Optimization of Nonlinear Singularly Perturbed Systems with

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BSU
Optimization of Nonlinear Singularly Perturbed
Systems with Hypersphere Control Restriction
A.I. Kalinin and J.O. Grudo
Belarusian State University, Minsk, Belarus
1. Introduction
Within the framework of the theory of optimal
control, great attention is given to the
singularly perturbed problems. As is known, the
numerical solution of optimal control problems
entails repeated integration of the original and
conjugate systems. In singularly perturbed
problems, these dynamical systems are stiff, and,
as a rule, the computations are associated with
serious difficulties resulting in large
computation time and accumulation of computation
errors. Therefore, the role of asymptotic methods
is growing, especially in view of the fact that
the use of these methods results in decomposition
of the original optimal control problem into
problems of lower dimension. In the report, we
consider a time-optimal problem for a nonlinear
singularly perturbed systems with
multidimensional control the values of
which are bounded in the Euclidean metrics An
algorithm for the construction of asymptotic
approximation to the solution to the problem in
question is proposed. The algorithm employs
solutions to two optimal control problems of
lower dimension than the original problem.
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2. Statement of the Problem
In the class of multidimensional controls
with piecewise-continuous components we
consider the time-optimal problem where
is a small positive parameter, is a
-dimensional vector, is a -dimensional
vector, the other elements of the problem have
the appropriate dimensions, and the following
assumptions are made Assumption 1. Matrix
is Hurwitz, that is, the real parts
of all its eigenvalues are negative. Assumption
2. All functions forming the problem are twice
continuously differentiable. Before proceeding,
we need to first define the asymptotic
approximations to the solution to problem (1),
(2). Definition. A control with
piecewise-continuous components and the
corresponding trajectory of system (1) are
said to be asymptotically suboptimal
(subextremal) if and the following
asymptotic equalities hold where is
the optimal time (the final time of a Pontryagin
extremal) in problem (1), (2). In this report, we
describe an algorithm by means of which an
asymptotically subextremal control can be
constructed for the problem in question.
(1)
(2)
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3. Algorithm
The calculations begin by solving the reduced
problem where Henceforth this will be called
the first basic problem. The purpose of using
numerical methods to solve nonlinear problems is
not so much to find an optimal control as to find
a Pontryagin extremum. We therefore assume that
the extremal has
been constructed as a result of solving the first
basic problem. The corresponding trajectories of
the direct and conjugate systems will be denoted
by According to the Pontryagin maximum
principle where Assumption 3. The vector of
conjugate variables which
corresponds to the extremal is
uniquely determined up to a positive multiplier,
and , When this assumption holds, as can
be seen from formula (3),
(3)
(4)
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The next stage of the algorithm is to solve the
optimal control problem with process of infinite
duration which henceforth will be called
the second basic problem. The specific feature of
this problem is as follows the point
is the equilibrium position of the
dynamical system for the control which makes the
integrand in the quality criterion vanish. In
particular, this implies that the second basic
problem has a solution if an optimal control
exists in a similar problem with a finite
sufficiently long process. Assumption 4. Problem
(6) has a solution and is
normal. In accordance with the maximum
principle where and is a
solution of the conjugate system If this
assumption is satisfied, as follows from formula
(7), the optimal control in the second basic
problem has the form
(5)
(6)
(7)
(8)
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We can prove, using the boundary functions
method, that under Assumptions 1  5 a Pontryagin
extremal exists in the original problem with
sufficiently small and the vector function
is an asymptotically subextremal control.
Note that it can be formed immediately after
solving the basic problems.
(9)
4. Conclusions
The proposed algorithm asymptotically solves the
problem under consideration. It is essential that
its realization presupposes the decomposition of
the original problem into two problems of lower
dimension, and what is more, the algorithm does
not contain integrations of stiff systems.