Delay compensation in packet-switching network controlled systems

Transcription

1 Delay compensation in packet-switching network controlled systems Antoine Chaillet and Antonio Bicchi EECI - L2S - Université Paris Sud - Supélec (France) Centro di Ricerca Piaggio - Università di Pisa (Italy) Groupe SDH, Paris 05/02/2009 A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

2 Overview 1 Context 2 Problem statement 3 Main results 4 Conclusive remarks A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

3 Overview 1 Context More and more interaction between network and control General structure of a NCS Challenges to face Packet-based feedforward 2 Problem statement 3 Main results 4 Conclusive remarks A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

4 Networks and Control: more and more interaction Very low bandwidth (water medium) Alternated communication (interferences with sonar) Underwater vehicles formation (LIRMM, France) Sporadic communication (autonomy, scalability) Time-varying topology (vehicle motion) Cooperative mobile robots (Pisa, Italy) A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

5 General structure of a NCS General structure of a Network Controlled System: cont. SYSTEM 1... SYSTEM N hyb. NETWORK cont. CONTROLER = Sensor = Hybrid behavior. Actuator A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

6 Challenges to face Problems posed by network communication Delays: data processing and transmission. Sampling of the data transferred. Partial access: physical distribution of nodes. Data losses: packet dropouts. Both measurement and control may be sporadic. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

7 Challenges to face Problems posed by network communication Delays: data processing and transmission. Sampling of the data transferred. Partial access: physical distribution of nodes. Data losses: packet dropouts. Both measurement and control may be sporadic. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

8 Challenges to face û Plant y Network Network u Controller ŷ A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

9 Packet-based feedforward Plant y u Embedded controller ȳ ū Network Network Idea: Central controller Transmit a model-based prediction of the control signal [Montestruque et al., Polushin et al., Quevedo et al.,...] Use time-stamping for on-board re-synchronization. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

10 Packet-based feedforward Plant y u Embedded controller ȳ ū Network Network Idea: Central controller Transmit a model-based prediction of the control signal [Montestruque et al., Polushin et al., Quevedo et al.,...] Use time-stamping for on-board re-synchronization. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

11 Overview 1 Context 2 Problem statement Assumptions Internal model prediction Delay compensation NCS hybrid form 3 Main results 4 Conclusive remarks A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

12 Assumptions τj m (resp. τj c ), j N: time instants at which a measurement (resp. control) is sent over the network. T m j (resp. τ c j ): measurement (resp. control) data transfer delays. Assumption 1: MATI [Walsh et al.] The maximum time between two consecutive measurement (resp. control) accesses is smaller than some constant h m 0 (resp. h c 0). i.e. τj+1 m τ j m h m, (resp. τj+1 c τ j c h c ), j N. Assumption 2: Bounded delays The measurement (resp. control) transmission delays do not exceed some constant T m 0 (resp. T c 0). i.e. T m j T m, (resp. T c j T c ), j N. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

13 Assumptions τj m (resp. τj c ), j N: time instants at which a measurement (resp. control) is sent over the network. T m j (resp. τ c j ): measurement (resp. control) data transfer delays. Assumption 1: MATI [Walsh et al.] The maximum time between two consecutive measurement (resp. control) accesses is smaller than some constant h m 0 (resp. h c 0). i.e. τj+1 m τ j m h m, (resp. τj+1 c τ j c h c ), j N. Assumption 2: Bounded delays The measurement (resp. control) transmission delays do not exceed some constant T m 0 (resp. T c 0). i.e. T m j T m, (resp. T c j T c ), j N. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

14 Assumptions τj m (resp. τj c ), j N: time instants at which a measurement (resp. control) is sent over the network. T m j (resp. τ c j ): measurement (resp. control) data transfer delays. Assumption 1: MATI [Walsh et al.] The maximum time between two consecutive measurement (resp. control) accesses is smaller than some constant h m 0 (resp. h c 0). i.e. τj+1 m τ j m h m, (resp. τj+1 c τ j c h c ), j N. Assumption 2: Bounded delays The measurement (resp. control) transmission delays do not exceed some constant T m 0 (resp. T c 0). i.e. T m j T m, (resp. T c j T c ), j N. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

15 Assumptions Assumption 3: Invariably UGES protocol The network protocol is invariably UGES with some parameters (a, a, ρ 0, c). Definition: Invariable UGES Given a, a, c > 0 and ρ (0; 1), the protocol defined by the discrete dynamics z(k + 1) = h k (z k ) is said to be invariably UGES with parameters (a, a, ρ, c) if, for any increasing sequence {σ k } k N N, there exists a differentiable function W, such that a z W (k, z) a z W (k + 1, h σk (z)) ρ W (k, z) W (k, z) z c. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

16 Assumptions Assumption 3: Invariably UGES protocol The network protocol is invariably UGES with some parameters (a, a, ρ 0, c). Definition: Invariable UGES Given a, a, c > 0 and ρ (0; 1), the protocol defined by the discrete dynamics z(k + 1) = h k (z k ) is said to be invariably UGES with parameters (a, a, ρ, c) if, for any increasing sequence {σ k } k N N, there exists a differentiable function W, such that a z W (k, z) a z W (k + 1, h σk (z)) ρ W (k, z) W (k, z) z c. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

17 Assumptions Invariable UGES ensures that, whatever the sequence of time accesses, the protocol tends to reduce the error between the measured output and the data actually received by the controller. Examples: Round Robin ( cyclic access to each node ) is not Invariably UGES (although it is UGES) Try-once-discard [Walsh et al., 06] ( update the node with greatest error ) is invariably UGES with W (z) = z, a = a = 1 and n 1 ρ = n. Due to the non-constant delays and transmission times, both on the measurement side and the control side, Assumption 3 is needed to guarantee that all nodes are correctly updated A more involved update of state estimate would allow to overpass this assumption. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

18 Assumptions Invariable UGES ensures that, whatever the sequence of time accesses, the protocol tends to reduce the error between the measured output and the data actually received by the controller. Examples: Round Robin ( cyclic access to each node ) is not Invariably UGES (although it is UGES) Try-once-discard [Walsh et al., 06] ( update the node with greatest error ) is invariably UGES with W (z) = z, a = a = 1 and n 1 ρ = n. Due to the non-constant delays and transmission times, both on the measurement side and the control side, Assumption 3 is needed to guarantee that all nodes are correctly updated A more involved update of state estimate would allow to overpass this assumption. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

19 Assumptions Invariable UGES ensures that, whatever the sequence of time accesses, the protocol tends to reduce the error between the measured output and the data actually received by the controller. Examples: Round Robin ( cyclic access to each node ) is not Invariably UGES (although it is UGES) Try-once-discard [Walsh et al., 06] ( update the node with greatest error ) is invariably UGES with W (z) = z, a = a = 1 and n 1 ρ = n. Due to the non-constant delays and transmission times, both on the measurement side and the control side, Assumption 3 is needed to guarantee that all nodes are correctly updated A more involved update of state estimate would allow to overpass this assumption. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

20 Assumptions Invariable UGES ensures that, whatever the sequence of time accesses, the protocol tends to reduce the error between the measured output and the data actually received by the controller. Examples: Round Robin ( cyclic access to each node ) is not Invariably UGES (although it is UGES) Try-once-discard [Walsh et al., 06] ( update the node with greatest error ) is invariably UGES with W (z) = z, a = a = 1 and n 1 ρ = n. Due to the non-constant delays and transmission times, both on the measurement side and the control side, Assumption 3 is needed to guarantee that all nodes are correctly updated A more involved update of state estimate would allow to overpass this assumption. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

21 Assumptions Dynamics of the plant dynamics to be controlled: ẋ = f (x, u). Assumption 4: Nominal closed-loop is GES There exists a state feedback κ and a function V satisfying, for all x R n, α x 2 V (x) α x 2 V x (x)f (x, κ(x)) α x 2 V x (x) d x, where α, α, α and d denote positive constants. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

22 Assumptions We assume we know an approximation ˆf of the plants dynamics f. Assumption 5: Enough regularity The vector fields f and ˆf and the feedback law κ are continuously differentiable and their gradients are bounded by positive constants λ f, λˆf and λ κ respectively. In addition, ˆf (0, κ(0)) = 0. Note: this global Lipschitz assumption can be strongly relaxed if only local (or semiglobal) properties are required. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

23 Assumptions We assume we know an approximation ˆf of the plants dynamics f. Assumption 5: Enough regularity The vector fields f and ˆf and the feedback law κ are continuously differentiable and their gradients are bounded by positive constants λ f, λˆf and λ κ respectively. In addition, ˆf (0, κ(0)) = 0. Note: this global Lipschitz assumption can be strongly relaxed if only local (or semiglobal) properties are required. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

24 Internal model prediction The internal-model state prediction is obtained by the following dynamics: ˆx j (t) = ˆf (ˆx j (t), κ(ˆx j (t))), j N, with a reset at each reception of a new measurement: ˆx j (τ m γ(j) + T m γ(j) ) = x(τ m γ(j) ) + h γ(j)(ˆxγ(j 1) (τ m γ(j) ) x(τ m γ(j) )). These state predictions generate the packet-encoded control signals: u j (t + T c j ) = κ(ˆx j (t)), t τ m γ(j) + T m γ(j), j N. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

25 Internal model prediction The internal-model state prediction is obtained by the following dynamics: ˆx j (t) = ˆf (ˆx j (t), κ(ˆx j (t))), j N, with a reset at each reception of a new measurement: ˆx j (τ m γ(j) + T m γ(j) ) = x(τ m γ(j) ) + h γ(j)(ˆxγ(j 1) (τ m γ(j) ) x(τ m γ(j) )). These state predictions generate the packet-encoded control signals: u j (t + T c j ) = κ(ˆx j (t)), t τ m γ(j) + T m γ(j), j N. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

26 Delay compensation Each measurement is sent together with a time stamping: an embedded device may then resynchronize the received control packet with the present time. The applied signal is then: u(t) = u j (t + Tj c + Tγ(j) m c ), t [τj + Tj c ; τj+1 c + T j+1 c ]. This signal is fully described by two dynamical variables x c1 and x c2 : { ẋc1 = ˆf (x c1, κ(x c1 )) ẋ c2 = ˆf (1) (x c2, κ(x c2 )) x c1 (τ m γ(j)+ ) = { xc1 (τ m γ(j) ), if j / 2N x(τ m γ(j) ) + h γ(j)( xc2 (τ m γ(j) )), if j 2N x c2 (τ m γ(j)+ ) = { x(τ m γ(j) ) + h γ(j) ( xc1 (τ m γ(j) )), if j / 2N x c2 (τ m γ(j) ), if j 2N A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

27 Delay compensation Each measurement is sent together with a time stamping: an embedded device may then resynchronize the received control packet with the present time. The applied signal is then: u(t) = u j (t + Tj c + Tγ(j) m c ), t [τj + Tj c ; τj+1 c + T j+1 c ]. This signal is fully described by two dynamical variables x c1 and x c2 : { ẋc1 = ˆf (x c1, κ(x c1 )) ẋ c2 = ˆf (1) (x c2, κ(x c2 )) x c1 (τ m γ(j)+ ) = { xc1 (τ m γ(j) ), if j / 2N x(τ m γ(j) ) + h γ(j)( xc2 (τ m γ(j) )), if j 2N x c2 (τ m γ(j)+ ) = { x(τ m γ(j) ) + h γ(j) ( xc1 (τ m γ(j) )), if j / 2N x c2 (τ m γ(j) ), if j 2N A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

28 Delay compensation Let the time-sequence {t i } i N be given by: { t0 := τ m γ(0) } t i+1 := min j N {τ m γ(j) : τ m γ(j) > t i, i N 1, Let the index ν(i) be defined as { } ν(i) := min j N : t i = τ m γ(j), i N. And define J n := ( In I n ). A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

29 Delay compensation Let the time-sequence {t i } i N be given by: { t0 := τ m γ(0) } t i+1 := min j N {τ m γ(j) : τ m γ(j) > t i, i N 1, Let the index ν(i) be defined as { } ν(i) := min j N : t i = τ m γ(j), i N. And define J n := ( In I n ). A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

30 Delay compensation Let the time-sequence {t i } i N be given by: { t0 := τ m γ(0) } t i+1 := min j N {τ m γ(j) : τ m γ(j) > t i, i N 1, Let the index ν(i) be defined as { } ν(i) := min j N : t i = τ m γ(j), i N. And define J n := ( In I n ). A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

31 NCS hybrid form Then the closed-loop dynamics can be summarized as the following hybrid dynamical system [Teel,Nesic]: where e = ( e1 e 2 ) := ẋ = F(t, x, e) (2a) ė = G(t, x, e) (2b) e(t + i ) = H(i, e(t i )), (2c) ( xc1 x x c2 x F := f (x, u a (t, e + J n x)) ) (ˆf (e1 + x, κ(e G := 1 + x)) f (x, u a (t, e + J n x)) ˆf (e2 + x, κ(e 2 + x)) f (x, u a (t, e + J n x)) ( h H := ν(i) (e 2 ) + [ e 1 h ν(i) (e 2 ) ] ) η(i) h ν(i) (e 1 ) + [ e 2 h ν(i) (e 1 ) ]. (1 η(i)) ) A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

32 NCS hybrid form Then the closed-loop dynamics can be summarized as the following hybrid dynamical system [Teel,Nesic]: where e = ( e1 e 2 ) := ẋ = F(t, x, e) (2a) ė = G(t, x, e) (2b) e(t + i ) = H(i, e(t i )), (2c) ( xc1 x x c2 x F := f (x, u a (t, e + J n x)) ) (ˆf (e1 + x, κ(e G := 1 + x)) f (x, u a (t, e + J n x)) ˆf (e2 + x, κ(e 2 + x)) f (x, u a (t, e + J n x)) ( h H := ν(i) (e 2 ) + [ e 1 h ν(i) (e 2 ) ] ) η(i) h ν(i) (e 1 ) + [ e 2 h ν(i) (e 1 ) ]. (1 η(i)) ) A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

33 Overview 1 Context 2 Problem statement 3 Main results The nonlinear case The linear case Sampling effects 4 Conclusive remarks A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

34 The nonlinear case Theorem 1: General case Assume that Assumptions 1-5 hold and that there exists a constant ε > 0 such that h m + h c + T m + T c < 1 ( ) L ln L + γ, (ρ 0 + ε)l + γ where L := 2(1 + ε)c a min{1, ε} max { λˆf (1 + λ κ ); 2λ f λ κ } γ := 2dλ f λ k c(1 + ε)(λ f + λˆf )(1 + λ κ ) α a min{1, ε} Then, the origin of the NCS (2) is UGES.. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

35 The nonlinear case The proof relies on the following result: Corollary 2 [Nesic and Teel, TAC 2004] Under the following assumptions, the NCS is UGES. 1 The protocol e(i + 1) = h(i, e(i)) is UGES with parameters (a, a, ρ) and Lyapunov function W satisfying W (i, e)g(t, x, e) LW (i, e) + H(x). e 2 ẋ = F(t, x, e) is L p -stable from W to H with gain γ > 0 3 The NCS is L p to L p detectable from (H, W ) to (x, e) 4 The NCS is UGFTIS with linear gain, i.e. T, ρ > 0: x(t; t 0, x 0 ) ρ x 0, t [t 0 ; t 0 + T ], t 0 0, x 0 R n. ( ) 5 MATI 1 L ln L+γ ρl+γ. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

36 The nonlinear case The proof relies on the following result: Corollary 2 [Nesic and Teel, TAC 2004] Under the following assumptions, the NCS is UGES. 1 The protocol e(i + 1) = h(i, e(i)) is UGES with parameters (a, a, ρ) and Lyapunov function W satisfying W (i, e)g(t, x, e) LW (i, e) + H(x). e 2 ẋ = F(t, x, e) is L p -stable from W to H with gain γ > 0 3 The NCS is L p to L p detectable from (H, W ) to (x, e) 4 The NCS is UGFTIS with linear gain, i.e. T, ρ > 0: x(t; t 0, x 0 ) ρ x 0, t [t 0 ; t 0 + T ], t 0 0, x 0 R n. ( ) 5 MATI 1 L ln L+γ ρl+γ. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

37 The nonlinear case The proof relies on the following result: Corollary 2 [Nesic and Teel, TAC 2004] Under the following assumptions, the NCS is UGES. 1 The protocol e(i + 1) = h(i, e(i)) is UGES with parameters (a, a, ρ) and Lyapunov function W satisfying W (i, e)g(t, x, e) LW (i, e) + H(x). e 2 ẋ = F(t, x, e) is L p -stable from W to H with gain γ > 0 3 The NCS is L p to L p detectable from (H, W ) to (x, e) 4 The NCS is UGFTIS with linear gain, i.e. T, ρ > 0: x(t; t 0, x 0 ) ρ x 0, t [t 0 ; t 0 + T ], t 0 0, x 0 R n. ( ) 5 MATI 1 L ln L+γ ρl+γ. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

38 The nonlinear case The proof relies on the following result: Corollary 2 [Nesic and Teel, TAC 2004] Under the following assumptions, the NCS is UGES. 1 The protocol e(i + 1) = h(i, e(i)) is UGES with parameters (a, a, ρ) and Lyapunov function W satisfying W (i, e)g(t, x, e) LW (i, e) + H(x). e 2 ẋ = F(t, x, e) is L p -stable from W to H with gain γ > 0 3 The NCS is L p to L p detectable from (H, W ) to (x, e) 4 The NCS is UGFTIS with linear gain, i.e. T, ρ > 0: x(t; t 0, x 0 ) ρ x 0, t [t 0 ; t 0 + T ], t 0 0, x 0 R n. ( ) 5 MATI 1 L ln L+γ ρl+γ. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

39 The nonlinear case The proof relies on the following result: Corollary 2 [Nesic and Teel, TAC 2004] Under the following assumptions, the NCS is UGES. 1 The protocol e(i + 1) = h(i, e(i)) is UGES with parameters (a, a, ρ) and Lyapunov function W satisfying W (i, e)g(t, x, e) LW (i, e) + H(x). e 2 ẋ = F(t, x, e) is L p -stable from W to H with gain γ > 0 3 The NCS is L p to L p detectable from (H, W ) to (x, e) 4 The NCS is UGFTIS with linear gain, i.e. T, ρ > 0: x(t; t 0, x 0 ) ρ x 0, t [t 0 ; t 0 + T ], t 0 0, x 0 R n. ( ) 5 MATI 1 L ln L+γ ρl+γ. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

40 The nonlinear case The proof relies on the following result: Corollary 2 [Nesic and Teel, TAC 2004] Under the following assumptions, the NCS is UGES. 1 The protocol e(i + 1) = h(i, e(i)) is UGES with parameters (a, a, ρ) and Lyapunov function W satisfying W (i, e)g(t, x, e) LW (i, e) + H(x). e 2 ẋ = F(t, x, e) is L p -stable from W to H with gain γ > 0 3 The NCS is L p to L p detectable from (H, W ) to (x, e) 4 The NCS is UGFTIS with linear gain, i.e. T, ρ > 0: x(t; t 0, x 0 ) ρ x 0, t [t 0 ; t 0 + T ], t 0 0, x 0 R n. ( ) 5 MATI 1 L ln L+γ ρl+γ. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

41 The nonlinear case In our case: 1 From Assumption 3 and the global Lipschitz of f and ˆf : W e G H + LW where H(x) x 2 Exploiting the Lyapunov function V associated to the feedback κ(x), the NCS is L 2 -stable from W to H with gain γ 3 The NCS is L 2 to L 2 detectable from (W, H) to (e, x) as W e and H x 4 The NCS is UGFTIS in view of Proposition 2 in [Nesic and Teel, 04] and the global Lispchitz of f and ˆf 5 The MATI condition holds by assumption. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

42 The nonlinear case In our case: 1 From Assumption 3 and the global Lipschitz of f and ˆf : W e G H + LW where H(x) x 2 Exploiting the Lyapunov function V associated to the feedback κ(x), the NCS is L 2 -stable from W to H with gain γ 3 The NCS is L 2 to L 2 detectable from (W, H) to (e, x) as W e and H x 4 The NCS is UGFTIS in view of Proposition 2 in [Nesic and Teel, 04] and the global Lispchitz of f and ˆf 5 The MATI condition holds by assumption. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

43 The nonlinear case In our case: 1 From Assumption 3 and the global Lipschitz of f and ˆf : W e G H + LW where H(x) x 2 Exploiting the Lyapunov function V associated to the feedback κ(x), the NCS is L 2 -stable from W to H with gain γ 3 The NCS is L 2 to L 2 detectable from (W, H) to (e, x) as W e and H x 4 The NCS is UGFTIS in view of Proposition 2 in [Nesic and Teel, 04] and the global Lispchitz of f and ˆf 5 The MATI condition holds by assumption. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

44 The nonlinear case In our case: 1 From Assumption 3 and the global Lipschitz of f and ˆf : W e G H + LW where H(x) x 2 Exploiting the Lyapunov function V associated to the feedback κ(x), the NCS is L 2 -stable from W to H with gain γ 3 The NCS is L 2 to L 2 detectable from (W, H) to (e, x) as W e and H x 4 The NCS is UGFTIS in view of Proposition 2 in [Nesic and Teel, 04] and the global Lispchitz of f and ˆf 5 The MATI condition holds by assumption. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

45 The nonlinear case In our case: 1 From Assumption 3 and the global Lipschitz of f and ˆf : W e G H + LW where H(x) x 2 Exploiting the Lyapunov function V associated to the feedback κ(x), the NCS is L 2 -stable from W to H with gain γ 3 The NCS is L 2 to L 2 detectable from (W, H) to (e, x) as W e and H x 4 The NCS is UGFTIS in view of Proposition 2 in [Nesic and Teel, 04] and the global Lispchitz of f and ˆf 5 The MATI condition holds by assumption. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

46 The linear case Focusing now on linear dynamics: f (x, u) = Ax + Bu, ˆf (x, u) = Âx + ˆBu, κ(x) = Kx, the NCS dynamics becomes ẋ = (A + BK )x + BK (e 1 P(t) + e 2 (1 P(t))) (à + BK )x + ( + BK )e 1 P(t) + [( + ˆBK ] )e 1 BKe 2 (1 P(t)) ė = (à + BK )x + ( + BK )e 2 (1 P(t)) + [( + ˆBK ] )e 2 BKe 1 P(t) ( e(t + h i ) = ν(i) (e 2 ) + [ e 1 h ν(i) (e 2 ) ] ) η(i) h ν(i) (e 1 ) + [ e 2 h ν(i) (e 1 ) ] (1 η(i)) where à :=  A and B := ˆB B. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

47 The linear case Focusing now on linear dynamics: f (x, u) = Ax + Bu, ˆf (x, u) = Âx + ˆBu, κ(x) = Kx, the NCS dynamics becomes ẋ = (A + BK )x + BK (e 1 P(t) + e 2 (1 P(t))) (à + BK )x + ( + BK )e 1 P(t) + [( + ˆBK ] )e 1 BKe 2 (1 P(t)) ė = (à + BK )x + ( + BK )e 2 (1 P(t)) + [( + ˆBK ] )e 2 BKe 1 P(t) ( e(t + h i ) = ν(i) (e 2 ) + [ e 1 h ν(i) (e 2 ) ] ) η(i) h ν(i) (e 1 ) + [ e 2 h ν(i) (e 1 ) ] (1 η(i)) where à :=  A and B := ˆB B. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

48 The linear case Corollary: the LTI case Assume that Assumptions 1, 2 and 3 hold and that there exists matrices K R n m and P = P T > 0 such that (A + BK ) T P + P(A + BK ) = I. Assume that there exists a positive constant ε > 0 such that the following bound holds h m + h c + T m + T c < 1 ( ) L ln L + γ, (ρ 0 + ε)l + γ where L := (1 + ε)c 2 { min{1, ε}a max  + ˆBK ; BK +  + BK } 2c(1 ε) PBK à + BK γ :=. a min{1, ε} Then, the origin of the linear NCS is UGES. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

49 Sampling effects The size of packets being limited (say N bits/packet), sampling needs to be taken into account. The NCS then results in a sampled-data version of the previous results. Theorem 2: with sampling effects Assume that Assumptions 1-5 hold and that T m + T c < 1 ( ) L ln L + γ. (ρ 0 + ε)l + γ Assume also that the state estimate is perfectly initialized, such that e 0 = 0. Then there exist κ 1, κ 2 > 0 such that, given any M > δ > 0, there exists a packet size N such that, for all x 0 M, the solution of the NCS with sampling-based packets satisfies x(t, t 0, x 0, e 0 = 0) δ + κ 1 x 0 e κ 2(t t 0 ), t t 0. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

50 Sampling effects The size of packets being limited (say N bits/packet), sampling needs to be taken into account. The NCS then results in a sampled-data version of the previous results. Theorem 2: with sampling effects Assume that Assumptions 1-5 hold and that T m + T c < 1 ( ) L ln L + γ. (ρ 0 + ε)l + γ Assume also that the state estimate is perfectly initialized, such that e 0 = 0. Then there exist κ 1, κ 2 > 0 such that, given any M > δ > 0, there exists a packet size N such that, for all x 0 M, the solution of the NCS with sampling-based packets satisfies x(t, t 0, x 0, e 0 = 0) δ + κ 1 x 0 e κ 2(t t 0 ), t t 0. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

51 Sampling effects The proof is based on sampled data results. Qualitative result: semiglobal practical stability is obtained, with packet size as tuning parameter. Trade-off: embedded computation capabilities / communication bandwidth / performance. Perfect initialization of the internal model (e 0 = 0) is required. Direct extension to the LTI case. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

52 Overview 1 Context 2 Problem statement 3 Main results 4 Conclusive remarks A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28

53 Summary A packet-based strategy to compensate for delays in sufficiently regular nonlinear NCS: Network effects in both measurement and actuation loops Unknown bounded delays Explicit bound on the tolerable delays and MATI Exploits the nominal feedback and an imprecise model of the plant Qualitative result for sampling effects Focus on LTI systems. A. Chaillet, A. Bicchi (Paris, Pisa) Delay compensation for NCS SDH, 05/02/ / 28