Observer design for classes of nonlinear networked ... - CiteSeerX

Observer design for classes of nonlinear networked control systems Romain Postoyan∗, Tarek Ahmed-Ali†, Fran¸coise Lamnabhi-Lagarrigue‡ May 5, 2010

Abstract Assuming a class of continuous-time observers is known for a given nonlinear system, an observation structure is derived when sensors information is subject to network-induced communication constraints. The approach is based on the recent results on hybrid observers for sampled-data systems by Karafyllis and Kravaris [5]. Considering two classes of protocols, some asymptotic stability properties are shown to hold for the observation error, under some conditions, and explicit bounds on the maximum allowable transmission interval are given.

1

Introduction

Networked control systems (NCS) are becoming more and more prevalent in the industry, because of their great flexibility, easy maintenance and low cost. They are characterized ∗

Univ Paris-Sud, LSS-CNRS, SUPELEC, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France ([email protected]) † Univ Caen Basse Normandie, GREYC-CNRS, 6 boulevard du Mar´echal Juin, 14050 Caen Cedex 9, France ([email protected]) ‡ LSS-CNRS, SUPELEC, EECI, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France ([email protected])

1

by the fact that some sensor and / or control data are transmitted over a serial communication channel. Consequently, only a part of the system measurements, for instance, is sporadically available for control or estimation purpose. Physically, the system output is decomposed into groups of sensors, depending of their spatial proximity, that are associated to a sensor node. Only one sensor node sends its packet-based message over the network at each transmission instant. Consider the observer design problem for NCS, the networkinduced constraints on system measurements make available results in continuous-time not directly applicable and therefore require the development of appropriate estimation tools. The rule that decides which node transmits its packet, called a protocol, plays a key role in the stability analysis. There exist two families of protocols whether they are static or dynamic. In the static case, the sequence of nodes is predetermined, contrary to dynamic protocols where the node selection depends on some system variables. Dynamic protocols usually require a higher cost of implementation but are expected to provide better results than the static ones, notably in term of the bound on the maximum allowable transmission interval (MATI). This study addresses the observer design problem for nonlinear NCS. Although the control of NCS attracts a lot of attention nowadays (see [12, 2, 7] and the references therein to mention a few), only few papers treat the observer design problem. Most of the existing results focus on linear systems in a stochastic context, see [3, 6, 10, 15, 16]. For a deterministic study see [16], where some conditions are given in order to maintain the observability of discrete-time linear systems under network communications. A methodology for the mutual design of dynamical protocols and observers for linear systems is proposed in [1]. Sufficient conditions for quadratic stability properties are then given in terms of matrix inequalities on the observer gains, the MATI and the protocol weights. Recently, a general framework for the observer design for nonlinear NCS has been proposed in [9, 8], via an emulation-like approach. Provided that the continuous-time observer is sufficiently 2

robust to measurement errors, sufficient conditions are given to guarantee the global convergence of the observation error for various in-network processing implementations and Lyapunov uniformly globally exponentially stable (UGES) protocols. Easy computable MATI bounds are obtained and the results have been applied to derive linear observers [9] and high-gain observers [8] for NCS for a range of network configurations. In this paper, we propose an alternative analysis to [9, 8] for the case where the observer is implemented using predictive-type in-network processing implementations. This study can be seen as an extension of the observer design for sampled-data systems in [5] to the case where system measurements are sent through a network. Contrary to [9, 8], we focus on a single type of in-network processing algorithm but we consider classes of protocols that are not necessarily UGES. Inspired by [5], the stability analysis relies on the use of the small gain theorem for hybrid systems in [4] that allows us to derive different MATI bounds compared to [9, 8].

After having defined some mathematical notations and properties, the main result of [5] is presented for multi-output systems in Section 3. In Section 4, a class of persistently excited protocols is considered and the proposed observation structure is shown to ensure global asymptotic convergence properties hold for the observation error, when the MATI is less than a given bound. Afterwards, similar results are derived for a dynamic protocol.

2 2.1

Preliminaries Nomenclature

Let R = (−∞, ∞), R>0 = (0, ∞), R≥0 = [0, ∞), N0 = {0, . . . , ∞}, N = {1, . . . , ∞}. The notation Jp, qK denotes the set {p, . . . , q} with p, q ∈ N0 and p ≤ q. The notation | · | is used to denote the Euclidean norm. Let C(Rp , Rq ), p, q ∈ N, denote the space of all continuous mapping Rp → Rq and C i (Rp , Rq ), i ∈ N, the space of continuous functions with 3

continuous ith order partial derivatives. For vector fields f = [f1 , . . . , fn ]T ∈ C(Rn , Rn ) n P and h ∈ C 1 (Rn , Rp ), the notation Lf h denotes (∂i h)fi where ∂i h is the first derivative of i=1

th

h with the respect to its i

argument, n, p ∈ N. A continuous function γ : R≥0 −→ R≥0 is

of class-K if it is zero at zero and strictly increasing and of class-K∞ if it is also unbounded. A continuous function σ : R≥0 × R −→ R≥0 is of class-KL if, for all s ∈ R≥0 σ(s, ·), is decreasing and σ(s, t) → 0 when t → ∞, and, for all t ∈ R, σ(·, t) is of class-K. A function ε : R → Rp belongs to the set Lloc (R, Rp ), p ∈ N, if it is Lebesgue measurable and locally bounded. The notation supτ ∈[t1 ,t2 ] |ε(τ )| denotes the supremum of ε over [t1 , t2 ], where t1 ≤ t2 ∈ R. Considering a mapping f : R → Rn , n ∈ N and a ∈ R, the notation f (a− ) represents limt→a, t0 such that, for every k ∈ N0 , υ < τk+1 − τk < T and τ0 ≤ T . Remark.

Constant υ is arbitrary and is used to prevent from Zeno phenomena, see

[7]. Note that the transmission instants sequence {τk }k∈N0 is allowed to be non-uniform.

The following proposition can be found in [5] for single output systems. Its proof follows immediately from Theorem 4.1 as explained later. Proposition 3.1 Assuming Hypotheses 3.1-3.3 hold, suppose that, for any i ∈ J1, pK there exist Ki ∈ R≥0 , a class-KL function σ ¯i such that for any (x0 , z0 , w0 ) ∈ Rn×m×p and ε ∈ Lloc (R, Rp ), along solutions to (1) with y = h(x) + ε, ε ∈ Lloc (R, Rp ), and (9)-(10), this holds, for all t ∈ R≥0 : |Lf hi (ˆ x) − Lf hi (x)| ≤ σ ¯i (|x0 | + |z0 |, t) + Ki sup |ε(τ )|, τ ∈[0,t]

6

(11)

if T

P

Ki < 1, then the hybrid system defined by, for all k ∈ N0 , t ∈ [0, τ0 ) ∪ [τk , τk+1 ),

i∈J1,lK

z(t) ˙ = F (z(t), w(t))

(12)

xˆ(t) = Ψ(z(t))

(13)

w˙ i (t) = Lf hi (ˆ x(t)), i ∈ J1, pK,

(14)

for t = τk+1 , − z(τk+1 ) = z(τk+1 )

(15)

xˆ(τk+1 ) = Ψ(z(τk+1 ))

(16)

wi (τk+1 ) = hi (x(τk+1 )), i ∈ J1, pK,

(17)

where w = [w1 , . . . , wp ]T ∈ Rp , is a robust observer for (1)-(2). Remark.

The variable w plays a key role in the observation structure (12)-(17) since

it somehow replaces the unavailable system output between two successive transmission instants. Introduced in [5] for the design of sampled-data observers, this technique can be seen as a particular type of in-network processing algorithm as shown in [9, 8].

Remark.

Condition (11) holds for globally Lipschitz observers like linear and high

gains observers, see Section IV in [5].

4

Observer design for NCS

In the remaining part of this paper, system outputs are considered to be subject to networkinduced communication constraints . The observer design problem for system (1)-(2) is

7

addressed for two types of protocols. The system output is decomposed as follows y = n P [y1T , . . . , ylT ]T = [H1 (x)T , . . . , Hl (x)T ]T where yi ∈ Rni for i ∈ J1, lK, ni ∈ N, and ni = i=1

p, where l ∈ N denotes the number of nodes. Similarly to the sampled-data case, the

observation structure is the following, for all k ∈ N0 , for t ∈ [0, τ0 ) ∪ [τk , τk+1), z(t) ˙ = F (z(t), w(t))

(18)

xˆ(t) = Ψ(z(t))

(19)

w˙ i (t) = Lf (ˆx(t)) Hi (ˆ x(t)), i ∈ J1, lK,

(20)

for t = τk+1 , − z(τk+1 ) = z(τk+1 )

(21)

xˆ(τk+1 ) = Ψ(z(τk+1 ))

(22)

− − ))), ), h(x(τk+1 w(τk+1) = ς(τk+1 , w(τk+1

(23)

where ς = [ς1T , . . . , ςlT ]T , with ςi : {τk }k∈N0 × Rp × Rp → Rni , is defined according to the protocol. Remark.

It is implicitly assumed throughout this study that considered observers need

the whole output vector, y, to work: no output vector component can be ignored for observation purpose.

4.1

For a class of persistently excited protocols

In this subsection, we focus on a class of persistently excited scheduling protocols. Let denote the sequence of instants where access is granted to node i by {τki }k∈N0 , i ∈ J1, lK. The following statements are supposed to apply.

8

Hypothesis 4.1 (i) Each sequence {τki }k∈N0 is unbounded. (ii) For all i ∈ J1, lK, there i exist υi , Ti ∈ R>0 such that, for all k ∈ N0 , υi < τk+1 − τki < Ti and τ0i ≤ Ti .

Remark. Like in Section 3, constants υi can be taken arbitrarily small since they are only introduced for preventing from Zeno phenomena.

Condition (i) ensures that no node is ignored after any instant, whereas (ii) guarantees the existence of common upper bound on the interval between two successive access instants of a node. Statements (i)-(ii) together prevent from the situation where the information of a node is or tends to be forgotten by the protocol as time grows. Remark. Round-Robin (RR) protocols [12], that consists in granting access to each node after l transmission instants, are a particular case of the protocol configuration (i)-(ii). We called uniform RR protocol when transmission instants occur at a fixed period T ∈ R>0 , i.e. Ti = lT .

The class of functions ς = [ς1T , . . . , ςlT ]T : {τk }k∈N0 ×Rp ×Rp → Rp is here, for i ∈ J1, lK: ςi : {τk }k∈N0 × Rp × Rp → Rni    Hi if τ ∈ {τ i }k∈N 0 k (τ, w, h) 7→   wi otherwise.

(24)

Note that {τki }k∈N0 = {τk : ςi (τk , w(τk− ), h(x(τk ))) = Hi (x(τk ))}. Theorem 4.1 Assuming conditions of Hypotheses 3.1-3.3, 4.1 hold, suppose that, for any i ∈ J1, lK there exist Ki ∈ R≥0 and a class-KL function σ ¯i , such that for any (x0 , z0 ) ∈ Rn×m and ε ∈ Lloc (R, Rp ), along solutions to (1) with the perturbed output y = h(x) + ε, and

9

(9)-(10), this holds, for all t ∈ R≥0 : |Lf (ˆx(t)) Hi (ˆ x(t)) − Lf (x(t)) Hi (x(t))| ≤ σ ¯i (|x0 | + |z0 |, t) + Ki sup |ε(τ )|

(25)

τ ∈[0,t]

if

l P

Ti Ki < 1, then the hybrid system (18)-(24), is a robust observer for (1)-(2).

i=1

Proof. The requirement that, for any system initial condition, there exists an observer initial condition such that, when the system output is not perturbed, xˆ(t) = x(t) for all t ∈ [0, tmax ) - where tmax ∈ (0, ∞) be such that [0, tmax ) is the maximum interval existence of solutions to (18)-(24),(1)-(2) follows immediately from Hypothesis 3.2. T T p Let (x0 , z0 , w0 ) ∈ Rn × Rm × Rp , ε = [εT 1 , . . . , εl ] ∈ Lloc (R, R ) and i ∈ J1, lK. In view of

(25), for all t ∈ [0, tmax ) |Lf (ˆx(t)) Hi (ˆ x(t)) − Lf (x(t)) Hi (x(t))| ≤ σ ¯i (|x0 | + |z0 |, t) +Ki sup |w(x(τ )) − h(x(τ ))|.

(26)

τ ∈[0,t]

i Let k ∈ N0 . For all t ∈ [τki , τk+1 ) ∩ [0, tmax ), according to (26) this holds:

|wi (t) − Hi (x(t))| = |

Z

t

τki

(Lf Hi (ˆ x) − Lf Hi (x)) ds + εi (t)|

≤ Ti σ ¯i (|x0 | + |z0 |, τki ) + Ti Ki sup |w(τ ) − h(x(τ ))| + sup |εi (τ )|. 0≤τ ≤t

Consider now the case where t ∈ [0, τ0i ) ∩ [0, tmax ): |wi (t) − Hi (x(t))| = |wi (0) − Hi (x0 ) +

Z

0

t

(Lf Hi (ˆ x) − Lf Hi (x)) ds|

≤ τ0i σ ¯i (|x0 | + |z0 |, 0) + τ0i Ki sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+ρi (|x0 | + |w0|).

10

0≤τ ≤t

where ρi is a class-K∞ function such that |wi − hi (x)| ≤ ρi (|wi | + |x|) for all (wi , x) ∈ Rni × Rn . Defining the following class-KL function    (Ti σ ¯i (s, 0) + ρ(s)) eTi −t σ ˆi (s, t) =   Ti σ ¯i (s, t − Ti ) + ρ(s)eTi −t

if t < Ti

(27)

if t ≥ Ti ,

for all t ∈ [0, tmax ),

|wi (t) − Hi (x(t))| ≤ σ ˆi (|x0 | + |z0 | + |w0 |, t) + Ti Ki sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+ sup |εi (τ )|.

(28)

0≤τ ≤t

It can be deduced from (28) that, for all t ∈ [0, tmax ): |w(t) − h(x(t))| ≤ σ ˘ (|x0 | + |z0 | + |w0 |, t) + + sup γ˘ (|ε(τ )|).

l X i=1

Ti Ki sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

(29)

0≤τ ≤t

where σ ˘=

n P

σ ˆi ∈ KL and γ˘ ∈ K. Then, since

i=1

sup |w(τ ) − h(x(τ ))| ≤

0≤τ ≤t

1− +

1 l P

Ti Ki

l P

Ti Ki < 1,

i=1

σ˘ (|x0 | + |z0 |, 0)

i=1

1−

1 l P

sup γ˘ (|ε(τ )|). Ti Ki

(30)

0≤τ ≤t

i=1

On the other hand, for all t ∈ [0, tmax ), in view of Hypothesis 3.2: |x(t) − xˆ(t)| ≤ σ ˆ (|x0 | + |z0 |, t) + sup γ(|w(τ ) − h(x(τ ))|), 0≤τ ≤t

11

(31)

and, according to Hypotheses 3.1-3.2, for t ∈ [0, tmax ):   |z(t)| + |x(t)| ≤ µ ˜(t) α(|x ˜ 0 | + |z0 |) + sup γ˜ (|w(τ ) − h(x(τ ))|) + µ(t)α(|x0 |). (32) 0≤τ ≤t

It can be shown, in view of (29), that for t ∈ [0, tmax ), by appropriately defining ρ˜ ∈ K∞ : |w(t)| ≤ σ ˘ (|x0 | + |z0 | + |w0 |, 0) +

l X i=1

Ti Ki sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+ sup γ˘ (|ε(τ )|) + ρ˜(µ(t)) + ρ˜(α(|x0 |)).

(33)

0≤τ ≤t

Combining (32) and (33),

|z(t)| + |x(t)| + |wi (t)| ≤ σ ˘ (|x0 | + |z0 | + |w0 |, 0) +

l X i=1

Ti Ki sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+ sup γ˘ (|ε(τ )|) + ρ˜(µ(t)) + ρ˜(α(|x0 |)) 0≤τ ≤t   +˜ µ(t) α ˜ (|x0 | + |z0 |) + sup γ˜ (|w(τ ) − h(x(τ ))|) 0≤τ ≤t

+µ(t)α(|x0 |).

(34)

Consequently, in view of Hypothesis 3.1, (30), (34), it can be shown system (18)-(24) is robustly forward complete. Moreover, invoking the small gain theorem for hybrid systems l P in [4] and since Ti Ki < 1, it can be concluded that there exist σ ˜ ∈ KL, γ¯ ∈ K such that i=1

for any (x0 , z0 , w0 , ε) ∈ Rn × Rp × Lloc (R, Rp ) such that, along solutions to (1) with the output y = h(x) + ε, (18)-(24), for all t ∈ [0, ∞): |w(t) − h(x(t))| ≤ σ ˜ (|x0 | + |z0 | + |w0 |, t) + sup γ¯ (|ε(τ )|).

(35)

0≤τ ≤t

Finally, by using the same arguments than in the proof of Theorem 3.1 in [5], in view of (31), (34), (35), and the small gain theorem in [4] guarantees that there exist σˇ ∈ KL and γˇ ∈ K such that for any (x0 , z0 , w0 , ε) ∈ Rn × Rm × Rp × Lloc (R, Rp ) such that, along 12

solutions to (1) with the output y = h(x) + ε, (18)-(24), for all t ∈ [0, ∞): |x(t) − xˆ(t)| ≤ σ ˇ (|x0 | + |ξ0 | + |w0 |, t) + sup γˇ (|ε(τ )|). τ ∈[0,t]

This completes the proof.



Remark. Condition (25) can be shown to be satisfied by linear and high gain observers, see Section 6.5 in [8].

Remark.

By setting l to 1 (i.e. sampled-data case), we see that Proposition 3.1 is a

particular case of Theorem 4.1.

Two results are now derived from Theorem 4.1. Corollary 4.1 Assuming Hypotheses 3.1-3.3, 4.1 and condition (25) hold and consider system (1)-(2) when system measurements are sent through a network that is ruled by a l P uniform RR protocol. Then system (18)-(24) is robust observer for (1)-(2) if lT Ki < 1. i=1

Corollary 4.2 Assuming Hypotheses 3.1-3.3, 4.1 hold and, suppose that, for any i ∈ J1, lK there exist Ki ∈ R≥0 and a class-KL function σ ¯i , such that for any (x0 , z0 ) ∈ Rn×m and ε = T T p [εT 1 , . . . , εl ] ∈ Lloc (R, R ), along solutions to (1) with the perturbed output y = H(x) + ε,

and (9)-(10), for all t ∈ R≥0 : |Lf (ˆx(t)) Hi (ˆ x(t)) − Lf (x(t)) Hi (x(t))| ≤ σ ¯i (|x0 | + |z0 |, t) + Ki sup |εi (τ )|,

(36)

τ ∈[0,t]

then system (18)-(24) is a robust observer for (1)-(2) if Ti Ki < 1 for all i ∈ J1, lK. Remark. Condition (36) is less general than (25) since the right-hand side only depends on a sub-vector of ε. This usually means that the system is composed of l independent sub13

systems corresponding to each node. For such particular classes of systems, the obtained bound on MATI is less conservative than in Theorem 4.1.

4.2

For an error discrepancy protocol

In this section, an error discrepancy, also called TOD (Try-Once-Discard), protocol is considered, that relies on the difference between the fictitious and the real output wi and yi at each node. It has been shown in [11], that TOD protocols do not generally satisfy Hypothesis 4.1. Moreover, when the network is ruled by a TOD protocol and the observer is implemented using the predictive-type in-network processing algorithm, like it is the case here, the use of smart sensors is required [9]. Indeed, the implementation is the following: identical observers are placed at each node and also at the ‘end’ of the network where the whole observation is realised, see Fig. 2 in [1]. Therefore, at each transmission instant, the error between wi and yi , is compared at each node. The node where this value is the biggest is selected to transmit its data to all observers. A counterpart of such a methodology is that it requires the use of l + 1 observers contrary to Section 4.1. Because of the complexity of this protocol in terms of sensors and hardware implementations, it can be used in variants of control area network (CAN) but not over wireless channels and some wired networks (see [14]) over CAN, as explained in [13]. The following hypothesis ensures the synchronisation of the l + 1 observers implemented over the network, for all time, since at each transmission instant, all observers receive the same input (for more details, see [1, 9] and Chapter 4.3 in [8]). Hypothesis 4.2 All identical observers placed at each node have the same initial conditions.

14

The protocol gives access to the node i0 = arg max {|wj − Hj |}. When several nodes j∈J1,lK

correspond to the maximum error, a preference order is supposed to apply. The function ς is defined as follows, i ∈ J1, lK: ςi : Rp × Rp → Rni    Hi if i = arg max {|wj − Hj |} j∈J1,lK (w, h) 7→   w otherwise. i

(37)

For the sake of clarity, it is supposed here that the transmission instants occur uniformly at a constant period T ∈ R>0 . The following lemma will be useful in the sequel. Lemma 4.1 Assuming Hypotheses 3.1-3.3, 4.2 and (25) hold, for all i ∈ J1, lK, for all k ≥ l − 1, for all t ∈ [τk , τk+1 ) ∩ [0, tmax ), the following inequality is satisfied along solutions to (1)-(2),(18)-(23) and (37):

|wi (t) − Hi (x(t))| ≤ lT

l X j=1

σ ¯j (|x0 | + |z0 |, τk−l+1)

+lT max {Kj } sup |w(τ ) − h(x(τ ))| + l max { sup |εj (τ )|}. j∈J1,lK

j∈J1,lK 0≤τ ≤t

0≤τ ≤t

(38)

Proof. Consider i ∈ J1, lK and k ≥ l − 1. If ςi (τk , w(τk− ), h(x(τk ))) = Hi (x(τk )), the desired result holds. Let focus on the case where ςi (τk , w(τk− ), h(x(τk ))) 6= Hi (x(τk )). For all t ∈ [τk , τk+1 ) ∩ [0, tmax ), in view of (25): |wi (t) − Hi (x(t))| ≤ T σ ¯i (|x0 | + |z0 |, τk ) + T Ki sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+|wi (τk ) − Hi (x(τk ))| + sup |εi (τ )|.

(39)

0≤τ ≤t

Two cases have to be investigated. (a) If for all j ∈ Jk − l + 1, kK, ςi (τj , w(τj− ), h(x(τj ))) 6= Hi (x(τj )), then, since k ≥ l − 1, at least one node, denoted i0 , has been activated at least twice over the interval [τk−l+1 , τk ] 15

at instants denoted j1 < j2 . Consequently, for t ∈ [τj2 , τj2 +1 ) ∩ [0, tmax ), ¯i (|x0 | + |z0 |, τj2 ) |wi (t) − Hi (x(t))| ≤ |wi (τj2 ) − Hi (x(τj2 ))| + T σ +T Ki sup |w(τ ) − h(x(τ ))| + sup |εi (τ )| 0≤τ ≤t

≤

|wi0 (τj−2 )

0≤τ ≤t

− Hi0 (x(τj2 ))| + T σ ¯i (|x0 | + |z0 |, τj2 )

+T Ki sup |w(τ ) − h(x(τ ))| + sup |εi (τ )|. 0≤τ ≤t

(40)

0≤τ ≤t

On the other hand for t ∈ [τj2 −1 , τj2 ) ∩ [0, tmax ), j2 −1

|wi0 (t) − Hi0 (x(t))| ≤ T

X

q=j1

σ ¯i0 (|x0 | + |z0 |, τq ) + (j2 − j1 )T Ki0 sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+(j2 − j1 ) sup |εi0 (τ )|,

(41)

0≤τ ≤t

hence, for t ∈ [τj2 , τj2 +1 ) ∩ [0, tmax ), j2 −1

|wi (t) − Hi (x(t))| ≤ T

X

q=j1

σ ¯i0 (|x0 | + |z0 |, τq ) + (j2 − j1 )T Ki0 sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+(j2 − j1 ) sup |εi0 (τ )| + T σ ¯i (|x0 | + |z0 |, τj2 ) 0≤τ ≤t

+T Ki sup |w(τ ) − h(x(τ ))| + sup |εi (τ )|. 0≤τ ≤t

0≤τ ≤t

16

(42)

Consider now t ∈ [τk , τk+1 ) ∩ [0, tmax ), in view of (39), if j2 < k, |wi (t) − Hi (x(t))| ≤ T σ ¯i (|x0 | + |z0 |, τk ) + T Ki sup |w(τ ) − h(x(τ ))| + sup |εi (τ )| 0≤τ ≤t

0≤τ ≤t

j2 −1

+T

X

q=j1

σ ¯i0 (|x0 | + |z0 |, τq ) + (j2 − j1 )T Ki0 sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+(j2 − j1 ) sup |εi0 (τ )| + T 0≤τ ≤t

k−1 X

q=j2

σ ¯i (|x0 | + |z0 |, τq )

+T Ki (k − j2 ) sup |w(τ ) − h(x(τ ))| + (k − j2 ) sup |εi (τ )| 0≤τ ≤t

≤ lT

l X j=1

0≤τ ≤t

σ ¯j (|x0 | + |z0 |, τk−l+1) + lT max {Kj } sup |w(τ ) − h(x(τ ))| j∈J1,lK

0≤τ ≤t

+l max { sup |εj (τ )|}.

(43)

j∈J1,lK 0≤τ ≤t

If j2 = k, |wi (t) − Hi (x(t))| ≤ T σ ¯i (|x0 | + |z0 |, τk ) + T Ki sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+ sup |εi (τ )| + T 0≤τ ≤t

k−1 X

q=j1

σ ¯i0 (|x0 | + |z0 |, τq )

+(k − j1 )T Ki0 sup |w(τ ) − h(x(τ ))| + (k − j1 ) sup |εi0 (τ )|. 0≤τ ≤t

0≤τ ≤t

In both cases, the desired result is verified. (b) If there exists j ∈ Jk − l + 1, k − 1K, such that ςi (τj , w(τj− ), h(x(τj ))) = Hi (x(τj )), for all t ∈ [τj , τj+1 ) ∩ [0, tmax ), |wi (t) − Hi (x(t))| ≤ T σ ¯i (|x0 | + |z0 |, τj ) + T Ki sup |w(τ ) − h(x(τ ))| + sup |εi (τ )|, 0≤τ ≤t

17

0≤τ ≤t

then, for t ∈ [τk , τk+1 ) ∩ [0, tmax ), k−1 X

|wi (t) − Hi (x(t))| ≤

q=j

Tσ ¯i (|x0 | + |z0 |, τq ) + (k − j)T sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+(k − j) sup |εi (τ )|.

(44)

0≤τ ≤t

Each possible case has been considered: the proof is completed.



Theorem 4.2 Assume Hypotheses 3.1-3.3 and 4.2 and condition (25) apply and suppose 3

that l 2 T max {Ki } < 1. Then the hybrid system (18)-(23), (37) is a robust observer for i∈J1,lK

(1)-(2). Proof. Like in the proof of Theorem 4.1, it can be proved that there exist an observer initial condition such that xˆ and x are synchronised on [0, tmax ). Let (x0 , z0 , w0) ∈ Rn × Rm × Rp , T T ε = [εT ∈ Lloc (R, Rp ) and i ∈ J1, lK. According to Lemma 4.1, for all t ∈ 1 , . . . , εl ]

[τk , τk+1 ) ∩ [0, tmax ), k ≥ l − 1, it can be shown that 3 2

|w(t) − h(x(t))| ≤ l T 3 2

l X j=1

σ ¯j (|x0 | + |z0 |, τk−l+1)

+l T max {Kj } sup |w(τ ) − h(x(τ ))| j∈J1,lK

0≤τ ≤t

3 2

+l max { sup |εj (τ )|}.

(45)

j∈J1,lK 0≤τ ≤t

For t ∈ [0, τl−1 ) ∩ [0, tmax ), the following bound can be obtained, for all i ∈ J1, lK: |wi (t) − Hi (x(t))| ≤ lT σ ¯i (|x0 | + |z0 |, 0) + lT Ki sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+(l − 1) sup |εi (τ )| + |wi (0) − Hi (x(0))| 0≤τ ≤t

≤ lT σ ¯i (|x0 | + |z0 |, 0) + lT Ki sup |w(τ ) − h(x(τ ))| 0≤τ ≤t

+(l − 1) sup |εi (τ )| + ρi (|wi (0)| + |x(0)|), 0≤τ ≤t

18

(46)

where ρi is a class-K∞ function such that |wi − hi (x)| ≤ ρi (|wi | + |x|) for all (wi , x) ∈ Rni × Rn . It can then be deduced that, for all t ∈ [0, τl−1 ) ∩ [0, tmax ), 3 2

|w(t) − h(x(t))| ≤ l T

l X j=1

3

σ ¯j (|x0 | + |z0 |, 0) + l 2 T max {Kj } sup |w(τ ) − h(x(τ ))| j∈J1,lK

0≤τ ≤t

l X √ √ + l(l − 1) sup |εj (τ )| + lρ(|w(0)| + |x(0)|), j=1

(47)

0≤τ ≤t

where ρ ∈ K∞ . Defining the function σ ˘ as:   l  √ P    l lT σ¯i (s, 0) + ρ(s) elT −t  i=1  σ ˘ (s, t) = l √ P  lT −t   l lT σ¯i (s, t − lT ) + ρ(s)e i=1

if t < lT ,

(48)

if t ≥ lT

this holds for all t ∈ [0, tmax ):

3

|w(t) − h(x(t))| ≤ σ ˘ (|x0 | + |z0 | + |w0 |, t) + l 2 T max {Kj } sup |w(τ ) − h(x(τ ))| j∈J1,lK

0≤τ ≤t

3 2

+l (l − 1) sup |ε(τ )|.

(49)

0≤τ ≤t

The proof is completed by following the same lines than in the proof of Theorem 4.1, 3

knowing that l 2 T max {Ki } < 1



i∈J1,lK

Remark. We are not able to say whether Theorem 4.2 always provides a less conservative estimate of the bound on MATI than Theorem 4.1, from this analysis. However, performed simulations confirm that the dynamic protocols generally allow to consider larger MATI, see Chapter 6.6 in [8].

Corollary 4.3 Considering system (1)-(2), assuming Hypotheses 3.1-3.3 and 4.2 and (36) hold, system (18)-(23), (37) is a robust observer for (1)-(2) if lT maxi∈J1,lK {Ki } < 1.

19

Sketch of Proof. The proof follows the same line than the proof of of Theorem 4.2 but using small gain arguments on max |wi − hi (x)| instead of |w − h(x)|. i∈J1,lK

Remark.



A slightly modified implementation can be done when (36) is ensured for the

case described in Remark 4.1: only a copy of the appropriate ‘sub-observer’ corresponding to the ‘sub-system’ associated to a node can be placed at the node.

As mentioned in the Introduction, results of Sections 4.1 and 4.2 can be used to derive new bounds on MATI bounds for linear observers and high-gain observers. Due to space limitations, these results are not presented in this paper but can be found in Chapter 6 in [8].

5

Conclusion

In this paper, observers have been designed for nonlinear NCS considering two classes of protocols. By extending the method of [5], sufficient conditions on observers are derived that ensure some global asymptotic stability properties for the observation error under networked-induced constraints.

References [1] D. Daˇci´c and D. Neˇsi´c. Observer design for linear networked control systems using matrix inequalities. Automatica, 44(1):2840–2848, 2008. [2] J. Hespanha, P. Naghshtabrizi, and Y. Xu. A survey of recent results in networked control systems. IEEE Special Issue on Technology of Networked Control Systems, 95(1):138–162, 2007.

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[3] Z. Jin, V. Gupta, and R.M. Murray. State estimation over packet dropping networks using multiple description coding. Automatica, 42:1441–1452, 2006. [4] I. Karafyllis and Z.-P. Jiang. A small-gain theorem for a wide class of feedback systems with control applications. SIAM Journal on Control and Optimization, 46(4):1483– 1517, 2007. [5] I. Karafyllis and C. Kravaris. From continuous-time design to sampled-data design of observers. IEEE Transactions on Automatic Control, 54(9):2169–2174, 2009. [6] A. Matveev and A. Savkin. The problem of state estimation via asynchronous communication channels with irregular transmission times. IEEE Trans. on Automatic Control, 48:670–676, 2003. [7] D. Neˇsi´c and A.R. Teel. Input-output stability properties of networked control systems. IEEE Transactions on Automatic Control, 49:1650–1667, 2004. [8] R. Postoyan. Commande et construction d’observateurs pour les syst`emes non lin´eaires `a donn´ees ´echantillonn´ees et en r´eseau. PhD thesis, Univ Paris-Sud (in French), 2009. [9] R. Postoyan and D. Neˇsi´c. A framework for the observer design for networked control systems. In ACC’10 (American Control Conference), Baltimore, U.S.A. (to appear), 2010. [10] B. Sinopoli, L. Schenato, M. Franschetti, K. Poolla, M.I. Jordan, and S.S. Sastry. Kalman filtering with intermittent observations. IEEE Transactions on Automatic Control, 49:1453–1464, 2004. [11] M. Tabbara, D. Neˇsi´c, and A.R. Teel. Stability of wireless and wireline networked control systems. IEEE Transactions on Automatic Control, 52(9):1615–1630, 2007.

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[12] G. C. Walsh and H. Ye. Scheduling of networked systems. IEEE Control Systems Magazine, 21(1):57–65, 2001. [13] G.C. Walsh, O. Beldiman, and L.G. Bushnell. Asymptotic behavior of nonlinear networked control systems. IEEE Transactions on Automatic Control, 46:1093–1097, 2001. [14] G.C. Walsh and H. Ye. Scheduling of networked control systems. IEEE Control Systems Magazine, 21(1):57–65, 2001. [15] J.K. Yook, D.M. Tilbury, and N.R. Soparkar. Trading computation for bandwith: reducing communication in distributed control systems using state estimators. IEEE Transactions on Control Systems Technology, 10(4):503–518, 2002. [16] L. Zhang and D. Hristu-Varsakelis. Stabilization of networked control system: designing effective communication sequence. In 16th IFAC World Congress, Prague, Czech Republic, 2005.

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