed, accelerate as quickly as possible to maximum speed, then at the end decelerate as quickly as possible to reach the target with zero velocity. You can compute the deceleration distance as$frac 12frac v^2a$ so you start decelerating that far from the end.
– Ross Millikan
Jul 27 at 15:37
Can I see this problem only with a variation calculation problem? Since I could get a function $T(v)=int_A^Bfracdxv$ where $v$ I would get it from the free-body diagram of the bus, considering that the mobile is on an inclined plane and there is a presence of friction.
– Alex Pozo
Jul 28 at 14:43
That is a completely different problem than you have been talking about. Earlier somebody was choosing the acceleration within certain constraints. Now you have a mechanics problem driven by gravity. Please figure out what you are asking.
– Ross Millikan
Jul 28 at 14:51
add a comment |Â
I think they do not understand the concerns I have. If we are driving the bus and we want to travel in the shortest possible time, the idea would be to press the accelerator thoroughly, but we have a speed limit of $40km/h$, remember that we want to reach stop B with zero speed, that is, we want to know how far we should go stop pressing the accelerator, with this I think we should have $v = v (x)$. I think that using equations of uniformly varied linear motion is not useful for this kind of problem.
– Alex Pozo
Jul 27 at 12:30
1
Without information on other constraints, the equation for uniform motion gives the optimum. You start from zero speed, accelerate as quickly as possible to maximum speed, then at the end decelerate as quickly as possible to reach the target with zero velocity. You can compute the deceleration distance as$frac 12frac v^2a$ so you start decelerating that far from the end.
– Ross Millikan
Jul 27 at 15:37
Can I see this problem only with a variation calculation problem? Since I could get a function $T(v)=int_A^Bfracdxv$ where $v$ I would get it from the free-body diagram of the bus, considering that the mobile is on an inclined plane and there is a presence of friction.
– Alex Pozo
Jul 28 at 14:43
That is a completely different problem than you have been talking about. Earlier somebody was choosing the acceleration within certain constraints. Now you have a mechanics problem driven by gravity. Please figure out what you are asking.
– Ross Millikan
Jul 28 at 14:51
I think they do not understand the concerns I have. If we are driving the bus and we want to travel in the shortest possible time, the idea would be to press the accelerator thoroughly, but we have a speed limit of $40km/h$, remember that we want to reach stop B with zero speed, that is, we want to know how far we should go stop pressing the accelerator, with this I think we should have $v = v (x)$. I think that using equations of uniformly varied linear motion is not useful for this kind of problem.
– Alex Pozo
Jul 27 at 12:30
I think they do not understand the concerns I have. If we are driving the bus and we want to travel in the shortest possible time, the idea would be to press the accelerator thoroughly, but we have a speed limit of $40km/h$, remember that we want to reach stop B with zero speed, that is, we want to know how far we should go stop pressing the accelerator, with this I think we should have $v = v (x)$. I think that using equations of uniformly varied linear motion is not useful for this kind of problem.
– Alex Pozo
Jul 27 at 12:30
1
1
Without information on other constraints, the equation for uniform motion gives the optimum. You start from zero speed, accelerate as quickly as possible to maximum speed, then at the end decelerate as quickly as possible to reach the target with zero velocity. You can compute the deceleration distance as$frac 12frac v^2a$ so you start decelerating that far from the end.
– Ross Millikan
Jul 27 at 15:37
Without information on other constraints, the equation for uniform motion gives the optimum. You start from zero speed, accelerate as quickly as possible to maximum speed, then at the end decelerate as quickly as possible to reach the target with zero velocity. You can compute the deceleration distance as$frac 12frac v^2a$ so you start decelerating that far from the end.
– Ross Millikan
Jul 27 at 15:37
Can I see this problem only with a variation calculation problem? Since I could get a function $T(v)=int_A^Bfracdxv$ where $v$ I would get it from the free-body diagram of the bus, considering that the mobile is on an inclined plane and there is a presence of friction.
– Alex Pozo
Jul 28 at 14:43
Can I see this problem only with a variation calculation problem? Since I could get a function $T(v)=int_A^Bfracdxv$ where $v$ I would get it from the free-body diagram of the bus, considering that the mobile is on an inclined plane and there is a presence of friction.
– Alex Pozo
Jul 28 at 14:43
That is a completely different problem than you have been talking about. Earlier somebody was choosing the acceleration within certain constraints. Now you have a mechanics problem driven by gravity. Please figure out what you are asking.
– Ross Millikan
Jul 28 at 14:51
That is a completely different problem than you have been talking about. Earlier somebody was choosing the acceleration within certain constraints. Now you have a mechanics problem driven by gravity. Please figure out what you are asking.
– Ross Millikan
Jul 28 at 14:51
add a comment |Â
You need to specify the constraints as otherwise the optimum is infinite speed until you reach the destination then stop. A typical one is a limit on acceleration, which might be different between acceleration and deceleration. Another is a limit to speed. If there are special speed limits at places along the way, for curves or hills, you need to say that.
– Ross Millikan
Jul 26 at 23:37
@amWhy I made a first sketch of this model based on the classical theory of optimal control ece.gmu.edu/~gbeale/ece_620/notes_620_minimum_time.pdf But then I realized that what really should be interesting is when the accelerator should be released in order to arrive with speed and also that the speed would be a function of distance and not of time.
– Alex Pozo
Jul 27 at 0:00
@RossMillikan Could you recommend books about this problem?
– Alex Pozo
Jul 27 at 2:08
No, I don't know any.
– Ross Millikan
Jul 27 at 3:05
1
@AlexPozo Could you specify the required dynamics? This knowledge is essential in order to formulate conveniently the problem.
– Cesareo
Jul 27 at 12:57