Computational Intelligence - August 2013 - 20
Given a set of continuous control inputs U and the initial
state X (0) = X 0 , the state of the system at any time t ! (0, T ]
can be determined uniquely in the form:
X (t) = X (0) +
#0 t
-
f (x, X (x), U (x)) dx .
(4)
This means that given the initial state X(0), the state X(t)
can be specified only by the control inputs U in the form
X (t|U ) .
B. Objective Function and Constraints
It is well-known that the canonical form of the objective function can be expressed as [33]:
J (U ) = U 0 (X (T eU )) +
#0 T L 0 (t, X (teU ), U (t)) dt .
(5)
The problem may also be subject to a variety of other constraints, generally in the form:
g i (U ) = U i (X (x i eU )) +
#0
6i ! " 1, g, M , .
xi
L i (t, X (teU ), U (t)) dt # 0
(6)
For a single system, the optimal control problem can be formulated as finding the continuous control inputs U and terminal time T that minimize the objective function J(U ):
min g min J (U )
(7)
min J (U, T ).
(8)
u 1, T
uN , T
The function U and time T are normally constrained by the
following equation.
U min # U (t) # U max, 6t ! [0, T ), T 2 0.
(9)
Defining the mth UAV as the formation center, the free terminal constraint is given by:
N
g 1 (U, Dt) = / {[(x i (T ) - x m (T )) - x mi ] 2
i =1
+ [(y i (T ) - y m (T )) - y mi ] 2
+ [(z i (T ) - z m (T )) - z mi ] 2} = 0,
(10)
where m ! " 1, g, N ,, [x mi , y mi , z mi ] T represents the desired relative coordinates of the ith UAV with respect to the mth UAV.
The distance between any two UAVs i and j is defined to be:
d i, j (x i (t), x j (t)) =
^x i (t) - x j (t) h2 + ^y i (t) - y j (t) h2
. (11)
+ ^z i (t) - z j (t)h2
In order to avoid collision, d i, j (x i (t), x j (t)) must be greater
than the safety collision distance D safe .
d i, j (x i (t), x j (t)) $ D safe,
6t ! [0, T ], 6 i ! j i, j ! " 1, g, N , .
20
IEEE ComputatIonal IntEllIgEnCE magazInE | august 2013
(12)
In order for real-time communication between the
UAVs to update one another on the combat situation of the
formation, d i, j (x i (t), x j (t)) must be smaller than the communication distance.
d i, j (r (t), m (t)) $ D comm 6t ! [0, T ]
6 i ! j i, j ! " 1, g, N , .
(13)
IV. Hpsoga Based Formation
Reconfiguration Time-optimal Controller
PSO and GA are global optimization algorithms and are
suitable for solving optimization problems with linear or
non-linear objective functions; Therefore, they are suitable
for solving non-linear formation reconfiguration problem.
However, the control inputs of each flight unit are continuous and the HPSOGA cannot solve the continuous control
input problem. In order to solve this problem, the control
inputs of each flight unit are piecewise linearized, and the
approximation piecewise linearization control inputs are
used to substitute the continuous inputs, then HPSOGA is
used to find the global optimal solution. Based on the above
ideas, this paper adopts the CPTD method, obtaining the
approximate objective function and constraints condition,
simplifying the problem in description and handling, and
then using HPSOGA to find the approximate solution
t (t; n p, X) until satisfying the constraints of Eq. (12), (13),
U
(18), (19) and (20).
A. Formation Reconfiguration Time-Optimal
Control Discrete Based on CPTD Method
The continuous control inputs u i are approximated by a piecewise function with a set of static parameters (in practice these
static parameters are constants). The terminal time T is first partitioned into n p time intervals. Partitioning is conducted to
introduce a piecewise function with n p constants that substitute the continuous control inputs.
The terminal time T is formulated as a function of time
interval Dt p , which is used for numerical integration.
The static control parameter is set and the time intervals are
found by minimizing the objective function with a standard
non-linear parametric optimization method.
The proposed method takes three steps to derive this
approximate solution of the problem. The following subsections describe these steps.
1) The division of the terminal time T: The terminal time T
is par titioned into n p ! {1, 2, g} inter vals, each
Dt p ! 0 + , so
T = n p Dt p .
(14)
At each time interval Dt p , according to the corresponding
control inputs, equation (3) does numerical integration.
2) The piecewise linearization of control inputs: For the n p
intervals, define ri # n p constants for the ith UAV as
X i = " v ij ! 0 ri|6j ! " 1, g, n p ,,, 6i ! " 1, g, N , . Then,
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