Hello,
I am a bit familiar already with MPC and optimization routine by defining the cost function.
My question is, generally we see MPC has cost function (if we talk about linear MPC) of the form (see the picture attached.)
Is there any possibility to add or alter this cost function.
For example, I am solving an optimal control problem where the time of arrival of vehicles at a particular location is also important. How can I change or modify this to include time as well in my optimization routine?.
Can anybody help me with this?
Hey.
Yes its possible to add time in your cost function. In fact, minimum time cost functions are very common. First, you need to know if the arrival time is constraint or cost? Its better to keep it in the cost function because it can help reduce your control effort.
I suggest designing a time optimal control with bounded control efforts as constraints. Meaning you can specify bounds on the control effort and simply use a time optimal control problem. Like bang-bang for example.
You can find more on time optimal cost function in the attached pdf.
Hi Madhur Tiwari Thank you for the response.
The time of arrival is a constraint in my scenario. Moreover, I see the paper you sent, I understand you recommend looking at different options such as Pontryagin's maximum principle or bang-bang control approach rather than MPC right?.
If that is the case, I know it can be solved using alternate (analytical) optimal control theory, however, I am looking if this can be solved using standard MPC?.
The idea is we have terminal states in MPC, however, is there any possibility to add time such that these terminal states are reached within a specific time?.
I believe you can modify your cost function so that you have total time as a cost in your MPC. Can you give me a little detail on your problem? For example: its like at x(T) = xf. but you also have a value for T? Basically time is not free in your case, right?
Hi Madhur Tiwari ,
The problem I am dealing with is related to merging of an on-ramp vehicle into the platoon on highways. The idea is this on-ramp vehicle has to reach a desired states (for ex- pos, vel, acc) in a certain time say at time T, before joining the platoon. Therefore, in my case the problem is not only restricted to certain states but also bounded by the time at which these states are reached.
I hope that gives you an insight?
Prateek Tiwari thank you for clarification.
the main issue is that MPC is a finite horizon controller. Unless your prediction horizon time = final time (T) you cannot constrain the problem. You can add it as a cost but not constrain. can you do this? Is your final time T small? if it’s too long then it Will take longer for the MPC to converge.
the heart of the problem is that the Constraints must be satisfied at each MPC iteration.
However, you can work around this (maybe, I haven’t tried) by constraining the MPC in the final iteration? You think that would work?
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