16 Application of Multi-agent Optimization Methods …
231
S(t) =
k S E n O n
O n O n
, Q(t) = E n ,
=
k E n O n
O n O n
, k S > 0, k > 0.
Solution search algorithm
Step 1
Set the method parameters: NP is the number of agents in the population, P max is the maximum number of passes, N M AX is the number
of iterations per pass, h is the step of integrating differential equations,
and k S , k are the coefficients for matrices defining quality criteria. Let
k = 0 (iteration count), v
0
= o, t 0 = 0.
Step 2
Generate the initial population on set D using the uniform distribution law: x
1
, . . . , x
N P . Calculate the values of the objective function
f (x
1
), . . . , f (x
N P
).
Step 3
Order the population consisting of NP agents by the value of the objective
function.
Step 4
Choose a leader agent and the corresponding best objective function
value:x
best,k , f
best,k .
Step 5
Divide all other (N P − 1) agents arbitrarily into four groups.
For each agent, create a differential equation
dX
dt
= AX (t) + B u(t), ,X (t k ) =
x
k
− x
best,k
v
k
(16.34)
where
A =
O n E n
O n O n
, B =
O n
E n
Step 6
Move the agents of the first group (optimal control with finite horizon is
applied).
Step 6.1 Find the solution of the Riccati differential equation P(t), ∀t ∈
t k , t k+1
using Eq. 16.35, where t k+1 = t k + N M AX · h.
˙
P(t) = −A T (t)P(t) − P(t)A(t) + P(t)B(t)Q −1 (t)B T (t)P(t) − S(t), P(t k+1 ) =
(16.35)
Step 6.2 Find the optimal feedback control u
∗
(t, ,X ) using Eq. 16.36, where
F(t) = Q
−1
(t)B
T
(t)P(t).
u
∗
(t, ,X ) = −Q
−1
(t)B
T
(t)P(t))X = −F(t))X
(16.36)
Step 6.3 For each agent of the first group, execute: