Ellipsoidal and Interval Techniques for State Estimation in Linear - - PowerPoint PPT Presentation

Ellipsoidal and Interval Techniques for State Estimation in Linear Dynamical Systems under Model Uncertainty

Sergey Nazin Institute of Control Science, Moscow, Russia CESAME – UCL, Louvain-la-Neuve, Belgium

Une s´ eminaire ` a CESAME, 21 mars 2006 ` a 14.00

Research interests Guaranteed state estimation for uncertain dynamic systems

Ellipsoidal estimation Interval estimation

Interval analysis, Robust linear algebra

Interval systems of linear algebraic equations

Invariant sets

in state estimation and control problems

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Today: Ellipsoidal Techniques for State Estimation in Linear Dynamical Systems under Model Uncertainty Outline:

• 1. Introduction to set-membership estimation, application to state

estimation problem, classical ellipsoidal method

• 2. State estimation for dynamic systems under model uncertainty

via method of ellipsoids

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• I. Set-membership State Estimation:

State vector x(t) = (x1(t), . . . , xn(t))T Discrete-time model x(k + 1) = f (x(k), v(k), k), x(0) = x0 y(k) = h (x(k), w(k), k) Estimation:

• x – approximation of state vector

Stochastic models: variables are random with known distribution (Kalman filter) Deterministic models: variables are unknown-but-bounded (set-membership estimation)

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Names and References:

Bertsekas, D. P., & Rhodes I. B. (1971). Recursive state estimation for a set-membership description of uncertainty. IEEE TAC, 16, 117–128. Schweppe, F. C. (1973). Uncertain Dynamic Systems. Englewood Cliffs, NJ: Prentice Hall. Fogel, E., & Huang, Y. F. (1982). On the value of information in system identification – bounded noise case. Automatica, 18, 229–238. Milanese, M., Norton, J. P., Piet-Lahanier, H., & Walter, E., Eds. (1996). Bounding Approaches to System Identification. New York: Plenum. Boyd, S., El Ghaoui, L., Ferron, E., & Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM.

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State estimation: problem statement

Linear discrete-time model: xk+1 = Ak xk + Bk vk, yk = Ck xk + wk

xk ∈ Rn – state vector, vk ∈ Rr – perturbation, yk ∈ Rm – measurements, wk ∈ Rq – noise.

Uncertainty: x0 ∈ X0, vk ∈ Vk, wk ∈ Wk

The problem is to find the guaranteed estimate for xk under given measurements y1, ..., yk and initial state x0 .

Techniques: ellipsoidal, interval, polyhedral, etc...

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Classical ellipsoidal method

(without model uncertainty) xk+1 = Ak xk + Bk vk yk = Ck xk + wk Ak, Bk, Ck are known; x0 ∈ E0 – bounded, vk ≤ 1, wk ≤ 1.

Ellipsoidal estimation: Prediction phase

— sum of ellipsoids

Correction phase

— intersection of ellipsoids Ellipsoid —

E(c, P) =

• x : (x − c)T P (x − c) ≤ 1, P ≥ 0
• Size of ellipsoid:

determinant criterion fdet = − ln det P trace criterion ftr = tr P −1

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Prediction phase

xk+1 = Ak xk + Bk vk; x0 ∈ E(c0, P0), P0 > 0, vk ≤ 1.

Lemma 1 (Durieu et al., 2001): Let SN = N

i=1 E(ci, Pi) and

AN =

• α :

αi > 0,

• αi = 1
• .

Then SN ⊆ E(cα, Pα), ∀ α ∈ AN with cα =

• ci,

P −1

α

=

• α−1

i

P −1

i

. fdet(Pα) = − ln det Pα and f tr (Pα) = tr P −1

α

— convex on AN.

• −2.5

−2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5

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Correction phase

xk ∈ E(ck, Pk), Pk > 0; yk = Ck xk + wk; wk ≤ 1.

Lemma 2 (Durieu et al., 2001): Let IN = N

i=1 E(ci, Pi),

AN = {α : αi ≥ 0, αi = 1}.

Then IN ⊆ E(cα, Pα/(1 − δα)) with Pα =

• αiPi,

cα = P −1

α

• αiPici,

δα =

• αicT

i Pici − cT αPαcα

∀ α ∈ AN provided that Pα > 0. fdet(Pα) = − ln det Pα and f tr (Pα) = tr P −1

α

— convex on AN.

• −3

−2 −1 1 2 3 −4 −3 −2 −1 1 2 3

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Set-membership filtering: Prediction step Correction step

Boyd et al. (1994), El Ghaoui & Calafiore (1999)

− → LMI approach My approach − → reduce the problem to one-dimentional

• ptimization

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• II. Ellipsoidal state estimation under model uncertainty

xk+1 = (Ak + ∆Ak) xk + vk yk = (Ck + ∆Ck) xk + wk

∆Ak2 ε2

Ak

+ vk2 δ2

vk

≤ 1, ∆Ck2 ε2

Ck

+ wk2 δ2

wk

≤ 1

x0 ∈ E(c0, P0), P0 > 0 Lemma 3

• Hx + w :

H2 ε2

+ w2

δ2

≤ 1

• =
• z : z2 ≤ ε2x2 + δ2

.

Ellipsoidal estimation: Prediction phase Correction phase

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Reachability Sets:

xk+1 = (Ak + ∆Ak) xk + vk, ∆Ak2 ε2

Ak

+ vk2 δ2

vk

≤ 1 x0 ∈ E(c0, P0), P0 > 0

−50 −40 −30 −20 −10 10 20 30 40 50 −25 −20 −15 −10 −5 5 10 15 20 25

D2

D3

D4

D5

−20 −10 10 20 30 40 50 60 70 80 −20 −10 10 20 30 40

D2

D3

D4

D5

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Prediction phase (model uncertainty)

• x = (A + ∆A) x + v,

x ∈ E(c, P), ∆A2 ε2 + v2 δ2 ≤ 1. Reachability set: F =

• x = A x + z :

x ∈ E(c, P), z2 ≤ ε2x2 + δ2 F ⊂ E(d, Q)

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Theorem 1 (prediction)

Each ellipsoid in the family E(d(τ), Q(τ)) with d(τ) = (1 − δ2τ) A Q−1

τ P c,

Q(τ) =

• A Q−1

τ AT + τ −1I

−1 1 − ξ(τ) , where Qτ = (1 − δ2τ) P − τε2I, ξ(τ) = (1 − δ2τ) cT P c − (1 − δ2τ)2 cT P Q−1

τ P c,

contains F for all τ such that 0 < τ < τ ∗ =

λmin δ2λmin+ε2 ,

where λmin = min eig P.

• τmin = arg

min

0<τ<τ ∗ f(Qτ) 13

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• x = (A + ∆A) x + v,

x ∈ E(c, P), ∆A2 ε2 + v2 δ2 ≤ 1. Example:

c =

• 1.5

2

• , P =
• 1/9

1

• ,

A =

• 1

1

• , ε = 1, δ = 0.5

−6 −4 −2 2 4 6 8 10 −4 −2 2 4 6 8

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• x = (A + ∆A) x + v,

x ∈ E(c, P), ∆A2 ε2 + v2 δ2 ≤ 1. Example: Optimal outer ellipsoid in the case when c = 0.

c = 0, P =

• 1/9

1

• ,

A =

• 1

5 1

• ,

ε = δ = 0.5

−8 −6 −4 −2 2 4 6 8 −4 −3 −2 −1 1 2 3 4 F E(0,P) E(0,Q)

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Correction phase (model uncertainty) x ∈ E(c, P) y = (C + ∆C) x + w, ∆C2 ε2 + w2 δ2 ≤ 1. y − Cx = ∆C x + w, y − Cx2 ≤ ε2x2 + δ2 (x − d)TM(x − d) ≤ 1, with

M =

R yT CR−1CT y−yT y+δ2 ,

d = R−1CT y, R = CT C − ε2I.

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Theorem 2 (correction)

If x ∈ E(c, P), and y = (C + ∆C)x + w,

∆C2 ε2

+ w2

δ2

≤ 1, then x ∈ E(g(τ), Q(τ)) where Q(τ) = (1 − ν)−1Qτ, Qτ = (1 − τ)P + τM, g(τ) = Q−1

τ [(1 − τ)Pc + τMd],

ν = (1 − τ) cT Pc + τdT Md − g(τ)T Qτg(τ),        ∀τ : 0 ≤ τ < τ ∗ = min

• 1,

1 1 − λmin

• ,

λmin = min eig(M, P). τmin = arg min

0≤τ<τ ∗ f(Qτ) 17

★ ✧ ✥ ✦ x ∈ E(c, P) y = (C + ∆C) x + w, ∆C2 ε2 + w2 δ2 ≤ 1. Example:

y = 3, C = (−1.5, 3) , ε = 1.5 and δ = 0.5. E(c, P ) : c = 0, P =

• 1

1/9

• .

−4 −3 −2 −1 1 2 3 −3 −2 −1 1 2 3 4

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Some extensions to

• 1. other matrix norms that specifies model uncertainty
• 2. separate uncertainties

Conclusions

• 1. One-step sub-optimal ellipsoidal estimates for prediction and correction.
• 2. Difficulties:

convexity of cost functions, stable calculations,

• utliers

Paper: Polyak, Nazin, Durieu, & Walter (2004). Ellipsoidal parameter or state estimation under model uncertainty. Automatica, 40(7).

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