16 Application of Multi-agent Optimization Methods …
237
The quality criterion is set by the Mayer functional (Eq. 16.51). The desired
optimal open-loop control is sought in the form of a saturation function, which
should guarantee the fulfillment of the parallelepiped type constraints on control
vector. The saturation function has a relay structure, and it is proposed to search for
its arguments in the form of a linear combination of given basis functions [12, 13].
The search algorithm of optimal open-loop control using expansion in a system
of basis functions is the following.
Step 1 Initialization. Select a method from the group of multi-agent algorithms and
set its parameters. Set the initial time truncation scale L = 1, the range of
possible values of the decomposition coefficient c 0 ∈
c 0 1 , c 0 2
.
Step 2 Generate the initial population (controls) of NP individuals, which determinate by coefficients c i of expansion g(t), using Eq. 16.55 where c i ∈
c i 1 , c i 2
, i ∈ 0, L − 1.
c
( j)
0 , c
( j)
1 , . . . , c
( j)
L−1
, j ∈ 1, N P
(16.55)
Step 3 Using the generated coefficients, form the control in the form of a saturation
function sat that guarantees the fulfillment the constraints on control vector:
u
(m)
j (t) = sat
g j (t)
, j ∈ 1, q,
(16.56)
where
∀t ∈ T, sat g j (t) =
a j g j (t) ≤ 0
b j g j (t) > 0
, g j (t) =
L−1
i=0
c
( j)
i p i (t).
As the basis function p i (t), we can take the system of nonstationary cosine
curves orthonormalized on the time interval T = [t 0 , t 1 ] with t 0 = 0 in the
form of Eq. 16.57.
p i (t) =
⎧
⎨
⎩
1
t 1
i = 0
2
t 1
cos
i π t
t 1
i = 1, 2, . . . , L − 1
(16.57)
Step 4 Integrate NP systems of differential equations (Eq. 16.52) with controls
u
1
(t), . . . , u
N P
(t) using the 4th order Runge–Kutta method. For any individual, obtain the corresponding trajectories x
1
1 , . . . , x
N P
1 , . . . , x
1
n , . . . , x
N P
n
and calculate the values of the cost functional I
1
, . . . , I
N P
.
Step 5 Fulfill the next iteration of the selected method of minimizing the functional
(Eq. 16.51). Obtain new positions of individuals 1
, . . . , N P
(coefficient
values). Go to Step 3.
237
The quality criterion is set by the Mayer functional (Eq. 16.51). The desired
optimal open-loop control is sought in the form of a saturation function, which
should guarantee the fulfillment of the parallelepiped type constraints on control
vector. The saturation function has a relay structure, and it is proposed to search for
its arguments in the form of a linear combination of given basis functions [12, 13].
The search algorithm of optimal open-loop control using expansion in a system
of basis functions is the following.
Step 1 Initialization. Select a method from the group of multi-agent algorithms and
set its parameters. Set the initial time truncation scale L = 1, the range of
possible values of the decomposition coefficient c 0 ∈
c 0 1 , c 0 2
.
Step 2 Generate the initial population (controls) of NP individuals, which determinate by coefficients c i of expansion g(t), using Eq. 16.55 where c i ∈
c i 1 , c i 2
, i ∈ 0, L − 1.
c
( j)
0 , c
( j)
1 , . . . , c
( j)
L−1
, j ∈ 1, N P
(16.55)
Step 3 Using the generated coefficients, form the control in the form of a saturation
function sat that guarantees the fulfillment the constraints on control vector:
u
(m)
j (t) = sat
g j (t)
, j ∈ 1, q,
(16.56)
where
∀t ∈ T, sat g j (t) =
a j g j (t) ≤ 0
b j g j (t) > 0
, g j (t) =
L−1
i=0
c
( j)
i p i (t).
As the basis function p i (t), we can take the system of nonstationary cosine
curves orthonormalized on the time interval T = [t 0 , t 1 ] with t 0 = 0 in the
form of Eq. 16.57.
p i (t) =
⎧
⎨
⎩
1
t 1
i = 0
2
t 1
cos
i π t
t 1
i = 1, 2, . . . , L − 1
(16.57)
Step 4 Integrate NP systems of differential equations (Eq. 16.52) with controls
u
1
(t), . . . , u
N P
(t) using the 4th order Runge–Kutta method. For any individual, obtain the corresponding trajectories x
1
1 , . . . , x
N P
1 , . . . , x
1
n , . . . , x
N P
n
and calculate the values of the cost functional I
1
, . . . , I
N P
.
Step 5 Fulfill the next iteration of the selected method of minimizing the functional
(Eq. 16.51). Obtain new positions of individuals 1
, . . . , N P
(coefficient
values). Go to Step 3.