Set-Membership Method for Discrete Optimal Control

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Set-Membership Method for Discrete Optimal Control

Rémy Guyonneau, Sébastien Lagrange, Laurent Hardouin, Mehdi Lhommeau

To cite this version:

Rémy Guyonneau, Sébastien Lagrange, Laurent Hardouin, Mehdi Lhommeau. Set-Membership Method for Discrete Optimal Control. 10th International Conference on Informatics in Control, Automation and Robotics, ICINCO, 2013, Reykjavik, Iceland. 2013, �10.5220/0004458001930200�.

�hal-01113457�

Set-Membership Method for Discrete Optimal Control

Rémy GUYONNEAU, Sébastien LAGRANGE, Laurent HARDOUIN, Mehdi LHOMMEAU

1. Problem

Considered System

We consider a control system, defined by the differential equa- tion

˙

x(^t) = ^f(^x(^t), u(^t)) ⁽¹⁾

x(^t) ∈ Rⁿ the state vector u(^t) ∈ ^U the control vector

The system is studied over [^t₀, ^tf]

t_k = ^t₀ + ^k × δ^t, ^tk ≤ ^tf, ^k ∈ {¹, · · · , ^m} ⁽²⁾

It is assumed that u(^tk) is bounded over [^tk, ^tk₊₁] so it is possible to determinate a box [^uk] ^{such that u}(^tk) ∈ [^uk] ^over [^tk, ^tk+1]

The flow map of the system is defined as

ϕ(^t₀, ^tk; ^x₀, u(^t)) = ^x(^t) ⁽³⁾

The reachable set of the system at time t_k is

ϕ(^t₀, ^tk; ^X₀, U) = {ϕ(^t₀, ^tk; ^x₀, u(^t))|ϕ(^t₀, ^t₀; ^x₀, u(^t)) = ^x₀

and ϕ : [^t₀, ^tk] × ^X₀ × ^U → R^{n is a}

solution of (1) for some u(^t) ∈

U

where

U

, ^tk−1] → ^U|^u is continuous over [^tk, ^tk+1]} ^de-

notes the set of admissible controls and X₀ a set of possible initial values x₀

Objective

Evaluate Ct₀

,^tf the subset of initial states of K (state constraint) from wich there exists at least one solution of (¹) reaching the tar- get T in finite time t_f starting at a time t₀:

Ct₀

,^tf = {^x₀ ∈ ^K|∃^u(^t) ∈

U

, ϕ(^t₀, ^tf; ^x₀, u(^t)) ∈ ^T} ⁽⁵⁾

Using interval analysis to compute an inner and an outer caracterisations of C^t_0,^t_f

C⁻_t

0,^tf ⊆ ^Ct_0,^tf ⊆ ^C+t₀

,^tf (6)

2. Caracterisation computation

Proposed approach

For each time t_k the algorithm computes a gridding of K (a slice), noted S(^tk)^. The resolution of the gridding is δ_K = (δ^x₁, · · · , δ^x_i, · · · , δ^x_n) ^where δ^x_i corresponds to the i^th dimension of K

t

x₁ x₂

t_f

t₁ t₀

δ^t

δ^x₁

δ^x₂

x₂

x₁

x1 x₁

x₂

x1

S(^tk)

T

Slice computation

We propose an iterative algorithm that classifies the cells of each slice in three categories:

- unreachable (blue), no state inside the cell allows the system to reach the target at time t_f

- reachable (red), all the states inside the cell allow the system to reach the target at time t_f

- indeterminate (yellow), neither reachable nor unreachable

t_k+1

t_k

t x₁ x₂

[^x₁]^∗ [^x₁]

[^x₂]^∗ [^x₂]

[^x₃]^∗ [^x₃]

S(^tk)

S(^tk₊₁)

The slices are built from S(^tf) ^{to S}(^t₀)

3.Optimal discrete path evaluation

Slice modification and graph building

For each cell s_i ∈ ^S(^tk) is defined a set of input vectors U(^si) ^{that leads si to}

reachable or indeterminate cells of S(^tk₊₁)

Gather the cells into nodes and build a graph

S(^t₀) ^S(^t₁) ^S(^t₂) ^S(^t₃) ^S(^tf )

n₀

n₁

n₂

n₁₁

n₁₂

n₂₁

n₁₁₁

n₁₂₁

n₂₁₁

n₂₁₂

n_T

[³, 4] [¹, 2]

[¹, 2] [¹, 2]

[¹, 2] [¹, 2] [¹, 2] [⁶, 7]

[⁶, 7] [⁶, 7] [¹, 2] [¹, 2]

[², 3]

J(^P(ⁿ₀, ⁿ₁, ⁿ₁₁, ⁿ₁₁₁, ⁿ_T)) = [⁶, 10]

J(^P(ⁿ₀, ⁿ₁, ⁿ₁₂, ⁿ₁₂₁, ⁿ_T)) = [¹¹, 15]

J(^P(ⁿ₀, ⁿ₂, ⁿ₂₁, ⁿ₂₁₁, ⁿ_T)) = [⁵, 9]

J(^P(ⁿ₀, ⁿ₂, ⁿ₂₁, ⁿ₂₁₂, ⁿ_T)) = [¹⁴, 18]

Obtained paths

Using the graph and a shortest path algorithm (e.g. Interval Dijkstra) it is possi- ble to compute:

- an enclosure (^P∗) of the optimal discrete control vector to reach the target from an initial state [^x₀] ∈ ^Ct_0,^tf

- an evaluation of the cost (^J(^P∗)) ^{of this}

control vector

For instance

P^∗ = {^P(ⁿ₀, ⁿ₁, ⁿ₁₁, ⁿ₁₁₁, ⁿ_T),

P(ⁿ₀, ⁿ₂, ⁿ₂₁, ⁿ₂₁₁, ⁿ_T)}

J(^P∗) = [⁶, 10] ∪ [⁵, 9] = [⁵, 9]

s_i S(^tk₊₁)

S(^tk) [^u_2,^k] [^u_1,^k] [^u_3,^k]

[email protected] - LISA, Université d’Angers (FRANCE) - ICINCO 2013