Tijs Donkers

Maurice Heemels

Nathan Van De Wouw

Laurentiu Hetel

Stability Analysis of Networked Control Systems Using a Switched Linear Systems Approach

published or not. The documents may come

Abstract-In this paper, we study the stability of Networked Control Systems (NCSs) that are subject to time-varying transmission intervals, time-varying transmission delays and communication constraints. Communication constraints impose that, per transmission, only one node can access the network and send its information. The order in which nodes send their information is orchestrated by a network protocol, such as, the Round-Robin (RR) and the Try-Once-Discard (TOD) protocol. In this paper, we generalise the mentioned protocols to novel classes of socalled 'periodic' and 'quadratic' protocols. By focussing on linear plants and controllers, we present a modelling framework for NCSs based on discrete-time switched linear uncertain systems. This framework allows the controller to be given in discrete time as well as in continuous time. To analyse stability of such systems for a range of possible transmission intervals and delays, with a possible nonzero lower bound, we propose a new procedure to obtain a convex overapproximation in the form of a polytopic system with norm-bounded additive uncertainty. We show that this approximation can be made arbitrarily tight in an appropriate sense. Based on this overapproximation, we derive stability results in terms of Linear Matrix Inequalities (LMIs). We illustrate our stability analysis on the benchmark example of a batch reactor and show how this leads to tradeoffs between different protocols, allowable ranges of transmission intervals and delays. In addition, we show that the exploitation of the linearity of the system and controller leads to a significant reduction in conservatism with respect to existing approaches in the literature.

Index Terms-Communication Constraints, Networked Control Systems, Switched Systems, Stability, Time-Varying Systems, Uncertain Systems

I. INTRODUCTION

N ETWORKED Control Systems (NCSs) are systems in which control loops are closed over a real-time communication network. The fact that controllers, sensors, and actuators are not connected through point-to-point connections, but through a multipurpose network offers advantages, such as increased system flexibility, ease of installation and maintenance, and decreased wiring and cost. However, networking This work is supported by the Dutch Science Foundation (STW) and the Dutch Organisation for Scientific Research (NWO) under the VICI grant "Wireless controls systems: A new frontier in automation", by the European Community through the FP7-ICT-2007-2 thematic programme under the WIDE-224168 project, and by the Embedded Systems Institute, Eindhoven, the Netherlands. Tijs Donkers, Maurice Heemels, and Nathan van de Wouw are with the Mechanical Engineering department of Eindhoven University of Technoloy, the Netherlands, {m.c.f.donkers, m.heemels, n.v.d.wouw}@tue.nl.

Laurentiu Hetel is with the Laboratoire d'Automatique, Génie Informatique et Signal (LAGIS) in Lille, France, laurentiu.hetel@ec-lille.fr. i.e., type (iii), (iv) and (v) phenomena.

Stability of NCSs subject to communication constraints, time-varying transmission intervals and transmission delays has already been considered in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], [START_REF] Chaillet | Delay compensation in packet-switching networked controlled sytems[END_REF]. The communication constraints impose that, per transmission, only one node can access the network and send its information and, hence, a protocol is needed to orchestrate when a certain communication node is given access to the network. Given a protocol, such as the Round-Robin (RR) and the Try-Once-Discard (TOD) protocol, the mentioned papers provide criteria for computing the so-called Maximum Allowable Transmission Interval (MATI) and the Maximum Allowable Delay (MAD). Stability is guaranteed as long as the actual transmission intervals and delays are always smaller than the MATI and MAD, respectively. The difference between the work in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF] and [START_REF] Chaillet | Delay compensation in packet-switching networked controlled sytems[END_REF], is that in the latter a delay compensation scheme is proposed. This delay compensation requires time stamping of the messages and sending future control signals in larger packets, which is not needed in the more basic emulation based approach, as in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF] and the earlier work without transmission delays in [START_REF] Nešić | A unified approach to controller design for systems with quantization and time scheduling[END_REF], [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF]- [START_REF] Walsh | Asymptotic behavior of nonlinear networked control systems[END_REF], [START_REF] Tabbara | Stability of wireless and wireline networked control systems[END_REF]. Furthermore, the results in [START_REF] Chaillet | Delay compensation in packet-switching networked controlled sytems[END_REF] have the drawback that they are not applicable to the commonly used Round-Robin protocol, while [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF] is.

The work presented in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], [START_REF] Chaillet | Delay compensation in packet-switching networked controlled sytems[END_REF] both apply to general nonlinear plants and controllers and are based on a continuoustime modelling paradigm related to hybrid systems as in [START_REF] Goebel | Hybrid dynamical systems[END_REF]. However, neither [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], nor [START_REF] Chaillet | Delay compensation in packet-switching networked controlled sytems[END_REF] include the possibility that the controller is formulated in discrete time. The case of discretetime controllers has been considered in [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], where, however, a fixed transmission interval and no delay are assumed. Another feature of [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], [START_REF] Chaillet | Delay compensation in packet-switching networked controlled sytems[END_REF] is that, in these works, zero lower bounds on the transmission intervals h k and delays τ k are considered (i.e., h k ∈ (0, h MATI ], τ k ∈ [0, τ MAD ]). The ability to handle discrete-time controllers and nonzero lower bounds on the transmission intervals and delays is highly relevant from a practical point of view, because controllers are typically implemented in a digital and, thus, discrete-time form. Furthermore, finite communication bandwidth always introduce nonzero lower bounds on the transmission intervals and transmission delays. This motivates the need for studying these situations as well, preferably in a nonconservative manner. Although the work presented in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], [START_REF] Chaillet | Delay compensation in packet-switching networked controlled sytems[END_REF] is very general and can accommodate for many nonlinear NCSs, their results cannot reduce conservatism when a certain structure is present in the NCS, such as linearity of the controller and plant.

In this paper, we focus on linear plants and linear controllers and study the stability of the corresponding NCS in the presence of communication constraints, time-varying transmission intervals and time-varying delays, where the latter two possibly have a nonzero lower bound. Moreover, we allow for both a continuous-time as well as a discrete-time controller, which requires a different modelling paradigm than in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], [START_REF] Chaillet | Delay compensation in packet-switching networked controlled sytems[END_REF], and in the work without transmission delays, [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF]- [START_REF] Walsh | Asymptotic behavior of nonlinear networked control systems[END_REF]. In particular, we provide techniques for assessing stability of the NCS with time-varying transmission intervals h k ∈ [h, h] and timevarying transmission delays τ k ∈ [τ , τ ] for two well-known protocols, namely, the Round-Robin (RR) protocol and the Try-Once-Discard (TOD) protocol, and their generalisations.

These generalisations consist of the classes of 'periodic' and 'quadratic' protocols, which are formally introduced here. In contrast with [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], [START_REF] Chaillet | Delay compensation in packet-switching networked controlled sytems[END_REF], we apply a discrete-time modelling framework that leads to a switched linear system model with exponential uncertainty. To properly handle this exponential uncertainty, we provide a polytopic overapproximation for this system. This overapproximation is obtained using a novel procedure that combines ideas from gridding [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF], [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF] and norm bounding [START_REF] Balluchi | Controller synthesis on non-uniform and uncertain discrete-time domains[END_REF]- [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF]. Unlike other methodologies for obtaining a convex overapproximation, see, e.g., [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF]- [START_REF] Gielen | On polytopic inclusions as a modeling framework for systems with time-varying delays[END_REF] and the overview paper [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF], we provide a proof that the newly proposed procedure can be made arbitrarily tight in an appropriate sense. Using this overapproximated system, we can assess stability using newly developed conditions based on Linear Matrix Inequalities (LMIs). We will show the effectiveness of the presented approach on the benchmark example of a batch reactor as used in [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF]- [START_REF] Walsh | Stability analysis of networked control systems[END_REF], [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], [START_REF] Tabbara | Stability of wireless and wireline networked control systems[END_REF], as well. Moreover, we will show that the linearity of plant and controller can indeed be exploited, which leads to a significant reduction of conservatism with respect to the existing approaches.

The remainder of this paper is organised as follows. After introducing the necessary notational conventions, we introduce the model of the NCS in Section II and propose a method to write it as a discrete-time switched linear uncertain system. We also state a precise problem formulation. Subsequently, in Section III, we provide a procedure to overapproximate the NCS model by a polytopic system with norm-bounded uncertainty. In Section IV, we provide conditions for stability of the NCS in terms of LMIs and reflect in Section V on the conservatism this approach introduces. Finally, we illustrate the stability results using a numerical benchmark example in Section VI and draw conclusions in Section VII. The appendix contains the proofs of the more technical lemmas and theorems.

A. Nomenclature

The following notational conventions will be used. diag(A 1 , . . . , A n ) denotes a block-diagonal matrix with the entries A 1 , . . . , A n on the diagonal and A ∈ R m×n denotes the transposed of matrix A ∈ R n×m . For a vector x ∈ R n , we denote by x i the i-th component and x :=

√

x x its Euclidean norm. We denote by A := λ max (A A) its the spectral norm, which is the square-root of the maximum eigenvalue of the matrix A A. For brevity, we sometimes write symmetric matrices of the form

A B B C , as A B
C . Finally, by lim s↓t and lim s↑t , we denote the limit as s approaches t from above and below, respectively, and the convex hull and interior of a set A are denoted by coA and intA, respectively.

II. NCS MODEL

dt x p (t) = A p x p (t) + B p û(t) y(t) = C p x p (t), (1)
where x p ∈ R np denotes the state of the plant, û ∈ R nu the most recently received control variable, y ∈ R ny the (measured) output of the plant and t ∈ R + the time. The controller, also an LTI system, is assumed to be given in either continuous time by

d dt x c (t) = A c x c (t) + B c ŷ(t) u(t) = C c x c (t) + D c ŷ(t), (2a)
or in discrete time by

x c k+1 = A c x c k + B c ŷk u(t k ) = C c x c k + D c ŷ(t k ). (2b)
In these descriptions, x c ∈ R nc denotes the state of the controller, ŷ ∈ R ny the most recently received output of the plant and u ∈ R nu denotes the controller output. At transmission instant t k , k ∈ N, (parts of) the outputs of the plant y(t k ) and controller u(t k ) are sampled and are transmitted over the network. We assume that they arrive at instant r k , called the arrival instant. The situation described above is illustrated in Fig. 1. In the case we have a discretetime controller (2b), the states of the controller x c k+1 are updated using ŷk := lim t↓r k ŷ(t), i.e., as in [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], directly after ŷ is updated. Note that in this case, the update of x c k+1 in (2b) has to be performed in the time interval (r k , t k+1 ].

Let us now explain in more detail the functioning of the network and define these 'most recently received' ŷ and û exactly, see also [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF]- [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]. The plant is equipped with sensors and actuators that are grouped into N nodes. At each transmission instant t k , k ∈ N, one node, denoted by σ k ∈ {1, . . . , N }, obtains access to the network and transmits its corresponding values. These transmitted values are received and implemented on the controller or the plant at arrival instant r k . As in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], a transmission only occurs after the previous transmission has arrived, i.e., t k+1 > r k t k , for all k ∈ N. In other words, we consider the sampling interval to be lower bounded and the delays to be smaller than the transmission interval. After each transmission and reception, the values in ŷ and û are updated with the newly received values, while the other values in ŷ and û remain the same, as no additional information is received. This leads to the constrained data exchange expressed as

ŷ(t) = Γ y σ k y(t k ) + (I -Γ y σ k )ŷ(t k ) û(t) = Γ u σ k u(t k ) + (I -Γ u σ k )û(t k ) (3)
for all t ∈ (r k , r k+1 ], where Γ σ k := diag(Γ y σ k , Γ u σ k ) is a diagonal matrix, given by

Γ i = diag(γ i,1 , . . . , γ i,ny+nu ). (4)
when σ k = i. In (4), the elements γ i,j , with i ∈ {1, . . . , N } and j ∈ {1, . . . , n y }, are equal to one, if plant output y j is in node i, elements γ i,j+ny , with i ∈ {1, . . . , N } and j ∈ {1, . . . , n u }, are equal to one, if controller output u j is in node i, and are zero elsewhere. The value of σ k ∈ {1, . . . , N } in (3) indicates which node is given access to the network at transmission instant t k , k ∈ N. Indeed, [START_REF] Tipsuwan | Control methodologies in networked control systems[END_REF] reflects that the values in û and ŷ corresponding to node σ k are updated just after r k , with the corresponding transmitted values at time t k , while the others remain the same. A scheduling protocol determines the sequence (σ 0 , σ 1 , . . .) and particular protocols will be made explicit later.

The transmission instants t k , as well as the arrival instants r k , k ∈ N are not necessarily distributed equidistantly in time. Hence, both the transmission intervals h k := t k+1 -t k and the transmission delays τ k := r k -t k are varying in time, as is also illustrated in Fig. 1. We assume that the variations in the transmission interval and delays are bounded and are contained in the sets [h, h] and [τ , τ ], respectively, with h h > 0 and τ τ 0. Since we assumed that each transmission delay τ k is smaller than the corresponding transmission interval h k , we have that (h k , τ k ) ∈ Θ, for all k ∈ N, where

Θ := (h, τ ) ∈ R 2 | h ∈ [h, h], τ ∈ [τ , min{h, τ }) . (5)
Remark II.1 In the above reasoning, we implicitly assumed that packet loss does not occur, similar to, e.g., [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], [START_REF] Carnevale | A Lyapunov proof of improved maximum allowable transfer interval for networked control systems[END_REF], [START_REF] Walsh | Stability analysis of networked control systems[END_REF], [START_REF] Walsh | Asymptotic behavior of nonlinear networked control systems[END_REF]. However, we could accommodate for packet dropouts by modelling them as prolongations of the transmission interval, as done in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]. This means that if we assume that there is a bound δ ∈ N on the maximum number of successive dropouts, and we have stability of the NCS for (h k , τ k ) ∈ Θ, for all k ∈ N, in the case without dropouts, then the NCS with dropouts is still guaranteed to be stable for (h k , τ k ) ∈ Θ , for all k ∈ N, where

Θ := (h, τ ) ∈ R 2 | h ∈ [h, h ], τ ∈ [τ , min{h, τ }) (6)
in which h := h δ+1 .

A. The NCS as a discrete-time switched uncertain system

To analyse stability of the NCS described above, we transform it into a discrete-time model. In this framework, we need a discrete-time equivalent of (1) and also of (2a) in case a continuous-time controller is used. To arrive at this description, let us first define the network-induced error as e y (t) := ŷ(t) -y(t) e u (t) := û(t) -u(t).

The discrete-time switched uncertain system can now be obtained by describing the evolution of the states between t k and t k+1 = t k +h k . In order to do so, we define x p k := x p (t k ), u k := u(t k ), ûk := lim t↓r k û(t) and e u k := e u (t k ). Since û, as in (3), is a piecewise constant left-continuous signal, i.e.,

= A h k + E h k BDC E h k BD -E h k -τ k BΓ σ k C(I -A h k -E h k BDC) I -D -1 Γ σ k + C(E h k -τ k BΓ σ k -E h k BD) =: Ãσ k ,h k ,τ k xk ( 12
)
lim s↑t û(s) = û(t), we can write ûk-1 = lim t↓r k-1 û(t) = û(r k ) = û(t k ). This allows us to write the exact discretisation of (1) as follows:

x p k+1 = e A p h k x p k + h k 0 e A p (h k -s) B p û(t k + s)ds = e A p h k x p k + τ k 0 e A p (h k -s) dsB p ûk-1 + h k τ k e A p (h k -s) dsB p ûk . ( 8
)
As ( 3) and ( 7)

yield ûk-1 = u k + e u k and ûk-1 -ûk = Γ u σ k e u k , (8)
can be rewritten as

x p k+1 = e A p h k x p k + h k h k -τ k e A p s dsB p ûk-1 + h k -τ k 0 e A p s dsB p ûk = e A p h k x p k + h k 0 e A p s dsB p ûk-1 + h k -τ k 0 e A p s dsB p (û k -ûk-1 ) = e A p h k x p k + h k 0 e A p s dsB p (u k + e u k ) - h k -τ k 0 e A p s dsB p Γ u σ k e u k . (9)
A discretised equivalent of (2a) is obtained in a similar fashion by defining x c k := x c (t k ), y k := y(t k ), e y k := e y (t k ), ŷk := lim t↓r k ŷ(t), and observing ŷk-1 = ŷ(t k ), and is given by

x c k+1 = e A c h k x c k + h k 0 e A c s dsB c (y k + e y k ) - h k -τ k 0 e A c s dsB c Γ y σ k e y k . (10)
We now present three different models, each describing a particular NCS. The first and the second model cover the situation where both the plant and the controller outputs are transmitted over the network, differing by the fact that the controller is given by (2a) and (2b), respectively. In the third model, it is assumed that the controller is given by (2a) and that only the plant outputs y are transmitted over the network and u are sent continuously via an ideal nonnetworked connection. We include this particular case, because it is often used in examples in NCS literature (see, e.g., the benchmark example in [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF]- [START_REF] Walsh | Stability analysis of networked control systems[END_REF], [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]) and it allows us to compare our methodology to the existing ones.

1) The NCS model with controller (2a): For an NCS having controller (2a), the complete NCS model is obtained by combining (3), ( 7), [START_REF] Nair | Stabilizability of stochastic linear systems with finite feedback data rates[END_REF], and (10) and defining

xk := x p k x c k e y k e u k . (11)
This results in the discrete-time model [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF], as shown on the top of this page, in which

Ãσ k ,h k ,τ k ∈ R n×n , with n = n p + n c + n y + n u , and
A h k := diag(e A p h k , e A c h k ), B := 0 B p B c 0 , (13a)
C := diag(C p , C c ), D := I 0 D c I , (13b)
E ρ := diag( ρ 0 e A p s ds, ρ 0 e A c s ds), ρ ∈ R. (13c)
2) The NCS model with controller (2b): For an NCS having controller (2b), the complete NCS model is obtained by combining (2b), ( 3), [START_REF] Heemels | Control of quantized linear systems: an l 1 -optimal control approach[END_REF], and ( 9), also resulting in [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF], in which now

A h k := diag(e A p h k , A c ), B := 0 B p B c 0 , (14a)
C := diag(C p , C c ), D := I 0 D c I , (14b)
E ρ := diag( ρ 0 e A p s ds, I), ρ ∈ R. ( 14c
)
3) The NCS model if only y is transmitted over the network: In this case we assume that only the outputs of the plant are transmitted over the network and the controller communicates its values continuously and without delay. We therefore have that u(t) = û(t), for all t ∈ R + , which allows us to combine (1) and (2a), yielding

ẋp (t) ẋc (t) = A p B p C c 0 A c x p (t) x c (t) + B p D c B c ŷ(t). ( 15
)
Since ŷ is still updated according to (3), we can describe the evolution of the states between t k and t k+1 = t k + h k in a similar fashion as in [START_REF] Nair | Stabilizability of stochastic linear systems with finite feedback data rates[END_REF]. In this case, (11) reduces to

xk := x p k x c k e y k , (16)
resulting in [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF], in which

A h k := e A p B p C c 0 A c h k , B := B p D c B c , (17a)
C := C p 0 , D := I, (17b)
E ρ := ρ 0 e A p B p C c 0 A c s ds, ρ ∈ R. (17c)

B. Protocols as a Switching Function

Based on the previous modelling steps, the NCS is formulated as a discrete-time switched uncertain system [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF]. In this framework, protocols are considered as the switching function determining σ k . We consider two commonly used protocols, see [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF]- [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], [START_REF] Tabbara | Stability of wireless and wireline networked control systems[END_REF], namely the Try-Once-Discard (TOD) and the Round-Robin (RR) protocol and generalise these into two novel classes of protocols, named 'quadratic' and 'periodic' protocols.

1) Quadratic Protocols: A quadratic protocol is a protocol, for which the switching function can be written as

σ k = arg min i=1,...,N x k P i xk , (18)
where P i , i ∈ {1, . . . , N }, are certain given matrices. In case two nodes have the same maximal values, one of them can be chosen arbitrarily. In fact, the well-known TOD protocol, sometimes also called Maximum Error First (MEF) protocol, belongs to this class of protocols. In this protocol, the node that has the largest network-induced error, i.e., the difference between the most recently transmitted values and its current values of the signals corresponding to the node, is granted access to the network. We can arrive at the TOD protocol by adopting the following structure in the P i matrices:

P i = P -diag(0, Γ i ), (19)
in which Γ i , i ∈ {1, . . . , N }, is given by ( 4). Furthermore, if we define ẽi k := Γ i e k , where e k := [e y k , e u k ] , [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] becomes

σ k = arg min -e k Γ 1 e k , . . . , -e k Γ N e k = arg max ẽ1 k , . . . , ẽN k , (20)
which is the TOD protocol.

2) Periodic Protocols: Another class of protocols that is considered in this paper is the class of so-called periodic protocols. A periodic protocol is a protocol that satisfies for some Ñ ∈ N

σ k+ Ñ = σ k (21)
for all k ∈ N. Ñ is then called the period of the protocol. Actually, the well-known RR protocol belongs to this class and is defined by

{σ 1 , . . . , σ N } = {1, . . . , N }, (22)
and period Ñ = N , i.e., during each period of the protocol every node has access to the network exactly once. The above modelling approach now provides a description of the NCS system in the form of a discrete-time switched linear uncertain system given by [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] and one of the protocols, characterised by [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF]. The system switches between N linear uncertain systems and the switching is due to the fact that only one node accesses the network at each transmission instant. The uncertainty is caused by the fact that the transmission intervals and the transmission delays (h k , τ k ) ∈ Θ are varying over time.

C. Stability of the NCS

The problem studied in this paper is to determine the stability of the continuous-time NCS, given by ( 1), (2a) or (2b), (3), and [START_REF] Heemels | Control of quantized linear systems: an l 1 -optimal control approach[END_REF], with protocols satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF] given the bounds [h, h] and [τ , τ ], or to find bounds that guarantee stability. Let us now formally define stability for this continuous-time NCS.

Definition II.2 The continuous-time NCS given by ( 1), (2a) or (2b), (3), and [START_REF] Heemels | Control of quantized linear systems: an l 1 -optimal control approach[END_REF], with protocols satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF], having states x(t) := x p (t) x c (t) e y (t) e u (t) ∈ R n , is said to be Uniformly Globally Exponentially Stable (UGES) if there exist c 0 and λ > 0, such that for any initial condition x(0), any sequence of transmission intervals (h 0 , h 1 , . . .), and any sequence of transmission delays

(τ 0 , τ 1 , . . .), with (h k , τ k ) ∈ Θ, for all k ∈ N, it holds that x(t) c x(0) e -λt , ∀ t ∈ R + . ( 23
)
Stability of the continuous-time NCS can be analysed by assessing stability of the discrete-time uncertain switched linear system [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] with switching functions satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or ( 21), as we will show. Let us now formally define stability of this discrete-time system.

Definition II.3 System (12) with switching sequences satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or ( 21) is said to be Uniformly Globally Exponentially Stable (UGES) if there exist c 0 and 0 λ < 1, such that for any initial condition x0 ∈ R n , any sequence of transmission intervals (h 0 , h 1 , . . .), and any sequence of transmission delays (τ 0 , τ 1 , . . .), with (h k , τ k ) ∈ Θ, for all k ∈ N, it holds that

xk c x0 λ k , ∀ k ∈ N. ( 24
)
Since the discrete-time switched uncertain linear system [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] with switching sequences satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF] is formulated in discrete time, we can only assess stability at the transmission instants. However, states of the plant (1) and controller (2a) actually evolve in continuous time. In the next lemma, we state that UGES of the discrete-time NCS model implies UGES of the continuous-time NCS.

Lemma II.4 Assume the discrete-time system [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] with switching sequences satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or ( 21) is UGES, then the continuous-time NCS given by (1), (2a) or (2b), (3), and (7), with protocols satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or (21) is also UGES.

Proof: The proof is given in the Appendix. This lemma states that it suffices to consider the discretetime model [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] with switching sequences satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF] to assess UGES of the continuous-time NCS system.

III. OBTAINING A CONVEX OVERAPPROXIMATION

In the previous section, we obtained an NCS model in the form of a switched uncertain system. However, the form as in [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] is not really convenient to develop efficient techniques for stability analysis due to the nonlinear dependence of Ãσ k ,h k ,τ k on the uncertain parameters h k and τ k . Therefore, we will Limited circulation. For review only Preprint submitted to IEEE Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST provide a procedure that overapproximates system (12) by a polytopic system with a norm-bounded additive uncertainty of the form

xk+1 = L l=1 α l k Āσ k ,l + B∆ k Cσ k xk , (25)
where Āσ,l ∈ R n×n , B ∈ R n×m , Cσ ∈ R m×n , for σ ∈ {1, . . . , N } and l ∈ {1, . . . , L}, with L the number of vertices of the polytope. Furthermore,

α k = [α 1 k . . . α L k ] ∈ A, k ∈ N,
denotes an unknown time-varying vector with

A = α ∈ R L L l=1 α l = 1, α l 0, l ∈ {1, . . . , L} (26)
and

∆ k ∈ ∆, k ∈ N,
where ∆ is a norm-bounded set of matrices in R m×m that describes the additive uncertainty. This additive uncertainty can have some specific structure, as we will see below. The model ( 25) should be an overapproximation of ( 12) in the sense that for all σ ∈ {1, . . . , N }, it holds that

Ãσ,h,τ | (h, τ ) ∈ Θ ⊆ L l=1 α l Āσ,l + B∆ Cσ | α ∈ A, ∆ ∈ ∆ . ( 27
)
In this paper, we use the gridding idea of [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF], [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF] to obtain, for a fixed σ, Āσ,l by evaluating Ãσ,h,τ of ( 12) at a collection of selected pairs of transmission intervals and transmission delays ( hl , τl ) ∈ Θ, l ∈ {1, . . . , L}. Hence, we take Āσ,l := Ãσ, hl ,τ l in [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], with l ∈ {1, . . . , L}. However, contrary to [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF], [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF], we choose to allow for convex combinations of the vertices, whereas in [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF], [START_REF] Fujioka | A discrete-time approach to stability analysis of systems with aperiodic sample-and-hold devices[END_REF] the system switches between the vertices only. Moreover, we construct a norm-bounded additive uncertainty ∆ ∈ ∆ to capture the remaining approximation error, as done in, e.g., [START_REF] Balluchi | Controller synthesis on non-uniform and uncertain discrete-time domains[END_REF]- [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF]. By comparing Ãσ,h,τ with the convex combinations of the vertices instead of with the vertices alone, we obtain smaller bounds on the additive uncertainty than in [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF]- [START_REF] Balluchi | Controller synthesis on non-uniform and uncertain discrete-time domains[END_REF], [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF].

By specifying ( hl , τl ), l ∈ {1, . . . , L}, and thereby determining Āσ,l , it only remains to show how to choose B∆ Cσ in [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF] and ∆ in order to satisfy [START_REF] Hristu | Limited communication control[END_REF]. This additive uncertainty is used to capture the approximation error between the original system (12) and the polytopic system

xk+1 = L l=1 α l k Āσ k ,l xk . ( 28
)
In order for [START_REF] Hristu | Limited communication control[END_REF] to hold, for each triple (σ, h, τ ), with σ ∈ {1, . . . , N } and (h, τ ) ∈ Θ, there should exist some α ∈ A and ∆ ∈ ∆, such that

Ãσ,h,τ - L l=1 α l Āσ,l = B∆ Cσ . (29)
Hence, we should determine the worst-case distance between the real system (12) and the polytopic system [START_REF] Rehbinder | Scheduling of a limited communication channel for optimal control[END_REF], leading to an upper bound on the approximation error. To obtain such an upper bound, we partition Θ into M triangles S 1 , . . . , S M , see Fig. 2, and we compare Ãσ,h,τ , for (h, τ ) ∈ S m , with where ( hl m j , τl m j ), j = {1, 2, 3}, denote the vertices (with vertex index l m j ∈ {1, . . . , L}, j ∈ {1, 2, 3} and m ∈ {1, . . . , M }) of triangle S m . This allows us to construct the right-hand side of ( 29) by computing the worst-case distance. Note that it is always possible to partition Θ into triangles, as Θ is a convex polytope. We will, however, also provide a systematic procedure to obtain a suitable partitioning.

A specific feature of the overapproximation presented in this paper is that, contrary to [START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF]- [START_REF] Gielen | On polytopic inclusions as a modeling framework for systems with time-varying delays[END_REF], it can be made arbitrarily tight, i.e., besides that [START_REF] Hristu | Limited communication control[END_REF] holds, it also holds that

L l=1 α l Āσ,l + B∆ Cσ | α ∈ A, ∆ ∈ ∆ ⊆ co Ãσ,h,τ | (h, τ ) ∈ Θ + { ∆ | ∆ ε}, (30)
for each σ ∈ {1, . . . , N }, in which ε > 0 can be chosen arbitrarily small. This can be achieved by increasing the number of pairs ( hl , τl ) ∈ Θ, l ∈ {1, . . . , L}, in a welldistributed fashion. The fact that (30) can be ensured to hold for an arbitrarily small ε > 0 is important, as it allows us to show that the existence of a Lyapunov function of a particular type for ( 12) is equivalent to the existence of a Lyapunov function of the same type for [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF]. Since we will indeed show that (30) can be guaranteed for any choice of ε > 0, we can let the introduced conservatism in the overapproximation vanish. We will formalise this result in Section V.

We now formalise the procedure to obtain a convex overapproximation as outlined above. The procedure results in a tight overapproximation, by adding pairs ( hl , τl ) ∈ Θ until ε ε u is achieved for an user-specified threshold ε u > 0, such that (30) holds with ε ε u .

for all σ ∈ {1, . . . , N } and ( hl , τl ) ∈ G, l ∈ {1, . . . , L}.

Step 3 To bound the approximation error, first construct the matrix Λ, that, depending on the NCS model defined in Section II-A, is given by 12) is as in Section II-A1, diag(A p , 0), if [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] is as in Section II-A2, 12) is as in Section II-A3.

Λ =          diag(A p , A c ), if (
A p B p C c 0 A c , if (
(33) Decompose the matrix Λ into its real Jordan form [START_REF] Horn | Matrix Analysis[END_REF], i.e., Λ := T ΛT -1 , where T is an invertible matrix and

Λ = diag(Λ 1 , . . . , Λ K ) (34)
with Λ i ∈ R ni×ni , i ∈ {1, . . . , K}, the i-th real Jordan block of Λ.

Step 4 Compute for each real Jordan block Λ i , i ∈ {1, . . . , K}, the worst-case approximation error of all triangles S m ∈ H, m ∈ {1, . . . , M }, i.e., For a detailed explanation of the origin of the approximation error bounds, see the proof of Theorem III.2.

δ A i = max m∈{1,...,M } max 3 j=1 αj =1, αj 0 δA i,m, α, (35a)
δ E h i = max m∈{1,...,M } max 3 j=1 αj =1, αj 0 δE h i,m, α, (35b) δ
Step 5 Define

Cσ :=   T -1 0 T -1 BDC T -1 BD 0 -T -1 BΓ σ   (37)
and

B := T T T -CT -CT -CT • diag(δ A 1 I 1 , . . . , δ A K I K , δ E h 1 I 1 , . . . , δ E h K I K , δ E h-τ 1 I 1 , . . . , δ E h-τ K I K ) (38)
with I i the identity matrix of size n i , complying with the i-th real Jordan Block, and compute

ε = B max σ∈{1,...,N } { Cσ }. ( 39
)
Step 6 In case that ε > ε u , meaning that the user-specified tightness of the overapproximation in the sense of ( 30) is not achieved we add a pair ( hL+1 , τL+1 ) ∈ Θ to G. In order to determine the specific pair to be added, compute the point (h, τ ) ∈ S m , where the maximum approximation error is achieved by solving

(m , α ) ∈ arg max m∈{1,...,M }, 3 j=1 αj =1, αj 0 δj i ,m, α (40)
in which

(i , j ) ∈ arg max i∈{1,...,K}, j∈{A,E h ,E h-τ } {δ j i }, (41)
and add this new pair ( hL+1 , τL+1 ) = ) to the set G, i.e., update G according to

G := G ∪ {( hL+1 , τL+1 )}, (42)
and redefine L := L + 1. Furthermore, subdivide the corresponding triangle S m into smaller triangles and replace S m by the smaller triangles in the set H, i.e.,

H := H\S m ∪ co{( hL+1 , τL+1 ), ( hl m 1 , τl m 1 ), ( hl m 2 , τl m 2 )} ∪ co{( hL+1 , τL+1 ), ( hl m 1 , τl m 1 ), ( hl m 3 , τl m 3 )} ∪ co{( hL+1 , τL+1 ), ( hl m 2 , τl m 2 ), ( hl m 3 , τl m 3 )}, (43)
redefine 1 M := M + 2, and repeat the procedure from Step 2.

Step 7 In case ε ε u , the user-specified tightness of the overapproximation is achieved and the resulting additive uncertainty set ∆ ⊆ R 3(np+nc)×3(np+nc) is given by

∆ = diag(∆ 1 , . . . , ∆ 3K ) | ∆ i+jL ∈ R ni×ni , ∆ i+jL 1, i ∈ {1, . . . , K}, j ∈ {0, 1, 2} . (44)
Theorem III.2 Consider the NCS given by [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] where (h k , τ k ) ∈ Θ, k ∈ N, with Θ as in (5). If system (25) is obtained by following Procedure III.1 for some user-specified ε u > 0, then (27) holds and thus (25) is an overapproximation of (12). Furthermore, the overapproximation is ε-tight, in the sense that (30) holds, with ε given by (39) and ε ε u .

Proof: The proof is given in the Appendix.

Remark III.3 In the special case that h = h or that τ = τ , Procedure III.1 has to be modified slightly. This is because we proposed to form triangles S m , m ∈ {1, . . . , M } having the property that intS m = ∅, which is not possible when h = h or τ = τ . Instead, in this case, we partition Θ

S m = co{( hl m 1 , τl m 1 ), ( hl m 2 , τl m 2 )}, (45)
where ( hl m j , τl m j ), j ∈ {1, 2}, now denote the vertices of the line segment S m . All other properties of S m , m ∈ {1, . . . , M } still hold and the remainder of the procedure can be applied mutatis mutandis.

UGES of the NCS system given by ( 1), (2a) or (2b), (3), and [START_REF] Heemels | Control of quantized linear systems: an l 1 -optimal control approach[END_REF], with protocols satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF], with (h k , τ k ) ∈ Θ, k ∈ N, can now be guaranteed by proving UGES of (25), with switching sequences satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF], α k ∈ A, and ∆ k ∈ ∆, k ∈ N, using the result of Lemma II.4 and the fact that ( 25) is a (tight) overapproximation of [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF].

IV. STABILITY OF SWITCHED SYSTEMS WITH

PARAMETRIC UNCERTAINTY In the previous sections, we discussed the NCS model and introduced a way to overapproximate it by a switched polytopic system with norm-bounded uncertainty. Given this switched uncertain system, we can analyse whether a switching sequence, as induced by a protocol, renders the switched system UGES.

We will start with so-called quadratic protocols that include the well-known TOD protocol as a particular case. The analysis is based on extensions of the ideas in [START_REF] Geromel | Stability and stabilization of discrete time switched systems[END_REF], in which only switched linear systems without any form of uncertainty are considered. Hence, generalisations are needed to include switched polytopic systems with norm-bounded uncertainties as in [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF]. After the stability analysis for the quadratic protocols and the TOD protocol as a special case, we will also show how we can analyse stability for periodic protocols, having the RR protocol as a special case.

For proving stability of system [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], with switching sequences satisfying [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] or ( 21), we will employ the so-called full block S-procedure [START_REF] Scherer | Robust mixed control and LPV control with full block scalings[END_REF], which is applied in the following lemma.

Lemma IV.1 Let P be given and consider the set ∆ as in [START_REF] Goebel | Hybrid dynamical systems[END_REF]. Then, it holds that

I Ā + B∆ C P I Ā + B∆ C ≺ 0. ( 46
)
for all ∆ ∈ ∆, if

I 0 Ā B P I 0 Ā B + 0 I C 0 -R 0 0 R 0 I C 0 ≺ 0, (47) where
R ∈ R = diag(r 1 I 1 , . . . , r K I K , r K+1 I 1 , . . . , r 2K I K , r 2K+1 I 1 , . . . , r 3K I K ) ∈ R 3(np+nc)×3(np+nc) | r i > 0 , ( 48
)
where I i is an identity matrix of size n i .

Proof: It follows directly from the full-block S-procedure [START_REF] Scherer | Robust mixed control and LPV control with full block scalings[END_REF] and the block-diagonal structure of [START_REF] Goebel | Hybrid dynamical systems[END_REF].

By choosing a suitable P , ( 46) can lead to a sufficient condition for stability of [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF].

A. Quadratic Protocols

In this section, we assume that the switching function is given by [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF]. To analyse the stability of ( 25) having this switching function, we introduce the non-quadratic Lyapunov function

V (x k ) = min i=1,...,N x k P i xk = min ν∈N x k N i=1 ν i P i xk , (49)
where

N := ν ∈ R N N i=1 ν i = 1, ν i 0, i ∈ {1, . . . , N } . ( 50
)
Furthermore, we introduce the class M of so-called Metzler matrices Π = {π ji } given by

M := Π ∈ R N ×N N j=1 π ji = 1, π ji 0, i, j ∈ {1, . . . , N } . ( 51
)
The main result of this section is presented in the following theorem.

Theorem IV.2 Assume that there exist a matrix Π = {π ji } ∈ M, positive definite matrices P i , and matrices R i,l ∈ R, i ∈ {1, . . . , N } and l ∈ {1, . . . , L}, satisfying

     P i 0 Ā i,l N j=1 π ji P j C i R i,l R i,l B N j=1 π ji P j 0 N j=1 π ji P j 0 R i,l      0, (52)
for all i ∈ {1, . . . , N } and l ∈ {1, . . . , L}. Then, the switching law (18) renders the system (25) UGES. Consequently, the NCS given by (1), (2a) or (2b), (3), and ( 7) is also UGES if the switching law (18) is employed as the protocol.

Proof: The proof is given in the Appendix.

Remark IV. [START_REF] Tipsuwan | Control methodologies in networked control systems[END_REF] The results of Theorem IV.2 can be exploited in two ways: (i) For the design of a stabilising protocol.

Then the conditions in [START_REF] Nešić | Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems[END_REF] are not LMIs, but Bilinear Matrix Inequalities (BMIs) due to the presence of the product of π ji and P j . Although literature on solving BMIs is available, see, e.g., [START_REF] Goh | A global optimization approach for the BMI problem[END_REF]- [START_REF] Iwasaki | The XY-centring algorithm for the dual LMI problem: a new approach to fixed-order control design[END_REF], solving BMIs is considered to be of a high numerical complexity. (ii) Stability analysis for a given protocol. In the situation that the matrices P i , i ∈ {1, . . . , N }, are completely given for a particular quadratic protocol, the conditions [START_REF] Nešić | Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems[END_REF] are LMIs in Π ∈ M and R i,l ∈ R, for all i ∈ {1, . . . , N } and l ∈ {1, . . . , L}.

B. The TOD Protocol

In Section II-B, we showed that by suitable choice of P i , i ∈ {1, . . . , N }, as in [START_REF] Cloosterman | Stability of networked control systems with uncertain time-varying delays[END_REF], the TOD protocol is a specific quadratic protocol. We can therefore use the result of Theorem IV.2 to determine the allowable range of transmission intervals and Limited circulation. For review only Preprint submitted to IEEE Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST transmission delays of the NCS using the TOD protocol. This result is formalised in the following corollary, in which

M := diag(0, N j=1 π ji Γ j ) ∈ R n×n N j=1 π ji = 1, π ji 0, i, j ∈ {1, . . . , N } . (53)
Corollary IV.4 Assume that there exist matrices Πi ∈ M, i ∈ {1, . . . , N }, a matrix P , matrices R i,l ∈ R, i ∈ {1, . . . , N } and l ∈ {1, . . . , L}, satisfying

    P -diag(0, Γ i ) 0 Ā i,l (P -Π i ) C i R i,l R i,l B (P -Π i ) 0 P -Π i 0 R i,l     0, ( 54
)
for all i ∈ {1, . . . , N } and l ∈ {1, . . . , L}, with Γ i , as in (4). Then, the system (25) with ( 20) is UGES. Consequently, the NCS, given by (1), (2a) or (2b), (3), and (7), with the TOD protocol (20) is also UGES.

Proof: The proof follows directly from Theorem IV.2 and the fact that P i is structured as in [START_REF] Cloosterman | Stability of networked control systems with uncertain time-varying delays[END_REF]. Therefore, it holds that

N j=1 π ji P j = P -diag(0, N j=1 π ji Γ j ) =: P -Π i .

C. Periodic protocols and the RR Protocol

We will now analyse another class of communication protocols, namely the periodic protocols, with the RR protocol as a special case. Hence, we need to analyse stability of the system [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF] with a switching sequence induced by [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF] or [START_REF] Naghshtabrizi | Stability of delay impulsive systems with application to networked control systems[END_REF]. This system is essentially a Ñ -periodic system. For this system, we introduce positive definite matrices P i , i ∈ {1, . . . , N }, and a time-dependent periodic Lyapunov function given by

V k (x k ) = x k P k xk , and V k+ Ñ (x k+ Ñ ) = V k (x k ). (55)
We can now present the main result of this section.

Theorem IV.5 Assume that there exist positive definite matrices P i , i ∈ {1, . . . , Ñ }, and matrices R i,l ∈ R, i ∈ {1, . . . , N } and l ∈ {1, . . . , L}, satisfying

    P i 0 Ā σi,l P i+1 C σi R i,l R i,l B P i+1 0 P i+1 0 R i,l     0, (56)
where P Ñ +1 := P 1 , for all i ∈ {1, . . . , Ñ } and l ∈ {1, . . . , L}. Then, the system (25) with ( 22) is UGES and consequently, the NCS as given by (1), (2a) or (2b), [START_REF] Tipsuwan | Control methodologies in networked control systems[END_REF], and (7) with a periodic protocol (21) is UGES.

Proof: The proof follows the same lines of reasoning as the proof of Theorem IV.2.

V. NONCONSERVATIVENESS OF THE OVERAPPROXIMATION

Given the results of the previous sections, it is now natural to ask if and how conservative the presented methodology is. The answer is given by the following result, showing that if the original system (12) (without any overapproximation), with protocol ( 18) or [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF], is UGES in the sense that a Lyapunov function of a particular type exists, given by ( 49) or (55), respectively, the presented procedure based on the overapproximation will guarantee stability and will find a respective Lyapunov function, given that the overapproximation of ( 12) is sufficiently tight, i.e., [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF] holds for a sufficiently small ε > 0. Therefore, making a convex overapproximation, according to Procedure III.1, introduces no conservatism in the stability analysis as presented in the previous section.

In the following theorem, we will show the result for the NCS model [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] with protocol [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF]. A similar result holds for the NCS model [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] with protocol [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF].

Theorem V.1 Suppose system (12), with protocol (18), has a Lyapunov function of the form (49), i.e., there exist a matrix Π = {π ji } ∈ M and positive definite matrices

P i , i ∈ {1, . . . , N }, such that à i,h,τ N j=1 π ji P j Ãi,h,τ -P i -γI, (57)
for all i ∈ {1, . . . , N } and (h, τ ) ∈ Θ, and some γ > 0.

Then, there exists an ε 0 , such that for any ε-tight overapproximation satisfying (30), with 0 < ε ε 0 , the conditions of Theorem IV.2 hold.

Proof: The proof is given in the Appendix. This result states that the convex overapproximation does not introduce conservatism when analysing UGES using mode-dependent quadratic Lyapunov functions.

VI. ILLUSTRATIVE EXAMPLE

In this section, we illustrate the presented theory using a well-known benchmark example in the NCS literature, see, e.g., [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], [START_REF] Carnevale | A Lyapunov proof of improved maximum allowable transfer interval for networked control systems[END_REF]- [START_REF] Walsh | Stability analysis of networked control systems[END_REF], [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], consisting of a model of a batch reactor. The linearised batch reactor is given by (1), with

A p B p C p =     
0 1 0 1 -1 0 1 0 0      . (58)
The continuous-time controller considered in [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF], [START_REF] Carnevale | A Lyapunov proof of improved maximum allowable transfer interval for networked control systems[END_REF]- [START_REF] Walsh | Stability analysis of networked control systems[END_REF], [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF] is given by (2a), with

A c B c C c D c =   0 0 0 1 0 0 1 0 -2 0 0 -2 0 8 5 0   . (59)
First, we will analyse the continuous-time NCS as also used in [START_REF] Carnevale | A Lyapunov proof of improved maximum allowable transfer interval for networked control systems[END_REF]- [START_REF] Walsh | Stability analysis of networked control systems[END_REF], [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]. As done in these references, we consider the TOD and RR protocol and assume that the controller is directly connected to the actuator, i.e., only the two outputs are transmitted via the network. Since communication delays

Limited circulation. For review only

Preprint submitted to IEEE Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST are only considered in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], and gives in absence of delays (i.e., τ = τ = 0) the same results as in [START_REF] Carnevale | A Lyapunov proof of improved maximum allowable transfer interval for networked control systems[END_REF], we compare our results with [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]. This will show that our results provide significantly less conservative bounds on the uncertain transmission intervals and transmission delays than earlier results in the literature. Secondly, we illustrate that our framework can equally well deal with discrete-time controllers, a larger number of nodes than used in previous examples in the literature, and a nonzero lower bound on the transmission interval.

A. Continuous-Time Controller

In order to assess the bounds on the allowable transmission intervals and delays, we first define our NCS model as in Section II-A3. This model appropriately describes the situation as discussed in this example, where only the plant outputs y are transmitted over the network and u are sent continuously via a nonnetworked connection. Then, we derive the uncertain polytopic system (25) that overapproximates the NCS model ( 12), using Procedure III.1. As in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], we try to obtain combinations of h and τ for which the NCS is stable, and we assume that τ = 0, and we take h = 10 -3 . We cannot make h too small, because, by doing so, Ãσ k ,h k ,τ k , as in [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF], approaches the identity matrix and LMI solvers run into numerical problems, as the system becomes close to unstable. Note that [START_REF] Carnevale | A Lyapunov proof of improved maximum allowable transfer interval for networked control systems[END_REF]- [START_REF] Walsh | Stability analysis of networked control systems[END_REF], [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF] also use nonzero lower bound on the transmission intervals to prevent Zeno behaviour, although, this lower bound can be taken arbitrarily small. Using Procedure III.1, we obtain a convex overapproximation, in which we choose ε u = 3 as decreasing ε u does not change the results in this example. Using the obtained overapproximation, we can check for which combinations of h and τ , the LMIs in Corollary IV.4 and Theorem IV.5 are feasible. This results for each τ in the maximum achievable h (or vice versa) for which the LMIs in Corollary IV.4 and Theorem IV.5 are satisfied. This results in tradeoff curves, as shown in Fig. 3. These tradeoff curves can be used to impose or select a suitable network with a certain communication delay and a certain allowable transmission interval.

Moreover, in Fig 3, also the tradeoff curves as obtained in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF] are given. We conclude that our proposed methodology is less conservative than the one in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]. More interestingly, in case there is no delay, i.e., τ = τ = 0, the maximum allowable transmission interval h obtained in [START_REF] Carnevale | A Lyapunov proof of improved maximum allowable transfer interval for networked control systems[END_REF], which provide the least conservative results known in literature so far, was h = 0.0108, while we obtain h = 0.0665. In [START_REF] Walsh | Stability analysis of networked control systems[END_REF], h was estimated (using simulations) to be approximately 0.08 for the TOD protocol. Furthermore, for the RR protocol, [START_REF] Carnevale | A Lyapunov proof of improved maximum allowable transfer interval for networked control systems[END_REF] provides the bound h = 0.009 in the delay-free case, while we obtain h = 0.0645. Also in [START_REF] Walsh | Stability analysis of networked control systems[END_REF], for a constant transmission interval, i.e. h = h, the bound 0.0657 was obtained for the RR protocol. The case where the transmission interval is constant, provides an upper bound on the true maximum allowable transmission interval (MATI). We can therefore conclude that for this example, our methodology reduces conservatism significantly in comparison to existing methodologies and even approximates known estimates of the true MATI closely.

B. Discrete-Time Controller

Next, we compute h, h, τ , and τ for the NCS with a discrete-time controller as in (2b). Contrary to the example presented above, and all examples considered in [START_REF] Carnevale | A Lyapunov proof of improved maximum allowable transfer interval for networked control systems[END_REF]- [START_REF] Walsh | Stability analysis of networked control systems[END_REF], [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], we now designate a node to each single sensor and actuator, resulting in an NCS with four nodes. By doing so, we try to point out that our methodology is also suitable to study more complex problems. In this example, the controller is given by an exact discretisation of the continuous-time controller (2a) with matrices (59) using a zero-order hold and assuming a nominal transmission interval h nom = 0.01 and a bounded variation h var around this nominal transmission interval. We assume that τ = 0 and h = 0, i.e., transmissions can be infinitely fast, but do not occur infinitely often. In this example, we select h = h nom -h var and h = h nom +h var , where 2h var > 0 determines the range of allowable transmission intervals and we only consider the RR protocol.

After obtaining a convex overapproximation using Procedure III.1, in which we have taken ε u = 0.02, and assessing stability using the results of Theorem IV.5, we can now plot for each τ , the largest range, determined by h = h nom -h var and h = h nom + h var , for which UGES is guaranteed. In this example, we take h nom = 0.01, which results in the tradeoff curve as shown in Fig. 4.

VII. CONCLUSIONS

In this paper, we studied the stability of Networked Control Systems (NCSs) that are subject to communication constraints, We analysed the stability of the NCS when the communication sequence is determined by one of the protocols in the newly introduced classes of quadratic protocols or periodic protocols, having the well-known Try-Once-Discard (TOD) and the Round-Robin (RR) as special cases. This analysis was based on a discrete-time switched linear uncertain system to model the NCS. A new and efficient convex overapproximation was proposed that allows us to analyse stability using a finite number of linear matrix inequalities. We presented an automated procedure to obtain the overapproximation and we formally showed that the convex overapproximation can be made arbitrarily tight and does not introduce conservatism. On a benchmark example, we illustrated the advantages and the effectiveness of the developed theory. In particular, we showed that stability can be guaranteed for a much larger maximum allowable transmission interval and maximum allowable transmission delay, when compared to the existing results in the literature. In addition, our results can be applied for stability analysis of NCS with discrete-time controllers and nonzero lower bounds on the transmission intervals and delays, which could not be analysed before even though they are highly relevant for practical implementations of networked controllers.

Future work focusses on studying the case where delays are not restricted to be smaller than the transmission interval, on the inclusion of quantisation effects of the sensor and actuator signals on the closed-loop stability and performance, and on co-design methods of the controller and the protocol.

APPENDIX PROOFS OF THEOREMS AND LEMMAS

Proof of Lemma II.4: In Theorem 4 of [START_REF] Nešić | Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems[END_REF], it was shown that a sampled-data system is UGES if and only if its corresponding discretised model is UGES and the intersample behaviour is so-called linearly uniformly globally bounded over T (LUGBT), where T is the sampling interval. This means, roughly speaking, that the intersample behaviour can be bounded by a linear function of the state of the system at the transmission instants. Since the discrete-time system is UGES by assumption, it only remains to show LUGBT. To do so, let us introduce an additional variable t := t -t k , for all t ∈ (t k , t k+1 ]. Solving the differential equation (1) on the interval t ∈ (0, τ k ] yields

x p (t k + t) = e A p tx p (t k ) + t 0 e A p ( t-s) dsB p û(t k ) (60)
and on the interval t ∈ (τ k , h k ]

x p (t k + t) = e A p tx p (t k ) + τ k 0 e A p ( t-s) dsB p û(t k ) + t τ k e A p ( t-s) dsB p lim t↓r k û(t). ( 61
)
Or equivalently, when expressed in states at the sample instants, for t ∈ (0, τ k ],

x p (t k + t) = e A p tx p k + t 0 e A p s dsB p (D c C p x p k + C c x c k + D c e y k + e u k ) (62)
and for t ∈ (τ k , h k ],

x p (t k + t) = e A p tx p k + t 0 e A p s dsB p (D c C p x p k + C c x c k + e u k + D c e y k ) - t-τ k 0 e A p s dsB p Γ u σ k e u k . (63)
Using ( 62) and (63), we can bound the intersample behaviour on the interval t ∈ (0, h k ] by

x p (t k + t) e A p t x p k + t 0 e A p s dsB p ( C c x c k + D c C p x p k + D c e y k + e u k ) + t-τ k 0 e A p s dsB p Γ u σ k e u k , (64)
Similar inequalities can be derived that bound the intersample behaviour for the state evolution x c (t) of (2a) and for the network-induced error given by [START_REF] Heemels | Control of quantized linear systems: an l 1 -optimal control approach[END_REF]. Therefore, by using the bounds on h k and τ k , the continuous-time NCS (1), (2a) or (2b), (3), and ( 7) is LUGBT. Consequently, Theorem 4 of [START_REF] Nešić | Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems[END_REF] implies that the continuous-time NCS is UGES.

Proof of Theorem III.2: The proof is based on showing that Procedure III.1 yields that system ( 25) is an overapproximation of [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] in the sense that ( 27) holds, and that this overapproximation is tight in the sense that (30) holds for a ε > 0, satisfying ε ε u for some ε u .

In order for [START_REF] Hristu | Limited communication control[END_REF] to hold, considering a fixed σ ∈ {1, . . . , N }, we should have that for all (h, τ ) ∈ Θ, there exist an α ∈ A and a ∆ ∈ ∆, such that (29) holds, i.e., Ãσ,h,τ = L l=1 α l Āσ,l + B∆ Cσ .

Therefore, given L distinct pairs ( hl , τl ) ∈ Θ, l ∈ {1, . . . , L}, and Āσ,l as in [START_REF] Cloosterman | Stabilization of networked control systems with large delays and packet dropouts[END_REF], we can write the approximation error between Ãσ,h,τ of (12) and L l=1 α l Āσ,l of (28) as

Ãσ,h,τ - L l=1 α l Āσ,l =   T -(CT ) T -(CT ) T -(CT )   =: B ∆α,h,τ   T -1 0 T -1 BDC T -1 BD 0 -T -1 BΓ σ   =: Cσ , (66)
where

∆α,h,τ = diag(T -1 (A h - L l=1 α l A hl )T, T -1 (E h - L l=1 α l E hl )T, T -1 (E h-τ - L l=1 α l E hl -τ l )T ), (67) in which A h , E h , E h-τ are
T -1 (A h - L l=1 α l A hl )T = e Λh - L l=1 α l e Λ hl , (68a)
T -1 (E h - L l=1 α l E hl )T = L l=1 α l h hl e Λs ds, (68b)
T -1 (E h-τ - L l=1 α l E hl -τ l )T = L l=1 α l h-τ hl -τ l e Λs ds (68c)
hold for all α ∈ A if A h , E h , and E h-τ are defined as in Section II-A1 and II-A3. Since E h and E h-τ , when defined as in Section II-A2, contain identity matrices in the lowerright part, the left-hand side of (68b) and (68c) contain zero blocks. Therefore, in case of E h and E h-τ being defined as in Section II-A2, equality in (68b) and (68c) do not automatically hold and we have to impose additional requirements on α ∈ A to ensure that the appropriate Jordan blocks of the righthand side of (68b) and (68c) also equal zero. These additional requirements are that L l=1 α l hl = h, and

L l=1 α l ( hl -τl ) = h -τ, (69)
since substituting (69) into (68) indeed results in zero-blocks at the appropriate places in the left-hand side of (68). Now, combining

provided that (69) holds.

As an intermediate step in the proof, we aim at finding a set ∆ of matrices, such that for all (h, τ ) ∈ Θ there is an α ∈ A such that ∆α,h,τ ∈ ∆. Since Θ = ∪ M m=1 S m , we wil perform the construction of ∆ per triangle S m , m ∈ {1 . . . , M }, and combine them later. Hence, for each m ∈ {1, . . . , M }, we now aim at constructing ∆m such that for all (h, τ ) ∈ S m , m ∈ {1, . . . , M }, there is an α ∈ A such that ∆α,h,τ ∈ ∆m . In particular, for (h, τ ) ∈ S m , m ∈ {1, . . . , M }, with S m as in [START_REF] Liberzon | Quantization, time delays, and nonlinear stabilization[END_REF], take ᾱl m j = αj , j = {1, 2, 3}, and ᾱi = 0, i / ∈ {l m 1 , l m 2 , l m 3 }, where

3 j=1 αj ( hm j , τ m j ) = (h, τ ), 3 j=1
αj = 1, and αj 0, j ∈ {1, 2, 3}. Let us now bound the norm of (70) for triangle S m , m ∈ {1, . . . , M }, and per Jordan block Λ i , i ∈ {1, . . . , K} using this particular choice ᾱ for α. Hence, for all (h, τ ) ∈ S m , ∆ᾱ,h,τ ∈ ∆m with

∆m := diag( ∆A 1 , . . . , ∆A K , ∆E h 1 , . . . , ∆E h K , ∆E h-τ 1 , . . . , ∆E h-τ K ) ∆A i max 3 j=1 αj =1, αj 0 δA i,m, α, ∆E h i max 3 j=1 αj =1, αj 0 δE h i,m, α, ∆E h-τ i max 3 j=1 αj =1, αj 0 δE h-τ i,m, α , i ∈ {1, . . . , K} , (71)
for m ∈ {1, . . . , M } in which δA i,m, α, δE h i,m, α, and δE h-τ i,m, α are given by [START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF]. This upper bound on the approximation errors To obtain ∆ independent of m, as in [START_REF] Goebel | Hybrid dynamical systems[END_REF], let us now introduce the scaling matrix

U := diag(δ A 1 I 1 , . . . , δ A K I K , δ E h 1 I 1 , . . . , δ E h K I K , δ E h-τ 1 I 1 , . . . , δ E h-τ K I K ) (73)
in which I i is an identity matrix of size n i , complying with the size of the ∆i and observe that ∆m ⊆ U ∆, with ∆ as in [START_REF] Goebel | Hybrid dynamical systems[END_REF]. Now due to [START_REF] Van De Wouw | Tracking control for sampled-data systems with uncertain sampling intervals and delays[END_REF] and ( 38), B = B • U , and this allows us to rewrite (72) as

Ãσ,h,τ | (h, τ ) ∈ S m ⊆ 3 j=1 αj Āσ,l m j + B∆ Cσ 3 j=1 αj = 1, αj 0, j ∈ {1, 2, 3}, ∆ ∈ ∆ ⊆ L l=1 α l Āσ,l + B∆ Cσ α ∈ A, ∆ ∈ ∆ , (74)
with A as in [START_REF] Brockett | Stabilization of motor networks[END_REF] and ∆ as in [START_REF] Goebel | Hybrid dynamical systems[END_REF], which is . By taking the convex hull over all m ∈ {1, . . . , M } in the left-hand-side and observing that the right-hand-side is independent of m, we obtain [START_REF] Hristu | Limited communication control[END_REF].

To show that (30) holds for ε as in [START_REF] Walsh | Stability analysis of networked control systems[END_REF], we consider a fixed σ ∈ {1, . . . , N } and show that for all α ∈ A and ∆ ∈ ∆, there exist a pair (h, τ ) ∈ Θ and a ∆, satisfying ∆ ε, such that

which holds for ε ε u as in [START_REF] Walsh | Stability analysis of networked control systems[END_REF], due to the fact that Procedure III.1 terminates not until ε ε u .

Proof of Theorem IV.2: The proof is based on showing that V as in ( 49) is a Lyapunov function for the switched uncertain system [START_REF] Dačić | Quadratic stabilization of linear networked control systems via simultaneous protocol and controller design[END_REF] with switching law [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF]. Note that V (x k ) = x k P i xk , with i = σ k , due to [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF]. Now, we obtain

is positive definite for all α ∈ A, ∆ ∈ ∆, and i ∈ {1, . . . , N }. A necessary and sufficient condition for positive definiteness of (79), for all α k ∈ A, is that G i,l 0 for all i ∈ {1, . . . , N } and l ∈ {1, . . . , L}. Using again a Schur complement, we can rewrite the condition G i,l 0 as follows:

for all ∆ ∈ ∆, i ∈ {1, . . . , N } and l ∈ {1, . . . , L}. As (81) has the form of (46) of Lemma IV.1, it is therefore implied by [START_REF] Geromel | Stability and stabilization of discrete time switched systems[END_REF]. Applying a Schur complement yields [START_REF] Nešić | Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems[END_REF] for all i ∈ {1, . . . , N } and all l ∈ {1, . . . , L}. Since (52) holds by the hypothesis of the theorem, we can conclude that V is strictly decreasing in spite of the presence of the uncertainty. Standard Lyapunov-based stability arguments now prove that (25) with [START_REF] Dritsas | Robust stability analysis of networked systems with varying delays[END_REF] is UGES. Using that ( 25) is an overapproximation of [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] as proven in Theorem III.2 and subsequently, using the result of Lemma II.4, it follows that the NCS system given by (1), (2a) or (2b), (3), and (7) is UGES.

Proof of Theorem V.1: Since (57) holds for all pairs (h, τ ) ∈ Θ and, therefore, for all ( hl , τl ) ∈ Θ, l ∈ {1, . . . , L}, we have that Ā i,l N j=1 π ji P j Āi,l -P i + γI 0 0 0 0,

for all i ∈ {1, . . . , N } and l ∈ {1, . . . , L}. Note that (82) holds irrespective of the choice of ( hl , τl ) ∈ Θ, l ∈ {1, . . . , L}. Now suppose that we would establish that there exist matrices R i,l ∈ R, i ∈ {1, . . . , N } and l ∈ {1, . . . , L}, such that

N j=1 π ji C i R i,l Ci -γI Ā i,l P j B B P j Āi,l B P j B -R i,l ≺ 0, (83)
for all i ∈ {1, . . . , N } and l ∈ {1, . . . , L}. Then, combining this expression with (82) yields, after taking a Schur complement, the conditions of Theorem IV.2. Hence, if the fact that (30) holds for a sufficiently small ε, implies that (83) holds for some R i,l ∈ R, we completed the proof. Therefore, it remains to show that there exists an ε 0 , such that for any 0 < ε ε 0 , (83) is satisfied for some R i,l ∈ R. Note that (83) holds if

C

i R i,l Ci + Ā i,l P j B R i,l -B P j B -1 B P j Āi,l ≺ γI, (84) and B P j B ≺ R i,l , for some R i,l ∈ R, and for all i, j ∈ {1, . . . , N } and l ∈ {1, . . . , L}. By choosing R i,l = rI, for all i ∈ {1, . . . , N } and l ∈ {1, . . . , L} with r > 0, we can observe that (84) is implied by

Pj Āi,l 2 B 2 r-λmin( B Pj B) < γ -r Ci 2 , (85)
where λ min ( B P j B) denotes the minimum eigenvalue of B P j B. Since it holds that λ min ( B P j B) B 2 P j , (85) is implied by

P j Āi,l 2 B 2 < (γ -r Ci 2 )(r -B 2 P j ), (86)
Furthermore, B P j B ≺ R i,l is implied by B 2 P j < r, for some r > 0 and all i, j ∈ {1, . . . , N } and l ∈ {1, . . . , L}. Now choosing r = γ 2 Ci 2 , multiplying the left-hand and the righthand side of (86) by Ci 2 , and realising that B Ci ε yields that (86), and thereby (84), is satisfied if

ε 2 P j Āi,l 2 < 1 4 γ 2 -1 2 γε 2 P j , (87)
and that B P j B ≺ R i,l is satisfied if ε 2 P j < 1 2 γ, which can be satisfied by choosing ε sufficiently small. Therefore, if (49) is a Lyapunov function for system [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF], with protocol [START_REF] Gielen | On polytopic inclusions as a modeling framework for systems with time-varying delays[END_REF], then there exists an ε 0 > 0, such that for any overapproximation satisfying [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF] with 0 < ε ε 0 , the conditions of Theorem IV.2 hold, which completes the proof.

2

 2

  AND PROBLEM STATEMENT In this section, we present the model describing the Networked Control Systems (NCSs), subject to communication constraints, time-varying transmission intervals and delays. Let us consider the linear time-invariant (LTI) continuous-time Limited circulation. For review only Preprint submitted to IEEE Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST

Fig. 1 :

 1 Fig. 1: Illustration of a typical evolution of y and ŷ.

  d

  Limited circulation. For review only Preprint submitted to IEEE Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST

  xk+1

Fig. 2 :

 2 Fig. 2: The partitioning of Θ into triangles S m .

Procedure III. 1 Step 1

 11 Choose a desired ε u > 0. Furthermore, select distinct pairs ( hl , τl ) ∈ Θ, l ∈ {1, . . . , L}, such that co G = Θ, where G = ∪ L l=1 {( hl , τl )}. Now partition Θ into M triangles S m , m ∈ {1, . . . , M }, such that, for each S m ∈ H, where H = {S 1 , . . . , S M }, it holds that S m = co{( hl m 1 , τl m 1 ), ( hl m 2 , τl m 2 ), ( hl m 3 , τl m 3 )}, (31) where l m j ∈ {1, . . . , L}, j ∈ {1, 2, 3}. Hence, ( hl m j , τl m j ) ∈ G, j ∈ {1, 2, 3} are the vertices of the triangle S m . Moreover, for all m, p ∈ {1, . . . , M } and p = m, intS p ∩ intS m = ∅, ∪ M m=1 S m = Θ, and Limited circulation. For review only Preprint submitted to IEEE Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST intS m = ∅, i.e., the triangles form a (nonoverlapping) partitioning of Θ and have nonempty interiors. Step 2 Define Āσ,l := Ãσ, hl ,τ l

e

  i,m, α , (35c) in which α = [α 1 α2 α3 ] and δA i,m, α = e Λis ds . (36c)

Fig. 3 :

 3 Fig. 3: Tradeoff curves between allowable transmission intervals and transmission delays for two different protocols.

Fig. 4 :

 4 Fig. 4: Tradeoff curves between allowable transmission intervals and transmission delays.

  Limited circulation. For review only Preprint submitted to IEEE Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST time-varying transmission intervals and time-varying delays.

  defined as in Section II-A1, II-A2 or II-A3. Using the real Jordan form of (33), we can observe Limited circulation. For review only Preprint submitted to IEEE Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST

  that

e

  and (68) yields ∆α,h,τ = L l=1 α l diag(e Λh -e Λ hl , Λs ds),

  allows us to writeÃσ,h,τ | (h, τ ) ∈ S m ⊆ , 2, 3}, ∆m ∈ ∆m . (72)

∈

  co Ãσ,h,τ | (h, τ ) ∈ Θ} + { ∆ | ∆ ε . Āσ, hl ,τ l ∈ co{ Āσ,h,τ | (h, τ ) ∈ Θ}, this inclusion is satisfied if B∆ Cσ ε,

  Limited circulation. For review only Preprint submitted to Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST using (49) and (25) thatV (x k+1 ) Āσ k ,l1 + B∆ k Cσ k ) N j=1 ν j P j ( L l2=1 α l2 k Āσ k ,l2 + B∆ k Cσ k )x k x k ( L l1=1 α l1 k Āi,l1 + B∆ k Ci ) N j=1 π ji P j ( L l2=1 α l2 k Āi,l2 + B∆ k Ci )x k . (77)To obtain UGES, it is sufficient to require that the Lyapunov function is strictly decreasing in the sense that (due to (77)) Āi,l2 + B∆ Ci )-P i ≺ 0. (78)for all α ∈ A, ∆ ∈ ∆, and i ∈ {1, . . . , N }. Here, we used that ∆ ⊆ ∆, as stated in Lemma IV.1. By taking a Schur complement, realising that N j=1 π ji P j 0, and using that α k ∈ A, we obtain that (78) is equivalent to stating that

P

  i -( Āi,l + B∆ Ci ) N j=1π ji P j ( Āi,l + B∆ Ci ) 0,

  Stability Analysis of Networked Control Systems Using a Switched Linear Systems Approach M.C.F. Donkers, Student Member, IEEE, W.P.M.H. Heemels Senior Member, IEEE, N. van de Wouw Member, IEEE, and L. Hetel

TABLE I :

 I References that study two networked induced imperfections simultaneously.

&(iv) (v) (i)[START_REF] Liberzon | Quantization, time delays, and nonlinear stabilization[END_REF] (ii)[START_REF] Cloosterman | Stabilization of networked control systems with large delays and packet dropouts[END_REF],[START_REF] Yu | An LMI approach to networked control systems with data packet dropout and transmission delays[END_REF] (iii)[START_REF] Naghshtabrizi | Stability of networked control systems with variable sampling and delay[END_REF],[START_REF] Van De Wouw | Tracking control for sampled-data systems with uncertain sampling intervals and delays[END_REF] [36]-[START_REF] Walsh | Asymptotic behavior of nonlinear networked control systems[END_REF] the control system also introduces new challenges, caused by the packet-based data exchange between different parts of the network. Therefore, control algorithms are needed that can handle the communication imperfections and constraints caused by the packet-based communication. The control community is widely aware of this fact, as is evidenced by the broad attention NCSs have received recently, see, e.g., the

Preprint submitted to IEEE Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST

Preprint submitted to Transactions on Automatic Control. Received: January 8, 2011 11:25:21 PST

PLACE PHOTO HERE

Maurice Heemels (M '06) received the MSc in mathematics (with honours) and the Ph.D. degrees (cum laude) from the Eindhoven University of Technology (TU/e), Eindhoven, the Netherlands, in 1995 and 1999, respectively.

From 2000 to 2004, he was with the Electrical Engineering Department, TU/e, as an Assistant Pro-


Andrea Espinel
email: andrea.espinel-rojas@irccyn.ec-nantes.fr

Ina Taralova Irccyn

René Lozi
email: lozi@unice.fr

NEW ALTERNATE LOZI FUNCTION FOR RANDOM NUMBER GENERATION

Introduction

The accelerated development of modern data transactions applications such as telecommunications requires encoding techniques with higher standards of security. Classically, these encoding sequences are obtained using Pseudo Random Number Generators (PRNG). As an efficient alternative, the chaotic-based generators are used to achieve even higher demanding encryption standards. The advantage to use chaotic systems lies in their extreme sensitivity to small parameter and initial conditions variations: in this way, as many different chaotic carriers as wanted can be generated.

However, the appropriate selection of a chaotic map that satisfies cryptographic applications requirements is a huge problem. Ideally, for cryptographic applications and higher security, an everywhere dense chaotic attractor is required, so all chaotic signal samples shall appear with the same probability. To evaluate the random properties of these generators the National Institute of Standards and Technology has developed a set of statisticalbased test known as NIST tests. A first coupled chaotic map confined to the 2D torus has already been proposed as a PRNG in [START_REF] Espinel | Dynamical and Statistical Analysis of a New Lozi Function for Random Numbers Generation[END_REF], which random characteristics have been validated using the NIST tests. Nevertheless, since the state variables were not equidistributed, the chaotic attractor exhibited holes delimited by the discontinuity lines and their forward iterates. Therefore, there have been values which the signal never took, and that impact deteriorates the randomness.

System Definition

To improve the latter results, in this paper we deal with the new Lozi system with alternate coupled maps, confined to the p-dimensional torus

[ ] 1 , 1 - = p p T by the map T T M p p p ⇒ : x k x x x k x x M x k x x n p p n p n n n n p n n n 1 1 3 2 2 2 1 2 1 1 1 1 2 1 2 1 : 2 1 × + - = × + - = × + - = + + +
where the parameters ( )

1 1 - = + i k i
. A previous model with non alternate coefficients has been proposed in [START_REF] Espinel | Dynamical and Statistical Analysis of a New Lozi Function for Random Numbers Generation[END_REF], with

1 = k i .
The state variables are contained on the torus:

2 1 2 1 2 1 2 1 1 1 1 1 substract x k x x if add x k x x if j n j j n j n j n j j n j n > × + - = - < × + - = + + + + |x n | denotes the absolute value of x n .
The alternate sign modification eliminates the holes from the previous model (with only positive signs), and therefore the resulting basin of attraction is everywhere dense, which is very satisfactory for the RNG applications, see fig. 1. (The transient of 10.000 points has been cut off).

Results and Discussion

The random properties validation of a 4dimensional system has been carried out. Additionally, the chaotic carrier output needs to be quantised and binarised (0 and 1) in order to be [START_REF] Espinel | Dynamical and Statistical Analysis of a New Lozi Function for Random Numbers Generation[END_REF] (2) X1 validated as being random using NIST tests. Therefore, different methods of binarisation (converting real signals to binary ones) have been implemented and compared.

A first 1-bit binarisation has been applied to the system (1) output:

0 1 0 = = ≥ b else b y if n n n
The results for the 4-dimentional system showed to be highly sensitive to the type of binarisation. Therefore, after testing several different methods, a 32-bit binarisation has been chosen as being the most suitable solution. Because the system is confined to the p-dimensional torus

[ ] 1 , 1 - = p p T
, 31 bits are assigned to represent the decimal part, and 1 bit to the sign.

To illustrate the results, the NIST test for the four dimensional Lozi system M 4 (1) with parameters

[ ] [ ] 1 , , 1 , 4 2 3 1 - = = k k k k
are in Table 1. The same conditions as in [START_REF] Espinel | Dynamical and Statistical Analysis of a New Lozi Function for Random Numbers Generation[END_REF] have been chosen:

Length of the original sequence: 10^8 bits Length of bit string: 1M Quantity of bit strings: 100

The output of the system has been arbitrary chosen as being:

4

x y = .

Furthermore, as the results show their independence from the initial conditions, every bit string in this first test is the resulting sequence of a different randomly chosen initial condition. The criterion for a successful test is that the p-value has to be superior to the significance level (0.01 for this case). For the present model ( 1), all tests were successful thus the sequence can be accepted as being random. Thus, the results demonstrate that the new system has better statistical performances than the initial system without alternate coefficients presented in [START_REF] Espinel | Dynamical and Statistical Analysis of a New Lozi Function for Random Numbers Generation[END_REF].

Finally, to improve the random properties of the signal, two possible strategies are suggested: undersampling of the output signal, or increasing the system order. Different under-sampling have been tested from which the "1 out of 10" showed to be particularly successful. The "1 out of 10" under-sampling strategy results are shown in [START_REF] Espinel | Dynamical and Statistical Analysis of a New Lozi Function for Random Numbers Generation[END_REF].

For the second method, the random properties validation of a 10-dimensional system has been carried out and the results are shown in Table . 2. The conditions for the NIST test are the same from the NIST test for the 4-dimentional Lozi system. In addition, the initial condition has been randomly chosen: x0 = [-0.3365, 0.9501, 0.8913, -0.7764, 0.0185, 0.4447, 0.7919, -0.9218, -0.9355, 0.0579] The output of the system has been arbitrary chosen as being:

10 x y = . Table 2.
Random improvement of both strategies has been corroborated by the experimental results.

Figure 1 .

 1 Figure 1. Map 2 (1) on the torus

Table 1 .

 1