An indirect numerical method for a time-optimal state-constrained control problem in a steady two-dimensional fluid flow Roman Chertovskih
Dmitry Karamzin
Nathalie T. Khalil
Fernando Lobo Pereira
Electrical and Computer Engineering Department, FEUP, Porto Portugal Motivation
Case Study
This work aims at the optimal motion planning of Autonomous Underwater Vehicles (AUVs) in underwater milieu subject to fluid vector fields.
1. U = D (unit disk in the plane). 2. Regularity condition is satisfied if |v1(x)| < 1 for all x. Governing System:
The Problem
( x˙ = u + v (x) ∂v1 ∂v ˙ ψ = −ψ ∂x + µ ∂x
I The waterway is defined on the plane by a given affine state constraints. I The AUV motion is determined by a linear control system that encompasses the vector field v (x) (describing the water flow) and the control u. I The initial and final positions of the AUV are given, respectively, by the points A and B. I The task is to solve the Minimum Time Problem between the points A and B.
where u1 = q
ψ1 − µ
u2 = q
,
(ψ1 − µ)2 + ψ22
and µ=
ψ2 (ψ1 − µ)2 + ψ22
|ψ2|v1 ψ1 + √ 2
for |x1| = 1
0
for − 1 < x1 < 1.
1−v1
The Algorithm Minimize T subject to x˙ = u + v (x) x(0) = A, x(T ) = B −1 ≤ x1 ≤ 1 u∈U
To solve the boundary-value problem (BVP) given by the PMP we integrate the governing system varying initial conditions for ψ on the unit circle: I backward in time from point B to find trajectories (red lines): . not meeting the boundaries, but satisfying the BVP, . meeting a boundary, where µ(t) is continuous at the boundary; I forward in time from point A to find trajectories (blue lines): . meeting the boundaries, where µ(t) is continuous at the boundary, . if forward and backward-time trajectories meet on a boundary, we check if µ(t) is continuous in the meeting point.
I State variable: x = (x1, x2), with A starting point, B final point. I Control variable: u = (u1, u2) with u(t) ∈ U. I Underwater vector field: smooth mapping v : R2 → R2. I The terminal time T is free and is supposed to be minimized.
Simulation Results v=(0,
-2
5.58
6.00
-1
-2
5.58
A
0
-1
5.52
-2
5.69 5.64
-3
4.30
4.30
-3
3.42
I Idea: a new Indirect Method using a standard shooting algorithm applied to the PMP above for optimal control problems with state constraints. (∗An extremal is a solution to the problem satisfying PMP.)
-3
-4
-4
-4
-5
-5
-5
-6 -1
B 0 x1
-6 1
The trajectory along the boundary is optimal
-1
4.00
-6
B 0 x1
1
A and B are closer to the left boundary. The trajectory in the left is optimal
Challenge and Approach I How to compute the entire field of extremals∗ providing an efficient numerical solution to the control problem?
5.19
x2
∈ x2
[0, 1], ψ
v=(0.5 sin(π x2), -x21)
2 -x1)
A
0
-1
x2
The regularity assumption proposed in [2] is assumed. Let (x ∗, u ∗) be an optimal process. Then, ∃λ ∈ 1,∞ 2 W ([0, T ]; R ), and µ a scalar function, such that: ∂v ∂v1 ˙ I ψ(t) = −ψ(t) ∂x + µ(t) ∂x , a.e. t ∈ [0, T ]; I ψ(0) and ψ(T ) are free; I u ∗(t) = argmaxu∈U {(ψ1(t) − µ(t))u1 + ψ2(t)u2}; ∗ Constant if − 1 < x 1 (t) < 1 I µ(t) = Increasing if x1∗(t) = −1 ∗ Decreasing if x1 (t) = 1, µ(t) is continuous; I λ + |ψ1(t) − µ(t)| + |ψ2(t)| > 0 ∀ t ∈ [0, T ].
v=(0,
A
0
Pontryagin Maximum Principle (PMP) [1, 2, 3]: Necessary Conditions
-x21)
5.84
B -1
0 x1
1
The water flow has a transversal component v1. The 2 trajectories that do not meet are not extremals
References On some continuity properties of the measure Lagrange multiplier from the Maximum Principle for state constrained problems. SIAM Journal on Control and
[1] A. V. Arutyunov, D. Yu. Karamzin.
Optimization, 53, no. 4, pp. 2514-2540 (2015). [2] R. V. Gamkrelidze, Time-optimal 125, 475-478 (1959).
process with bounded phase coordinates, Dokl. Akad. Nauk SSSR,
[3] D. Karamzin, F. L. Pereira. On a few questions regarding the study of state constrained in optimal control. To appear in Journal of Optimization Theory and Applications (2019).
problems
Novelties The continuity of µ(.): I Suitable for the computation of the multipliers. I Efficient algorithms to numerically solve state-constrained optimal control problems. http://www.nathaliekhalil.com
Acknowledgments The partial supports of FCT R&D Unit SYSTEC - POCI-01-0145-FEDER-006933/SYSTEC funded by ERDF — COMPETE2020 — FCT/MEC — PT2020 - extension to 2018, Project STRIDE - NORTE-01-0145-FEDER-000033, funded by ERDF — NORTE 2020, and project MAGIC - POCI-01-0145- FEDER-032485 - funded by FEDER via COMPETE 2020 - POCI, and by FCT/MCTES via PIDDAC are acknowledged.
[email protected] 