H Filtering of Uncertain LPV Systems with Time-Delay C.Briat, - - PowerPoint PPT Presentation

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H Filtering of Uncertain LPV Systems with Time-Delay C.Briat, - - PowerPoint PPT Presentation

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H Filtering of Uncertain LPV Systems with Time-Delay C.Briat, O.Sename and JF.Lafay August 2009 ECC09 - Budapest, Hungary C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 1/21 Outline Introduction Stability of Uncertain LPV

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H∞ Filtering of Uncertain LPV Systems with Time-Delay

C.Briat, O.Sename and JF.Lafay August 2009 ECC’09 - Budapest, Hungary

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 1/21

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Outline

Introduction Stability of Uncertain LPV Systems with Delays The filtering Problem Conclusion and Future Works

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 2/21

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Introduction

Considered Systems Filters Structures

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 3/21

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Considered Systems (1)

Uncertain LPV Systems with Delay   ˙ x(t) z(t) y(t)   = Σ(ρ(t), δ)   x(t) x(t − h(t)) w(t)   x(θ) = φ(θ), θ ∈ [−hM, 0] ρ ∈ Uρ ˙ ρ ∈ hull[Uν] δ ∈ Uδ h(t) ∈ [0, hM] ˙ h(t) < µ < 1

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 4/21

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Considered Systems (2)

System Matrix : Σ =   A(ρ) + ∆A(ρ, δ) Ah(ρ) + ∆Ah(ρ, δ) E(ρ) + ∆E(ρ, δ) C(ρ) Ch(ρ) F(ρ) Cy(ρ) + ∆Cy(ρ, δ) Cyh(ρ) + ∆Cyh(ρ, δ) Fy(ρ) + ∆Fy(ρ, δ)   where the uncertain part obeys ∆A ∆Ah ∆E ∆Cy ∆Cyh ∆Fy

  • (ρ, δ) =

H0 H1

  • (ρ)∆(δ)

F0 F1 F2 F3 F4 F5

  • (ρ)

with ||∆||2 ≤ 1

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 5/21

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Filters Structures

Filters with memory ˙ xF(t) zF(t)

  • =

AF(ρ) AhF(ρ) BF(ρ) CF(ρ) ChF(ρ) DF(ρ)   xF(t) xF(t − h(t)) y(t)   Memoryless filters ˙ xF(t) zF(t)

  • =
  • AF(ρ)

BF(ρ) CF(ρ) DF(ρ) xF(t) y(t)

  • These matrices are aimed to be chosen such that

||z − zF||L2 ≤ γ||w||L2 with a minimal L2-gain γ > 0.

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 6/21

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Stability of Uncertain LPV Systems with Delays

Lyapunov-Krasovskii Functional Asymptotic Stability Theorem Relaxed Version

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 7/21

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Lyapunov-Krasovskii Functional (1)

Delay-Dependent LKF : V(xt, ˙ xt, ρ) = V1(xt, ρ) + V2(xt) + V3( ˙ xt) V1(xt, ρ) = x(t)TP(ρ)x(t) V2(xt) = t

t−h(t)

x(θ)TQx(θ)dθ V3( ˙ xt) = hM

−hM

t

t+θ

˙ x(η)TR ˙ x(η)dηdθ whose derivative along the trajectories solution of the system satisfies : ˙ V1 = ˙ x(t)TP(ρ)x(t) + x(t)TP(ρ) ˙ x(t) + x(t)T

  • i

˙ ρi ∂ ∂ρi P(ρ)

  • x(t)

˙ V2 ≤ x(t)TQx(t) − (1 − µ)x(t − h(t))TQx(t − h(t)) ˙ V3 ≤ h2

M ˙

x(t)TR ˙ x(t) − hM t

t−h(t)

˙ x(θ)TR ˙ x(θ)dθ

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 8/21

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Lyapunov-Krasovskii Functional (2)

Using Jensen’s Inequality : −hM t

t−h(t)

˙ x(θ)TR ˙ x(θ)dθ ≤ − t

t−h(t)

˙ x(θ)dθ T R t

t−h(t)

˙ x(θ)dθ

  • we get

˙ V ≤ χ(t)T Ψ + h2

MATRA

PAh + R + h2

MATRAh

⋆ −(1 − µ)Q − R + h2

MAT h RAh

  • χ(t)

with Ψ = ATP + PA + Q − R +

i ˙

ρi ∂ ∂ρi P(ρ) and χ(t) = col(x(t), x(t − h(t)).

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 9/21

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Asymptotic Stability

Theorem The unertain LPV system is asymptotically stable for all h(t) ∈ [0, hM] such that ˙ h(t) < µ if there exist P : Uρ → Sn

++, Q, R ∈ Sn ++ such that

the LMI   Ψ(ρ, ν) P(ρ)Ah(ρ) + R hMA(ρ)TR ⋆ −(1 − µ)Q − R hMAh(ρ)TR ⋆ ⋆ −R   ≺ 0 holds for all (ρ, ν) ∈ Uρ × Uν with Ψ(ρ, ν) = A(ρ)TP(ρ) + P(ρ)A(ρ) + Q − R +

i νi

∂ ∂ρi P(ρ).

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 10/21

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Bounded Real Lemma

Theorem The unertain LPV system is asymptotically stable for all h(t) ∈ [0, hM] such that ˙ h(t) < µ if there exist P : Uρ → Sn

++, Q, R ∈ Sn ++ and γ > 0

such that the LMI       Ψ(ρ, ν) P(ρ)Ah(ρ) + R P(ρ)E(ρ) C(ρ)T hMA(ρ)TR ⋆ −(1 − µ)Q − R Ch(ρ)T hMAh(ρ)TR ⋆ ⋆ −γI F(ρ)T hME(ρ)TR ⋆ ⋆ ⋆ −γI ⋆ ⋆ ⋆ ⋆ −R       ≺ 0 holds for all (ρ, ν) ∈ Uρ × Uν with Ψ(ρ, ν) = A(ρ)TP(ρ) + P(ρ)A(ρ) + Q − R +

i νi

∂ ∂ρi P(ρ). Moreover, we have ||z||L2 ≤ γ||w||L2.

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 11/21

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Relaxed Version

Theorem The unertain LPV system is asymptotically stable for all h(t) ∈ [0, hM] such that ˙ h(t) < µ if there exist P : Uρ → Sn

++, X : Uρ → Rn×n,

Q, R ∈ Sn

++ and γ > 0 such that the LMI

          −(X + X T) P + X TA X TAh X TE X T hMR ⋆ Φ1 R CT ⋆ ⋆ Φ2 CT

h

⋆ ⋆ ⋆ −γI F T ⋆ ⋆ ⋆ ⋆ −γI ⋆ ⋆ ⋆ ⋆ ⋆ −P −hMR ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −R           ≺ 0 holds for all (ρ, ν) ∈ Uρ × Uν with Φ1 = P + Q − R +

i νi

∂ ∂ρi P(ρ) and Φ2 = −(1 − µ)Q − R. Moreover, we have ||z||L2 ≤ γ||w||L2.

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 12/21

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Filtering Problem

Augmented System Relaxation Example

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 13/21

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Augmented System

Interconnection between the system and filter ˙ xa(t) = Axa(t) + Ahxa(t − h(t)) + Ew(t) ze(t) = Cxa(t) + Chxa(t − h(t)) + Fw(t) xa(t) = col(x(t), xF(t)) ze(t) = z(t) − zF(t) with A =

  • A

A − BFCy AF

  • Ah =
  • Ah

Ah − BFCyh AFh

  • E =
  • E

E − BFFy

  • C =

C − DFCy − CF CF

  • Ch =

Ch − DFCyh − CFh CFh

  • F = F − DFFy

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 14/21

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Relaxation of Bilinear Terms

Bilinear Terms X TA, X TAh, X TE X TA =

  • X T

1

X T

3

X T

2

X T

4

A A − BFCy AF

  • =

(X1 + X3)TA − X T

3 BFCy

X T

3 AF

(X2 + X4)TA − X T

4 BFCy

X T

4 AF

  • Set X4 = X3 (both system and filter have the same order)

Linearization ˜ AF ˜ AhF ˜ BF

  • = X T

3

AF AhF BF

  • we get a LMI problem

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 15/21

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Example 1

Let ˙ x(t) =

  • −2

−0.9

  • x(t) +
  • −1

−1 −1

  • x(t − h(t)) +
  • 1
  • w(t)

z(t) = 1 2 x(t) y(t) = 1 x(t) We set hM = 1 and we study γ w.r.t. µ using a memoryless filter µ 0.4 0.8 Fridman [2003] 1.4086 1.8311 15.8414 This result 0.06484 0.10651 0.48661

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 16/21

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Example 2(1)

We consider the LPV system ˙ x(t) = 1 + 0.2ρ −2 −3 + 0.1ρ

  • x(t) +
  • 0.2ρ

0.1 −0.2 + 0.1ρ −0.3

  • x(t − h(t))

+

  • −0.2

−0.2

  • w(t)

z(t) = 0.3 1.5 −0.45 0.75

  • x(t) +

0.5ρ −0.5ρ

  • w(t)

y(t) =

  • 1

0.5

  • x(t) +
  • 1 + 0.1ρ
  • w(t)

ρ ∈ [−1, 1] ˙ ρ ∈ [−1, 1] we choose P(ρ) = P0 + P1ρ and we study γ w.r.t. the delay bound hM

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 17/21

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Example 2(2)

We get the following figures

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 18/21

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Example 2(3)

Adding uncertainties H0 = H1 = 0.1I, F0 = F1 = F3 = F4 = I F2 = F5 = 1 1

  • We get

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 19/21

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Conclusion et Future Works

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 20/21

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Conclusion et Future Works

Advantages (Stability/Performance Analysis) :

Simple and Fast Interesting results but still conservative despite of the use of the Jensen’s inequality.

Use a more complex LKF , e.g. V2 =

N

  • i=1

t−(i−1)hn(t)

t−ihn(t)

x(θ)TQix(θ)dθ, hn(t) = h(t)/N V3 = ¯ h

N

  • i=1

t−(i−1)¯

h t−i¯ h

t

t+θ

˙ x(η)TRi ˙ x(η)dηdθ, ¯ h = hM/N Tackle the delay knowledge uncertainty

C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 21/21