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problem itself can be reduced to the optimization
task with real variables, or the optimization task with
real and integer variables. If the time is fixed, the
criterion can be defined with the function
*
1
() () min
SS
S
FS xTx
=
=− →
%
%
%
,
(4)
where
()
SS
xT
=
%
is the system (1) state at the point
T
, and the control function is determined by S
%
. If
the time is free, then the criterion is
*
2
,
(,) () min
SS
ST
FST xTx
=
=− →
%
%
%
.
(5)
There are also inequality constraints for both criteria
____
1
,1,
iii
rrrR i k
+
+
<∈∀=,
(6)
which ensure every switch point to be inside the
[0, ]T interval. It means that we use P
%
instead of P
that gives one more constraint:
k
rT< .
Let introduce a special penalty function
,0
()
0, 0
xx
x
x
⎧>
ϕ=
⎨
≤
⎩
,
and weight coefficient
. Now the constrained
minimization problem becomes the unconstrained
optimization problem. After adding the penalty
function into criteria (4) and (5), we have,
respectively
31
() () ( ) min
k
S
FS FS r T=+α⋅ϕ−→
%
%%
%
,
(7)
42
,
(,) (,) ( ) min
k
ST
FST FST r T=+α⋅ϕ−→
%
%%
%
.
(8)
For the constraints
1ii
rr
< , we can add the sum of
penalty functions
1
1
1
()
k
ii
j
rr
−
+
=
β⋅ ϕ −
∑
with weight
coefficient
β
to every criterion (4), (5), (7), (8). This
sum gives a penalty for violation of the constraint
________
1
,1,1
ii
rri k
+
<=−. Adding it to criteria (4), (5), (7)
or (8) gives us unconstrained optimization problem.
3 MODIFIED EVOLUTIONARY
STRATEGIES ALGORITHM
Thus, the termination control problem was reduced
to the optimization task with one of objective
functions (4), (5), (7), (8) with real and integer
numbers. The objective function, in general case,
has no analytical form and has to be evaluated
numerically. This is why evolution-based
optimization algorithm has to be used. Genetic
algorithm (GA) does not fit to the given task,
because it needs the a priori known range for every
variable. Also, the discretization of real numbers for
GA adds extra troubles to the computation process.
The main principle of evolutionary strategies
(ES) is described in (Schwefel, 1995). Additionally,
we borrowed the operand definitions for integer
numbers from the GA. Our ES-based optimization
algorithm uses selection, recombination, mutation
and local optimization operands. The selected pair of
parents creates an offspring with given probability.
Then the offspring is mutated. The population size is
constant for all generations. The following GA
selection types were used: fitness proportional, rank
based and tournament based. Let every individual be
represented with a tuple
______
,, (),1,
ii i
ip
d op sp fitness op i N==
,
where
1
( ) , {1,...,4}
1()
q
fitness op q
Fop
=∈
+
is the fitness function for problems (4), (5), (7), (8),
respectively,
____
,1,
i
j
op R j k∈= is the set of objective
parameters,
____
,1,
i
j
pRj k
+
∈=
is the set of method
strategic parameters and
p
N is the size of
population.
The solution of any task with criteria (4), (5),
(7), (8) determines the set of objective parameters
4
1
j
j
op
=
ρ
U
, where the every criterion defines sets
____
,1,4
j
jρ=
. Here
____
1
{,1,}
j
tRj kρ= ∈ =
is the set of switch
points,
____
2
{,1,}
j
lNj kρ= ∈ = is the set of indexes,
3
{}TR
=∈ is the time and
______
4
{,1,}
u
uRj Nρ= ∈ =
is the
u
U set. According to the nature of given sets,
standard ES recombination and mutation can be used
for
134
,,
ρρ, and the standard GA recombination
(one-point, two-point and uniform crossover) can be
used for
2
.
The set of strategic parameters
1234
,sp sp
ρ+ρ+ρ+ρ , defines the mutation
operands.
Now we have to modify the mutation operation
for the ES adapting to our problems. Let
1
[0,1]
p
m ∈
be the mutation probability for every gene and
1
be
the Bernoulli distributed random value with
1
1
(1)
Pz m== . Then
ModifiedHybridEvolutionaryStrategiesMethodforTerminationControlProblemwithRelayActuator
335