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

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

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

Introduction Considered Systems Filters Structures C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 3/21

Considered Systems (1) Uncertain LPV Systems with Delay  ˙    x ( t ) x ( t ) z ( t ) = Σ( ρ ( t ) , δ ) x ( t − h ( t ))     y ( t ) w ( t ) x ( θ ) = φ ( θ ) , θ ∈ [ − h M , 0 ] ρ ∈ U ρ ρ ˙ ∈ hull [ U ν ] δ ∈ U δ h ( t ) ∈ [ 0 , h M ] ˙ h ( t ) < µ < 1 C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 4/21

Considered Systems (2) System Matrix :   A ( ρ ) + ∆ A ( ρ, δ ) A h ( ρ ) + ∆ A h ( ρ, δ ) E ( ρ ) + ∆ E ( ρ, δ ) Σ = C ( ρ ) C h ( ρ ) F ( ρ )   C y ( ρ ) + ∆ C y ( ρ, δ ) C yh ( ρ ) + ∆ C yh ( ρ, δ ) F y ( ρ ) + ∆ F y ( ρ, δ ) where the uncertain part obeys � ∆ A � � H 0 � � F 0 � ∆ A h ∆ E F 1 F 2 ( ρ, δ ) = ( ρ )∆( δ ) ( ρ ) ∆ C y ∆ C yh ∆ F y H 1 F 3 F 4 F 5 with || ∆ || 2 ≤ 1 C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 5/21

Filters Structures Filters with memory � ˙ �   x F ( t ) � � A F ( ρ ) x F ( t ) A hF ( ρ ) B F ( ρ ) = x F ( t − h ( t ))   z F ( t ) C F ( ρ ) C hF ( ρ ) D F ( ρ ) y ( t ) Memoryless filters � ˙ � � � � � x F ( t ) A F ( ρ ) B F ( ρ ) x F ( t ) = z F ( t ) C F ( ρ ) D F ( ρ ) y ( t ) These matrices are aimed to be chosen such that || z − z F || L 2 ≤ γ || w || L 2 with a minimal L 2 -gain γ > 0. C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 6/21

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

Lyapunov-Krasovskii Functional (1) Delay-Dependent LKF : V ( x t , ˙ V 1 ( x t , ρ ) + V 2 ( x t ) + V 3 ( ˙ x t , ρ ) = x t ) x ( t ) T P ( ρ ) x ( t ) V 1 ( x t , ρ ) = � t x ( θ ) T Qx ( θ ) d θ V 2 ( x t ) = t − h ( t ) � 0 � t x ( η ) T R ˙ V 3 ( ˙ ˙ x t ) = h M x ( η ) d η d θ − h M t + θ whose derivative along the trajectories solution of the system satisfies : �� � ∂ ˙ x ( t ) T P ( ρ ) x ( t ) + x ( t ) T P ( ρ ) ˙ x ( t ) + x ( t ) T ˙ V 1 = ρ i ˙ P ( ρ ) x ( t ) ∂ρ i i ˙ x ( t ) T Qx ( t ) − ( 1 − µ ) x ( t − h ( t )) T Qx ( t − h ( t )) V 2 ≤ � t ˙ h 2 x ( t ) T R ˙ x ( θ ) T R ˙ M ˙ ˙ V 3 ≤ x ( t ) − h M x ( θ ) d θ t − h ( t ) C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 8/21

Lyapunov-Krasovskii Functional (2) Using Jensen’s Inequality : � T � t �� t �� t � x ( θ ) T R ˙ ˙ ˙ ˙ − h M x ( θ ) d θ ≤ − x ( θ ) d θ R x ( θ ) d θ t − h ( t ) t − h ( t ) t − h ( t ) we get � Ψ + h 2 M A T RA PA h + R + h 2 M A T RA h � ˙ V ≤ χ ( t ) T χ ( t ) − ( 1 − µ ) Q − R + h 2 M A T ⋆ h RA h ∂ with Ψ = A T P + PA + Q − R + � i ˙ ρ i P ( ρ ) and ∂ρ i χ ( t ) = col ( x ( t ) , x ( t − h ( t )) . C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 9/21

Asymptotic Stability Theorem The unertain LPV system is asymptotically stable for all h ( t ) ∈ [ 0 , h M ] such that ˙ h ( t ) < µ if there exist P : U ρ → S n ++ , Q , R ∈ S n ++ such that the LMI  h M A ( ρ ) T R  Ψ( ρ, ν ) P ( ρ ) A h ( ρ ) + R  ≺ 0 h M A h ( ρ ) T R ⋆ − ( 1 − µ ) Q − R  ⋆ ⋆ − R holds for all ( ρ, ν ) ∈ U ρ × U ν with ∂ Ψ( ρ, ν ) = A ( ρ ) T P ( ρ ) + P ( ρ ) A ( ρ ) + Q − R + � i ν i P ( ρ ) . ∂ρ i C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 10/21

Bounded Real Lemma Theorem The unertain LPV system is asymptotically stable for all h ( t ) ∈ [ 0 , h M ] such that ˙ h ( t ) < µ if there exist P : U ρ → S n ++ , Q , R ∈ S n ++ and γ > 0 such that the LMI C ( ρ ) T h M A ( ρ ) T R   Ψ( ρ, ν ) P ( ρ ) A h ( ρ ) + R P ( ρ ) E ( ρ ) C h ( ρ ) T h M A h ( ρ ) T R ⋆ − ( 1 − µ ) Q − R 0    F ( ρ ) T h M E ( ρ ) T R  ⋆ ⋆ − γ I ≺ 0     ⋆ ⋆ ⋆ − γ I 0   ⋆ ⋆ ⋆ ⋆ − R holds for all ( ρ, ν ) ∈ U ρ × U ν with ∂ Ψ( ρ, ν ) = A ( ρ ) T P ( ρ ) + P ( ρ ) A ( ρ ) + Q − R + � i ν i P ( ρ ) . Moreover, ∂ρ i we have || z || L 2 ≤ γ || w || L 2 . C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 11/21

Relaxed Version Theorem The unertain LPV system is asymptotically stable for all h ( t ) ∈ [ 0 , h M ] such that ˙ h ( t ) < µ if there exist P : U ρ → S n ++ , X : U ρ → R n × n , Q , R ∈ S n ++ and γ > 0 such that the LMI − ( X + X T ) P + X T A X T A h X T E X T   0 h M R C T ⋆ Φ 1 R 0 0 0     C T ⋆ ⋆ Φ 2 0 0 0   h   F T ⋆ ⋆ ⋆ − γ I 0 0 ≺ 0     ⋆ ⋆ ⋆ ⋆ − γ I 0 0     ⋆ ⋆ ⋆ ⋆ ⋆ − P − h M R   ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ − R ∂ holds for all ( ρ, ν ) ∈ U ρ × U ν with Φ 1 = P + Q − R + � i ν i P ( ρ ) ∂ρ i and Φ 2 = − ( 1 − µ ) Q − R. Moreover, we have || z || L 2 ≤ γ || w || L 2 . C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 12/21

Filtering Problem Augmented System Relaxation Example C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 13/21

Augmented System Interconnection between the system and filter ˙ x a ( t ) = A x a ( t ) + A h x a ( t − h ( t )) + E w ( t ) z e ( t ) = C x a ( t ) + C h x a ( t − h ( t )) + F w ( t ) x a ( t ) = col ( x ( t ) , x F ( t )) z e ( t ) = z ( t ) − z F ( t ) with � � � � � � A 0 A h 0 E A = A h = E = A − B F C y A F A h − B F C yh A Fh E − B F F y � C − D F C y − C F � � C h − D F C yh − C Fh � C = C F C h = C Fh F = F − D F F y C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 14/21

Relaxation of Bilinear Terms Bilinear Terms X T A , X T A h , X T E � X T X T � � � A 0 X T A 1 3 = X T X T A − B F C y A F 2 4 � ( X 1 + X 3 ) T A − X T X T � 3 B F C y 3 A F = ( X 2 + X 4 ) T A − X T X T 4 B F C y 4 A F Set X 4 = X 3 (both system and filter have the same order) Linearization � ˜ ˜ ˜ = X T � � A F � A hF B F A F A hF B F 3 we get a LMI problem C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 15/21

Example 1 Let � � � � � � − 2 0 − 1 0 0 ˙ x ( t ) = x ( t ) + x ( t − h ( t )) + w ( t ) 0 − 0 . 9 − 1 − 1 1 � 1 2 � z ( t ) = x ( t ) � 1 0 � y ( t ) = x ( t ) We set h M = 1 and we study γ w.r.t. µ using a memoryless filter µ 0 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

Example 2(1) We consider the LPV system � 0 � � � 1 + 0 . 2 ρ 0 . 2 ρ 0 . 1 ˙ x ( t ) = x ( t ) + x ( t − h ( t )) − 2 − 3 + 0 . 1 ρ − 0 . 2 + 0 . 1 ρ − 0 . 3 � � − 0 . 2 + w ( t ) − 0 . 2 � 0 . 3 � 0 . 5 ρ � � 1 . 5 z ( t ) = x ( t ) + w ( t ) − 0 . 45 0 . 75 − 0 . 5 ρ � � � � 0 1 0 y ( t ) = x ( t ) + w ( t ) 0 . 5 0 1 + 0 . 1 ρ ρ ∈ [ − 1 , 1 ] ρ ∈ [ − 1 , 1 ] ˙ we choose P ( ρ ) = P 0 + P 1 ρ and we study γ w.r.t. the delay bound h M C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 17/21

Example 2(2) We get the following figures C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 18/21

Example 2(3) Adding uncertainties H 0 = H 1 = 0 . 1 I , F 0 = F 1 = F 3 = F 4 = I � 1 � F 2 = F 5 = 1 We get C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 19/21

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

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. � t − ( i − 1 ) h n ( t ) N � x ( θ ) T Q i x ( θ ) d θ, V 2 = h n ( t ) = h ( t ) / N t − ih n ( t ) i = 1 � t − ( i − 1 )¯ � t N h ¯ ¯ � x ( η ) T R i ˙ ˙ V 3 = h x ( η ) d η d θ, h = h M / N t − i ¯ h t + θ i = 1 Tackle the delay knowledge uncertainty C.Briat, O.Sename and JF.Lafay corentin.briat@briat.info 21/21