MarcPuche T.Reis Funnelcontrolfor boundarycontrolsystems...fakult t f rÊmathematik,Êinformatik...

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für mathematik, informatik

und naturwissenschaften

Marc PucheJoint work with T. Reis & F. L. Schwenninger

Funnel control forboundary control systems

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Motivation

Wave equation on the discFor t ≥ 0 and (r, θ) ∈ (0, 1] × [0, 2π) consider

∂tp(t, r, θ) = ∇r,θ · q(t, r, θ),∂tq(t, r, θ) = ∇r,θp(t, r, θ),

u(t) = (η(r, θ) · q(t, r, θ))|r=1,

y(t) =∫ 2π

0p(t, 1, θ)dθ.

u(t) is prescribed, y(t) is measured.

27.02.2019 | Marc Puche Joint work with T. Reis& F. L. Schwenninger - Funnel control for boundary control systems 2 / 17

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MotivationAbstract formulation

x :=(pq

) ẋ(t) = Ax(t), Ax =[

0 ∇ · q∇p 0

],

u(t) = Bx(t), Bx = (η · q)|Γ,

y(t) = Cx(t), Cx =∫

Γ p|Γdσ.

Power12ddt

∥x(t)∥2L2 =∫

Γp(t) η · q(t)︸︷︷︸

Bx(t)

dσ

= u(t)∫

Γp(t)dσ︸︷︷︸Cx(t)

= u(t)y(t)


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MotivationAbstract formulation

x :=(pq

) ẋ(t) = Ax(t), Ax =[

0 ∇ · q∇p 0

],

u(t) = Bx(t), Bx = (η · q)|Γ,

y(t) = Cx(t), Cx =∫

Γ p|Γdσ.

Power12ddt

∥x(t)∥2L2 =∫

Γp(t) η · q(t)︸︷︷︸

Bx(t)

dσ = u(t)∫

Γp(t)dσ︸︷︷︸Cx(t)

= u(t)y(t)


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Motivation

System

Controller

u(t) y(t)

yref(t)

Aim: Controller design so that “y(t) follows yref(t)”.Only “structural” assumptions on the model.


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Funnel control

System

u(t) = λ(t) − k(t)e(t) +

y(t)u(t)

−yref(t)e(t)

t

1/φ(t)

∥e(t)∥Km

k(t) = k01 − φ(t)2∥e(t)∥2

λ ∈ W 2,∞c (R≥0;Km)

First introduced byIlchmann, Ryan, Sangwin ’02


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System class

Boundary control systemX Hilbert space,m ∈ N. Consider

(BCS)

ẋ(t) = Ax(t), x(0) = x0 ∈ D(A) ⊂ X,u(t) = Bx(t),y(t) = Cx(t),

whereB,C : D(A) ⊂ X → Km. We denote it byS = (A,B,C).


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Controller design

DefinitionLet ϵ > 0. We consider 1/φ(·)

φ ∈ Φϵ :={

φ ∈ W 2,∞(R≥0)∣∣∣ φ(0) = ϵ;∀t > 0, φ(t) > ϵ}

to which we associateFφ := {(t, e) ∈ R≥0 × Km | φ(t)∥e∥ < 1}.

t

±ϕ(t )−1±λ−1e(t )

(0,e(0))


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Controller designAimApply toS = (A,B,C) the feedback law

u(t) =(u0 +

k01 − ϵ2∥e0∥2

e0

)p(t) − k0

1 − φ(t)2∥e(t)∥2e(t),

where

k0 > 0;yref ∈ W 2,∞(R≥0;Km);p ∈ W 2,∞c (R≥0) with p(0) = 1;

e := y − yref ;u0 := Bx0;φ ∈ Φϵ;

for ϵ > 0 such that ϵ∥e0∥ < 1, where e0 := e(0).


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Main result

Theorem (P. ’18)Assume that

∃α ∈ R such that ∀z ∈ D(A) it holds

Re ⟨z,Az⟩X ≤ Re ⟨Bx,Cx⟩Km + α∥z∥2X (Generalized Passivity)

A|kerB generates a C0-semigroup;[BC

]is onto, C : D(A|kerB) → Km continuous.


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Main result




A|kerB generates a C0-semigroup;

[BC



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Main result




A|kerB generates a C0-semigroup;[BC



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Main result

Theorem (P. ’18)Then ∃!x ∈ C1(R≥0;X) at a.e. t ≥ 0 such that

ẋ(t) = Ax(t), x(0) = x0 ∈ D(A)Bx(t) = (u0 + ψ(ϵ, e0))p(t) − ψ(φ(t),Cx(t) − yref(t)),

and u := Bx, y := Cx ∈ C(R≥0;Km) at a.e. t ≥ 0. Moreover, if α < 0, thenu, y ∈ L∞(R≥0;Km).


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Main result

Outline of the proofTransformation

x(t) = φ(t)−1z(t) +Qyref(t) + P (u0 + ψ(ϵ, e0))p(t),

where [BC

] [P Q

]=

[Im 0m0m Im

].


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Main resultOutline of the proofSo that

ż(t) = A(z(t)) + ω(t)z(t) + f(t),z(0) = z0 ∈ D(A),

where

A(z) := A|D(A)z,D(A) := {z ∈ D(A) | ∥Cz∥ < 1,Bz + ϕ(Cz) = 0},

ϕ(y) := k01 − ∥y∥2 y,

D(ϕ) := {y ∈ Kn | ∥y∥ < 1},

and ω := φ̇φ−1 ∈ W 1,∞(R≥0;R) and f ∈ W 1,∞(R≥0;X).


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Main resultOutline of the proofSo that

ż(t) = A(z(t)) + ω(t)z(t) + f(t),z(0) = z0 ∈ D(A),

whereA(z) := A|D(A)z,

D(A) := {z ∈ D(A) | ∥Cz∥ < 1,Bz + ϕ(Cz) = 0},

ϕ(y) := k01 − ∥y∥2 y,

D(ϕ) := {y ∈ Kn | ∥y∥ < 1},

and ω := φ̇φ−1 ∈ W 1,∞(R≥0;R) and f ∈ W 1,∞(R≥0;X).


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Main result

Outline of the proofKato-type Theorem:

∃! locally Lipschitz continuous z : R≥0 → X , such thatz(t) ∈ D(A) at every t ∈ R≥0;

at a.e. t ∈ R≥0 it holds that

ż(t) = A(z(t)) + ω(t)z(t) + f(t),z(0) = z0,

ż and A(z) are continuous at a.e. t ∈ R≥0.If α < 0 then z ∈ W 1,∞(R≥0;X).


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Main result


∃! locally Lipschitz continuous z : R≥0 → X , such thatz(t) ∈ D(A) at every t ∈ R≥0;at a.e. t ∈ R≥0 it holds that

ż(t) = A(z(t)) + ω(t)z(t) + f(t),z(0) = z0,



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Main result



ż(t) = A(z(t)) + ω(t)z(t) + f(t),z(0) = z0,

ż and A(z) are continuous at a.e. t ∈ R≥0.

If α < 0 then z ∈ W 1,∞(R≥0;X).


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Main result



ż(t) = A(z(t)) + ω(t)z(t) + f(t),z(0) = z0,



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Example: Lossytransmission line

Vζ(ζ, t) = −LIt(ζ, t) − RI(ζ, t),Iζ(ζ, t) = −CVt(ζ, t) − GV (ζ, t),

u(t) =(

V (a, t)V (b, t)

), y(t) =

(I(a, t)

−I(b, t)

).

−1.0

−0.5

0.0

0.5

1.0

y

yref,1y1

−1.0

−0.5

0.0

0.5

1.0

y

yref,2y2

−100

−75

−50

−25

0

25

50

75

100

u

u1

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00t (µs)

−40

−20

0

20

40

u

u2


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Example: 2D Waveequation on the unit disc

∂tp(t, r, θ) = ∇r,θ · q(t, r, θ),∂tq(t, r, θ) = ∇r,θp(t, r, θ),

u(t) = (η(r, θ) · q(t, r, θ))|r=1,

y(t) =∫ 2π

0p(t, 1, θ)dθ.

−0.0010

−0.0005

0.0000

0.0005

0.0010

e

e

0.0

0.2

0.4

0.6

0.8

1.0

1.2

y

yref

y

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0t (m)

−1.0

−0.5

0.0

0.5

1.0

u

u


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Example: 2D Heatequation on the unit disc

∂tx(t, r, θ) = ∆r,θx(t, r, θ),u(t) = (∂rx(t, r, θ))|r=1,

y(t) =∫ 2π

0x(t, 1, θ)dθ.

−1.0 −0.5 0.0 0.5 1.0−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00t = 1.25

−1.0 −0.5 0.0 0.5 1.0−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00t = 2.5

−1.0 −0.5 0.0 0.5 1.0−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00t = 3.75

−1.0 −0.5 0.0 0.5 1.0−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00t = 5.0

−0.03

0.00

0.03

0.06

0.09

0.12

0.15

0.18

0.00

0.01

0.02

0.03

0.04

0.05

0.06

−0.150

−0.125

−0.100

−0.075

−0.050

−0.025

0.000

−0.056

−0.048

−0.040

−0.032

−0.024

−0.016

−0.008

0.000

−0.010

−0.005

0.000

0.005

0.010

e

e

−1.0

−0.5

0.0

0.5

1.0

y

yref

y

0 1 2 3 4 5t (s)

−1.0

−0.5

0.0

0.5

1.0

u

u


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Conclusions

Summary

The funnel controller is feasible forS = (A,B,C);

The approach works for hyperbolic and parabolic PDEs;S = (A,B,C) not necessarily well-posed.

Further work

Weak formulation in parabolic scenarios.


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Conclusions

Summary

The funnel controller is feasible forS = (A,B,C);The approach works for hyperbolic and parabolic PDEs;

S = (A,B,C) not necessarily well-posed.

Further work



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Conclusions

Summary

The funnel controller is feasible forS = (A,B,C);The approach works for hyperbolic and parabolic PDEs;S = (A,B,C) not necessarily well-posed.

Further work



MarcPuche T.Reis Funnelcontrolfor boundarycontrolsystems...fakult t f rÊmathematik,Êinformatik...

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