16 Application of Multi-agent Optimization Methods …
235
Section 16.3.3 develops search algorithm of optimal open-loop control using expansion in a system of basis functions. Solving the problem of finding optimal open-loop
control is considered in Sect. 16.3.4.
16.3.1 Statement of the Problem
Let the behavior of the control object model be described by an ordinary differential equation in the form of Eq. 16.50, where x is the system state vector,
x = (x 1 , . . . , x n )
T
∈ R
n , u is the control vector, u = (u 1 , . . . , u q )
T
∈ U ⊆ R
q , U
is some given set of admissible control values determined by the direct product of
segments [a 1 , b 1 ] × · · · × [a q , b q ], t ∈ T = [t 0 , t 1 ] is the time interval, the start time
t 0 and terminal time t 1 are given, f (t, x, u) is the continuous vector function; R
n is
n-dimensional Euclidean space.
˙
x(t) = f (t, x(t), u(t))
(16.50)
The initial condition x(t 0 ) = x 0 sets the initial state of the system.
We define the set of admissible processes D(t 0 , x 0 ) as a set of pairs d = (x(·), u(·))
that include the trajectory x(·) and control u(·) (where ∀t ∈ T : x(t) ∈ R
n
, u(t) ∈
U , functions x(·) are continuous and piecewise-differentiable, and u(·) piecewisecontinuous) satisfying Eq. 16.50 with given initial condition.
On the set D(t 0 , x 0 ), we define the cost functional in the form of Eq. 16.51.
I (d) = F(x(t 1 ))
(16.51)
It is need to find such a pair d
∗
= (x
∗
(·), u
∗
(·)) ∈ D(t 0 , x 0 ) that I (d
∗
) =
min
d∈D(t 0 ,x 0 )
I (d).
16.3.2 Search Algorithm of Optimal Open-Loop Control
Using Switching Points
We consider Eq. 16.50 as a linear in control, which has the form of Eq. 16.52, where
A(x) is the nonlinear function and B(t) is the matrix (n × q) depending on time.
˙
x(t) = A(x(t)) + B(t)u(t)
(16.52)
In Eq. 16.52, the structure of optimal open-loop control is relay according to the
maximum principle; therefore, it is proposed to look for an approximate solution in a
parametric form determined by the number of control switching moments and their
values.
235
Section 16.3.3 develops search algorithm of optimal open-loop control using expansion in a system of basis functions. Solving the problem of finding optimal open-loop
control is considered in Sect. 16.3.4.
16.3.1 Statement of the Problem
Let the behavior of the control object model be described by an ordinary differential equation in the form of Eq. 16.50, where x is the system state vector,
x = (x 1 , . . . , x n )
T
∈ R
n , u is the control vector, u = (u 1 , . . . , u q )
T
∈ U ⊆ R
q , U
is some given set of admissible control values determined by the direct product of
segments [a 1 , b 1 ] × · · · × [a q , b q ], t ∈ T = [t 0 , t 1 ] is the time interval, the start time
t 0 and terminal time t 1 are given, f (t, x, u) is the continuous vector function; R
n is
n-dimensional Euclidean space.
˙
x(t) = f (t, x(t), u(t))
(16.50)
The initial condition x(t 0 ) = x 0 sets the initial state of the system.
We define the set of admissible processes D(t 0 , x 0 ) as a set of pairs d = (x(·), u(·))
that include the trajectory x(·) and control u(·) (where ∀t ∈ T : x(t) ∈ R
n
, u(t) ∈
U , functions x(·) are continuous and piecewise-differentiable, and u(·) piecewisecontinuous) satisfying Eq. 16.50 with given initial condition.
On the set D(t 0 , x 0 ), we define the cost functional in the form of Eq. 16.51.
I (d) = F(x(t 1 ))
(16.51)
It is need to find such a pair d
∗
= (x
∗
(·), u
∗
(·)) ∈ D(t 0 , x 0 ) that I (d
∗
) =
min
d∈D(t 0 ,x 0 )
I (d).
16.3.2 Search Algorithm of Optimal Open-Loop Control
Using Switching Points
We consider Eq. 16.50 as a linear in control, which has the form of Eq. 16.52, where
A(x) is the nonlinear function and B(t) is the matrix (n × q) depending on time.
˙
x(t) = A(x(t)) + B(t)u(t)
(16.52)
In Eq. 16.52, the structure of optimal open-loop control is relay according to the
maximum principle; therefore, it is proposed to look for an approximate solution in a
parametric form determined by the number of control switching moments and their
values.