1D couple’s state: Feedback investigation, lovebirds trajectories, and you will stabilizing
Given x0, the control paths (i = 1, 2) obtained from (SM1) constitute an open-loop numerical solution of the couple’s effort problem. The scheme (SM2) allows us to compare the stabilizing solution with the unperturbed solution.
Brand new 1D form of the fresh couple’s efforts problem is actually considered within the , which ended up the current presence of yet another solution toward “lovebirds situation”, that’s, considering a first feeling x(0) = x
A couple of sizes of couple’s efforts disease are believed 2nd. Very first, the new 1D make of the problem is analysed. This new viewpoints strategy provides right here valuable subservient suggestions to the (open-loop) control-theoretical steps involved in . After that, the research of one’s dyadic (2D) brand of the problem is handled.
A similar factor thinking are used in both numerical training. They truly are present in Desk step 1. The newest electric and disutility attributes useful the research, namely (14) are the same while the those people considered during the . This method is great to give their unlock-loop study of your own disease. The latest utility and you can disutility attributes a lot more than satisfy the design requisite expected in the last section. The newest mathematical abilities exhibited within this part is sturdy with respect to various criteria of your model enters.
Furthermore, for any initial feeling x0, the corresponding optimal trajectory (c ? (t), x ? (t)) converges towards the unique equilibrium of the following dynamical system, which is obtained from Pontryagin’s maximum principle, (15) The equilibrium is a saddle point, so the optimal trajectory lies on the stable manifold of the system (see or Theorem 1 in ). Continue reading →