Luca Greco

Antoine Chaillet

Antonio Bicchi

Exploiting Packet Size in Uncertain Nonlinear Networked Control Systems ⋆

This paper addresses the problem of stabilizing uncertain nonlinear plants over a shared limited-bandwidth packet-switching network. While conventional control loops are designed to work with circuit-switching networks, where dedicated communication channels provide almost constant bit rate and delay, many networks, such as Ethernet, organize data transmission in packets, carrying larger amount of information at less predictable rates. To avoid the bandwidth waste due to the relatively large overhead inherent to packet transmission, we exploit the packet payload to carry longer control sequences. To this aim we adopt a model-based approach to remotely compute a predictive control signal on a suitable time horizon, which leads to effectively reducing the bandwidth required to guarantee stability. We consider networks for which both the time between consecutive accesses to each node (MATI) and the transmission and processing delays (MAD) for measurements and control packets are bounded. Communications are assumed to be ruled by a rather general protocol model, which encompasses many protocols used in practice. As a distinct improvement over the state of the art, our result is shown to be robust with respect to sector-bounded uncertainties in the plant model. Namely, an explicit bound on the combined effects of MATI and MAD is provided as a function of the basin of attraction and the model accuracy. A case study is presented to appreciate the improvements induced by the packet-based control strategy over existing methods.

Introduction

Industrial manufacturing is witnessing an ever more extensive use of communication networks to support automated scheduling, control and diagnostic activities [START_REF] Moyne | The emergence of industrial control networks for manufacturing control, diagnostics, and safety data[END_REF], [START_REF] Zurawski | Special issue on industrial communication systems[END_REF]. The possibility offered by networks of replacing traditional point-to-point connections with more complex and dynamic schemes, opens unprecedented opportunities for factory control and management. Alongside allowing a pervasive adoption of decentralization and cooperation, networks convey many advantages in terms of flexibility, scalability and robustness. The adoption of a distributed networked architecture can induce a remarkable reduction of costs and delays for both installation and maintenance. These advantages justify the increas-⋆ A partial version of this paper has been submitted to the 18th IFAC World Congress. This work was partially supported by the HYCON2 NoE, contract number FP7-ICT-257462. Corresponding author L. Greco. Tel. +33 1 69 85 17 50, Fax +33 1 69 85 17 69.

Email addresses: lgreco@ieee.org (Luca Greco), chaillet@ieee.org (Antoine Chaillet), bicchi@ing.unipi.it (Antonio Bicchi). ing interest in control over networks (see for instance [START_REF] Bushnell | Special issue on networks and control[END_REF], [START_REF] Antsaklis | Special issue on networked control systems[END_REF], [START_REF] Antsaklis | Special issue on technology of networked control systems[END_REF], [START_REF] Di Benedetto | Special issue: Industrial control over wireless networks (ICWN08)[END_REF], [START_REF] Wang | Networked Control Systems: Theory and Applications[END_REF]).

In general terms, a Networked Control System (NCS) is a system in which sensors, actuators and controllers are spatially distributed and exchange information through a shared, digital, finite capacity channel. The use of the network as a communication medium and the distributed nature of the system make traditional control theory not always applicable. Issues such as quantization errors, data dropouts, variable transmission intervals, variable communication delays, and constrained access to the network, can no longer be ignored [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]. The NCS literature has separately addressed many of these problems, and sometimes the combinations thereof. An excellent discussion of the state-of-the-art is reported in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], and the reader is referred there for a detailed analysis of the literature towards the mentioned communication constraints. An essential aspect of NCS, not considered in detail in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF], is the packet-switching nature of many networks. As opposed to conventional control loops, which are designed to work with circuit-switching networks where dedicated communication channels provide almost constant bit rate and delay, networks such as Ethernet organize data transmission in packets, carrying larger amount of information at less predictable rates.

The organization of control information in data packets, which have relatively large transmission overhead, substantially alter the bandwidth/performance tradeoff of traditional design. For instance, important datarate theorems [START_REF] Hespanha | Towards the control of linear systems with minimum bit-rate[END_REF], [START_REF] Nair | Exponential stabilisability of finite-dimensional linear systems with limited data rates[END_REF], [START_REF] Nair | Stabilizability of stochastic linear systems with finite feedback data rates[END_REF] expressing a fundamental relationship between the degree of instability of a given physical system and the minimum bit rate required to stabilize it, do not account for the fact that data come in packets with a minimum size (e.g. 84 bytes in Ethernet). To simplify, transmitting a 16 bits record every millisecond requires as much bandwidth in average as sending a packet of 84 bytes every 48 milliseconds; however, the implications on the effective sampling rate and feedback control performance are apparent. How to recover part of this performance is an objective of this study.

A second aspect inherent to packet-switching networks is transmission overhead. For instance, every Ethernet packet carries 38 bytes of headers and interframe separations, and useless information is necessarily padded into the payload to reach the minimum required packet length. As a consequence, transmitting a few bits per packet has essentially the same bandwidth cost as transmitting hundreds of them. A new, specific trade-off hence arises between packet rate and packet dimension for a given estimation/control task.

While the above aspects have been observed and described in the early literature on NCS (see e.g. the surveys [START_REF] Walsh | Scheduling of networked control systems[END_REF], [START_REF] Lian | Performance evaluation of control networks: Ethernet, ControlNet, and DeviceNet[END_REF], [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]), only recently have appeared results which address them explicitly in controller design. The goal can be succinctly described as to decrease the network utilization (in terms of bandwidth, or packets per unit of time) without compromising control performance. To achieve this, [START_REF] Montestruque | Stability of model-based networked control systems with time-varying transmission times[END_REF] pioneered the idea of exploiting the empty portion of packet payload to carry feedforward control sequences, computed in advance on the basis of a model-based scheme. Following developments along these lines generalized the technique to address nonlinear systems [START_REF] Quevedo | Packetized predictive control over erasure channels[END_REF], time-varying delays and packet dropouts [START_REF] Polushin | On the modelbased approach to nonlinear networked control systems[END_REF], [START_REF] Pin | Networked predictive control of constrained nonlinear systems: Recursive feasibility and input-to-state stability analysis[END_REF], as well as the constraints imposed by communication protocols on state measurement access [START_REF] Chaillet | Delay compensation in packetswitching networked controlled systems[END_REF].

In this paper we also adopt the feedforward approach to send in a packet not only the control value to be applied at a specific instant, but also a prediction of the control law valid on a given time-horizon, so as to better exploit the payload. In the same spirit of other model-based approaches (e.g. [START_REF] Montestruque | Stability of model-based networked control systems with time-varying transmission times[END_REF], [START_REF] Quevedo | Packetized predictive control over erasure channels[END_REF], [START_REF] Polushin | On the modelbased approach to nonlinear networked control systems[END_REF], [START_REF] Pin | Networked predictive control of constrained nonlinear systems: Recursive feasibility and input-to-state stability analysis[END_REF]), the control sequence is obtained by simulating an (imprecise) model of the closed-loop plant. The internal state of the model is asynchronously updated by means of the measurements of the plant state provided by sensors. Due to their spatial distribution, only portions of the model state can be updated in each instant. Therefore, we consider the constrained access to the network to be ruled by a protocol deciding which sensor can communicate at each instant. The large control-packet, sent by the remote controller, is stored in an embedded memory on the plant side. Based on a local re-synchronization, made possible by a time-stamping of measurements, this strategy also allows to compensate the effect of bounded communication delays in the control loop. We build our model upon the powerful hybrid formalism introduced in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], and we consider network imperfections affecting both sides of the control loop. We provide explicit bounds on the Maximum Allowable Delay (MAD [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]) and on the Maximum Allowable Transfer Interval (MATI [START_REF] Walsh | Stability analysis of networked control systems[END_REF], i.e. the maximum duration between two successive communications) ensuring the semiglobal exponential stability of the NCS.

The main contribution of this paper is a control strategy for packet-switching networks ensuring the stability of an uncertain nonlinear NCS affected by varying transmission intervals, varying (and potentially large) delays, and constrained access to the network. Unlike the commonly assumed small-delays hypothesis (see for instance [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]), we can compensate for delays larger than the transmission interval. It should be noticed that the authors of [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF] consider the development of a model for NCSs accounting for variable transmission intervals, potentially large and varying delays and constrained network access, as a hard problem.

A line of work close to ours is reported in [START_REF] Polushin | On the modelbased approach to nonlinear networked control systems[END_REF], where the problem of stabilizing a nonlinear NCS with feedforward control sequences is addressed. Such sequences are computed by means of an approximate discrete-time plant model. Authors assume that the approximation algorithm is the only source of uncertainty in the model and that the inaccuracy of such a model can be reduced at will in order to achieve the desired MATI. In this paper, instead, we consider a robustness problem, where the plant uncertainty is a given, and we provide a bound on the MATI in terms of the model inaccuracy (measured through its local Lipschitz constant).

Preliminary results concerning our approach were presented in [START_REF] Chaillet | Delay compensation in packetswitching networked controlled systems[END_REF]. The present paper extends that in at least three relevant aspects. We consider here uniformly global exponentially stable (UGES) protocols, rather than the conservative class of invariably UGES protocols (which do not include, for instance, the very common Round Robin protocol) assumed in [START_REF] Chaillet | Delay compensation in packetswitching networked controlled systems[END_REF]. We significantly extend the class of nonlinear plants by imposing only local Lipschitz conditions, instead of global ones in [START_REF] Chaillet | Delay compensation in packetswitching networked controlled systems[END_REF]. Finally, we take directly into account, in the computation of the MATI, the accuracy of the model used to build the prediction.

Problem Statement

Notation: Given a set A ⊂ R and a ∈ A, A ≥a denotes the set {s ∈ A | s ≥ a}. Given a vector x = (x 1 , . . . ,

x n ) T ∈ R n , n ∈ N ≥1 , |x| denotes its Euclidean norm, i.e. |x| n i=1 x 2 i 1/2 . Given R ≥ 0, B R denotes the closed ball of radius R centered in zero: B R {x ∈ R n | |x| ≤ R}.
Given a locally essentially bounded signal u : R ≥0 → R n , u L ∞ ess sup t≥0 |u(t)|. We use mod to denote the modulo operator, i.e. given m, n ∈ N, m mod n = p if and only if there exists r ∈ N such that m = rn + p with p < n. We define the floor function ⌊•⌋ : R → Z as ⌊x⌋ max{m ∈ Z | m ≤ x}.

Network Model

We consider a NCS constituted of a remote controller receiving measurements from and sending commands to a physical plant through a shared communication channel (see Figure 1). Control sequences are sent over the digital network as packets. An elementary embedded control device receives, decodes, synchronizes these packets and applies control commands to the plant. Measurements are taken by physically distributed sensors and sent towards the controller as packets encoded with sufficient precision to neglect quantization effects. Sensors are assumed to be embedded with the plant and hence synchronized with it. Due to the distributed nature of the sensors, we also assume that the measurement part of the network is partitioned in ℓ nodes and only a unique node at a time can send its information (i.e. only partial knowledge of the plant state is available at each time instants).

We consider that measurements are taken and sent at instants {τ m i } i∈N , and are received by the remote controller at instants {τ m i + T m i } i∈N . In other words, {T m i } i∈N denote the (possibly time-varying) measurement data delays, which cover both processing and transmission delays on the measurement chain. In the same way, control commands are computed, encoded into packets and sent over the network at time instants {τ c j } j∈N . They reach the plant at instants {τ c j + T c j } j∈N , where {T c j } j∈N denote the (possibly time-varying) control data delays accounting both for computation and transmission delays from the remote controller to the plant.

Assumption 1 (Network) The communication network satisfies the following properties:

i) (MATI) There exist two constants τ m , τ c ≥ 0 such that τ m i+1 -τ m i ≤ τ m and τ c j+1 -τ c j ≤ τ c , ∀i, j ∈ N; ii) (MAD) There exist two constants T m , T c ≥ 0 such that T m i ≤ T m and T c j ≤ T c , ∀i, j ∈ N; iii) (No Zeno phenomenon) There exist constants ε m > 0 and ε c > 0 such that ε m ≤ τ m i+1 -τ m i , ∀i ∈ N and ε c ≤ τ c j+1 -τ c j , ∀j ∈ N.
Item i) in the previous assumptions imposes that the MATI between two consecutive accesses to the network is bounded both for measurements and control. Item ii) imposes that the MAD, both on measurements and control side, is bounded. Item iii) imposes that the minimum time interval between two consecutive accesses to the network by the nodes is lower bounded away from zero, and similarly for the control side. The objective of this paper is to provide explicit bounds on the MATIs (τ m and τ c ) and on the MADs (T m and T c ) to guarantee exponential stability of the closed-loop NCS based on a specific control procedure.

Protocol Model

The access to the network is ruled by a protocol choosing, at each instant τ m i , which node communicates its data. Decisions can be taken either according to the time index i (static protocol) or based on the value of the error e between the state estimate x and the available state measurements x from sensors (dynamic protocol). More precisely, in the spirit of [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], we model the network protocol as a time-varying discrete-time system involving the error R n ∋ e

x -x, n ∈ N ≥1 , that this type of communication generates:

e(i + 1) = h(i, e(i)) , ∀i ∈ N , (1)
where h : N×R n → R n . If the network were able to send the measurement of the whole state at each time instant τ m i , then the function h would be identically zero; this is an assumption commonly posed in the literature on NCSs (see for instance [START_REF] Branicky | Stability of networked control systems: Explicit analysis of delay[END_REF], [START_REF] Zhivoglyadov | Networked control design for linear systems[END_REF], [START_REF] Montestruque | Stability of model-based networked control systems with time-varying transmission times[END_REF], [START_REF] Seuret | Exponential stabilization of delay neutral systems under sampled-data control[END_REF], [START_REF] Yue | Networked-based robust H∞ control of systems with uncertainty[END_REF], [START_REF] Naghshtabrizi | Stability of networked control systems with variable sampling and delay[END_REF], [START_REF] Quevedo | Packetized predictive control over erasure channels[END_REF], [START_REF] Polushin | On the modelbased approach to nonlinear networked control systems[END_REF], [START_REF] Pin | Networked predictive control of constrained nonlinear systems: Recursive feasibility and input-to-state stability analysis[END_REF]) where network effects are mostly modeled as sampling and delays. This assumption may no longer be justified when sensors are physically distributed.

Purely static protocols involve a function h which takes as an argument the time index i only. An example of such protocols is the Round Robin (RR) protocol, which executes a cyclic inspection of each node. On the opposite, some network protocols purely rely on the current value of the error, in which case h is independent of i: this is the case of the Try-Once-Discard (TOD) protocol [START_REF] Walsh | Stability analysis of networked control systems[END_REF]. The objective of most communication protocols is to decrease some function of the transmission error e at each transmitted packet. A particularly relevant class of such protocols is the one that ensures an exponential decay of this error. We recall here a slightly modified version of the definition in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF] to focus on the class of protocols we deal with in this work.

Assumption 2 (UGES Protocol) The protocol modeled by the discrete-time system (1) is uniformly globally exponentially stable (UGES) and admits an associated Lyapunov function with bounded gradient. That is, there exist a function W 0 : N × R n → R ≥0 locally Lipschitz in the second argument, and constants a, a, c > 0 and ρ 0 ∈ [0, 1) such that, for all e ∈ R n and all i ∈ N,

a |e| ≤ W 0 (i, e) ≤ a |e| (2)
W 0 (i + 1, h(i, e)) ≤ ρ 0 W 0 (i, e) , (3)
and for almost all e ∈ R n and all i ∈ N

∂W 0 ∂e (i, e) ≤ c. (4)
It is worth stressing that the UGES protocols considered here are not necessarily invariably UGES, as assumed in [START_REF] Chaillet | Delay compensation in packetswitching networked controlled systems[END_REF]. The latter property is rather restrictive, as it excludes, for instance, the commonly adopted Round Robin protocol.

Remark 1 We do not explicitly consider packet dropouts here. However, its inclusion is possible without modifying the overall framework if some additional assumptions are made. Dropouts in a plant-to-controller channel governed by an invariably UGES protocol (see [START_REF] Chaillet | Delay compensation in packetswitching networked controlled systems[END_REF]) are easily dealt with by considering a scaled MATI, as proposed in Remark II.4 in [START_REF] Heemels | Networked control systems with communication constraints: Tradeoffs between transmission intervals, delays and performance[END_REF]. It should be noticed however that the MATI scaling approach does not apply if an UGES, but not invariably UGES, protocol is considered. Bounded packet dropouts in the controller-to-plant channel can be tolerated in our framework if consecutive feedforward control packets overlap sufficiently (this will be more clear in the sequel).

The plant and its model

We assume that a nominal feedback controller is given, which would be able, in the absence of the effects induced by the network, to globally exponentially stabilize the real plant. More precisely, we assume the following.

Assumption 3 (Nominal GES)

There exists a continuously differentiable function κ : R n → R m such that the closed-loop system

ẋ = f (x, u) (5) u = κ(x) (6)
is globally exponentially stable (GES), so that there exists a differentiable function V : R n → R ≥0 and constants α, α, α, d > 0 such that the following conditions hold for all

x ∈ R n α |x| 2 ≤ V (x) ≤ α |x| 2 ∂V ∂x (x)f(x, κ(x)) ≤ -α |x| 2 ∂V ∂x (x) ≤ d |x| .
In order to compute the control signal, the remote controller makes use of a state estimate based on an approximate model f of the plant f . Both the plant and its model are considered to be zero at the origin (f (0, κ(0)) = f(0, κ(0)) = 0). The strategy developed in this paper relies on the assumption that the plant, its model and the nominal controller are all locally Lipschitz.

Assumption 4 (Local Lipschitz) Given some constants R x , R u > 0, there exist some positive constants λ f and λ κ1 such that for all x 1 , x 2 ∈ B R x and all u 1 , u 2 ∈ B R u , the following inequalities hold

|f (x 1 , u 1 ) -f (x 2 , u 2 )| ≤ λ f (|x 1 -x 2 | + |u 1 -u 2 |) (7) |κ(x 1 ) -κ(x 2 )| ≤ λ κ |x 1 -x 2 | . (8)
It is worth noting that the previous assumption represents a further important relaxation with respect to [START_REF] Chaillet | Delay compensation in packetswitching networked controlled systems[END_REF], where all involved vector fields were assumed to be globally Lipschitz.

Finally, we assume that the plant model inaccuracy is sector-bounded.

Assumption 5 (Sector-Bounded Model Inaccuracy) Given R x , R u > 0, there exists a nonnegative constant λ f f such that for all x ∈ B Rx and all u ∈ B Ru ,

f (x, u) -f(x, u) ≤ λ f f (|x| + |u|) . (9)
The constant λ f f thus measures the model accuracy: the closer the model f is to the real system f , the smaller is λ f f (in the ideal case of perfect modeling, it would be zero). Note that Assumption 5 allows to cope with both parametric uncertainties and unmodeled dynamics.

3 A model-based strategy

Modeling the overall setup

We develop here a model-based strategy exploiting the relatively large payload of a packet. At each reception of a new measurement, the remote controller updates an estimate of the current state of the plant and computes a prediction of the control signal over a fixed time horizon T p 0 by numerically running the model f. This signal is then coded and sent in a single packet at the next network access. When received by the plant, it is decoded and re-synchronized by the embedded computer, based on the time-stamping of the original measurement. We assume here that the plant and its sensors have a common clock; however, we also stress that in our strategy there is no need for clock synchronization between the plant and the remote controller.

In order to guarantee that a relevant control signal is always available, the fixed time horizon on which each state prediction is achieved is chosen as

T p 0 ≥ T c + T m + τ m + τ c . (10)
This prediction horizon guarantees, in view of Assumption 1, that a control sequence corresponding to the present time is always loaded in the memory of the embedded controller.

For sake of mathematical rigor, we introduce first a model accounting for infinitely many state variables and infinitely many duplicates of the model f. In Section 3.2, we show how to properly reduce them to a finite number. Therefore, for any measurement taken at τ m i , i ∈ N, we consider a new estimate state variable xi , valid over the time interval [τ m i , τ m i + T p 0 ], whose evolution is given by

ẋi (t) = f (x i (t), κ(x i (t))), ∀t ∈ [τ m i , τ m i + T p 0 ] xi (τ m+ i ) = x(τ m i ) + h(i, xi-1 (τ m i ) -x(τ m i )). ( 11
)
Each variable is updated at time τ m+ i according to the protocol h. Usually, when dealing with a unique variable, the update of an estimate is performed by means of the error between the measurement and the variable itself. In our case, instead, a new estimate variable xi is created at each τ m i , with the previous variable xi-1 containing the latest value of the estimate. Hence, the error we compute at time τ m i is between the measurement made on x(τ m i ) and the previous estimate variable xi-1 (τ m i ). In this way i + T m i , i ∈ N at which the measurements x(τ m i ) reach the controller. It is important to remark that the dynamics [START_REF] Montestruque | Stability of model-based networked control systems with time-varying transmission times[END_REF] actually evolves in a virtual (simulated) time. The measurement x(τ m i ) reaches the controller only at τ m i + T m i and then triggers the simulation of the dynamics [START_REF] Montestruque | Stability of model-based networked control systems with time-varying transmission times[END_REF] for a virtual time interval [τ m i , τ m i + T p 0 ]. The actual time spent for this simulation and for the computation of the predicted control signal is, in fact, part of the delay T c j . What we have done in [START_REF] Montestruque | Stability of model-based networked control systems with time-varying transmission times[END_REF], is to consider the estimate dynamics 'stretched' on the real time as if it ran concurrently with the plant. This notation trick allows us to cast the overall system in a compact model similar to the one in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF].

m i m i T + τ m i m i T 1 1 + + + τ m i m i T 2 2 + + + τ m i m i T 3 3 + + + τ m i 1 + τ m i 2 + τ m i 3 + τ m i τ c j τ c j 1 + τ c τ t i x ˆ1 ˆ+ i x 2 ˆ+ i x 3 ˆ+ i x j u ˆ1 ˆ+ j
At each instant τ c j a new control signal u j (t) is computed. It is based on the estimate variable xγ(j) , where γ(j) denotes the index of the latest measurement received before τ c j . More precisely, the function γ : N → N is defined as

γ(j) max i ∈ N | τ m i + T m i < τ c j , ∀j ∈ N .
It can be easily verified that, in view of Assumption 1, the time horizon for the control signal has to satisfy

T c 0 ≥ T c + τ c ( 12
)
in order to guarantee that a valid control signal is always available to the embedded controller. Note that the required time horizon T c 0 for the control signal is smaller than the time horizon T p 0 used for prediction, as it does not need to account for measurement MAD and MATI.

We thus define an infinite number of feedforward control signals as

ûj (t) = κ(x γ(j) (t)), ∀t ∈ [τ c j , τ c j + T c 0 ], ∀j ∈ N .
At each reception of a new control packet (i.e. at instants τ c j + T c j ), the buffer of the embedded controller is updated. Consequently, the control signal applied to the plant is given by

û(t) = ûj (t), ∀t ∈ [τ c j + T c j , τ c j+1 + T c j+1 ). ( 13
)
Both the feedforward signals ûj and the control û are depicted at the bottom of Figure 2. Line styles are consistent with those of the estimate evolutions used to build the control signals. Vertical arrows show which estimate variable xγ(j) is chosen for the computation of the feedforward signal ûj at time instant τ c j , and which control signal ûj is used at τ c j + T c j to update the embedded controller. In the particular example of Figure 2, it can be noticed that ûj and ûj+1 are computed with respect to the same estimate xi since γ(j) = γ(j + 1) = i. On the other hand, xi+1 is not directly used by any control since γ(j + 2) = i + 2.

A reduced NCS model

The model considered so far makes use of infinitely many state estimate variables xi and control signals ûj . They can be reduced to a finite number by noticing that they are all defined over compact time intervals and that "old" variables are no longer used after a while. State estimates variables are stored in a finite memory and new values are cyclically written on dismissed variables. We must prevent that a variable is accidentally reset while still in use for the computation of a control signal. In particular, xγ(j) cannot be reset during the interval [τ m γ(j) , τ c j+1 + T c j+1 ]. Hence, the dimension of such a memory, in terms of number of variables, is given by the maximum number of measurements that can be received during the life horizon T p 0 of an estimate variable. Recalling that T p 0 accounts also for the interval during which no measurements are received, whose length is bounded by τ m , the dimension N of the memory is given by

N T p 0 -τ m ε m + 1. (14)
Therefore, we use only N state variables x c r , r ∈ {1, . . . , N }, to store the state estimates. They are cyclically updated according to the following relation

x c r (t) xi (t) iff η(t, i, r) = 1,
where η : R ≥0 × N × {1, . . . , N } → {0, 1} is the function defined as η(t, i, r)

1 if t ∈ (τ m i , τ m i+1 ] and µ(i) = r 0 otherwise, (15)
which identifies the index of the relevant state estimate, and µ : N → {1, . . . , N } is defined as

µ(i) ((i -1) mod N ) + 1 (16)
to make a cyclic update of the state estimates in the memory. By means of the vectors x, x c , e ∈ R Nn defined as

x [x T , . . . , x T ] T , x c [x T c1 , . . . , x T cN ]
T and e = [e T 1 , . . . , e T N ] T x c -x, the closed-loop dynamics of the NCS can be compactly written as

ẋ = F (t, x, e) (17a) ė = G(t, x, e) (17b) e(τ m+ i ) = H(i, e(τ m i )), (17c)
where

F (t, x, e) = f (x, u(t, e + x)) (18a)
G(t, x, e) =      f (e 1 + x, κ(e 1 + x)) -f(x, u(t, e + x)) . . . f (e N + x, κ(e N + x)) -f(x, u(t, e + x))      (18b)
H(i, e) =        e 1 + (h(i, e N ) -e 1 ) η(t, i , 1)
e 2 + (h(i, e 1 ) -e 2 ) η(t, i, 2) . . .

e N + (h(i, e N-1 ) -e N ) η(t, i, N )        . ( 18c
)
The control signal u in (18a) and (18b) is given by2 u(t, x c )

N k=1 κ(x c k )ν(t, j, k), ∀j ∈ N, (19)
where the function ν : R ≥0 × N × {1, . . . , N } → {0, 1} is defined as

ν(t, j, k)        1 if t ∈ (τ c j + T c j , τ c j+1 + T c j+1 ] and µ(γ(j)) = k 0 otherwise.
This compact notation carries the advantage to involve a finite number of state variables and to fit the framework of [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF]. Note that the control signal in (13) now reads û(t) = u(t, x c (t)).

Main results

We start by proving that the obtained protocol (17c) and (18c) inherits the UGES property from the original one [START_REF] Antsaklis | Special issue on networked control systems[END_REF]. All proofs are deferred to Section 6.

Proposition 1 Under Assumption 2, the protocol modeled by the discrete-time system (17c) and (18c) is UGES and admits an associated Lyapunov function

W : N × R Nn → R ≥0 given by W (k, e) N r=1 W 0 (k, e r )η(t, k, r) ,
where η is defined in [START_REF] Nair | Stabilizability of stochastic linear systems with finite feedback data rates[END_REF], and satisfying for all k ∈ N and all e ∈ R Nn :

a L |e| ≤ W (k, e) ≤ a H |e| (20)
W (k + 1, H(k, e)) ≤ ρ 0 W (k, e) (21)
∂W ∂e (k, e) ≤ c , (22)
with a L a for N = 1 and a L a N min 1, a a

1 ρ0

for N > 1, and a H a.

Let us now present a local result on the exponential stability of the NCS [START_REF] Pin | Networked predictive control of constrained nonlinear systems: Recursive feasibility and input-to-state stability analysis[END_REF]. It provides an explicit bound (cf.

(23) below) on the measurement MATI τ m in terms of the characteristic parameters of the network-free closedloop system, the protocol, the regularity assumptions on the dynamics and the model precision.

Theorem 1 Assume that Assumptions 1-3 hold. Given some R > 0, fix R x = R and R u = λ κ R and suppose that Assumptions 4-5 hold with these constants. Let a, a, ρ 0 , c, α, α, α, d, λ f f , λ f , λ κ , a L , a H be generated by these assumptions and by Proposition 1. Assume that the following conditions on τ m , τ c , T m , T c , ε m hold

τ m ∈ (0, τ ⋆ m ), τ ⋆ m 1 L ln Hγ 2 + a L L Hγ 2 + a L ρ 0 L (23)
N = T c + T m + τ c ε m + 1 ( 24
)
τ m ≥ ε m ( 25
)
where

L c a L √ N λ f f (1 + λ κ ) + √ Nλ f + √ N -1 + N -1 λ f λ κ (26) H cNλ f f (1 + λ κ ) (27)
γ 2 d α α α λ f λ κ . ( 28
)
Then, the origin of the NCS ( 17) is exponentially stable with radius of attraction

R = R K ( 29
)
where

K √ 2 1 -γ 1 γ 2 max {k 2 (1 + γ 1 ) , k 1 (1 + γ 2 )} (30) γ 1 exp (Lτ m ) -1 a L L (1 -ρ 0 exp (Lτ m )) H (31) k 1 a H ρ 0 a L (32) k 2 α α . (33)
It is important to remark that the bound ( 23) on the measurement MATI is also related to the dimension of the memory N , whose definition [START_REF] Wang | Networked Control Systems: Theory and Applications[END_REF], obtained by [START_REF] Nair | Exponential stabilisability of finite-dimensional linear systems with limited data rates[END_REF] for T p 0 = T c + T m + τ m + τ c , embeds the other relevant communication parameters: MADs and control MATI. The pair ( 23)-( 24) thus imposes a trade-off between the two MATIs and the MADs. The packet-based strategy aims at enlarging the control MATI τ c , but a larger τ c could require a larger memory N hence could produce a lower measurement MATI τ m . Moreover, conditions ( 24)-( 25) bind the four relevant parameters (i.e. T c , T m , τ c and τ m ) together and with the constant ε m , bounding the minimum time between two consecutive accesses to the network. In particular, they require that the communication MATI τ m is not smaller than ε m . Furthermore, depending on the parameter R for which Assumptions 4 and 5 hold, an explicit estimate R of the radius of attraction can be computed, cf. (29). Note that, since Theorem 1 guarantees only local properties, Assumption 3 could be relaxed to local exponential stability of the nominal plant, over a sufficiently large domain.

The following proposition establishes that the MATI and memory requirements of the previous theorem can always be satisfied.

Proposition 2 Given any R > 0, the parameters τ m , τ c , T m , T c , ε m can always be picked small enough to satisfy conditions ( 23), ( 24) and [START_REF] Yue | Networked-based robust H∞ control of systems with uncertainty[END_REF].

In general, the radius of attraction R of the resulting NCS guaranteed by Theorem 1 cannot be arbitrarily specified due to the possible dependency of the constants L and H (and consequently K) in the parameter R ruling the domain on which Assumptions 4 and 5 hold. To see this more clearly, consider, for instance, the case of K proportional to R. Relation (29) shows that, in this case, the radius of the initial condition R would be a constant irrespective of the amplitude of R. One could even imagine that, in some situations, R actually shrinks when R is enlarged. Hence, in order to ensure that the set of initial conditions can be arbitrarily enlarged, we must add some constraints on the growth rate of the constant K or, equivalently, on some of the Lipschitz constants. After reporting a definition of semiglobal exponential stability which is adapted to our NCS framework, we present our main result in this regard in Theorem 2.

Definition 1

The NCS ( 17) is said to be semiglobally exponentially stable if, for any R > 0, there exist positive constants

τ ⋆ m ( R), τ ⋆ c ( R), T ⋆ m ( R), T ⋆ c ( R) and ε ⋆ m ( R), as introduced in Assumption 1, such that its origin is ex- ponentially stable on B R.
Theorem 2 Suppose that Assumptions 1-4 hold for all R x , R u > 0 and that there exists σ ∈ (0, 1) such that

lim s→∞ λ f (s)λ κ (s) s σ < ∞. (34)
Then, the NCS ( 17) is semiglobally exponentially stable.

The above result guarantees that, provided sufficient regularity of the dynamics involved (i.e. Lipschitz constants sublinear in the size of the domain over which they are computed), any prescribed compact domain of attraction can be reached if MADs and MATIs are small enough.

Case Study

The exploitation of the packet payload and the modelbased predictive strategy presented in this paper can improve the MATI bounds obtained by sending a single control value in each packet. Let us illustrate such improvement by comparing our bounds with those computed in [START_REF] Tabbara | Stability of wireless and wireline networked control systems[END_REF] for a Ch-47 Tandem-Rotor Helicopter. The linearized model describing the helicopter can be written as

ẋ = Ax + Bu y = Cx with C = 0 1 0 0 0 0 0 57.3 , A =        -0.02 0.005 2.4 -32 -0.14 0.44 -1.3 -30 0 0.018 -1.6 1.2 0 0 1 0        , B =       
0.14 -0.12 0.36 -8.6

0.35 0.009

0 0        .
The exponentially stabilizing static output feedback is

given by u = Ky with K = -12.7177 -45.0824

63.5123 25.9144 . We assume the state x is transmitted by means of two links (ℓ = 2) ruled by the Round Robin protocol. The bound on the MATI provided in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF] is τ ⋆ [16] = 1.20 10 -5 s and the improvement in [START_REF] Tabbara | Stability of wireless and wireline networked control systems[END_REF] provides τ ⋆

[21] = 2.81 10 -4 s. In the ideal case of a linear system with equidistant transmissions and Round Robin protocol, the exact MATI can be computed following the argument in [16, Section VII-A]. For the present case study we get τ ⋆ single ≃ 1.13 10 -3 s.

In order to appreciate the improvement induced by the packet payload exploitation, let us compute the MATI bound according to expression [START_REF] Walsh | Stability analysis of networked control systems[END_REF]. For the comparison to be fair, we assume zero delays (T m = T c = 0). We also fix ε m ≃ τ ⋆ m and τ c slightly less than ε m (just to keep N = 1 in ( 24)). The relevant protocol parameters, according to Assumption 2, are a = 1, a = √ ℓ, ρ 0 = (ℓ -1)/ℓ, c = √ ℓ (see [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF]). As concerns the nominal stability parameters in Assumption 3, we have α = 9 10 -4 , α = 399.22, α = 1, d = 2α (see [START_REF] Tabbara | Stability of wireless and wireline networked control systems[END_REF]). The Lipschitz constants required by Assumption 4 are λ f = max{|A| , |B|} = 43.8904 and 3 λ κ = |KC| ≃ 2980. We assume also to have a perfect model: f = f and λ f f = 0 (see Assumption 5). With N = 1, the constants in [START_REF] Zhivoglyadov | Networked control design for linear systems[END_REF] and [START_REF] Zurawski | Special issue on industrial communication systems[END_REF]

become L = c aL λ f f (1 + λ κ ) + λ f = c a λ f and H = cλ f f (1 + λ κ ) = 0. Thus, the MATI bound (23) is given by τ ⋆ m = a cλ f ln 1 ρ 0 ≃ 5.58 10 -3 s,
i.e. about 20 times larger than τ ⋆ [21] -or, in other terms, our method would require sending ca. 20 times less packets to stabilize the system. In reality, the exploitation of the packet payload often induces a more substantial improvement than that shown by the bounds above. If we compare the exact MATI 4 achievable with our predictive control strategy with a classical single control value technique, we have τ ⋆ mult ≃ 1.3105 s, hence a theoretical improvement of

τ ⋆ mult τ ⋆ single ≃ 1160.

Proofs

Proof of Proposition 1

For N = 1, W (k, e) = W 0 (k, e) and the thesis follows from Assumption 2. For N > 1, let us consider any s ∈ {1, . . . , N -1} satisfying η(t, k, s) = 1 (with η(•) defined in ( 15)) for some t ≥ 0 and some k ∈ N ≥1 . Then W (k, e) = W 0 (k, e s ) and for k + 1 we have

η(t, k + 1, s + 1) = 1 and W (k + 1, H(k, e)) = W 0 (k + 1, h(k, e s )) ≤ ρ 0 W 0 (k, e s ) = ρ 0 W (k, e s ). The inequal- ity W (k, e) = W 0 (k, e s ) ≤ a|e s | ≤ a|e| is easily verified (thus a H = a)
, while the other inequality requires to consider the evolution of the system.

Recall that at step k the (s + 1)-th variable is updated, hence the s-th variable is left unchanged for N -1 periods, the (s -1)-th variable for N -2 periods, etc. Summarizing, the following relations hold:

e s-1 (k) = h(k -1, e s-2 (k -1)) e s-2 (k) = e s-2 (k -1) = h(k -2, e s-3 (k -2))
. . .

e 1 (k) = e 1 (k -1) = • • • = e 1 (k -s + 2) = h(k -s + 1, e N (k -s + 1)) e N (k) = e N (k -1) = • • • = e N (k -s + 1) = h(k -s, e N-1 (k -s)) . . . e s+1 (k) = e s+1 (k -1) = • • • = e s+1 (k -N + 2) = h(k -N + 1, e s (k -N + 1)) e s (k) = e s (k -1) = • • • = e s (k -N + 1) = h(k -N, e s-1 (k -N )).
This permanency allows us to write N -1 of the previous relations referring to the variables computed at instant k -1, and hence to suppress the dependency from k:

e s-1 = h(k -1, e s-2 ) • • • e s+1 = h(k -N +1, e s ).
By the previous relations we have

W 0 (k, e s-1 ) = W 0 (k, h(k - 1, e s-2 )) ≤ ρ 0 W 0 (k -1, e s-2
), and consequently W 0 (k -1, e s-2 ) ≥ W0(k,es-1) ρ0

. Analogously we can write

W 0 (k -N + 1, e s ) ≥ W 0 (k -N + 2, e s+1 ) ρ 0 ( 35
)
4 Even in this case we used the argument in [16, Section VII-A].

and iterating W 0 (k

-N + 1, e s ) ≥ W0(k-N+2,es+1) ρ0 ≥ W 0 (k-N+3,e s+2 ) ρ 2 0 • • • ≥ W 0 (k,e s-1 ) ρ N-1 0 .
Recalling that for all k we can write a|e| ≤ W 0 (k, e) ≤ a|e|, then for all e, q and k we have

W 0 (q, e) ≥ a a W 0 (k, e). (36)
Whereby, using ( 35) and (36) we have W 0 (k, e s ) ≥

ρ 0 W 0 (k, e s+1 ) • • • ≥ a a 2 1 ρ N-2 0 W 0 (k, e s-2 ) ≥ a a 1 ρ N-1 0 W 0 (k, e s-1
). Using the previous relations we can write the sought inequality and compute a L :

W (k, e) = W 0 (k, e s ) = 1 N N r=1 W 0 (k, e s ) ≥ 1 N W 0 (k, e s ) + a a 2 1 ρ 0 W 0 (k, e s+1 ) + • • • + a a 1 ρ N-1 0 W 0 (k, e s-1 ) ≥ a N |e s | + a a 2 1 ρ 0 |e s+1 | + • • • + a a 1 ρ N-1 0 |e s-1 | ≥ a N min 1, a a 2 1 ρ 0 , • • • , a a 1 ρ N-1 0 |e| ≥ a N min 1, a a 2 1 ρ 0 |e|,
where we used the fact that a a , ρ 0 < 1.

Finally we write

∂W ∂e (k, e) = ∂W 0 ∂e (k, e s ) = 0, • • • , 0, ∂W 0 ∂e s (k, e s ), 0, • • • , 0 T ≤ c.

Proof of Theorem 1

The proof consists of the following 4 steps:

(1) Show that the system (17b) is locally input-to-state exponentially stable with linear gain from x to e, provided that solutions remain inside the domain B R;

(2) Show that the system (17a) is locally input-to-state exponentially stable with linear gain from e to x, provided that solutions remain inside the domain B R;

(3) Show by means of small gain arguments that the overall system ( 17) is locally exponentially stable inside B R; (4) Show that solutions remain indeed inside B R.

Step 1 Let us consider any r ∈ {1, . . . , N } such that ν(t, j, r) = 1 for some t ≥ 0 and some j ∈ N. In view of [START_REF] Quevedo | Packetized predictive control over erasure channels[END_REF]

+ c N s=1 |f (e s + x, κ(e s + x)) -f(x, κ(e r + x))| ≤ c N s=1 λ f f (|e s | + |x| + λ κ (|e s | + |x|)) + c N s=1 λ f (|e s | + λ κ |e s -e r |) ≤ cN λ f f (1 + λ κ ) |x| + c √ N λ f f (1 + λ κ ) + √ N λ f + √ N -1 + N -1 λ f λ κ |e| ≤ |ỹ| + LW (i, e),
where L is given by ( 26) and ỹ Hx with H given by [START_REF] Zurawski | Special issue on industrial communication systems[END_REF]. In the light of footnote 8 in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF] we have that, for all i ∈ N and almost all t, d dt W (i, e(t)) ≤ LW (i, e(t))+ |ỹ(t)|. From Proposition 6 and the proof of Proposition 7 in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], we conclude the input-output stability of system (17b) from ỹ to W with exp-KL function and linear gain, provided that (x T (t), e T (t)) ≤ R at all time. More precisely, for any t ∈ [t s i , t s i+1 ] and i ≥ k ≥ 0 arbitrarily chosen, we have that t-t sk ≤ (i-k+1)τ m and, as long as (x T (t), e T (t)) ≤ R,

W (i, e(t)) ≤ exp(Lτ m )λ (k+1)→i W (k, e(t s k )) + exp(Lτ m ) -1 L (1 -ρ 0 exp(Lτ m )) ỹ[t sk , t] L∞ , with λ (k+1)→i (ρ 0 exp(Lτ m )) i-k ≤ exp ln (ρ 0 exp(Lτ m )) t-ts k τ m ρ 0 exp(Lτ m ) .
Note that [START_REF] Walsh | Stability analysis of networked control systems[END_REF] ensures in particular that ρ 0 exp(Lτ m ) < 1. The previous expression holds in particular for k = 0. Recalling the definition of ỹ and the inequalities (20), we can write, as long as (x T (t), e T (t)) ≤ R,

|e(t)| ≤ k 1 e -λ1(t-ts 0 ) |e 0 | + γ 1 x[t s0 , t] L ∞ , (37)
with γ 1 and k 1 given by ( 31) and (32) respectively, and λ 1 -ln(ρ 0 exp(Lτ m )) τm > 0.

Step 2 Let us consider the Lyapunov function V of Assumption 3. In view of Assumption 4, the total derivative of V along the solutions of (17a) yields for all

(x T (t), e T (t)) ≤ R ∂V ∂x F (t, x, e, u(t)) = ∂V ∂x f(x, κ(e r + x)) = ∂V ∂x f(x, κ(x)) + ∂V ∂x [f (x, κ(e r + x)) -f(x, κ(x))] ≤ -α |x| 2 + dλ f λ κ |x| |e r | ≤ -α |x| 2 + dλ f λ κ |x| |e r | + α 2 |x| - dλ f λ κ √ 2α |e r | 2 = - α 2 |x| 2 + d 2 λ 2 f λ 2 κ 2α |e r | 2 ≤ - α 2α V (x) + d 2 λ 2 f λ 2 κ 2α |e| 2 . ( 38
)
Whereby, applying the comparison lemma, we get, as long as

(x T (t), e T (t)) ≤ R, V (x(t)) ≤ e -α 2α (t-ts 0 ) V (x(t s 0 )) + d 2 λ 2 f λ 2 κ 2α t ts 0 e -α 2α (t-s) |e(s)| 2 ds ≤ e -α 2α (t-ts 0 ) V (x 0 ) + d 2 λ 2 f λ 2 κ α 2 1 -e -α 2α (t-ts 0 ) e[t s0 , t] L∞ .
Hence, recalling Assumption 3, we can write, as long as (x T (t), e T (t)) ≤ R,

|x(t)| ≤ k 2 e -λ2(t-ts 0 ) |x 0 | + γ 2 e[t s0 , t] L ∞ , (39)
with γ 2 and k 2 given by ( 28) and (33) respectively, and λ 2 α 4α > 0.

Step 3 By means of a local version of Corollary 1 in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], we can conclude, for the fixed R > 0, that the NCS system ( 17) is locally exponentially stable if the small gain condition γ 1 γ 2 < 1 is verified and (x T (t), e T (t)) ≤ R for all t. In view of (28) and (31), it is easy to see that the previous inequality is satisfied for every τ m ∈ (0, τ ⋆ m ) compatible with conditions ( 23), [START_REF] Wang | Networked Control Systems: Theory and Applications[END_REF] and [START_REF] Yue | Networked-based robust H∞ control of systems with uncertainty[END_REF]. The value of τ ⋆ m in (23) can be found by solving in τ m the small gain condition. Moreover, in order for the definition (14) (or equivalently [START_REF] Wang | Networked Control Systems: Theory and Applications[END_REF]) of the memory N to be consistent and for the meaning of τ m as MATI to be preserved, it must hold τ m ≥ ε m , namely condition [START_REF] Yue | Networked-based robust H∞ control of systems with uncertainty[END_REF].

Step 4 Finally, we can compute the set of initial conditions for which trajectories remain inside B R . Recalling the inequalities (37) and (39) we can write

x[t s 0 , t] L∞ ≤ k 2 |x 0 | + γ 2 e[t s 0 , t] L∞ e[t s 0 , t] L∞ ≤ k 1 |e 0 | + γ 1 x[t s 0 , t] L∞ , and
x[t s0 , t] e[t s0 , t] L∞ ≤ x[t s0 , t] L ∞ + e[t s0 , t] L ∞ ≤ K x 0 e 0
with K given by (30). Consequently, in order to ensure that the evolution of the system does not exit the ball B R , it is sufficient to impose that K (x T 0 , e T 0 ) < R, or equivalently that

(x T 0 , e T 0 ) ∈ B R ⊆ B R with R = R K (cf. ( 29
)).

Proof of Proposition 2

Let us name τ 1 m the value assumed by τ ⋆ m (see ( 23)) for N = 1. Conditions ( 23), ( 24) and ( 25) are satisfied for every τ m , T c , T m , τ c , ε m such that τ m ∈ (0, τ 1 m ), ε m ∈ (0, τ m ] and 0 < T c + T m + τ c < ε m , which is always feasible for sufficiently small values of these parameters.

Proof of Theorem 2

This proof strongly relies on that of Theorem 1. According to Definition 1, we must show that for any arbitrarily fixed set of initial conditions we can find suitable values for the parameters τ m , T c , T m , τ c , ε m ensuring the exponential stability of the NCS on the chosen set. Let us consider R > 0 as a free variable. By the small gain condition γ 1 (R)γ 2 (R) < 1 (cf. Step 3 of the proof of Theorem 1) and recalling expression (31) of γ 1 (R), we have that, for any δ ∈ (0, 1) independent of R, we can find a constant

τ ⋆ m (R, δ) 1 L(R) ln H(R)γ 2 (R) + (1 -δ)a L L(R) H(R)γ 2 (R) + (1 -δ)a L ρ 0 L(R) ,
such that for every τ m ∈ (0, τ ⋆ m (R, δ)] γ 1 (R)γ 2 (R) ≤ 1 -δ. In a way similar to Proposition 2 we can show that it is always possible, for any fixed R, to find a set of parameters τ m , T c , T m , τ c , ε m the previous condition and the conditions [START_REF] Wang | Networked Control Systems: Theory and Applications[END_REF] and [START_REF] Yue | Networked-based robust H∞ control of systems with uncertainty[END_REF].

All the Lipschitz constants of Assumptions 4 and 5 are non-decreasing functions of R. If lim R→∞ γ 2 (R) = γ2 constant, then τ ⋆ m (R, δ) ≥ τ ⋆ mγ2 (R, δ) with

τ ⋆ mγ 2 (R, δ) 1 L(R) ln H(R)γ 2 + (1 -δ)a L L(R) H(R)γ 2 + (1 -δ)a L ρ 0 L(R)
and for every τ m ∈ (0, τ ⋆ mγ 2 (R, δ)) we have γ 1 (R) < 1-δ γ2 . This means that K(R) of ( 30) is bounded by the constant K = ∞ as desired. Due to the ultimate nature of the condition (34), it can be violated for small values of R, say for R < R. Let us then define R ⋆ = max( R, R). If for every R < R ⋆ we fix all the Lipschitz constants to the value assumed for R = R ⋆ , then, for any fixed τ m ∈ (0, τ ⋆ m (R ⋆ , δ)), K(R ⋆ ) is a constant independent of R. We can fix as before the other parameters T c , T m , τ c . Let us define R⋆ = R ⋆ K(R ⋆ ) . For any radius R of the set of the initial conditions such that R ∈ (0, R⋆ ], the values required by Definition 1, are fixed independently of R in terms of R ⋆ (or equivalently in terms of R⋆ ). For any R > R⋆ we can find a solution R to the inequality R ≥

Conclusions

The problem of stabilizing nonlinear time-invariant plants over a limited-bandwidth packet-switching network has been considered. Traditional control schemes, designed for circuit-switching networks, send small pieces of data very frequently, thus driving the network towards bandwidth saturation. On the other hand, in a packet-switching network the adoption of feedforward control sequences allows to send larger packets less frequently. To this aim, we presented a model-based approach to remotely compute a predictive control signal on a given time horizon. We considered a robustness problem, where the plant uncertainty is given a priori, and we provided a bound on the combined effects of the MATI and MAD as a function of the basin of attraction and the model precision. The improvement to the MATI induced by our control strategy has been verified by means of a case study. Our future research will focus on the exploitation of the packetization of measurements to further reduce the bandwidth occupation and to better cope with model parameter variations.

Fig. 1 .

 1 Fig. 1. Networked Control System with packet-switching network and protocol.

2 .

 2 Excerpt of an infinite sequence of estimate variables and control feedforward signals along with the control signal applied to the plant. all measurements are used to continuously update the internal model. The infinite sequence of evolutions for the simulated dynamics (11) is schematically depicted at the top of Figure 2, above the time line. Each simulated evolution is represented by a straight line starting at times τ m i , i ∈ N (explicitly reported at their left). Different line styles represent different evolutions for the estimate variables. The time line reports the instants τ m

a a W 0 a 2 1 ρ0

 01 (k -N + 1, e s ) ≥ a aρ0 W 0 (k -N + 2, e s+1 ) ≥a W 0 (k, e s+1 ), and iterating W 0 (k, e s )

√ 2 δ 2 δ k 1 ( 1 + 2 δ k 1 (1+γ 2 ( 2 δ

 22112122 max k 2 1 + 1-δ γ2, k 1 (1 + γ2 ) , thus allowing the radius R of the set of initial conditions to be arbitrarily chosen. Indeed, once R is fixed, R can easily be computed as R = K R (cf. (29)). Such a value of R is a function of R, hence, it can be used to explicitly compute the parametersτ ⋆ m ( R), T ⋆ c ( R), T ⋆ m ( R), τ ⋆ c ( R), ε ⋆ m ( R) required by Definition 1. If, instead, lim R→∞ γ 2 (R) = ∞, we can chose τ m ∈ (0, τ ⋆ m (R, δ)) such that lim R→∞ γ 1 (R) = 0 to ensure that γ 1 (R)γ 2 (R) ≤ 1 -δ. By (28) we see that γ 2 (R) is a non-decreasing function of R. Hence, there exists an R such that for any R > R, max{k 2 (1 + γ 1 (R)) , k 1 (1 + γ 2 (R))} = k 1 (1 + γ 2 (R)).Let us consider the case of R > R. By the condition (29), for R to be arbitrary enlargeable, it must hold lim R→∞ R K(R) = ∞. By the previous relations we have that, for any R > R and any τ m ∈ (0, τ ⋆ m (R, δ)), K(R) ≤ √ γ 2 (R)). Using the condition (34) on the growth rate of λ f (R)λ κ (R) and the expression (28) of γ 2 (R), we see that lim R→∞ γ2(R) R σ < ∞ for some σ ∈ (0, 1). Hence, lim R→∞ R K(R) ≥ lim R→∞ R √ k1(1+γ2(R))

√ 2 δ k 1 ( 1 +

 211 γ 2 (R)) R, which is finite if R is finite. Once again, such R can be used to compute the parameters τ ⋆ m ( R), T ⋆ c ( R), T ⋆ m ( R), τ ⋆ c ( R), ε ⋆ m ( R) required by Definition 1.

  we can simply write u(t, e + x) = κ(e r + x). Since |e s | ≤ |e| for all s ∈ {1, . . . , N}, in light of Assumptions 4 and 5, we have for all (x T (t), e T (t)) ≤ + x, κ(e s + x)) -f (e s + x, κ(e s + x))

	R
		∂W ∂e	G(t, x, e, u(t)) ≤	∂W ∂e	|G(t, x, e, u(t))|
		N
	≤ c	s=1	f (e s + x, κ(e s + x)) -f (x, κ(e r + x))
		N
	≤ c	s=1	f (e s

We stress that λκ can be chosen independently of Ru.

Since ν(t, j, k) = 0 only when µ(γ(j)) = k, the control input in[START_REF] Quevedo | Packetized predictive control over erasure channels[END_REF] is independent of j contrarily to what the notation suggests.

In computing λκ we accounted for the fact that we send the state vector instead of the output vector.


Marion Mcallister
email: marion.mcallister@manchester.ac.uk

PhD Graham Dunn

PhD Chris Todd

Keywords: Empowerment: Qualitative Underpinning of a New Clinical Genetics-Specific Patient Reported Outcome Patient Reported Outcomes, empowerment, qualitative research, emotional regulation, grounded theory

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INTRODUCTION

Evaluating interventions in clinical genetics is difficult because there is lack of clarity about which are the best outcome domains to measure [START_REF] Wang | Assessment of genetic testing and related counseling services: current research and future directions[END_REF][START_REF] Williams | Quality in clinical genetics[END_REF][START_REF] Chou | Quality Assurance in Medical and Public Health Genetics Services: A Systematic Review[END_REF] , although efforts are underway to define and achieve consensus on outcomes [START_REF] Chou | Quality Assurance in Medical and Public Health Genetics Services: A Systematic Review[END_REF][START_REF] Silvey | Outcomes of Genetics Services: Creating an Inclusive Definition and Outcomes Menu for Public health and Clinical Genetics Services[END_REF] . Traditional measures of health status e.g. EQ-5D or SF36 may not be appropriate because many genetic conditions can be neither treated nor cured [START_REF] Jarrett | Genetic health technology and economic evaluation. A critical review[END_REF] . Often, the interventions offered in clinical genetics relate to provision of information about a diagnosis, or a genetic test result, or empiric risk information. One approach has been to measure patient knowledge, or information recall, but this approach has been criticized because of the substantial assumptions that specific items of knowledge are either valued by patients, or contribute to effective decision-making. In effect, these studies are tests of memory and understanding, not evaluations of patient benefit. Another approach has been to use generic measures of psychological constructs, e.g. anxiety, but these approaches have not been shown to discriminate effectively between different models of service delivery in clinical trials [START_REF] Mcallister | What process attributes of clinical genetics services could maximise patient benefits?[END_REF] .

A third, more recent approach is to measure effectiveness of decisions using measures of informed decision-making or decisional conflict. These approaches are limited to evaluating effectiveness of a single decision, and may not be relevant for evaluating clinical genetics services, where outcomes relate to the capacity to make many decisions 7 . Measures of satisfaction have also been used to evaluate clinical genetics [START_REF] Payne | Outcome measurement in clinical genetics services: a systematic review of validated measures[END_REF][START_REF] Shiloh | Satisfaction with genetic counseling: dimensions and measurement[END_REF][START_REF] Zellerino | Development, Testing and Validation of a Patient Satisfaction Questionnaire for Use in the Clinical Genetics Setting[END_REF] .

Although an important component in assessing healthcare quality, patient satisfaction is not sufficient to capture all the important patient benefits, and furthermore, is influenced by expectations [START_REF] Coulter | Tha autonomous patient: Ending paternalism in medical care[END_REF] .

Our recent systematic review of validated outcome measures used in clinical genetics identified that 67 different outcome measures have been used to evaluate clinical genetics. 46 of those measures were used in just one study each [START_REF] Payne | Outcome measurement in clinical genetics services: a systematic review of validated measures[END_REF] , highlighting the confusion about appropriate measures. Outcome domains captured by the measures ranged from health status through knowledge to quality of life. The review concluded that for clinical genetics services (a) there was no single validated outcome measure available to capture all potential patient benefits (b) some important aspects of patient benefit are not captured by available measures and (c) there is scope for new measures.

Our previous qualitative research [START_REF] Mcallister | What process attributes of clinical genetics services could maximise patient benefits?[END_REF]7,[START_REF] Mcallister | The emotional effects of genetic diseases: implications for clinical genetics[END_REF][START_REF] Mcallister | Improving service evaluation in clinical genetics: identifying effects of genetic diseases on individuals and families[END_REF] suggests that patient benefits from clinical genetics services could be summarised using a new construct labelled Empowerment 7 .

Empowerment was defined as a set of beliefs that enable a person from a family affected by a genetic condition, to feel that they have some control over and hope for the future. Empowerment included four dimensions: the beliefs that one (1) can make important life decisions in an informed way (Decision-Making), (2) has sufficient information about the condition, including risks to oneself and one's relatives, and any treatment, prevention and support available (Knowledge and Understanding), (3) can make effective use of the health and social care systems for the benefit of the whole family (Instrumentality) and ( 4) can look to the future with hope for a fulfilling family life, for oneself, one's family, and/or one's future descendents (Future Orientation) 7 .

When compared with outcome measures used in evaluations of clinical genetics, Empowerment was found to be conceptually most similar to the construct captured by the Perceived Personal Control (PPC) measure [START_REF] Berkenstadt | Percieved Personal Control (PPC) : A new concept in measuring outcome of genetic counseling[END_REF] robust evidence base for practice. However, PROMs development is difficult and timeconsuming [START_REF] Clark | Constructing validity: Basic Issues in Objective Scale Development[END_REF][START_REF] Streiner | Health measurement scales: A practical guide to their development and use[END_REF][START_REF] Mcallister | Outcome Measures in Clinical Genetics Services[END_REF] , and justifiable only if there is evidence that the new measure will capture a Patient Reported Outcome (PRO) that is novel, useful, relevant and important.

The aims in the present study were to (1) explore the validity, relevance and importance of the Empowerment construct and refine the construct as necessary and (2) inform the decision whether or not to develop a clinical genetics-specific PROM of Empowerment.

METHODS

To meet the research objectives, further qualitative data were collected. Qualitative methods are suitable for exploratory research in areas that are not well understood, and where the relevant variables have not been fully identified [START_REF] Miles | Qualitative data analysis: An expanded sourcebook[END_REF] . Since one purpose of this research was to explore whether the Empowerment model includes all relevant facets of patient benefit from using clinical genetics services, a qualitative approach was appropriate. Patients were recruited through Genetic Medicine, St Mary's Hospital, Manchester, UK. The term 'patient' is used here to refer to all users of a clinical genetics service, including unaffected parents and unaffected at risk families members.

Patients were asked to participate in a focus group, but also had the option to be interviewed. Genetics clinicians were recruited through professional networks, and were only asked to participate in a focus group. Previous research indicated that focus groups work well with genetics clinicians, whereas a combined focus group / interview approach works best with patients of clinical genetics services 4 5 11 12 . Commissioners (who purchase clinical genetics services on behalf of primary care) and patient representatives (members of patient support groups) were recruited through UK professional networks, to participate in an interview only because of geographical spread. Focus groups were conducted in Nowgen, Manchester. Interviews were conducted in person or by telephone, according to participant preference. Ethical approval was obtained from the NHS National Research Ethics Service.

Participants were asked to describe their views about the benefits and harms of using clinical genetics services, and then given a page describing the Empowerment construct, with its four dimensions, and the dimension definitions 7 . Participants were asked to read and critique this, in particular to identify any benefits / harms of using a clinical genetics service that were missing. Focus groups and interviews were transcribed in full, and analysed using tools of grounded theory methodology [START_REF] Glaser | The discovery of grounded theory[END_REF]20 including: (1) open coding: labels (codes) significant pieces of text as important (2) constant comparison: each new instance of a code is compared with every other instance of that code, and the concept definition revised accordingly and (3) questioning the data in relation to the research question. Glaser and Strauss, the developers of grounded theory methodology, recommend that any emerging grounded theory (model) should be "checked out" for usefulness with "experts" familiar with the substantive area under investigation [START_REF] Glaser | The discovery of grounded theory[END_REF]21 . In this study, the emerging construct (model) of Empowerment is checked out with patients, patient representatives, clinicians, and commissioners of clinical genetics services to clarify whether the construct provides a plausible and useful summary of patient benefit. Feedback of this kind can indicate whether or not the construct (model) has "grab" 21 i.e. whether it provides a novel and useful framework.

RESULTS

138 people were approached and asked to participate. 54 (39.2%) agreed to participate, and 18 (13%) declined. Of the 54 who agreed to participate, 13 (9.4%) were excluded because they could not be accommodated in a focus group because of scheduling difficulties, and did not wish to be interviewed instead. The final sample was composed of 41 participants (29.8%) (Tables 1&2). Due to ethical constraints, those approached were not asked to give reasons for their decision, so reasons for the low response rate amongst patients are not known. Three focus groups were conducted, two with genetics clinicians (n = 5 in both) and one with patients (n = 4). 27 interviews were conducted, 4 with service commissioners, 8 with patients, and 15 with patient representatives.

Participants' views about the 'Empowerment' construct

The qualitative study generated data confirming the importance of the Empowerment Results suggest that genetic diagnosis may engender a state of disempowerment, which throws families into a new state, perhaps best described as social alienation, which can be addressed in genetic counseling, as illustrated by this participant: "I just feel though that in a way the system of the diagnosis is what has disempowered parents in the first place, so it actually about re-establishing confidence rather than just giving it them for the first time." (patient representative, D/deaf charity) A good outcome from using a clinical genetics service could then be seen to involve learning about their new situation, and regaining some confidence and control over their lives, so that they "know what they are doing", which is experienced as empowering, expressed as follows:

"…the information empowers you to make decisions, and provided us with the opportunity to have a family, it gave us the confidence to try for another baby, which we would not have done otherwise." (patient, previously had a late termination at 23 weeks gestation of a pregnancy affected by Edwards syndrome)

The views expressed by patients, and patient representatives, were endorsed by the service commissioner participants, who confirmed that they want to commission clinical genetics services that can offer choice to patients: "… the patients […] have the choice to consider their own lifestyles, their own social decisions, the future of their own children and that chance of things if they wish to […] carry on with the pregnancy if they've already conceived, or to terminate or to not conceive in future, to adopt, to go through PGD (preimplantation genetic diagnosis) whatever, their decision, it is the parents decision." (service commissioner) One participant, although she validated the importance of the four dimensions of Empowerment, was unsure about the appropriateness of the word "Empowerment": 'empowerment' suggests that you are able to be proactive and that I don't think is always going to be the way or would be a potential outcome of a clinical genetic service, I don't think that's feasible. (patient representative, charity supporting couples who have terminated pregnancies for fetal abnormalities) Because all 41 participants, except for this one, endorsed use of the word "Empowerment" to summarise patient benefits from using clinical genetics services, and some did so strongly, it was decided to retain the term "Empowerment" as an overarching term to describe the model.

Refinement of the Empowerment construct

All participants endorsed the relevance and importance of the four dimensions of Empowerment when shown the definitions, but the analysis resulted in renaming three of these dimensions to further clarify the Empowerment construct. Data analysis confirmed that Knowledge and Understanding, as conceptualised in Empowerment, is more akin to "Cognitive Control", a dimension of PPC, than to information recall.

Knowledge about the condition is valued for its own sake because it provides an explanation for previously unexplained, unexpected and distressing events (recurrence of disease in the family, loss of a child, disease symptoms), and this in itself is a benefit. It appeared that labeling symptoms with a scientific name had the quality of a "latin incantation" which reduced uncertainty/anxiety, and validated the illness experience.

Similarly, the data confirmed that Decision-Making, as conceptualised in Empowerment, is more like "Decisional Control", a dimension of PPC, than to effectiveness of any single decision. Finally, Instrumentality, as conceptualised in Empowerment, is more similar to "Behavioural Control", a dimension of PPC, than to any specific action taken. Understanding the implications of the genetic condition to oneself, one's relatives, and future generations, and knowing about available medical interventions and support enables decisions to be made, and actions to be taken: "…it's the information that enables you to make the decisions, whether to go ahead Future-Orientation was also endorsed by all participants in this study as relevant and important to good patient outcomes. Understanding the genetic condition, and knowing about services and interventions available to help manage the effects of the condition enables families to orient themselves towards the future with more confidence: "… it comes back to this control isn't it that you are able to map out your future life you know, it's not that much of an unknown […] (and) feeling comfortable about one's children's future" (patient representative, child affected by an unbalanced chromosome translocation) One participant suggested that the Future-Orientation dimension be renamed "Hope" because she felt that providing Hope was an important part of any diagnosis of a lifelimiting condition. The word Hope came up a few times in interviews as an important benefit from using clinical genetics, and the decision was made to change the name of this dimension from Future-Orientation to Hope, partly because the term Future-Orientation required some explanation to participants, whereas the word Hope is more widely understood.

Further development of the Empowerment construct

The qualitative analysis resulted in identification of one further dimension of Empowerment: Emotional Regulation. Genetics clinicians in one focus group, and a number of patient representatives, some of whom were affected by genetic conditions, and some of whom were parents of affected people, commented in interviews that the model omitted any mention of emotions, which they felt were important. No other omissions were identified. Patient representatives, all of whom had a genetic condition in their own family, and were also able to provide an overview of the variability in experiences of their patient group members, described how a genetic diagnosis throws the family into emotional turmoil, and highlighted the need for families to be given In contrast to other research, for example exploring autism in France 22 , neither this nor our previous research [START_REF] Mcallister | What process attributes of clinical genetics services could maximise patient benefits?[END_REF]7,[START_REF] Mcallister | The emotional effects of genetic diseases: implications for clinical genetics[END_REF][START_REF] Mcallister | Improving service evaluation in clinical genetics: identifying effects of genetic diseases on individuals and families[END_REF] identified any differences between the views of healthy parents of a child affected by a condition, patients affected by a condition, and those at risk for developing a condition, or between adults at different stages of development.

There is scope to explore this further in future research.

The Empowerment PRO, if operationalised as a PROM, could be used for evaluation of whether some of the criteria in the recently developed EuroGentest "Instrument for quality self-assessment in provision of genetic counselling" (http://www.eurogentest.org/professionals/assessment_tool/assessment_tool/form.xhtm) have been met. For example, the EuroGentest instrument includes the criteria (1) "Counsellees are provided with decision-making support", which could provide counsellees with improved decisional control and (2) "Counsellees understand what they are being told" which could provide counsellees with improved cognitive control.

A PROM of Empowerment would enable these criteria to be measured by patient selfreport, rather than by service self-assessment.

The constituent dimensions of Empowerment identified in the present study also overlap with some of the outcomes identified in a consensus study recently reported by the US Western States Genetics Services Collaborative (WSGSC) [START_REF] Silvey | Outcomes of Genetics Services: Creating an Inclusive Definition and Outcomes Menu for Public health and Clinical Genetics Services[END_REF] , and the present study provides further support for the relevance and importance of these initiatives.

For example, the WSGSC outcome "Individuals and families feel supported in managing grief, stress, and emotional challenges of living with a genetic condition" could be interpreted as Emotional Regulation. The WSGSC outcome "Individuals and families are able to make informed health and life decisions based upon diagnosis" could be interpreted as Decisional Control. However, the present study goes further than identifying patient outcomes by integrating 5 different patient outcomes in the overarching patient-centred construct of Empowerment. The present study provides sufficient evidence to suggest that Empowerment is a novel PRO for clinical genetics services.

Identification of Emotional Regulation as an additional dimension of Empowerment, and as an important outcome from using clinical genetics services, is not surprising.

Good patient outcomes from using healthcare services could be conceptualised as alleviation of those disease effects experienced as problematic by patients and their families. From this perspective, the finding that Emotional Regulation is an important outcome in clinical genetics is consistent with research evidence that the emotional effects of genetic diseases are problematic for families [START_REF] Mcallister | The emotional effects of genetic diseases: implications for clinical genetics[END_REF] . These emotional effects may include feelings of powerlessness and a threatened sense of security and identity, previously described in relation to diagnosis of long-term chronic conditions [START_REF] Aujoulat | Reconsidering patient empowerment in chronic illness: a critique of models of self-efficacy and bodily control[END_REF] . This finding also supports models of genetic counselling practice, which emphasise a psycho-educational approach [START_REF] Biesecker | Process studies in genetic counseling: peering into the black box[END_REF] , rather than a purely educational model [START_REF] Harper | General Aspects of Genetic Counseling[END_REF] . The educational model emphasises 'educating' the patient, and is based on the assumption that patients come to clinical genetics services for information only. In contrast, the psycho-educational approach assumes that patients are seeking a therapeutic relationship in which they are supported in using genetic information in a personally meaningful way that minimizes psychological distress and increases personal control. 5) alleviate feelings of guilt that she had done something in pregnancy to cause her child's condition (Emotional Regulation). Clearly, it may not be possible to bring about positive change in all 5 dimensions for every patient who attends a clinical genetics service. However it is plausible to hypothesise that the 5 dimensions of Empowerment are facets of a single overarching construct.

The main weakness of this research is that it is limited to the perspectives of families living in the UK, who speak English. Empowerment is likely to be influenced by culture 7 , and so Empowerment would benefit from validation in other cultures and languages. A further weakness is that patient participation rates were low. However, because this work builds on extensive previous qualitative research 4 5 11 12 , and because care was taken to reach data saturation, it is unlikely that significant issues were missed.

The main strength of this research is that the new PRO identified -Empowerment -has been developed through consultation with (1) people struggling with the challenges of having a genetic condition in their family and (2) clinicians with experience of helping these families. They are the "experts" consulted in this research to identify the patient benefits from using clinical genetics services. The traditional approach to PROMs development is to use theoretical frameworks to identify PROs, and this approach was used in identifying PPC as a patient outcome from genetic counselling [START_REF] Berkenstadt | Percieved Personal Control (PPC) : A new concept in measuring outcome of genetic counseling[END_REF][START_REF] Shiloh | Mediating effects of Percieved Personal Control in coping with a health threat: The case of genetic counselling[END_REF] . This approach is useful, but as the present study shows, consulting patients and clinicians can enable new constructs to be developed, and existing constructs to be extended, to ensure that PROs include all benefits valued by patients.

A measure of PPC was developed and evaluated in Hebrew [START_REF] Berkenstadt | Percieved Personal Control (PPC) : A new concept in measuring outcome of genetic counseling[END_REF] . When the paper reporting development of the PPC questionnaire was published in a US journal, the PPC was translated into English [START_REF] Berkenstadt | Percieved Personal Control (PPC) : A new concept in measuring outcome of genetic counseling[END_REF] . Whilst the English translation of the PPC has been used with English speaking samples [START_REF] Davey | Evaluating genetic counselling: Client expectations, psychological adjustment and satisfaction with service[END_REF] , it has not yet been validated for use with a UK population.

A study is underway to operationalise Empowerment by developing a new geneticsspecific PROM of Empowerment, using the PPC measure and the qualitative data collected in this and previous studies 4 5 11 12 . Both PROMs and patient empowerment have been identified as key aspects of current UK health policy [START_REF] Darzi | High Quality Care for All: NHS Next Stage Review final report[END_REF] . In addition, there was a recent call for better measures to capture aspects of Health Related Quality of Life (HRQoL) in clinical trials in medical genetics [START_REF] Stevenson | Health-Related Quality of Life Measures in Genetic Disorders: An Outcome Variable for Consideration in Clinical Trials[END_REF] . These developments strengthen the argument in support of investing the time and resources needed to develop a new genetics-specific PROM to capture Empowerment. This PROM would then be available to capture HRQoL outcomes in clinical trials, and in clinical practice to generate data to support continuous quality improvement [START_REF] Gray | Peering into the Chasm: Improving the Quality of Clinical Genetics Services[END_REF] . As part of PROM development [START_REF] Streiner | Health measurement scales: A practical guide to their development and use[END_REF] , factor analysis will investigate the dimensional structure of Empowerment to confirm whether or not Empowerment is a single multidimensional construct.

  , which includes three dimensions: Cognitive Control, Decisional Control and Behavioural Control. However, the PPC questionnaire does not capture all the important patient benefits summarised in the Empowerment construct e.g. benefits to other relatives and to future generations. There is scope to develop a new Patient Reported Outcome Measure (PROM) to capture the Empowerment construct. PROMs are short validated questionnaires completed by patients to evaluate health interventions. A PROM of Empowerment could be useful in evaluating new interventions in clinical genetics, thus contributing to establishing a

  construct, and the previously identified four dimensions of Empowerment (Knowledge and Understanding, Decision-Making, Instrumentality and Future-Orientation). All 41 participants, except for one patient representative, endorsed use of the word "Empowerment" as a good overarching term to summarise patient benefits, and clarified further what Empowerment means in the context of clinical genetics: "I think that's the best word you could think of, yes it is, having empowermenthaving that knowledge is the biggest benefit that you can gain. And yes I would recommend a genetic service to anybody, well I do when they have a new baby and they're all sort of all bewildered […] 'empowerment,' yes that is […] a very good word to describe it." (patient, Congenital adrenal hyperplasia).

"

  I have slight reservations about the word but certainly in terms of those -what you've laid out there as the ... what underpins that I think all are extremely valid, you know, I think that would […] definitely [be] a useful way of measuring (outcomes) […]

  Simply having a diagnosis provides reassurance and enables families to prepare for the future. "[…] The benefits to me of having the diagnosis confirmed that what I was experiencing that it actually had a name, and despite the fact that there was very little research done in to the condition twelve years ago it enabled us to do some research ourselves and to keep up with the latest what was going on. So we could put a focus on it, you know... " (patient representative, affected by Porphyria)

  and have another baby, or whether to have more children, and what sort of tests to have." (patient, baby affected by chromosome abnormality) For these reasons, the PPC terminology of Cognitive, Decisional and Behavioural Control was adopted for the Empowerment dimensions previously known as Knowledge and Understanding, Decision-making and Instrumentality respectively.

  emotional support: "… for families to have emotional support around a point of diagnosis […] I would say that everybody is depressed around the time of using the service". (patient representative, charity supporting families affected by Battens disease) They felt strongly that clinical genetics services should be instrumental in addressing these difficult emotions, and in helping families to reach a new equilibrium: "… when you're first given a diagnosis it's like your whole life is spinning out of control you don't know where you're going, you don't know what's going to happen to you or your family, what the future holds, it's incredibly frightening having been through it myself […] a very important outcome (is) that the emotions are dealt with." (patient group representative, child has unbalanced chromosome translocation) Genetic counsellors commented that a large part of their work focuses on offering emotional support to patients, and so they appeared to recognise the importance to families of addressing and dealing with difficult emotions in genetic counselling. No other new issues were identified in the final 5 interviews, and so theoretical saturation was assumed, and recruitment stopped. The new Empowerment construct In summary, the qualitative analysis endorsed the Empowerment construct, and refined the construct by (a) adopting the PPC terminology of Cognitive, Decisional and Behavioural Control for the dimensions previously labeled Knowledge and Understanding, Decision-Making and Instrumentality respectively, (2) adopting the term "Hope" for the dimension previously labeled "Future-Orientation", and (3) extending the model to include a fifth dimension: Emotional Regulation. DISCUSSION This research has validated and further developed and refined the Empowerment construct, to summarise the patient benefits from using clinical genetics services 7 . A sample of patients and patient representatives, clinicians and commissioners of clinical genetics services in the UK endorsed use of the term "Empowerment" to summarise the patient benefits, and identified one further dimension of Empowerment, Emotional Regulation, that was missing from the four-dimensional model. Furthermore, the data have enabled the meaning of the previously identified four dimensions of Knowledge and Understanding, Decision-Making, Instrumentality and Future-Orientation to be refined. The revised construct is composed of 5 dimensions: Cognitive Control, Decisional Control, Behavioural Control, Hope and Emotional Regulation, and has clarified the relationship between Empowerment, and the concept of Perceived Personal Control (PPC) 7,12,20 . Findings support the importance of PPC as a PRO, and identified that Empowerment includes the three PPC dimensions (Cognitive, Decisional and Behavioural Control), and two further dimensions: Hope and Emotional Regulation.

  Hope has not previously been identified as a valued outcome from clinical genetics services, and its inclusion in the model is novel. On a theoretical level, Emotional Regulation is likely to interact with Hope, and with Perceived Personal Control, and it's three dimensions of Cognitive, Decisional and Behavioural Control. A genetic diagnosis, and a scientific explanation for the previously unexplained death of her child, information about risks to future pregnancies, and options for prenatal diagnosis could help a bereaved mother to (1) understand why her child died (Cognitive Control) (2) make decisions about whether to have further children (Decisional Control) (3) use prenatal testing in future pregnancies to avoid recurrence (Behavioural Control) (4) feel that it is possible to have healthy children and a rewarding family life in the future (Hope) and (

ACKNOWLEDGEMENTS

We are grateful to the patients and clinicians who participated. This research was funded by the Medical Research Council. Ethical approval was provided by Central Manchester NHS Research Ethics Committee (ref.07/H1008/226). The views expressed in this paper are those of the authors, and not those of the funding body.

This research was funded by the Medical Research Council.

CONFLICT OF INTEREST:

The authors declare no conflict of interest.