Online Identification of parameters in time dependent differential - - PowerPoint PPT Presentation

Online Identification of parameters in time dependent differential equations (using partial

• bservation)

Johann Baumeister

Goethe University, Frankfurt, Germany

IMPA – Rio de Janeiro / October 30th to November 3rd 2017

Supported by the Alexander von Humboldt Foundation

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Outline

Model reference adaptive systems–Introduction Model reference adaptive systems–Complete observation Addendum–Noisy data Model reference adaptive systems–Observation by a projection Concluding remarks

October 28, 2017 2 / 28

Introduction

Process equation (∗) Dtw + A0(w) + A(q, w) = f (t) in V∗, t ∈ (0, T); w(0) = ζ Dtw +cw

• A0(w)

−∇(q∇w)

• A(q,w)

= f (t), t ∈ (0, ∞); w(0) = ζ (V∗ = H−1(Ω))

• r

Dtw −∆w

A0(w)

+qw

• A(q,w)

= f (t), t ∈ (0, ∞); w(0) = ζ (V∗ = H−1(Ω))

• r . . . . . .

(Model data)                        V, H, P separable Hilbert spaces V ֒ → H ֒ → V∗ Gelfand triple , A0(·) : V − → V∗ linear w(t) ∈ V state at time, t ∈ [0, T) q ∈ Qad ⊂ P parameter (to be identified) A(·, ·) : Q × V − → V∗ bilinear (has to be specified) ζ ∈ H initial state f : (0, ∞) − → V∗

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Introduction: Weak solution and assumptions

Weak solution w : [0, ∞) − → H with w(t) ∈ V, t > 0, Dtw(t), v + A0(w(t)), v + A(q, w(t)), v = f (t), v for all t > 0, v ∈ V w(0) = ζ Assumption (A0)

(1) (Continuity of A0) |A0(u), v| ≤ c0uVvV, u, v ∈ V, with a constant c0 ≥ 0 . (2) (Continuity of A(·, ·)) A(q, u), v ≤ c0(q)uVvV for all q ∈ Qad, u, v ∈ V, where c0(q) ≥ 0 is a constant which may depend on q . (3) (Coercivity of A0(·) + A(q, ·)) A0(u), u+A(q, u), u+β(q)u2

H ≥ c1(q)u2 V for all q ∈ Qad, u ∈ V

where β(q), c1(q) are constants which may depend on q with β(q) ≥ 0, c1(q) > 0 . (4) f ∈ L2(0, T; V∗) for all T > 0 , ζ ∈ H .

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Introduction: Parameter to solution map

(∗) Dtw + A0(w) + A(q, w) = f (t) in V∗, t ∈ (0, T); w(0) = ζ W ((0, T); V) := {w : (0, T] − → V : Dtw(t) ∈ V∗ , t ∈ (0, T]} . W ((0, T); V) is a Hilbert space and each function in w ∈ W ((0, T); V) may be considered as w ∈ C([0, T]; H) . Fact Let the assumption (A0) hold. Then there exists for each q ∈ Qad a uniquely determined weak solution w of (∗) which satisfies: w ∈ W ((0, T); V) , w ∈ C([0, T]; H) for all T > 0 . (1) In other words: The parameter to solution map (PtS) from Qad into V ⊂ H is well defined.

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Introduction: Plant

Model equation (∗) Dtz + A0(z) + A(p, z) = f (t) in V∗, t ∈ (0, T); z(0) = ζ (p, z) is a plant iff z is a weak solution of (∗) |A(q, z(t)), v| ≤ c2qPvV , t ≥ 0, q ∈ P, v ∈ V, with a constant c2 ≥ 0 Identication problem with complete observation Given a plant (p, z), find the parameter p from an observation of the state z on [0, ∞) .

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(MRAS): Model reference adaptive systems

Idea Given a plant (p, z) consider a reference system with state t − → u(t) driven by a time dependent parameter t − → q(t) which is adapted by matching the state z of the modelled system and the state u of the reference system. Historically Used by engineers to control online a finite dimensional system whose parameters are not known in advance. Goal Find a reference system and an adaption rule such that lim

t→∞(u(t) − z(t)) = θ , lim t→∞(q(t) − p) = θ .

Analysis Stability arguments for linear nonautonomous systems. Realization Here, partial differential equations. Applicability to each type of a differential equations which contains parameters, even to their stationary equations (artificial dynamics). Difficulties In infinite dimensional systems the observation of the state z is crucial.

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(MRAS): References

In the next part we follow mainly

• W. Scondo

Ein Modellabgleichungs Verfahren zur adaptiven . . . Dissertation, Goethe Universit¨ at Frankfurt/Main, 1987

• J. Baumeister, W. Scondo, M.A. Demetriou and I.G. Rosen

On-line parameter estimation for infinitedimensional dynamical systems SIAM J. Control Optim. 35 (1997), 678-713

In the third part we are inspired by

• R. Boiger and B. Kaltenbacher

A online parameter identification method for time dependent partial differential equations Inverse Problems, 32 (2016), 28 pp.

See also

• P. K¨

ugler Online parameter identification without Ricatti-type equations in a class of time-dependent partial differential equations: an extended state approach with potential to partial observations Inverse Problems, 26 (2010)

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(MRAS): complete observation

Model equation and plant (∗) Dtz + A0(z) + A(p, z) = f (t) in V∗, t ∈ (0, T); z(0) = ζ (MRAS)        Dtu + C(u − z(t)) + A0(z(t)) + A(q, z(t)) = f (t) in V∗, t > 0, u(0) = u0 Dtq − b(z(t), u − z(t)) = θ in P∗, t > 0, q(0) = q0 What is b in the adaptation rule? It satisfies A(q, z(t)), v = q|b(z(t), v) for all parameters q ∈ P and all t ≥ 0 . Such a mapping b exists since A(·, z(t), v ∈ P∗ ≡ P . due to the fact that (p, z) is a plant.

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(MRAS): well-posedness of the system

e := u − z , r :== q − p . These error quantities are solutions of Dte + Ce + A(r, z(t)) = θ , t > 0, e(0) = e0 = u0 − ζ , Dtr − b(z(t), e) = θ , t > 0, r(0) = r0 = q0 − p . (A priori estimate) e(t)2

H + r(t)2 P + 2c5

t

t0

e(s)2

V ds ≤ e(t0)2 H + r(t0)2 P , t > t0 .

(Consequences) t − → L(e, r)(t) := e(t)2

H + r(t)2 P is decreasing

(L is a Ljapunov-type function). supt∈[0,∞) e(t)2

H + r(t)2 P + 2c5

∞

t0 e(s)2 V ds < c :=

e(0)2

H + r(0)2 P .

The error system has a uniquely determined solution (e, r) in W ((0, T), V × P) for all T > 0 . (MRAS) has a uniquely determined weak solution (u, q) in W ((0, T), V × P) for all T > 0 .

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(MRAS): identifiabilty

Fact (Output-identifiability) lim

t→∞ u(t) − z(t)H = 0

Fact (Parameter-identifiability) lim

t→∞ q(t) − pP = 0

if the plant (p, z) is asymptotically persistently excited. Definition Let (p, z) be a plant. The state z is asymptotically persistently excited if there exist numbers l > 0, µ > 0 and a sequence (tn)n∈I

N in

(0, ∞) with limn tn = ∞ such that the following condition holds: ∀ h ∈ P ∀ n ∈ I N ∃tn,1, tn,2 ∈ [tn, tn + l] ∃v ∈ V\{θ}

• tn,2

tn,1

A(h, z(t)), v ds

• ≥ µhPvV
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(MRAS): Richness

Persistently exciteness is the key property to ensure that adaptation can be achieved in a sequence of intervalls [tn, tn + l] with limn tn = ∞ . Notice Dtq|q = q|b(z(t), e) = A(q, z(t)), e Conditions like persistently exciteness are also called richness conditions. Here is stronger richness condition which plays a fundamental role in the next part. Definition Let (p, z) be a plant. The state z is uniformly persistently excited if there exist numbers l > 0, µ > 0 such that ∀ h ∈ P ∀ t0 ∈ (0, ∞) ∃t1, t2 ∈ [t0, t0 + l] ∃v ∈ V\{θ}

• t2

t1

A(h, z(t)), v ds

• ≥ µhPvV
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(MRAS): Richness/References

Observation The term t2

t1

A(h, z(t)), v ds is the variational right hand side of the Fr´ echet-derivative of the parameter to solution mapping in the parameter p . We should note that the excitation conditions are, in practice, difficult to

• verify. In

M.A. Demetriou and I.G. Rosen On the persistence of excitation in the adaptive estimation of distributed parameter systems IEEE Trans. on Autom. Control 39, 1994

a careful study and analysis of the excitation conditions for a parabolic equation with a constant parameter can be found. In the context of adaptive control of finite dimensional systems richness conditions are analyzed for instance in

K.S. Narendra and A.M. Annaswammy Persistent excitation in adaptive systems

• Int. J. Control 45, 1987

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Addendum: Noisy data

Up to now, the analysis of the (MRAS)-method for noisy data is not satisfying. Here are some considerations for noisy data which are smooth. The noise-model is There exists a normed subspace U of V such that the following conditions hold: (1) uV ≤ uU, u ∈ U , |A(q, u), v| ≤ c3qPuUvV , q ∈ P, u ∈ U, v ∈ V, for a constant c3 . (2) z ∈ C((0, ∞); U) ∩ L∞((0, ∞); U) . (3) η = ηε with ηε ∈ L∞(0, ∞; U) , ηε(t)U ≤ ε for all t ≥ 0 . Conditions (1),(2) are requirements of smoothnes for the state, (3) claims that the additive error ηε is smooth. One may argue that this noise-model can be used in practice if one considers the error ηε as the consequence of an error σδ in H where ηε is the result of a smoothing of σδ .

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Addendum: (MRAS)ε

Model equation (∗) Dtz + A0(z) + A(p, z) = f (t) in V∗, t ∈ (0, T); z(0) = ζ (MRAS)        Dtu + C(u − z(t)) + A0(z(t)) + A(q, z(t)) = f (t) in V∗, t > 0, u(0) = u0 Dtq − b(z(t), u − z(t)) = θ in P∗, t > 0, q(0) = q0 (MRAS)ε (t > 0)        Dtuε + C(uε − zε(t)) + A0(zε(t)) + A(qε, zε(t)) = f (t) in V∗; uε(0) = u0 Dtqε − b(zε(t), uε − zε(t)) = θ in P∗; qε(0) = q0

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Addendum: error system

Error quantities eε := uε − z , r ε := qε − p (Error system) Dteε + C(eε) + A(r ε, zε(t)) = f ε(t) , t > 0 eε(0) = eε

0 := u0 − ζ

Dtr ε − b(zε(t), eε) = g ε(t) , t > 0 r ε(0) = r e

0 ps := q0 − p

eε(t)2

H + r ε(t)2 P + 2c5

t eε2

V ds

≤ eε(0)2

H + r ε(0)2 P +

t f ε(s), eε(s) ds + t g ε(s)|r ε(s) ds , t f ε(s), eε(s) ds + t g ε(s)|r ε(s) ds = O(ε) t eε(s)V ds + t r ε(s)P ds

• .

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Addendum: estimate for the state

eε(t)2

H + r ε(t)2 P + 2c5

t e(s)2

V ds

≤ y0 + 2c1(ε) t e(s)V ds + 2c2(ε) t r(s)V ds ≤ y0 + c1(ε)t + c2(ε)t + c1(ε) t e(s)2

V ds + c2(ε)

t r(s)2

V ds

with c1(ε) = (c6 + c0 + pP)ε, c2(ε) = (c2 + c3ε)ε . If 2c5 − c1(ε) ≥ 0 r ε(t)2

P

≤ y0 + (c1(ε) + c2(ε))t + c2(ε) t r(s)2

V ds

and with a Gronwall lemma r ε(t)2

P

≤ y1 + exp(c2(ε)t) with y1 independent of ε, t .

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Addendum: output-identifiability

Fact If (2c5 − c1(ε)) ≥ 0 (notice c1(ε) = O(ε)) then eε(t)2

H + r ε(t)2 P + (2c5 − c1(ε))

t e(s)2

V ds

≤ κ for all t ≤ t(ε) where t(ε) := O(1/ε) . Fact If (2c5 − c1(ε)) ≥ 0 then eε(t)H ≤ κ exp(−(2c5 − c1(ε))t) , 0 ≤ t ≤ t(ε) . Notice: · H ≤ · V . Fact (Output-identifiability) If (2c5 − c1(ε)) > 0 then lim

ε→0 uε(t(ε)) − z(t(ε))H = 0 .

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Addendum: stopping rule

Observations Two error quantities describe the method: eε, rε . eε is available, rε is not. Stopping with eε at ts should imply an estimate for rε(ts) . A consideration of eε in H is not enough, one has to consider also eε in V . From the numerical point of view this is not so

• desirable. But one cannot avoid this even in the case of exact

data. What should be the goal? If the Ljapunov-function along the trajectory t − → (eε(t), rε(t)) changes from be decreasing to be increasing we should stop. How to find a stopping time? Again, the richness condition is helpful (and necessary).

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Addendum: parameter-identifiability

Consider with l > 0 (definition of u.p.e.) [jl, jl + l], j ∈ I N0, and choose the first m with jl

jl−1

C(eε(s), eε(s) ds − jl

jl−1

(f ε(s), eε(s) + g ε|r ε) ds > 0, j ≤ m, ml+l

ml

C(eε(s), eε(s) ds − ml+l

ml

(f ε(s), eε(s) + g ε|r ε) ds ≤ Then 2c5 ml+l

ml

eε(s)2

V ds

≤ 2 ml+l

ml

C(eε(s), eε(s) ds ds ≤ c5 ml+l

ml

eε2

V ds + O(ε)

and ml+l

ml

eε(s)2

V ds = O(ε) .

With the knowledge concerning the behavior t − → eε(t), the stopping rule can be realized.

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(MRAS) with observation by a projection (MRASpro)

Model equation (∗) Dtz + A0(z) + A(p, z) = f (t) in V∗, t ∈ (0, T); z(0) = ζ Observation Observation operator O : V − → Y, linear, continuous Observation space Y (Hilbert space) Observation y(t) := Oz(t), t > 0 Observable part ˆ z ∈ ker(O)⊥ Unobservable part ˘ z ∈ ker(O) ˆ z = O−1(y) ???

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(MRASpro): observable part

Fact Suppose that O is linear, continuous and surjective. Then: The pseudo inverse O† exists O†O : V − → V is linear and continuous ran(O†O) = ker(O)⊥ P := O†O is an orthogonal projection onto ker(O)⊥ Q := I − O†O is an orthogonal projection onto ker(O) But: In distributed systems of parabolic type observation operators are not surjective in general (Smoothing property of the forward

• perator).

Remedy Approximate O†O by regularization. Not considered here.

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(MRASpro): identification from a projection of the state

Problem Find the parameter in the model equation from a projected part P, Q : V − → V be a linear, continuous, orthogonal projection with Q := I − P . (1) Subspaces ˆ V, ˘ V such that V = ˆ V ⊕ ˘ V , ran(P) = ˆ V, ran(Q) = ˘ V (2) Mappings P∗, Q∗ : V∗ − → V∗ with P∗(λ), v = λ, Pv , Q∗(λ), v = λ, Qv , λ ∈ V∗, v ∈ V (3) P∗ ◦ P∗ = P∗, Q∗ ◦ Q∗ = Q∗ , P∗ ◦ Q∗ = Θ w ∈ V can be decomposed as follows: w = ˆ w + ˘ w ∈ ˆ V ⊕ ˘ V , ˆ w = Pw, ˘ w = Qw Problem Find the parameter p of the plant (p, z) from the observable part ˆ z : [0, ∞) ∋ t − → Pz(t) ∈ V by a model reference adaptive system

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Realization

(MRASpro)                    Dtz + A0(z) + A(p, z) = f (t) in V∗, z(0) = ζ Dtu + C(Pu − Pz(t)) + P∗A0(Qu + Pz(t)) +P∗A(q, Qu + Pz(t)) + Q∗M(Qu) = f (t) in V∗, u(0) = ζ Dtq − b(Qu + Pz(t), Pu − Pz(t)) = θ in P∗, q(0) = q0 Compare with (MRAS)        Dtu + C(u − z(t)) + A0(z(t)) + A(q, z(t)) = f (t) in V∗, t > 0, u(0) = ζ Dtq − b(z(t), u − z(t)) = θ in P∗, t > 0, q(0) = q0

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Outline of the analysis

(MRASpro)                    Dtz + A0(z) + A(p, z) = f (t) in V∗, z(0) = ζ Dtu + C(Pu − Pz(t)) + P∗A0(Qu + Pz(t)) +P∗A(q, Qu + Pz(t)) + Q∗M(Qu) = f (t) in V∗, u(0) = ζ Dtq − b(Qu + Pz(t), Pu − Pz(t)) = θ in P∗, q(0) = q0 (1) Choose C and M in an appropriate way in order to find a decoupling for computing ˆ u, ˘ u . Crucial point: Choice of C . (2) The property we want to exploit is the fact Cw ′, w ′′ = 0 for w ′ ∈ ˆ V, w ′′ ∈ ˘ V . (3) Once ˘ u is computed find a solution of the resulting computational scheme for ˆ u and q . (4) Analyze the assymptotic properties of ˆ u, q .

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Outline of the analysis

We say that assumption (B1) is satisfied if the following conditions hold: (1) Let C be chosen as follows: C : V − → V∗ , C(v), v ′ := γ0v|v ′V , v, v ′ ∈ V with γ0 > 0 . (2) M : V − → V∗ is coercive on ˘ V, i.e. M(w ′), w ′ ≥ γ1w ′2

V , w ′ ∈ ˘

V with a constant γ1 > 0 . (3) M : V − → V∗ is continuous, i.e. M(u), v ≤ γ2uVvV , u, v ∈ V, with γ2 ≥ 0 .

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Decoupling

(MRASpro)                    Dtz + A0(z) + A(p, z) = f (t) in V∗, z(0) = ζ Dtu + C(Pu − Pz(t)) + P∗A0(Qu + Pz(t)) +P∗A(q, Qu + Pz(t)) + Q∗M(Qu) = f (t) in V∗, u(0) = ζ Dtq − b(Qu + Pz(t), Pu − Pz(t)) = θ in P∗, q(0) = q0 Computation of the unobservable part ˘ u Dt˘ u + M(˘ u) = Q∗f (t) in V∗, u(0) = Q∗ζ Computation of the observable part ˆ u and and the parameter q            Dtˆ u + C(ˆ u − Pz(t)) + A0(˘ u + Pz(t)) +A(q, ˘ u + Pz(t)) = P∗f (t) in V∗, u(0) = P∗ζ Dtq − b(˘ u + Pz(t), ˆ u − Pz(t)) = θ in P, q(0) = q0

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Assumptions, results, supplements

1) Assumptions similar to the noisy (MRAS)ε since the missing part ˘ u may be considered as an ”error“ in the state z . 2) Argumentation for output-identifiability as in the case of exact data for (MRAS) (”Replace V by ˆ V “). 3) Argumentation for parameter-identifiability with the richness condition in the space ˆ V . 4) Open: The case of noisy data for (MRASpro). 5) Open: How to get results for approximating P by an unbounded mapping from the observation space Y into V . 6) All the methods descripted above may be considered as on-line methods in a finite interval by using the Kaczmarz idea. 7) The adaptation rules can be generalized for smooth parameters. This is important if there are isolated points where no adaptation is guaranteed. 8) Missing: New numerical tests for the methods descripted above.

The End

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