I'm reading about optimal control problems and I found the following case.
Suppose a motorist needs to drive a fixed distance from point A to point B in the minimum possible time, commencing and ending with zero velocity. Then it is clear that the maximum possible acceleration must be applied until an intermediate point between A and B is reached. Then maximum braking is applied to bring the vehicle to rest at point B. If the driver is regarded as the controller, then positive control inputs are implemented through the accelerator pedal and negative control inputs are implemented through the brake pedal. In this illustration, the control variable consists of an extreme positive value followed by a switch to an extreme negative value.
My questions:
1. How to formulate the optimal control model to minimize travel time from point A to point B?
For this question, it should be addressed with the time-optimal control section.
2.. It is mentioned that the maximum acceleration must be reached at the intermediate point between A and B. How would this result be obtained?
3. According to the classical theory of optimal control, we must give a functional cost that represents the time we want to minimize subject to a dynamic. If this is the way to approach this problem, the functional cost would be min T= ∫dt from 0 to T.
But my tutor tells me that this is not the functional one, but that I must have something based on the position, speed, controller, friction among others.
Could you help me with this please
Dear Alex,
It seams to me that your problem can be formulated as optimal control minimising time, energy or consumption that is why you will need integrating time and energy or consumption in fonctionnel cost.
For more details and information about this subject take a look at links and attached files in topic.
-Energies | Free Full-Text | A Pontryagin Minimum Principle-Based ...
www.mdpi.com/1996-1073/10/9/1379/htm
-Optimal Ecodriving Control: Energy-Efficient ... - Archive Ouverte HAL
https://hal-ifp.archives-ouvertes.fr/hal-01252197/document
-Deterministic Optimal Control - SIAM
www.siam.org/books/dc13/dc13AppendixA.pdf
-Analytical Solution of Optimized Energy Consumption ... - Science Direct
www.sciencedirect.com/science/article/pii/.../pdf?md5...pid...
-Design and Validation of Real-Time Optimal Control with ECMS to ...
https://www.hindawi.com/journals/mpe/2017/3095347/
-Optimal control - Wikipedia
https://en.wikipedia.org/wiki/Optimal_control
Best regards
Dear Alex Eduardo Pozo ,
if the problem is exactly as you described it, then regarding your first question your tutor is wrong. You simply write min T as cost functional (which probably your program won't like, and which is probably why your tutor is not looking for that answer).
Regarding your second question, you can prove this by contradiction. Just assume that for a short period you do not accelerate/break with the maximal force. As the travel distance is fixed, it will take you longer, which is a contradiction to time minimality. So it has to be bang-ban. Next you assume that there exists more than one switching point. Again by contradiction, you get that it has to be unique. Or you can apply Pontryagin's Maximumprinciple.
Regarding your last question,you can extend your dynamics by the running cost function (the 1 under the integral), and then minimise that last state. Please don't use and combination of the other states/controls/whatever, that is NOT an equivalent transformation.
I hope that helps solving your problem.
Best,
Jürgen
For question #2 I think you meant that the maximum *velocity* is reached at the intermediate point, given that the maximum acceleration is applied from the beginning and reversed at the intermediate point. This assumes, of course, that there is no friction and no terminal velocity to be reached along the way.
Thank you for your answers and clarifications to the proposed items.
How can I find that maximum speed and know correctly that this speed is reached at the midpoint. Also, the dynamics for this problem must come from the free body diagram of the vehicle?
The midpoint is determined by logical proof as described by the other researchers. The maximum speed is then determined computationally by the acceleration and the distance to the midpoint.
I have proposed the following optimal control problem. It is written in Spanish
http://www.ee.ic.ac.uk/ICLOCS/ExampleBangBang.html
..this might be of interest for you!
Hi Alex,
I would not like to confuse you (and at this stage, the adviser's opinion is most relevant for you) yet the very common bang-bang acceleration solution is the optimal solution for a simple second order system xdoddot=u, or m*xdotdot=u, which represents simple mass, with no friction (or damping) and no oscillations whatsoever. Otherwise, while your command ends at the desired time, the sudden accelerations jumps may keep the mass moving and oscillating for along time, if not forever.
Taking into account other terms, like friction and natural frequencies, makes the computation pretty complex.
In my experience, for complex systems I ended with smooth trajectories, such as polynomials, which start and end at zero speed, may take a bit longer then the bang-bang, yet do not excite the oscillation and so, practically end in shortest time possible.
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