Time Optimization for Non-Linear Systems A. E. Gil Garc´ıa
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1
Department of Automatic Control. CINVESTAV Av. Instituto Polit´ecnico Nacional 2508. 07300 M´exico D.F., M´exico
Bang-Bang control:
Abstract— . . .
Keyword: Optimal Control, Time-optimization.
u∗1 = sign(p1 x1 ),
...
We are using this approximation for sign function: p1 x1 u∗1 ≈ p , 2 + (p1 x1 )2 (10) p2 x2 u∗2 ≈ p . 2 + (p2 x2 )2
II. Preliminaries and some Basic Facts ... III. A Numerical Example Let the non-linear system: 2 x˙ 1 −x1 + x2 + x1 u1 ˙ X := = x˙ 2 x21 + x2 u2 with initial and final conditions: x1 (0) −7 x1 (T ) 1 = ; = x2 (0) 3 x2 (T ) −1
(1)
Our goal it’s to minimize time, our criteria: Z T min J := 1 dt = min T
We define x3 := p1 , x4 = p2 then our augmented system is:
x˙ 1 =
−x21 + x2 + x1
x˙ 2 =
x21 + x2
(2)
The family of control (constrained control): U := {u1 , u2 ∈ R| − 1 ≤ u1 ≤ 1, −1 ≤ u2 ≤ 1}
(3)
(4)
∂H X˙ = ∂P
(6)
∂H ; p1 (T ) = 0 ∂x1 ∂H p˙2 = − ; p2 (T ) = 0 ∂x2
(7)
Co-state equations: p˙1 = −
Our system is linear with respect to u, the control that maximize the Hamiltonian is: u∈U
1 The
corresponding author:
[email protected]
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p
!!
x1 x3 p 2 + (x1 x3 )2 !! x2 x4
2 + (x2 x4 )2
x˙ 3 =
2x1 x3 − 2x1 x4 − x3
x˙ 4 =
−x3 − x4
0
The Hamiltonian associated to the system (1) and criteria (4): p1 H = 1 + p1 (−x21 + x2 + x1 u1 ) + p2 (x21 + x2 u2 ); P = p2 (5) We need to satisfies the optimal nessesary conditions:
u∗ := arg maxH
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u∗2 = sign(p2 x2 ).
I. Introduction
x1 x3 p
2 + (x1 x3 )2 !!
x2 x4 p 2 + (x2 x4 )2
IV. Concluding Remarks ...
!!
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