### Optimization and Systems Theory Seminar

Tuesday, October 16, 2001, 11.00-12.00, Room 3721, Lindstedtsvägen 25

**Natalia Balashevich**

Department of Control Processes Theory

Institute of Mathematics

National Academy of Sciences of Belarus

220072 Minsk, Belarus

E-mail: balash@im.bas-net.by

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Real-time optimization of control systems

A problem of real-time optimization of a control system with nonlinear
dynamics subject to control end endpoint state constraints is
considered. At first, algorithms of constructing an open-loop solution
is described. This algorithm consists of two procedures.
The first procedure includes a piecewise linear approximation of a
nonlinear element of the system. Although after such approximation the
optimal control problem remains nonlinear, it allows to develop an
effective method based on fast algorithms of optimization of linear
control systems. The second procedure of the algorithm consists in
asymptotic correction of the solution obtained by the first
procedure. The suggested method of correction is based on asymptotic
expansions of switching points of control and accompanying
elements. This is the main difference of the method from known
asymptotic methods which are based on asymptotic expansions of primal
and dual variables.

The idea of closed-loop solution to the problem is based on
constructing a realization of optimal feedback in any concrete control
process under unknown but bounded disturbances. The realization of the
algorithm of open-loop solution is oriented on fast corrections of
optimal open-loop control subject to small variations of initial
state. This is possible due to storage a small amount of additional
information allowing to avoid the complete integration of primal or
adjoint system. Such a realization is a basis of an algorithm of
operating an asymptotically optimal controller which generated signals
of optimal feedback in actual control processes in real time.

An application of this approach to constructive solution of
nonextremal control problems is discussed.

This talk is based on joint work with Professors Rafail Gabasov, Faina
Kirillova and Anatolii Kalinin.

Calendar of seminars

*Last update: October 11, 2001 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*