Set-Membership Method for Discrete Optimal Control

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

) ∈ R

ⁿ the state vector

u

(

t

) ∈

U the control vector

The system is studied over

[

t₀

,

t_f

]

t_k

=

t₀

+

k

× δ

_t

,

t_k

≤

t_f

,

k

∈ {

1

, · · · ,

m

}

(2) It is assumed that u

(

t_k

)

is bounded over

[

t_k

,

t_k+1

]

so it is possible to determinate a box

[

u_k

]

such that u

(

t_k

) ∈ [

u_k

]

over

[

t_k

,

t_k+1

]

The flow map of the system is defined as

ϕ(

t₀

,

t_k

;

x₀

,

u

(

t

)) =

x

(

t

)

(3) The reachable set of the system at time t_k is

ϕ (

t₀

,

t_k

;

X₀

,

U

) = { ϕ (

t₀

,

t_k

;

x₀

,

u

(

t

))| ϕ (

t₀

,

t₀

;

x₀

,

u

(

t

)) =

x₀ and

ϕ : [

t₀

,

t_k

] ×

X₀

×

U

→ R

^{n is a}

solution of (1) for some u

(

t

) ∈ U ^}

⁽⁴⁾

where

U ^{= {}

^u

^{: [}

^t0

^,

^tk−1

^{] →}

^U

^|

^u is continuous over

[

t_k

,

t_k+1

]}

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

Objective

Evaluate C_t0,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:

C_t0,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_t0,tf

C⁻_t

0,t_f

⊆

C_t0,tf

⊆

C⁺_t

0,t_f (6)

2. Caracterisation computation

Proposed approach

For each time t_k the algorithm computes a gridding of K (a slice), noted S

(

t_k

)

. The resolution of the gridding is

δ

_K

= (δ

_x1

, · · · , δ

_xi

, · · · , δ

_xn

)

where

δ

_xi 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

(

t_k

)

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

(

t_k

)

S

(

t_k+1

)

The slices are built from S

(

t_f

)

to S

(

t₀

)

3.Optimal discrete path evaluation

Slice modification and graph building

For each cell s_i

∈

S

(

t_k

)

is defined a set of input vectors U

(

s_i

)

that leads s_i to reachable or indeterminate cells of S

(

t_k+1

)

Gather the cells into nodes and build a graph

S

(

t₀

)

S

(

t₁

)

^S

(

t₂

)

S

(

t₃

)

^S

(

t_f

)

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

] ∈

C_t0,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

(

t_k+1

)

S

(

t_k

)

[

u_1,k

] [

u_2,k

]

[

u_3,k

]

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