Trajectory planning for the control of distributed parameter systems - - PDF document

Trajectory planning for the control of distributed parameter systems with lumped controls

Joachim Rudolph

Institut für Regelungs- und Steuerungstheorie TU Dresden

En l’honneur du 60e anniversaire de Michel Fliess

Joachim Rudolph (TU Dresden) 1 / 50

Aim: Transfer of a distributed parameter system with lumped controls

◮ between two regimes (most often stationary) ◮ following a reference trajectory.

Tasks: Calculating possible trajectoires, including the (open-loop) control, (close the loop by feedback). Approach followed: extension of the flatness based control design for nonlinear finite dimensional systems (ode’s) and the module theoretic approach to linear systems. Work initiated together with H. Mounier, M. Fliess, P . Rouchon (1995), and done at Dresden with A. Lynch, F. Woittennek, J. Winkler, . . .

Joachim Rudolph (TU Dresden) 2 / 50

Contents Wave Eq. and its Relatives (Hyperbolic Systems) Torsion of an Elastic Rod with a Tip Load Heavy Ropes in Horizontal Motion Heat Eq. and its Relatives (Parabolic Systems) Fundamentals on a Heat Conduction Example Examples of Beams and Plates Formal Framework for Linear Systems with Lumped Controls VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Outline of a Flatness Based Approach and Simulation Flatness Based Parameterization for the Crystallization Stage

Joachim Rudolph (TU Dresden) 3 / 50 Wave eq./hyperbolic syst. Ex.: Rod under torsion

Elastic rod with tip load

q(x, t) x

• Math. model (wave eq.):

ρ ∂2q ∂t2 (x, t) = G ∂2q ∂x2 (x, t), 0 ≤ x ≤ l, t ≥ 0 Initial cond.: q(x, 0) = 0, ∂q ∂t (x, 0) = 0 Boundary cond. (Torque µ(t) at x = 0): GI ∂q ∂x(0, t) = −µ(t), GI ∂q ∂x(l, t) = −Θ ∂2q ∂t2 (l, t) Aim: Turn the load in finite time following a trajectory for the angle while avoiding final oscillations.

Joachim Rudolph (TU Dresden) 5 / 50

Wave eq./hyperbolic syst. Ex.: Rod under torsion

With the derivation operator s one gets the boundary value pb. in x : G d2ˆ q dx2 (x) − ρ s2 ˆ q(x) = 0, GI dˆ q dx (0) = −ˆ µ(s), GI dˆ q dx (l) = −Θ s2 ˆ q(l) Solution (with σ =

• ρ/G) :

ˆ q(x) = A exp (σ s x) + B exp (−σ s x)

• r, better,

ˆ q(x) = C1 cosh (σ s (x − l)) + C2 sinh (σ s (x − l)) One wants to follow the angular trajectory t → y(t) = q(l, t) : ˆ y = ˆ q(l) = C1 Boundary cond.s at x = l give GI dˆ q dx (l) = GI C2 σ s = −Θ s2 ˆ q(l) = −Θ s2 ˆ y whence the parameterization ˆ q(x) =

• cosh (σ s (x − l)) − Θ s

GIσ sinh (σ s (x − l))

• ˆ

y(s)

Joachim Rudolph (TU Dresden) 6 / 50 Wave eq./hyperbolic syst. Ex.: Rod under torsion

With the shift (or delay) operator exp(−σ(x − l) s) : ˆ q(x) = 1 2

• eσ(x−l)s + e−σ(x−l)s −

Θ GIσ

• eσ(x−l)s − e−σ(x−l)s

s

• ˆ

y Solution in t q(x, t) =1 2

• y(t + σ(x − l)) + y(t − σ(x − l))

− Θ GIσ

• ˙

y(t + σ(x − l)) − ˙ y(t − σ(x − l))

• The control follows from the gradient ˆ

µ = −GI dˆ q dx (0) at x = 0 : µ(t) = − GIσ 2

• ˙

y(t − σl) − ˙ y(t + σl)

• − Θ

2

• ¨

y(t − σl) + ¨ y(t + σl)

• ⇒

Choice of the trajectory for load position y by T = σl ahead! Lumped variable y is the free parameter: “basic (or flat) output”.

Joachim Rudolph (TU Dresden) 7 / 50

Wave eq./hyperbolic syst. Ex.: Heavy Ropes

Heavy rope

x x = 0 w(x, t) u(t) x = L

• Math. model:

∂ ∂x

• gx ∂w

∂x (x, t)

• − ∂2w

∂t2 (x, t) = 0, x ∈ [0, L], t ≥ 0 Initial cond.: w(x, 0) = 0, ∂w ∂t (x, 0) = 0, x ∈ [0, L] Boundary cond.: w(L, t) = u(t), ∂w ∂x (0, t) = 0, t ≥ 0 Aim: Horizontal transport without residual oscillations at the arrival

Joachim Rudolph (TU Dresden) 8 / 50 Wave eq./hyperbolic syst. Ex.: Heavy Ropes

P .d.e. (recall): ∂ ∂x

• gx ∂w

∂x (x, t)

• − ∂2w

∂t2 (x, t) = 0, x ∈ [0, L], t ≥ 0 Replacing derivation w.r.t. t by s yields o.d.e. d dx

• gx dˆ

w dx (x)

• − s2 ˆ

w(x) = 0, x ∈ [0, L] with boundary cond. ˆ w(L) = ˆ u, dˆ w dx (0) = 0. Transformation of the independent variable z = 2s

• x/g, x ∈ (0, L],

(and limit for x = 0) yields modified Bessel equation z2 d2ˆ w dz2 + z dˆ w dz − (z2 + n2) ˆ w = 0, with n = 0

Joachim Rudolph (TU Dresden) 9 / 50

Wave eq./hyperbolic syst. Ex.: Heavy Ropes

• Modif. Bessel eq. (n = 0, recall):

z2 d2ˆ w dz2 + z dˆ w dz − z2 ˆ w = 0 Solution formula (with z = 2s

• x/g)

ˆ w(x) = C1 I0(2s

• x/g) + C2 K0(2s
• x/g)

where I0 and K0 are Bessel fct.s of order 0 of the 1st and 2nd kind Boundary condition at x = 0 : 0 = C1 I1(0) + C2 K1(0), I1(0) = 0, K1(x) → ∞ for x ↓ 0 ⇒ C2 = 0 Position of free end ˆ y = ˆ w(0) ⇒ C1 = ˆ y, thus: ˆ w(x) = I0(2s

• x/g) ˆ

y Condition at x = L : ˆ u = I0(2s

• L/g)ˆ

y System equation, relating ˆ w(x) and ˆ u : I0(2s

• L/g) ˆ

w(x) = I0(2s

• x/g) ˆ

u

Joachim Rudolph (TU Dresden) 10 / 50 Wave eq./hyperbolic syst. Ex.: Heavy Ropes

Recall: ˆ w(x) = I0(2s

• x/g) ˆ

y Representation of the function I0 by Poisson integral I0(x) = 1 π

• π

2

− π

2

exp (x sin θ) dθ yields I0(2s

• x/g)ˆ

y = 1 π

• π

2

− π

2

exp (2s

• x/g sin θ)ˆ

y dθ With the exponential fct. as shift operator of amplitude 2

• x/g sin θ :

w(x, t) = 1 π

• π

2

− π

2

y(t + 2

• x/g sin θ) dθ

where the basic output y(t) = w(0, t) is the position of the free end. We need values of y between t − 2

• x/g and t + 2
• x/g

⇒ distributed delay and advance (prediction).

Joachim Rudolph (TU Dresden) 11 / 50

Wave eq./hyperbolic syst. Ex.: Heavy Ropes

Simulation of the heavy rope

Joachim Rudolph (TU Dresden) 12 / 50 Wave eq./hyperbolic syst. Ex.: Heavy Ropes

Two ropes attached on one trolley

• Math. model, with lengths L1 = L2 and deviations from vertical w1, w2

I0(2s

• L1/g) ˆ

w1(x) = I0(2s

• x/g) ˆ

u I0(2s

• L2/g) ˆ

w2(x) = I0(2s

• x/g) ˆ

u Introduce ˆ w(x) for a ficticious rope of length L1 + L2 and ˆ y = ˆ w(0), then ˆ w(x) = I0(2s

• x/g) ˆ

y Now ˆ w(L2) = I0(2s

• L2/g) ˆ

y =: ˆ w1(0), ˆ w(L1) = I0(2s

• L1/g) ˆ

y =: ˆ w2(0) whence ˆ w1(x) = I0(2s

• x/g) I0(2s
• L2/g) ˆ

y, ˆ w2(x) = I0(2s

• x/g) I0(2s
• L1/g) ˆ

y ˆ u = I0(2s

• L1/g)I0(2s
• L2/g) ˆ

y, and ˆ uL = I0(2s

• (L1 + L2)/g) ˆ

y

Joachim Rudolph (TU Dresden) 13 / 50

Wave eq./hyperbolic syst. Ex.: Heavy Ropes

Simulation of two ropes

Joachim Rudolph (TU Dresden) 14 / 50 Wave eq./hyperbolic syst. Ex.: Heavy Ropes

Related problems

◮ Telegrapher’s equation ◮ Timoshenko beam equation ◮ Heat Exchangers ◮ Networks of strings and Timoshenko-beams ◮ etc.

Joachim Rudolph (TU Dresden) 15 / 50

Parabolic syst./beams/plates Ex.: Heat Eq.

Heat conduction problem

u(t) x = −L x = 0 x T(x, t)

• Math. Modell:

λ ∂2 T ∂ x2 (x, t) = (ρc) ∂ T ∂ t (x, t) , −L ≤ x ≤ 0, t ≥ 0 Initial cond.: T(x, 0) = 0 Boundary cond.: ∂ T ∂ x (0, t) = 0, λ ∂ T ∂ x (x, t) = β

• T(−L, t) − u(t)
• Task: Finite time transition between stationary temperatures,

using the boundary control u.

Joachim Rudolph (TU Dresden) 17 / 50 Parabolic syst./beams/plates Ex.: Heat Eq.

Using derivation operator s yields ordinary boundary value pb.: λ d2ˆ T dx2 (x) − (ρc) s ˆ T(x) = 0, dˆ T dx (0) = 0, −λ dˆ T dx (−L) + β ˆ T(−L) = β ˆ u Solution: ˆ T(x) = C1 cosh

• (ρc)s

λ x

• + C2 sinh
• (ρc)s

λ x

• Evaluating boundary cond.:

dˆ T dx (x) = C1

• (ρc)s

λ sinh

• (ρc)s

λ x

• + C2
• (ρc)s

λ cosh

• (ρc)s

λ x

• dˆ

T dx (0) = 0 ⇔ C2 = 0 − λ dˆ T dx (−L) + β ˆ T(−L) = β ˆ u ⇔ C1

• −λ
• (ρc)s

λ sinh

• (ρc)s

λ (−L)

• + β cosh
• (ρc)s

λ (−L)

• = β ˆ

u

Joachim Rudolph (TU Dresden) 18 / 50

Parabolic syst./beams/plates Ex.: Heat Eq.

Choice of free parameter C1 as a basic (or flat) output ˆ y = C1 gives ˆ T(x) = cosh

• (ρc)s

λ x

• ˆ

y, i.e. ˆ y = ˆ T(0) ˆ u =

• −λ

β

• (ρc)s

λ sinh

• (ρc)s

λ (−L)

• + cosh
• (ρc)s

λ (−L)

• ˆ

y Series expansion of hyperb. functions, e.g., cosh(x√s) = ∞

k=0

x2k (2k)! sk, yields ˆ T(x) =

∞

• k=0

ρc λ k x2k (2k)!sk ˆ y, ˆ u =

∞

• k=0

ρc λ k (−L)2k (2k)!

• sk +

(ρc)L β(2k + 1) s(k+1)

• ˆ

y

Joachim Rudolph (TU Dresden) 19 / 50 Parabolic syst./beams/plates Ex.: Heat Eq.

Solution (time domain) Temperature: T(x, t) =

∞

• k=0

ρc λ k x2k (2k)! y(k)(t) Control: u(t) =

∞

• k=0

ρc λ k (−L)2k (2k)!

• y(k)(t) +

(ρc)L β(2k + 1) y(k+1)(t)

• Reference traj. for flat output y is chosen such that series converge !

Stationary profiles at t = 0 and t = t∗ ⇔ y(n)(0) = y(n)(t∗) = 0, n > 0 ⇒ t → y(t) are C∞, but not analytic → Gevrey function of an order < 2 Fast convergence: Implementation with truncation after few terms

Joachim Rudolph (TU Dresden) 20 / 50

Parabolic syst./beams/plates Ex.: Heat Eq.

A Gevrey function (of order 1 + 1/p) yp(t) =        for t < 0

1 2 + 1 2 tanh

2

• 2 t

t ∗ − 1

• 4 t

t ∗

• 1 − t

t ∗

p for t ∈ [0, t∗] 1 for t > t∗ y ∈ C∞ is a Gevrey-Funktion

• f order (or class) α ≥ 1,

if ∀ K ⊂ R sup

τ∈K

|y(n)(τ)| ≤ m(n!)α γn , n ≥ 0 for some α, m, γ ∈ R dep. on K

1 1 y t p=1.1 p=10

Joachim Rudolph (TU Dresden) 21 / 50 Parabolic syst./beams/plates Ex.: Beams & Plates

Euler-Bernoulli beam

x u(t) x = 1 x = 0 w(x, t)

• Math. model (dimensionless):

∂4w ∂x4 (x, t) + ∂2w ∂t2 (x, t) = 0, 0 ≤ x ≤ 1, t ≥ 0 Initial cond.: w(x, 0) = 0, ∂w ∂t (x, 0) = 0 Boundary cond.: w(0, t) = 0, ∂w ∂x (0, t) = 0, ∂2w ∂x2 (1, t) = u(t), ∂3w ∂x3 (1, t) = 0 Task: Bend in finite time, without residual vibrations.

Joachim Rudolph (TU Dresden) 22 / 50

Parabolic syst./beams/plates Ex.: Beams & Plates

Using derivation operator s yields o.d.e. in x : s2ˆ w(x) + d4ˆ w dx4 (x) = 0, 0 ≤ x ≤ 1 with boundary cond. ˆ w (0) = 0, dˆ w dx (0) = 0, d2ˆ w dx2 (1) = ˆ u, d3ˆ w dx3 (1) = 0 Solution (i = √ −1): ˆ w(x) = C1 cosh √ isx

• + C2 sinh

√ isx

• + C3 cos

√ isx

• + C4 sin

√ isx

• Bound. cond. provide linear eq. Q C = D ˆ

u for C : C1 + C3 = 0 C2 + C4 = 0 C1 is cosh √ is + C2 is sinh √ is − C3 is cos √ is − C4 is sin √ is = ˆ u C1 sinh √ is + C2 cosh √ is + C3 sin √ is − C4 cos √ is = 0

Joachim Rudolph (TU Dresden) 23 / 50 Parabolic syst./beams/plates Ex.: Beams & Plates

Linear eq. for C : Q C = D ˆ u Ansatz for ˆ w(x) : ˆ w(x) = W(x) C C = Q−1D ˆ u = (adj Q)D det Q ˆ u Avoid division by det Q by introducing basic output ˆ y via ˆ u = (det Q)ˆ y In addition this means: C = (adj Q)Dˆ y ˆ w(x) = W(x) (adj Q)Dˆ y

Joachim Rudolph (TU Dresden) 24 / 50

Parabolic syst./beams/plates Ex.: Beams & Plates

We get ˆ w(x) = P(x) ˆ y, ˆ u = det Q ˆ y with det Q = is

• 2 + cosh

√ is cos √ is

• and

P(x) =

• cosh

√ isx

• − cos

√ isx cosh √ is + cos √ is

• −
• sinh

√ isx

• − sin

√ isx sinh √ is − sin √ is

• Using Addition Theorems and power series expansion yields

ˆ w(x) = 2

∞

• k=0

(−1)k (4k + 2)!

• (1 − x)4k+2+Re
• (x + i)4k+2

+Im

• (x − i)4k+2

s2kˆ y ˆ u = 2

• 1 +

∞

• k=0

1 (4k) !(2s)2k

• ˆ

y ⇒ Use Gevrey functions analogous to heat conduction pb.

Joachim Rudolph (TU Dresden) 25 / 50 Parabolic syst./beams/plates Ex.: Beams & Plates

Piezo-electric actuator: An experimental result

• controller

laser sensor piezo−electric bender

Task: Bend in finite time without residual vibrations Flat output: abstract quantity from operational calculus

Joachim Rudolph (TU Dresden) 26 / 50

Parabolic syst./beams/plates Ex.: Beams & Plates

Simulation of a piezo-electric bender with 3 sections

Joachim Rudolph (TU Dresden) 27 / 50 Parabolic syst./beams/plates Ex.: Beams & Plates

Simulation of the bending of a ring

Joachim Rudolph (TU Dresden) 28 / 50

Parabolic syst./beams/plates Ex.: Beams & Plates

Symmetric ring-shaped piezo-plate

• Math. model (Kirchhoff plate, rotational inertia neglected):

1 r ∂r (r∂r) 2 w(r, t) + hρD−1∂ttw(r, t) = 0

• n

[ri, ra] × R+ Boundary cond. (clamped at inner, free with control u at outer bd.y): w(ri, t) = 0, ∂rw(ri, t) = 0 kmu(t) = 1 r ∂r (r∂r) w(ra, t) − (1 − ν)1 r ∂rw(ra, t) kf u(t) = ∂r 1 r ∂r (r∂r)

• w(ra, t)

Initial cond.: w(r, 0) = 0, ∂rw(r, 0) = 0 Task: Find control voltage t → u(t) that bends the ring in finite time. Solution via operational calculus, introduction of flat output, series expansion

• f operators, recursive calculation of series coeff.s by integration, Gevrey fcts.

Joachim Rudolph (TU Dresden) 29 / 50 Parabolic syst./beams/plates Ex.: Beams & Plates

A plate with angular dependence

Joachim Rudolph (TU Dresden) 30 / 50

Formal Framework for Linear Systems with Lumped Controls

System of linear pdes For distributed variables w = (w1, . . . , wp) and lumped controls v = (v1, . . . , vr) A(x) w(x, t) = D(x) v(t)

• x ∈ Ω = [0, L] ⊂ R, t ∈ R+, A(x) ∈
• C

∂

∂x, ∂ ∂t

p×p , D(x) ∈

• C

∂

∂t

p×r with (appropriate) boundary conditions B0w(x, t)|x=0 + BLw(x, t)|x=L + C v(t) = 0

• t ≥ 0, B0, BL ∈
• C

∂

∂x, ∂ ∂t

q×p , C ∈

• C

∂

∂t

q×r and stationary initial conditions w(x, 0) = w0(x), ∂kw ∂tk (x, 0) = 0, v(0) = v0, dkv dtk (0) = 0, k > 0, x ∈ Ω, which means (w0, v0) satisfies A(x) w0(x) = D(x) v0, B0w0(x)|x=0 + BLw0(x)|x=L + C v0 = 0.

Joachim Rudolph (TU Dresden) 32 / 50 Formal Framework for Linear Systems with Lumped Controls

Operational calculus Introducing the operator s yields ordinary system ˆ A(x) ˆ w(x) = ˆ D(x)ˆ v, x ∈ Ω, ˆ A(x) ∈

• C

∂

∂x, s

p×p , ˆ D(x) ∈ (C[s])p×r with boundary conditions ˆ B0ˆ w(x)|x=0 + ˆ BLˆ w(x)|x=L + ˆ C ˆ v = 0, ˆ B0, ˆ BL ∈

• C

∂

∂x, s

q×p , ˆ C ∈ (C[s])q×r Given a fundemantal system of operational fcts. ˆ Wi = (ˆ wi,1, . . . , ˆ wi,p), i = 1, . . . , q

• f the homogenous diff eq. and a particular solution ˆ

P : Ω → Mp×r of ˆ A(x)ˆ Y(x) = ˆ D(x) the linear combinations ˆ w(x) =

q

• i=1

ˆ Ki ˆ Wi(x) + ˆ P(x)ˆ v, ˆ Ki ∈ M, x ∈ Ω are solutions of the diff.eq. (with values in Mp), too.

Joachim Rudolph (TU Dresden) 33 / 50

Formal Framework for Linear Systems with Lumped Controls

Evaluating boundary cond. gives ˆ B0 q

• k=1

ˆ Wk(x)ˆ Kk + ˆ P(x)ˆ v

• x=0

+ ˆ BL q

• k=1

ˆ Wk(x)ˆ Kk + ˆ P(x)ˆ v

• x=L

= −ˆ Cˆ v

q

• k=1

ˆ Wk(x) ˆ Kk + ˆ P(x)ˆ v = ˆ w(x) which can be rewritten in matrix form (if controls v are independent): ˆ S ˆ W

• ˆ

K =

• ˆ

T ˆ v ˆ w − ˆ Pˆ v

• This yields

ˆ K = ˆ S−1ˆ T ˆ v and ˆ w = ( ˆ Wˆ S−1ˆ T + ˆ P)ˆ v. Equivalently: (det ˆ S) ˆ w −

• ˆ

W(adj ˆ S)ˆ T + (det ˆ S)ˆ P

• ˆ

v = 0 with coeff.s from R = C[s, S, T , P, W], generated by the entries of ˆ W, ˆ S, ˆ P, ˆ T.

Joachim Rudolph (TU Dresden) 34 / 50 Formal Framework for Linear Systems with Lumped Controls

System equations: (det ˆ S) ˆ w −

• ˆ

W(adj ˆ S)ˆ T + (det ˆ S)ˆ P

• ˆ

v = 0 Introduce another family of (lumped) variables, ˆ y = (ˆ y1, . . . ,ˆ yr), called a basic output: ˆ w(x) =

• ˆ

W(x)(adj ˆ S)ˆ T + (det ˆ S)ˆ P(x)

• ˆ

y ˆ v = (det ˆ S)ˆ y Interpretation of the operators:

◮ using power series in s, which yields series in derivatives of y ◮ or power series in 1/s yielding integrals with compact support kernels

− → „distributed delays“ Remark: Systems considered may be uncontrollable, meaning ˆ W(adj ˆ S)ˆ T + (det ˆ S)ˆ P and (det ˆ S)I have a (non-unimodular) common left divisor (over R).

Joachim Rudolph (TU Dresden) 35 / 50

VGF Crystal Growth: A Quasi-linear Problem with Free Boundary

Vertical-Gradient-Freeze Method

Crystal Heater 3 vg Heater 1 Melt Heater 2 Crystallization− front

Problem: Produce high quality semi-conductor single-crystals. Method: Cool the melt in a temperature field with (low) vertical gradient. Partial problem: heat conduction pb. for two phase system with moving boundary Process in 2 steps:

◮ Cool down the melt, ◮ grow the crystal (from bottom to top).

Joachim Rudolph (TU Dresden) 37 / 50 VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Outline of a Flatness Based Approach and Simulation

Control Problem and Flatness Based Approach 2 Controls: temperatures of heaters at top and bottom of crucible Model: 2 coupled heat eqs. with free boundary, nonlinear (temperature dependent parameters, radial heat transfer) Solution: flatness based trajectory planning and control based on power series Flat output: 1st Stage: temperature and gradient at the bottom 2nd Stage: velocity of phase interface and temperature gradient in the melt at the interface

Joachim Rudolph (TU Dresden) 38 / 50

VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Outline of a Flatness Based Approach and Simulation

Simulation of Heat Conduction in the VGF Process

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4

Melt C r y s t a l

z ub

T

ut t

Joachim Rudolph (TU Dresden) 39 / 50 VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Outline of a Flatness Based Approach and Simulation

Model of the Heat Conduction Problem

• H

TM TI TC vI r ζ ζ = − zI (t) ζ = H − zI (t) r ∗ z R

Crystallization front Top heater Crystal Bottom heater Crucible Melt

Important parameters:

◮ temperatures

TM, TC

◮ mass densities

ρM, ρC

◮ conductivities

λM, λC

◮ heater temp.s

ub, ut Heat equation:

∂ ∂t

• ρ
• T(z, t)
• T(z, t)
• = ∂

∂z

• λ(T(z, t)) ∂T

∂z (z, t)

• Boundary cond.:

λ(T) ∂T

∂z = α(T − u) + εσS

• T4 − u4

Initial cond.: TM(z, 0) = TM,0, z ∈ [0, H]

Joachim Rudolph (TU Dresden) 40 / 50

VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Outline of a Flatness Based Approach and Simulation

Conditions at the Phase Interface Temperatures equal to melting (or crystallization) temperature TI: T(zI(t), t) = TM(zI(t), t) = TC(zI(t), t) = TI with zI(t) the location of the interface at time t. Energy balance yields λM(TI)∂TM ∂z (zI(t), t) − λC(TI)∂TC ∂z (zI(t), t) = hvI(t) with vI(t) = ˙ zI(t) the velocity of the interface.

Joachim Rudolph (TU Dresden) 41 / 50 VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Flatness Based Parameterization for the Crystallization Stage

The Crystallization Stage Assume ρM and λM constant and ρC(TC) = ρC,0 + ρC,1TC, λC(TC) = λC,0 + λC,1TC Using a moving frame with origin at the interface, with ζ = z − zI(t), ϑ(ζ, t) = T(z, t) we get the p.d.e. ρM,0 ∂ϑM ∂t − vI ∂ϑM ∂ζ

• = λM,0

∂2ϑM ∂ζ2 (ρC,0+2ρC,1ϑC) ∂ϑC ∂t − vI ∂ϑC ∂ζ

• = (λC,0+λC,1ϑC)∂2ϑC

∂ζ2 + λC,1 ∂ϑC ∂ζ 2

Joachim Rudolph (TU Dresden) 42 / 50

VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Flatness Based Parameterization for the Crystallization Stage

Boundary cond.: λM,0 ∂ϑM ∂ζ = αb(ϑM − ub) + εbσS

• ϑ4

M − u4 b

• ,

at ζ = −zI −(λC,0 + λC,1ϑC)∂ϑC ∂ζ = αt(ϑC − ut) + εtσS

• ϑ4

C − u4 t

• ,

at ζ = H − zI Compatibility cond.: ϑC = ϑM = TI, at ζ = 0 vIh = λM,0 ∂ϑM ∂ζ − (λC,0 + λC,1TI)∂ϑC ∂ζ , at ζ = 0

Joachim Rudolph (TU Dresden) 43 / 50 VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Flatness Based Parameterization for the Crystallization Stage

Formal Power Series Parameterization Replacing ϑM and ϑC in the model p.d.e. by the power series ϑM(ζ, t) =

∞

• n=0

bn(t)ζn n! ϑC(ζ, t) =

∞

• n=0

cn(t)ζn n! and comparing coefficients of ζn yields the recursion formulae bn+2 = ̺M,0 λM,0 ˙ bn − vIbn+1

• ,

n ≥ 0 cn+2 = 1 λC,0 + λC,1c0

• ρC,0 (˙

cn − vIcn+1) +

n

• l=0

n l

• 2ρC,1 (cl˙

cn−l − vIclcn+1−l) − λC,1 n + 1 l + 1 cl+1cn+1−l

• Joachim Rudolph (TU Dresden)

44 / 50

VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Flatness Based Parameterization for the Crystallization Stage

Choosing a “Flat Output” The 4 “initial” parameters b0, b1, c0, c1 are related by the compatibility conditions at ζ = 0: b0 = c0 ≡ TI ˙ zIh = λM,0b1 − (λC,0 + λC,1TI)c1 ⇒ There exist 2 free parameters (flat output components). One possible choice is y(t) = (zI(t), b1(t)) =

• zI(t), ∂ϑM

∂ζ (0, t)

• ,

the interface location and the gradient in the melt. The trajectories of all other variables are calculated from t → y(t) using the compatibility condition, the series with the recursions, and the boundary conditions. The latter yield the control trajectory.

Joachim Rudolph (TU Dresden) 45 / 50 VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Flatness Based Parameterization for the Crystallization Stage

Parameterization of the Crystallization Stage Transition from a regime with interface at rest to another stationary regime (constant profile in moving frame). Flat output trajectory: t → y(t) = Φσ,t2(t) v2t ∆TM,2

• with

Φσ,T(t) = 1 2

• 1 + tanh
• 2t/T − 1

(4t/T (1 − t/T))σ

• For σ > 1 this choice guarantees series convergence.

(Gevrey order is α = 1 + 1/σ and σ a slope parameter.) Alternatively, modifying the trajectory of Stage 1, one may start the crystallization at t = t1 with a constant initial velocity v2.

Joachim Rudolph (TU Dresden) 46 / 50

VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Flatness Based Parameterization for the Crystallization Stage

Numerical Implementation Numerical implementation may use series truncation T(z, t) ≈ TN(z, t) =

N

• n=0

cn(t)zn n! As an alternative, as proposed by Meurer/Wagner/Zeitz, instead of TN, the so called (N, ξ)-approximate k-sum ˆ TN,k,ξ(z, t) = N

n=0 ξn Γ(1+nk)

n

j=0 cj(t) zj j!

N

n=0 ξn Γ(1+nk)

with the parameters N, k, and ξ could be used. With an appropriate choice of k and ξ lim

N→∞

ˆ TN,k,ξ(z, t) = T(z, t) with increased rate of convergence.

Joachim Rudolph (TU Dresden) 47 / 50 VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Flatness Based Parameterization for the Crystallization Stage

Conclusion Shown: Boundary control of (lin. and nonlin.) distributed parameter systems with lumped (mostly boundary) controls. The parameterization of trajectories uses

◮ power series involving derivatives of a so-called flat output for parabolic

systems,

◮ integrals with finite support kernels corresponding to finite distributed

delays and advances of the flat output for hyperbolic systems. We have a far reaching theory for linear systems (also with coeffs. depending

• n the spatial variable) via operational calculus and module theory.

We have some results for

◮ higher dimensional parabolic problems ◮ quasi-linear parabolic (and specific hyperbolic) problems ◮ design of feedback control and observers on the basis of truncated

series.

Joachim Rudolph (TU Dresden) 48 / 50

VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Flatness Based Parameterization for the Crystallization Stage

References

◮ Wave eq.: Mounier/Rud./Fliess/Petitot (ECC’95),

Fliess/Mounier/Rouchon/Rud. (CDC’95, COCV’98)

◮ Linear Parabolic eqs.: Fliess/Mounier/Rouchon/Rud. (CESA’98, CDC’98),

Laroche/Martin/Rouchon (IJRNC’98),

◮ Nonlin. Parab. Eqs.: Lynch/Rud. (at’00,...) ◮ Raleigh- a. Euler-Bernoulli-Beam-Eqs.: Fliess/Mounier/Rouchon/Rud.

(ESAIM Proc.’97), Aoustin/Fliess/Mounier/Rouchon/Rud. (Syroco’97), Haas/Rud. (ECC’99), Rud./Woittennek (at’02)

◮ Telegraphers Eq.: Mounier/Rouchon/Rud. (CESA’96) (R = G = 0),

Fliess/Martin/Petit/Rouchon (Traitm. Sign. 98)

◮ Heat exchangers: Rud. (at’00) ◮ Heavy chains (ropes): Petit/Rouchon (NCN’00,SIAM’02), Rud. (02) (n r.) ◮ Timoshenko-Beams: Woittennek/Rud (at’02,ZAMM’02,COCV’03) ◮ nonlin. hyperbol. tubular reactor: Rouchon/Rud. (hermes ’01) ◮ . . .

Stabilization etc.: Meurer and Zeitz, Kugi et al. Other “flatness like” approaches: Martin/Laroche, Ollivier/Sedoglavic, ...

Joachim Rudolph (TU Dresden) 49 / 50 VGF Crystal Growth: A Quasi-linear Problem with Free Boundary Flatness Based Parameterization for the Crystallization Stage

Instead of further references :-) Advertisement

◮ J. Rudolph,

Beiträge zur flachheitsbasierten Folgeregelung linearer und nichtlinearer Systeme endlicher und unendlicher Dimension, Habilitationsschrift, Shaker Verlag, 2003.

◮ J. Rudolph,

Flatness Based Control of Distributed Parameter Systems, Notes for a course at the Max-Planck-Institute for Dynamics of Complex Technical Systems at Magdeburg, Germany, on February 24-28, 2003, Shaker Verlag, 2003.

◮ J. Rudolph, J. Winkler, and F. Woittennek,

Flatness Based Control of Distributed Parameter Systems: Examples and Computer Exercises from Various Technological Domains, Notes for exercises in a course at the Max-Planck-Institute for Dynamics

• f Complex Technical Systems at Magdeburg, Germany, including a

CDROM, Shaker Verlag, 2003.

Joachim Rudolph (TU Dresden) 50 / 50