I consider a practical optimal control problem with controls entering linearly both the right-hand sides and the cost function. Thus the optimal control must be bang-bang, i.e. a sequence of pulses of amplitude u_max.
Some preliminary analysis indicates that the width of pulses in the optimal control will be rather small. However, due to technical limitations we cannot realize pulses of width less than Delta_min (e.g. it is too expensive to start up the actuator for a short time).
Thus the question is: can one expect that a suboptimal control will have a similar pattern, i.e. a sequence of pulses of width more or equal to Delta_min and with amplitudes possibly less than u_max?
Or is it not necessarily the case and we can get a better (in terms of cost function) control in a different class of functions for which this restriction will not apply?
Also, is it possible to consider this restriction in a formal way, say in the framework of Pontryagin's maximum principle?
Thanks.
There is no guarantee that if you change the control policy, you arrive at a (local) optimum. Regarding the amplitude constraint, it should simple for you to restate the optimization problem. For the pulse width, have you been considering discretizing the system with a required sample step?
Hi Pavel:
The idea here is not to change the optimal policy, but rather restrict the class of admissible control functions. When the class of admissible controls is fixed, I can always hope to find an optimal solution within it (which will be suboptimal when seen globally).
My question was if I can stay with a similar control pattern (= class of admissible controls) and hope to have a small optimality degradation or there is no evidence for that?
Concerning your last advice, the problem is that I do not know the starting points for these pulses and thus I cannot expect that they will begin at the mesh points.
Danke trotzdem.
The theoretical question regarding optimality is not stated completely. You ask whether an optimal solution given a class of control functions (bang bang with minimal pulse width), is optimal for your system, which allows a wider class of control functions. But it is not known what this wider class is. Clearly switching from maximal to minimal input is not always allowed, but what rate of input change is allowed in these situations?
If you are 'just' looking for the best control signal for your system, this immediately suggests a practical answer: model the actuator, for instance as a simple first or second order linear system. Also, potentially add a cost for a high rate of change of the input.
"Also, potentially add a cost for a high rate of change of the input."
I was thinking about such a suggestion to Dmitry, but couldn't come up with any nice penalty formulation. Everything looks pretty ugly. I also didn't manage to find a good reference. Perhaps, your suggestion to model the actuator and incorporate into the optimization problem is more suitable.
Meanwhile, the penalty I was thinking about is something like fading after a rising (falling) edge. The more time passes by, the lower the penalty becomes.
I'd also recommend to take a look at reinforcement learning.
Wouter, the larger class of admissible control functions could be, e.g., a set of piece-wise continuous functions. But I do not think that this would help us a lot. When checking optimality of a candidate control we use local (needle) variations. And thus the result is also of local nature: we maximize the Hamiltonian at any time instant.
Any restriction on the width of pulse would imply that we need to memorize previous values of the control etc.
When I asked my question, I thought that there could be some intuition for this particular class of optimal control problems (linear in control). Well, let's see...
The idea of having a model of the actuator is good indeed, but in my case the "actuator" is a highly complex system which is out of my reach (let's say we consider the model of disease propagation and the actuator is the medical department which implements some treatment).
Anyway, thank you for your comments. I'll write later when there will be some progress on this issue.
I would use our CRAB optimisation and use the Bang-Bang-like pulses as a truncated bases. The constraints on amplitude and bandwidth comes almost for free. Have a look for the details at T. Caneva, T. Calarco, and S. Montangero, Phys. Rev. A 84, 1 (2011) and at S. Lloyd and S. Montangero, Phys. Rev. Lett. 113, 010502 (2014) for the theoretical insight.
Simone, thank you for pointing out these interesting papers. I'll read them and get back to you if I decide to use this approach.
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