[PPT] - About prior saturation points for affine control systems T erence PowerPoint Presentation

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About prior saturation points for affine control systems

T´ erence Bayen (Universit´ e de Montpellier) (collaboration avec O. Cots)

JOURN´

EES MODE 2018 - AUTRANS 28 mars 2018

T. Bayen (Universit´

e de Montpellier) 1 / 25

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Outline

1

Preliminary results for affine controlled systems

2

Saturation phenomenon in a fed-batch model

3

Determination of prior saturation points and the notion of bridge

T. Bayen (Universit´

e de Montpellier) 2 / 25

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Preliminary results for affine controlled systems

Outline

1

Preliminary results for affine controlled systems

2

Saturation phenomenon in a fed-batch model

3

Determination of prior saturation points and the notion of bridge

T. Bayen (Universit´

e de Montpellier) 3 / 25

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Preliminary results for affine controlled systems

Minimal time control problem

Given smooth vector fields f, g : Rn ! Rn and a closed subset T ⇢ Rn, consider v(x0) := infu2U Tu s.t. xu(Tu) 2 T 8 > < > : ˙ x = f(x) + u(t)g(x), |u|  1, x(0) = x0.

Goal

Synthesize an optimal feedback control x0 7! u[x0] by: u[x0] := u⇤(0, x0).

We suppose classical hypotheses ensuring existence of an optimal control.

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e de Montpellier) 4 / 25

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Preliminary results for affine controlled systems

Application of the Pontryagin Maximum Principle

Hamiltonian condition: H = ⌦p, f(x)↵ + u ⌦p, g(x)↵ + p0

) for a.e. t 2 [0, Tu] 8 > > > < > > > : (t) > 0 ) u(t) = +1 (t) < 0 ) u(t) = 1 (t) = 0, 8t 2 [t1, t2] ) Singular arc where t 7! p(t) satisfies ˙ p(t) = rxH(x(t), p(t), 1, u(t)).

By differentiating the switching function t 7! (t) := ⌦p(t), g(x(t))↵:

˙ (t) = ⌦p(t), [f, g](x(t))↵ , ¨ (t) = ⌦p(t), [f, [f, g]](x(t))↵ + u(t) ⌦p(t), [g, [f, g]](x(t))↵

The singular control us : [t1, t2] ! R is

us(t) := ⌦p(t), [f, [f, g]](x(t))↵ ⌦p(t), [g, [f, g]](x(t))↵ (for a singular arc of order 1).

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e de Montpellier) 5 / 25

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Preliminary results for affine controlled systems

Saturation phenomenon

Question

Consider a singular arc of order 1 that satisfies Legendre-Clebsch optimality condition

@ @u d2 dt2 Hu > 0 (i.e. it is a turnpike), and such that the singular arc becomes non-admissible i.e.

9t 2 [t1, t2], |us(t)| > 1,

Should we follow the singular arc until the saturation point such that us(t⇤) = 1 ?
How does it affect the synthesis ?
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e de Montpellier) 6 / 25

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Preliminary results for affine controlled systems

Related works about saturation and prior-saturation

A. Rapaport, T. B., M. Sebbah, A. Donoso, A. Torrico, Dynamical modelling and optimal control of landfills,

Mathematical Models and Methods in Applied Sciences, issue No 05, vol. 26, pp. 901-929, 2016.

T. B., F

. Mairet, M. Mazade, Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc, DCDS, series B, vol. 20, 1, pp.39–58, 2015.

B. Bonnard, O. Cots, S. Glaser, M. Lapert, D. Sugny, Y. Zhang, Geometric optimal control of the contrast

imaging problem in nuclear magnetic resonance, IEEE Trans. Automat. Control, 57 (2012), no 8, pp. 1957–1969.

H. Schaettler and M. Jankovic, A synthesis of time-optimal controls in the presence of saturate singular

arcs, Forum Mathematicum, 5, (1993), pp. 203–241.

U. Ledzewicz, H. Sch¨

attler, Anti-angiogenic therapy in cancer treatment as an optimal control problem, SIAM J. on Control and Optimization, 46, 3, pp. 1052–1079, 2007.

U. Ledzewicz, H. Sch¨

attler, Geometric Optimal Control : Theory, Methods and Examples, Springer, 2012.

B. Bonnard, M. Chyba, Singular Trajectories and their role in Control Theory, Springer, SMAI, vol. 40, 2002.
T. Bayen (Universit´

e de Montpellier) 7 / 25

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Preliminary results for affine controlled systems

Existence of a prior saturation point in the plane

˙ x = f(x) + u(t)g(x)

Assumption

The target T is reachable from every point x0 2 R2 and det(f(x), g(x)) < 0 for any x 2 R2 (i.e.

∆0 = ;).

Along ∆SA, the strict Legendre-Clebsch condition

@ @u d2 dt2 Hu > 0 is satisfied.

The singular locus ∆SA := {x 2 R2 ; det(g(x), [f, g](x) = 0)} has exactly one saturation point

x? s.t. ✓(x?) = 1 where for x 2 ∆SA ✓(x) := ⌦g(x)?, [f, [f, g]](x)↵ ⌦g(x)?, [g, [f, g]](x)↵

Moreover, there exists a neighborhood V of x? s.t. ✓ is increasing in V.
If Γ := {xm(t, x?) ; t 0} where xm(·, x?) is the unique solution of the system starting from

x? at time 0 with the control u = 1, then T \ Γ = ;.

T. Bayen (Universit´

e de Montpellier) 8 / 25

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Preliminary results for affine controlled systems

Existence of a prior saturation point

Theorem

Under the previous assumptions:

there exists a prior-saturation point xa 2 ∆SA : any admissible singular arc losses its
ptimality at xa (say at time ta) such that ✓(xa) 2] 1, 1[.
an optimal control satisfies u = +1 in a right neigborhood of ta.

Sketch of proof. The switching function t 7! (t) := ⌦p(t), g(x(t))↵ satisfies the ODE: ˙ (t) = u(t)(t) + ↵(xu(t)) a.e. t 2 [0, Tu] where ↵(x) := det(g(x), [f, g](x)) det(f(x), g(x)) and (x) := det(f(x), [f, g](x)) det(f(x), g(x)) , and u(t) := (xu(t)) ↵(xu(t))u(t), t 2 [0, Tu].

Switching rule : switching points in {x 2 R2 ; det(f(x), [f, g](x)) > 0} occur from 1 to 1.
If xu(·) is optimal until x? at t = t? ) (t) < 0, 8t > t? ) contradiction with the PMP

.

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e de Montpellier) 9 / 25

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Saturation phenomenon in a fed-batch model

Outline

1

Preliminary results for affine controlled systems

2

Saturation phenomenon in a fed-batch model

3

Determination of prior saturation points and the notion of bridge

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e de Montpellier) 10 / 25

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Saturation phenomenon in a fed-batch model

Fed-batch reactor with one species

X,S Outlet pump when V = vmax Feed pump Sin Agitation system,

X, S, V

Temperature θ,... sensors

Biological reaction : S ! X

8 > > > < > > > : ˙ x = µ(s)x u

v x,

˙ s = µ(s)x + u

v (sin s),

˙ v = u,

X, S : micro-organisms, substrate

concentrations ; V: volume ; u: dilution rate sin: input substrate concentration

Objective : transformation of sin into

sref s.t. sref ⌧ sin.

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e de Montpellier) 11 / 25

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Saturation phenomenon in a fed-batch model

The two-dimensional optimal control problem

Target set:

T := {(s, v) 2 [0, sin] ⇥ R⇤

+ ; s  sref and v = ¯

v} = [0, sref ] ⇥ {¯ v}.

Notice that M := v(x + s sin) is conserved ) x = M

v + sin s

Admisible control set U := {u : [0, +1) ! [0, 1] ; u meas.}

Optimal control problem

min

u2U Tu,

8 > > < > > : ˙ s = µ(s) h M

v + sin s

i + u

v (sin s),

˙ v = u, s.t. ( (s(Tu), v(Tu)) 2 T , (s(0), v(0)) = (s0, v0).

Kinetics of Haldane type : µ(s) =

¯ µs k+s+k0s2 with a

unique maximum ¯ s 2 [0, sin] s.t. µ0(¯ s) = 0.

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e de Montpellier) 12 / 25

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Saturation phenomenon in a fed-batch model

Pontryagin Maximum Principle

Hamiltonian : H := sµ(s)

h M

v (s sin)

i + u "s(sin s) v + v # | {z }

+0.
Switching function t 7! (t)

8 > < > : (t) < 0 =) u(t) = 0 (No feeding), (t) > 0 =) u(t) = 1 (Maximal feeding),

Lemma

(i) The singular locus is s = ¯ s and the singular control is us[v] := µ(¯ s) " v + M sin ¯ s # (ii) In the set {s > ¯ s}, resp. {s < ¯ s}, only a switching from 0 to 1, resp. from 1 to 0 is possible

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e de Montpellier) 13 / 25

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Saturation phenomenon in a fed-batch model

Synthesis without saturation

Theorem

If the singular arc is admissible i.e. 8v 2 [0, ¯ v], us[v]  1, the optimal feedback control is: u[s, v] = 8 > > > < > > > : 1 if s < ¯ s and v < ¯ v, us[v] if s = ¯ s and v  ¯ v, if s > ¯ s

r

v = ¯ v.

2 4 6 1 3 5 2 1 3 0.5 1.5 2.5

Substrate Volume

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e de Montpellier) 14 / 25

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Saturation phenomenon in a fed-batch model

Synthesis with saturation

Assumption : 9 v⇤ < ¯ v s.t. us[v⇤] = 1 : v⇤ := 1 µ(¯ s) M sin ¯ s

Corollary

Any singular optimal trajectory leaves the singular arc before v⇤ with u = 1 i.e. there exists

va < v⇤ such that a singular arc is not optimal for v > va.

A switching curve emanates from (¯

s, va). Sketch of proof. verify the assumptions of the previous proposition.

Definition

The point (¯ s, va) is a prior saturation point ; it is a frame point of type (CS)2 at the intersection of a singular arc and a switching curve emanating from this point.

U. Boscain, B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Springer SMAI, vol. 43,

2004.

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e de Montpellier) 15 / 25

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Saturation phenomenon in a fed-batch model

Switching curve and frame point of type (CS)2

The switching curve is computed by minimization of the time of trajectories B B+ B

starting at (sin, v0).

2 4 6 1 3 5 2 4 6 1 3 5 7

Substrate Volume

Figure: Singular arc until va < v⇤ (in red) and switching curve emanating from (¯ s, va) (in green).

Lemma

(i) The switching curve connects (¯ s, va) and (¯ s, ¯ v). (ii) The switching curve C can be parameterized by a curve v 2 [va, ¯ v] 7! sc(v).

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e de Montpellier) 16 / 25

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Saturation phenomenon in a fed-batch model

Synthesis with saturation

Theorem

The optimal feedback control is: u[s, v] = 8 > > > < > > > : 1 if s < sc(v) and v < ¯ v, us[v] if s = ¯ s and v  va, if s > sc(v)

r

v = ¯ v.

2 4 6 1 3 5 2 4 6 1 3 5 7

Substrate Volume

Remark : The singular arc is indeed necessarily optimal if v is small !

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e de Montpellier) 17 / 25

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Saturation phenomenon in a fed-batch model

Synthesis with saturation (with v⇤ < 0)

20 10 2 4 6 8 12 14 16 18 2 4 6 1 3 5 7

Substrate Volume

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e de Montpellier) 18 / 25

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Determination of prior saturation points and the notion of bridge

Outline

1

Preliminary results for affine controlled systems

2

Saturation phenomenon in a fed-batch model

3

Determination of prior saturation points and the notion of bridge

T. Bayen (Universit´

e de Montpellier) 19 / 25

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Determination of prior saturation points and the notion of bridge

Synthesis 1 : from D to (sref, ¯ v)

1 2 3 4 5 6 s 1 2 3 4 5 6 7 8 v

The Bang arc u = 1 is tangent to the switching curve at (¯ s, va).

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e de Montpellier) 20 / 25

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Determination of prior saturation points and the notion of bridge

Synthesis 2 : from (¯ s, va) to D

1 2 3 4 5 6 s 1 2 3 4 5 6 7 8 v

The switching curve is tangent to the singular locus at (¯ s, v⇤)

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e de Montpellier) 21 / 25

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Determination of prior saturation points and the notion of bridge

Bridge between a singular arc and the extended target

Definition

We call bridge the bang arc u = +1 which connects the prior saturation point ∆SA to the extended target set T := [0, sin] ⇥ {¯ v} Write the system ˙ x = f(x) + u(t)g(x). Finding va amounts to solve

8 > > > > > < > > > > > : ⌦p(0), g(x(0))↵ = 0, ⌦p(0), [f, g](x(0))↵ = 0, ⌦p(tc), g(x(tc))↵ = 0 and x(tc) 2 T , H = 0

with unknown: (va, p(0), tc) 2 R4.

1 2 3 4 5 6 s 1 2 3 4 5 6 7 8 v

Synthesis 1: in blue : initial condition in D ;

target point (sref , ¯ v)

Synthesis 2: in green : initial condition

(s(tc), va) ; “free” terminal condition

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e de Montpellier) 22 / 25

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Determination of prior saturation points and the notion of bridge

Bridge between two singular arcs

Definition

We call bridge the bang arc u = +1 which connects two singular arcs (from a prior-saturation point).

B. Bonnard, O. Cots, J. Rouot, T. Verron, Working Notes on the Time Minimal Saturation of a Pair of Spins

and Application in Magnetic Resonance Imaging https://hal.archives-ouvertes.fr/hal-01721845/

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e de Montpellier) 23 / 25

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Determination of prior saturation points and the notion of bridge

Example in MRI with a single spin

min|u|1 Tu s.t. (y(Tu), z(Tu)) = (0, 0) with ( ˙ y = Γy u(t)z ˙ z = (1 z) + u(t)y y(0) z(0) ! 2 R2

∆SA = {z = zs} [ {y = 0}
Determination of the prior-saturation point:

8 > > > > > > < > > > > > > : ⌦(0), g(x(0))↵ = 0, ⌦(0), [f, g](x(0))↵ = 0, ⌦(tc), g(x(tc))↵ = 0, ⌦(tc), [f, g](x(tc))↵ = 0, H = 0

where x = (y, z) and unknown: (z(0), (0), tc, y(tc)) 2 R5.

A bridge (u=+1) connects the two singular arcs.
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e de Montpellier) 24 / 25

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Determination of prior saturation points and the notion of bridge

Concluding remarks

In the setting of affine systems in the plane with a single input:

Synthesis in blue
A switching curve emanates from the prior saturation point xa.
The switching curve connects two singular arcs or a singular arc to T .
The bridge is tangent to the switching curve at the prior saturation point (to be done...)
Synthesis in green
A switching curve emanates from xsat.
The switching curve connects two singular arcs or a singular arc to T .
The switching curve is tangent to the singular locus at the saturation point (to be done...)

and in higher dimension?

Merci pour votre attention

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e de Montpellier) 25 / 25