Abstract
A solution of a nonlinear perturbed unconstrained point-to-point control problem, in which the unperturbed system is differentially flat, is considered in the paper. An admissible open-loop control in it is constructed using the covering method. The main part of the obtained admissible control correction in the limit problem is found by expanding the perturbed problem solution in series by the perturbation parameter. The first term of the expansion is determined by A.N. Tikhonov’s regularization of the Fredholm integral equation of the first kind. As shown by numerical experiments, the found structure of an admissible control allows one to find the final form of high precision point-to-point control based on the solution of an auxiliary variational problem in its neighborhood.
Highlights
Federal Research Center “Computer Science and Control” of Russian Academy of Sciences (FRC CSC RAS), Department of Mathematical Modelling, Bauman Moscow State Technical University (BMSTU), 2nd Baumanskaya Str. 5, 105005 Moscow, Russia
One of them is associated with the use of algorithms for solving the introduced optimal control problems and application of the penalty function method to deal with the given terminal equality constraints
The introduction of a small parameter and the application of asymptotic expansions may lead to the construction of extrapolation procedures that make it possible to obtain open-loop and closed-loop controls with lower stiffness of calculations that translate the system into a given state with greater accuracy [1]
Summary
Let us consider a perturbed nonlinear point-to-point control problem:. 0 < ε ≤ ε 0 1. It is necessary to find such an open-loop control u1 (t, ε) = u0 (t) + εu (t), where εu (t) is the main part of the u0 (t) control correction, which solves perturbed nonlinear control problem (1) more precisely than u0 (t). Experiments show that when solving optimal control problems with a small parameter by traditional methods, asymptotic approximations, which can contain only part of the qualitative information about the exact solution (see in [11], for example), are of great help. This means that even with the help of local procedures, solutions can be found that are close to the points of the global extremum. Finding ω (t) means finding the missing part of the exact value of the admissible point-to-point control, the main part of which u1 (t, ε) has already been found
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