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A multi-objective synthesis of optimal control system by

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Title: A multi-objective synthesis of optimal control system by


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A multi-objective synthesis of optimal control
system by the network operator method
A.I. Diveev
Institution of Russian Academy of Sciences
Dorodnicyn Computing Centre of RAS, Moscow,
Russia (e-mail aidiveev_at_mail.ru).
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Classical problem of optimal control
The mathematical model of object of control is set
Initial values
Terminal condition
Functional is criterion optimization
A solution is optimal control
Synthesizing function
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Synthesis of optimal control
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Problem statement
Initial values are in closed bounded domain
It is necessary to find the admissible control
that satisfies the restrictions
or
is vector of parameters
where
Pareto set is considered to be the solution of
problem
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Theorem 1. Assume
is a finite denumerable set
Suppose
is the solution of the multi-objective synthesis
problem.
and
is the solution of the same problem, but with the
set of
initial condition
Then
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The network operator
The set of variables
The set of parameters
The set of unary operations
The set of binary operations
Commutative
Unit element
Associative
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Binary operations
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The network operator
Network operator is a directed graph with
following properties
  • graph should be circuit free
  • b) there should be at least one edge from the
    source node to any non-source node
  • c) there should be at least one edge from any
    non-source node to sink node
  • d) every source node corresponds to the item of
    set of variables or of parameters
  • e) every non-source node corresponds to the item
    of
  • binary operations set
  • f) every edge corresponds to the item of unary
    operations set .

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Definition 1. Program notation of mathematical
equation is a notation of equation with the help
of elements of constructive sets X, Q,O1,O2.
Definition 2. Graphic notation of mathematical
equations is the notation of program notation
that fulfills the following conditions a) binary
operation can have unary operations or unit
element of this binary operation as its
arguments b) unary operation can have binary
operation, parameter or variable as its
argument c) binary operation cannot have unary
operations with equal constants or variables as
its arguments.
Theorem 2. Any program notation can be
transformed in graphic notation.
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An example
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Program notation
Graphic notation
Network operator
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Network operator matrix (NOM)
Definition 4 Network operator matrix (NOM) is an
integer upper-triangular matrix that has as its
diagonal elements numbers of binary operations
and non-diagonal elements are zeros or numbers of
unary operations, besides if we replace diagonal
elements with zeros and nonzero non-diagonal
elements with ones we shall get an vertex
incident matrix of the graph that satisfies
conditions a-c of network operator definition..
vertex incidence matrix
NOM
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Network operator matrix (NOM)
vector of numbers of nodes for input variables
vector of numbers of nodes for parameters
vector of numbers of nodes for output variables
a node vector
if
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Theorem 3. If we have the NOM
and vectors of
numbers of nodes for variables
parameters
and outputs
then it is sufficient to calculate
the mathematical expression is described by NOM
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Small variations of network operator 0 -
replacement of unary operation by the edge of the
graph 1 - replacement of binary operation in
the node 2 - addition of the edge with unary
operation 3 - removal of unary operation with
the edge of the graph.
Vector of variations
specifies the number of variation, is the
number of node that the edge comes out, is the
number of node that the edge comes in, is
number of unary or binary operation.
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The principle of basic solution
When solving optimization problems, initially we
set the basic solution that is one of admissible
solutions, then define small variations of basic
solution and create search algorithm that
searches for the optimal solution on the set of
small variations
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The genetic algorithm
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The basic solution is a basic NOM
A structural part of chromosome
NOM for chromosome i
A parametrical part of chromosome
is number of bit for integer part
is number of bit for fractional part
Grey code
Vector of parameters
ltgt
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An example
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Pareto set
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the solution
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