236
A. V. Panteleev and M. M. S. Karane
The search algorithm of optimal open-loop control using switching points is the
following.
Step 1 Initialization. Select a method from the group of multi-agent algorithms and
set its parameters. Set the number of switching p = 0 in the control u(t);
wherein t 0 ∈ {t 0 , t 1 }.
Step 2 Generate the initial population (controls) of N P individuals on the time
interval t ∈ [t 0 , t 1 ]. The resulting sequences of values 1, . . . , N P are the
switching points t ∈ [t 0 , t 1 ] in the control u(t).
Step 3 Generate control by generating the switching point values
u
j
p (t) = a p χ(t 0 ) + (a p − b p )
p
k=0
(−1)
k
χ(t − t k ),
(16.53)
where
χ(t) =
0 t ≤ 0
1 t > 0
, j ∈ 1, N P, p ∈ 1, q, a p ≤ u ≤ b p .
Step 4 Integrate NP systems of differential equations (Eq. 16.52) with controls
u
1
(t), . . . , u
N P
(t) using the fourth-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
(switching
point values). Go to Step 3.
Step 6 The loop (Step 3–Step 5) ends when a certain number of iterations are
reached. The best individual is selected (set of control switching points).
The corresponding control and trajectory, as well as, the value of the cost
functional I
∗
p , are taken as an approximate solution of the problem with the
number of switching equal to p.
Step 7 If I
∗
p < I
∗
p−1 (condition is checked under p ≥ 1), then let p = p + 1 and
go to Step 2. If I
∗
p ≥ I
∗
p−1 , then the search procedure for optimal open-loop
control is completed and control with p switching is selected.
16.3.3 Search Algorithm of Optimal Open-Loop Control
Using Expansion in a System of Basis Functions
A class of nonlinear continuous deterministic dynamical systems linear in bounded
control is considered in the form of Eq. 16.54.
˙
x(t) = A(x) + B(x)u
(16.54)
A. V. Panteleev and M. M. S. Karane
The search algorithm of optimal open-loop control using switching points is the
following.
Step 1 Initialization. Select a method from the group of multi-agent algorithms and
set its parameters. Set the number of switching p = 0 in the control u(t);
wherein t 0 ∈ {t 0 , t 1 }.
Step 2 Generate the initial population (controls) of N P individuals on the time
interval t ∈ [t 0 , t 1 ]. The resulting sequences of values 1, . . . , N P are the
switching points t ∈ [t 0 , t 1 ] in the control u(t).
Step 3 Generate control by generating the switching point values
u
j
p (t) = a p χ(t 0 ) + (a p − b p )
p
k=0
(−1)
k
χ(t − t k ),
(16.53)
where
χ(t) =
0 t ≤ 0
1 t > 0
, j ∈ 1, N P, p ∈ 1, q, a p ≤ u ≤ b p .
Step 4 Integrate NP systems of differential equations (Eq. 16.52) with controls
u
1
(t), . . . , u
N P
(t) using the fourth-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
(switching
point values). Go to Step 3.
Step 6 The loop (Step 3–Step 5) ends when a certain number of iterations are
reached. The best individual is selected (set of control switching points).
The corresponding control and trajectory, as well as, the value of the cost
functional I
∗
p , are taken as an approximate solution of the problem with the
number of switching equal to p.
Step 7 If I
∗
p < I
∗
p−1 (condition is checked under p ≥ 1), then let p = p + 1 and
go to Step 2. If I
∗
p ≥ I
∗
p−1 , then the search procedure for optimal open-loop
control is completed and control with p switching is selected.
16.3.3 Search Algorithm of Optimal Open-Loop Control
Using Expansion in a System of Basis Functions
A class of nonlinear continuous deterministic dynamical systems linear in bounded
control is considered in the form of Eq. 16.54.
˙
x(t) = A(x) + B(x)u
(16.54)