234
A. V. Panteleev and M. M. S. Karane
Step 9.2.3 Among all the positions of the agent during the passage, choose the best
for the entire period of movement, which corresponds to the best value
of the objective function.
The result of Steps 6–8 are the positions of NP agents in the population,
of which (N P − 1) agents with a new position as a result of movement
under the action of control within a fixed group and the leader agent x
best ,
which has not changed its position during the last pass.
Step 10 Check the global search stop conditions.
If k < P max − 1, then continue search, go to Step 3 (order the population
of agents, identify the current leader, shuffle groups) and let k = k + 1.
If k ≥ P max − 1, then finish search, go to Step 11.
Step 11 Select the solution from the last population.
Stop the algorithm. As an approximate solution to the problem f (x
∗
) =
max
x∈D
f (x) select the agent with the best value of the objective function
from the current population x
∗ ∼ = ˜
x
∗
= arg max
j=1,...,N P
f (x
j
).
Recommendations on the parameters selection. The size of population NP determines the number of calculations of the objective function at each iteration. The value
N P − 1 must be divisible by four. For a task with a large range of admissible solutions, it is recommended to take a larger value of the parameter NP. Recommended
parameter value is N P ∈ [501, 1601].
The number of iterations N M AX during the passage determines how long
the search for new solutions per passage will continue. As the parameter NMAX
increases, the accuracy of the solution increases. Recommended values for the
considered set of standard functions depending on the complexity of the function
are N M AX ∈ [20, 80].
The maximum number of passes P max determines how long the search for new
solutions will continue. As the number of passes increases, the accuracy of the
solution increases. Recommended value for this parameter is P max ∈ [10, 30].
Coefficient k S for matrix S defines the criterion of quality of agent trajectories
control. Recommended value for this parameter is k S ∈ [0.1, 1].
Coefficient k for matrix defines the criterion of quality of agent trajectories
control. Recommended value for this parameter is k ∈ [1, 5].
Step of integrating Riccati differential equations is denoted by h. Recommended
value for this parameter is h = 0.0001.
16.3 Application of Multi-agent Methods for Optimal
Open-Loop Control Problems
Hereinafter, Sect. 16.3.1 involves a statement of the problem. Search algorithm
of optimal open-loop control using switching points is presented in Sect. 16.3.2.
A. V. Panteleev and M. M. S. Karane
Step 9.2.3 Among all the positions of the agent during the passage, choose the best
for the entire period of movement, which corresponds to the best value
of the objective function.
The result of Steps 6–8 are the positions of NP agents in the population,
of which (N P − 1) agents with a new position as a result of movement
under the action of control within a fixed group and the leader agent x
best ,
which has not changed its position during the last pass.
Step 10 Check the global search stop conditions.
If k < P max − 1, then continue search, go to Step 3 (order the population
of agents, identify the current leader, shuffle groups) and let k = k + 1.
If k ≥ P max − 1, then finish search, go to Step 11.
Step 11 Select the solution from the last population.
Stop the algorithm. As an approximate solution to the problem f (x
∗
) =
max
x∈D
f (x) select the agent with the best value of the objective function
from the current population x
∗ ∼ = ˜
x
∗
= arg max
j=1,...,N P
f (x
j
).
Recommendations on the parameters selection. The size of population NP determines the number of calculations of the objective function at each iteration. The value
N P − 1 must be divisible by four. For a task with a large range of admissible solutions, it is recommended to take a larger value of the parameter NP. Recommended
parameter value is N P ∈ [501, 1601].
The number of iterations N M AX during the passage determines how long
the search for new solutions per passage will continue. As the parameter NMAX
increases, the accuracy of the solution increases. Recommended values for the
considered set of standard functions depending on the complexity of the function
are N M AX ∈ [20, 80].
The maximum number of passes P max determines how long the search for new
solutions will continue. As the number of passes increases, the accuracy of the
solution increases. Recommended value for this parameter is P max ∈ [10, 30].
Coefficient k S for matrix S defines the criterion of quality of agent trajectories
control. Recommended value for this parameter is k S ∈ [0.1, 1].
Coefficient k for matrix defines the criterion of quality of agent trajectories
control. Recommended value for this parameter is k ∈ [1, 5].
Step of integrating Riccati differential equations is denoted by h. Recommended
value for this parameter is h = 0.0001.
16.3 Application of Multi-agent Methods for Optimal
Open-Loop Control Problems
Hereinafter, Sect. 16.3.1 involves a statement of the problem. Search algorithm
of optimal open-loop control using switching points is presented in Sect. 16.3.2.