Alexandre Kruszewski
email: alexandre.kruszewski@ec-lille.fr

Wenjuan Jiang
email: wenjuan.jiang@ec-lille.fr

Emilia Fridman
email: emilia@eng.tau.ac.il

Jean-Pierre Richard
email: jean-pierre.richard@ec-lille.fr

Armand Toguyeni
email: armand.toguyeni@ec-lille.fr

W.-J Jiang

J.-P Richard

W.-J Jiang

J.-P Richard

J.-P Richard

A

Keywords: Networked control systems, time delay approach, gain scheduling, Lyapunov-Krasovskii method, LMI I

come

to stability without additional performance evaluation. In [START_REF] Seuret | Networked control using gps synchronization[END_REF] Lyapunov-Krassovskii Functionals (LKF) are applied and exponential stabilization is considered. The structure of NCS that we consider is depicted on Figure 1. It is a single feedback loop which corresponds to the classical simplification of more general NCS, while still capturing many important characteristics of NCS (see [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]). There, a plant (namely, the Slave) is controlled by a remote controller and observer (the Master) via a network. The two communication delays are denoted τ c 1 (t) for the control channel (from Master to Slave) and τ c 2 (t) for the measurement channel (Slave to Master). Inspired by [START_REF] Seuret | Networked control using gps synchronization[END_REF], we limit the waiting delay strategy (aiming at obtaining a constant, virtual delay as recalled above [START_REF] Estrada-Garcia | Master-slave synchronization for two inverted pendulums with communication time-delay[END_REF], [START_REF] Lelevé | Telerobotics over IP networks: Towards a low-level real-time architecture[END_REF], [START_REF] Ploplys | Closed-loop control over wireless networks[END_REF]) to the Slave side, while on the side of Master there is no such additional delay. In this way, the measurement channel data can be used directly by the Master as soon as received from the Slave. It will be shown that this structure allows the Master to reconstruct the present Slave's state despite the delays. Note that it could be possible to get rid of the waiting strategy in both the control and measurement channel, and thus to allow a more reactive communication. Such a solution was proposed in [START_REF] Seuret | Control of a remote system over network including delays and packet dropout[END_REF] at the price of an additional complexity of the LMI involved in the observer design. In the present paper, we prefer to avoid this additional complexity and to focus on the switched gain control, which effects on the performance more essentially.

Controller

A main novelty of the present paper lays in the introduction of a QoS-dependent controller structure. In the existing works [START_REF] Gao | A new delay system approach to network-based control[END_REF], the stabilization is obtained on the basis of the two bounds h min and h max , disregarding the way the delays vary between these extrema. If the actual delays stay a long time near to h min , the robust control design keeps taking into account the worst case h max (highest delay, lowest QoS). Such a cautious solution generally yields poor speed performance and we prefer to design a switching controller that adapts to different intervals of delay variation.

In this case, the usual techniques for time-delay systems cannot be directly applied and one has to go to switched systems techniques. Switched systems are dynamical hybrid systems consisting of a family of continuous-time subsystems and a logical rule that orchestrates the switching between them [START_REF] Hirche | Performance oriented control over networks -switching controllers and switched time delay[END_REF], [START_REF] Liberzon | Switching in Systems and Control[END_REF]. It is well established that the stability of the subsystems themselves is not sufficient for the stability of the overall system and Lyapunov techniques constitute the basis of the global analysis. In what concerns switched time-delay systems, single [START_REF] Jiang | A gain scheduling strategy for the control and estimation of a remote robot via internet[END_REF], [START_REF] Sun | Stability and l 2 -gain analysis for switched delay systems[END_REF] and multiple [START_REF] Nilsson | Real-time control systems with delays[END_REF] Lyapunov-Krasovskii Functionals (LKFs) methods are the most frequently used for the stabilization of the whole system. A common LKF can ensure the stability with arbitrary switching, but it does not necessarily exist. As an alternative, dwell-time-based switching is considered in [START_REF] Hespanha | stability of switched systems with average dwell-time[END_REF], [START_REF] Sun | Stability and l 2 -gain analysis for switched delay systems[END_REF], [START_REF] Yan | Stability analysis of switched time delay systems[END_REF]. These results on switched time-delay systems consider switched linear systems with the same delays in all the modes.

In this paper, we consider a two-modes switched system with a time-varying delay. We assume that the control input delay δ may belong to 2 intervals: it is either small (belonging to [h 1 , h 2 ] or big (belonging to [h 2 , h 3 ]). Similarly, we assume that the measurement time-varying delay is either small or big. We derive novel conditions for the exponential stability of switched systems with switched delays and coefficients. Next, we solve the output-feedback stabilization problem by designing switched gains for the controller and for the observer. These gains are switched correspondingly to delay switches (such delays are known from the Round-Trip-Time (RTT) measurement and a time-stamp strategy. Of course, more numerous modes could be considered with an increased, off-line computation effort, but this is not to be developed in the present paper.

Finally, differently from the existing works on NCS, we illustrate the efficiency of our theoretical results by experimental ones, where a light-inertia robot is controlled via Internet.

The paper is organized as follows. The features of NCS as well as the problem statement are described in Section II. Section III addresses the exponential stability analysis and the design of switched time-delay systems under arbitrary switching. In Section IV, Multiple LKFs are adapted in the exponential stabilization theory to get a better performance and the global stability is guaranteed by adding the minimum dwell-time into the switched system. The experimental example is given in Section V and confirms the theoretical results.

II. PROBLEM STATEMENT

A. Description of the considered system

In the NCS of Figure 1, the process ("Slave") is a low energy consumption system, the embedded computer of which has a limited computation power. Thus, the control and observation complexity is concentrated on the remote Master computer (the "Master"). This structure makes our theory and application adaptable to sensor and actuator NCSs.

The transmission protocol UDP [START_REF] Postel | User datagram protocol[END_REF] is applied to communicate the data packets between Master and Slave. It is known that, compared to TCP/IP which retransmits the dropped out data packets, UDP suffers from a lack of reliability. However, in our situation, transmitting fresh data (control data and sampled measurement data) is preferable to retransmitting old ones. Finally, UDP is consuming much less time and is preferred to TCP.

B. The three delay sources

In Figure 1, the control channel total delay (Master-to-Slave) δ con (t) results from the addition of three (variable) delay sources: 1) The communication through the Internet; 2) The datasampling (see [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]); 3) The possible packet losses (which can be seen as longer sampling in the case of UDP, see [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]). The same phenomenon stands for the measuring channel delay (Slave-to-Master) which total value is denoted δ obs (t).

1) Communication:

The time-delays of communication τ c i (t) are variable and not bounded, it is reasonable to assume an upper-bound value since if the time-delay surpasses this value, the packet can be treated as lost.

2) Sampling and packet dropout: The real remote system, including Master, Slave and Network, must involve some data sampling. However, following [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF], [START_REF] Seuret | Sampled-data exponential stabilization of neutral systems with input and state delays[END_REF], this phenomenon is equivalent to a time-varying, discontinuous delay. If the sampling period is not negligible, it constitutes a disturbance that should be considered in the stabilization design [START_REF] Yu | An LMI approach to network control systems with data packet dropout and transmission delays[END_REF]. If some packet p t k containing the Slave's output (or Master's control data) sampled at t k is lost, or arrives later than the packet p t k+1 , then the other part only considers the most recent data (i.e., those from p t k+1 ).

The phenomena acts as a longer sampling period. N is the maximum number of consecutive loss of packet. The packet loss sampling and the sampling induced delay is τ s i (t) ≤ (N + 1)T, i = 1, 2.

In conclusion, the total delays are treated as variable and bounded:

δ con (t) = τ c 1 (t) + τ s 1 (t) δ obs (t) = τ c 2 (t) + τ s 2 (t) (1)
C. Output stabilization 1) Stabilization for closed-loop system: The master-slaves exchanges are network packets composed by a data and a time-stamp. From the master to slave, the time stamps represent the instant when the master would like to apply the control input. In the other way, the time stamps represent the time t o that the measurement has been done. The clocks of the master and the slave are synchronized before powering up the system. The NTP (Network Time Protocol) [START_REF] Mills | Improved algorithms for synchronizing computer network clocks[END_REF] is used for this purpose.

The Slave system, is constituted by the plant and a communication system. The communication system is in charge of reordering the packets and applying the control value in the right moment t c given by the master. It also in charge of acquiring the measurement and send them periodically to the master. The measurement time is noted t o . by:

   ẋ(t) = Ax(t) + Bu(t c ), y(t) = Cx(t), (2)
where x(t) ∈ R n , u(t) ∈ R m , y(t) ∈ R p are respectively the state, the input and the output vector.

t c is the time of the last update of the control value.

The master part is an observer-based state-feedback controller with a communication structure.

This latter compares the newest measurements with its respective estimation done at the time t o , i.e. when the data was acquired, and send it to the observer structure. The same method is used to send the input value to the observer, ensuring the synchronization of the observer and the plant. This is possible thanks to the buffer in the salve part allowing the master to choose the instant t c that the plant will apply the control value. The delay induced by this buffer strategy must be larger than the maximum delay induced by the communication delay τ c 1 (t), i.e.

t c = t + buffer = t + max(τ c 1 (t))
. The last task is to send the control values and the corresponding appliance time to the slave.

The output controller (Master) is based on the following observer:

         ẋ(t) = A x(t) + Bu(t c ) + L(y(t o ) -ŷ(t o )), ŷ(t) = C x(t), u(t) = K x(t) + ky c (t), (3)
where y c (t) is the desired setpoint

The estimation error e(t) = x(t)x(t) is given by:

ė(t) = Ae(t) -LCe(t o ). (4)
Note that the error system does not depend on u nor x. This ensures that the separation principle is applicable and allows for reducing the study of the closed-loop dynamic to two smaller stabilization problems. These two problem can be formulated as delay-system stabilization problems [START_REF] Seuret | Networked control using gps synchronization[END_REF]:

1) ẋ(t) = Ax(t) + BKx(t -δ con (t)), (5) 2) ė
(t) = Ae(t) -LCe(t -δ obs (t)). (6)
where δ con (t) = t -t c , δ con (t) ∈ [h min , h max ], and δ obs (t) = tt o , δ obs (t) ∈ [h min , h max , ] being the resulting delays described by .

2) QoS adaptation with switched controller: In order to enhance the performance of the system and take into account the network QoS into the control, the controller switches to the mode i according to the value of the delay. The controller has the information to which predefined subset S i of [h min , h max ] belongs the delays. In the following, it is assumed that the two delays {δ con (t), δ obs (t)} are always in the same subset S i i.e. δ con (t) ∈ S i ⇔ δ obs (t) ∈ S i . The number of modes considered is two i.e. i ∈ {1, 2}. The two subsets are defined as:

S i = [h i , h i+1 ]
where δ (t) ∈ S 1 represents a small delay, and δ (t) ∈ S 1 represents a big delay. To cover all possible delays, the bounds of the sets are chosen as h 1 = h min and

h 3 = h max then S 1 ∩ S 2 = [h min , h max ].
Remark 1: Only two modes are considered and the same delay subsets are chosen for the controller and the observer for sake of clarity. It is obvious that the results presented here can be extended to more complex cases where the subsets are different for the two delays and where more modes are considered. This is not detailed in this paper due to the heavy burden it may cause to the comprehension.

The resulting controller differs slightly from the non switching one. In the i th mode, the time t c of appliance of the control value is based on the maximum of the set S i , h i+1 . It is given by t c = t + h i+1 -T . The delay is detected by analyzing the time stamps in the packets, or by measuring the RTT. This introduce a delay in the delay measurement. According to this measure, the switching strategy have the following properties:

1) As soon as a bigger delay is detected the controller must switch to the corresponding mode to minimize disturbances due to a bad size of buffer (buffer < delay).

2) The controller has to follow the dwell time requirements when the detected delay becomes smaller, i.e. in a mode i smaller than the current mode. During the dwell time, the controller artificially enlarge the delay to meet the properties of the current mode.

In its i th mode, the controller is given by:

         ẋ(t) = A x(t) + Bu(t -δ i con (t)) + L i (y(t -δ i obs (t)) -ŷ(t -δ i obs (t))), ŷ(t) = C x(t), u(t) = K i x(t) + ky c (t), (7)
where δ i con (t) is predicted control delay implied by the buffer. δ i con (t)) = δ i con (t)) when the communication delay is less or equal to the buffer, otherwise δ i con (t)) = δ i con (t)) and is not in the same set than δ i con (t). This latter case occurs when the delay moves from the set S 1 to S 2 , making the buffer smaller that the delay, disturbing the observer [START_REF] He | Parameter-dependent lyapunov functional for stability of time-delay systems with polytopic-type uncertainties[END_REF].

Remark 2: An important feature of this controller is that it has the ability to maintain artificially the delay at an higher value simply by increasing the buffer size. So even if the communication delay is small, we can force δ con to be big, i.e it belongs to S 2 . The global stability will be ensured by a correct use of this ability.

Since a delay is either in S 1 or in S 2 , one naturally introduces χ : R → {0, 1}, the characteristic function of S 1 , defined by:

χ S 1 (s) = 1, if s ∈ S 1 0, otherwise. (8)
The characteristic function of S 2 is 1 -χ. Since it is assumed that the two delays are always in the same subsets we have χ(δ con (t)) = χ(δ obs (t)).

Now, considering the control problem [START_REF] Seuret | Networked control using gps synchronization[END_REF] together with this 2-modes gain switching strategy, the stabilization problems of observer and controller can be rewritten as follows:

1) ẋ(t) = Ax(t) + χ S 1 (δ con (t))BK 1 x(t -δ con (t)) + (1 -χ S 1 (δ con (t)))BK 2 x(t -δ con (t)), (9)
2)

ė(t) = Ae(t) -χ S 1 (δ obs (t))L 1 Ce(t -δ obs (t)) -(1 -χ S 1 (δ obs (t)))L 2 Ce(t -δ obs (t)) -Bu(t -δ i con (t)) + Bu(t -δ i con (t)). (10)
It must be notified that the separation principle does not hold in this case because of the term-Bu(t -δ i con (t)) + Bu(t -δ i con (t)). This term equals zero except when a switch from mode 1 to mode 2 occurs. This means that the separation principle holds if there are no switches.

III. EXPONENTIAL OUTPUT-FEEDBACK STABILIZATION OF SWITCHED TIME-DELAY SYSTEM

This part provides two results in the general framework of time delay system with switches.

The first one gives sufficient conditions for the stability of time delay system with arbitrary switches. The second provides a controller design procedure guaranteeing exponential stability when no switches occurs. These two results are important to design a switching strategy guaranteeing stability and performances.

A. Exponential stability of switched time-delay systems

Consider the switched system:

ẋ(t) = Ax(t) + χ S 1 (τ)A 1 x(t -τ(t)) + (1 -χ S 1 (τ))A 2 x(t -τ(t)), (11)
where the delays τ ∈ S 1 S 2 are assumed to be fast-varying (no restrictions on the delayderivative) within these bounds.

The easy way to guarantee the stability of a switched system with arbitrary switches is to find a common LKF for each modes. This has a negative effect on the decay rate α which will be the same in all the modes (the worse one) and it is not the goal of QoS based controller. Another way to deal with switch system is to consider a LKF for each mode and compute some minimum dwell time requirement for stability and performances. This allows performances adaptation according to the current QoS and has a reduced conservatism (in term of finding gains and LKF for some α).

For the present problem, the stability analysis is performed by using a pair of LKFs for each modes. In order to be able to compute a dwell time condition, they must be easy to compare so they have the same structure:

V j (t, x t , ẋt ) = x T (t)P j x(t) + ∑ 2 i=0 t-h i t-h i+1 e 2α j (s-t) x T (s)S i j x(s)ds + ∑ 2 i=0 (h i+1 -h i ) -h i -h i+1 t t+θ e 2α j (s-t) ẋT (s)R i j ẋ(s)dsdθ , (12)
where j represents the mode and h 0 = 0.

Remark 3: These functionals are taking into account the two possible subsets for the delay.

It has no repercussions on the conservatism even in the case where the delay is considered to be in a given subset (for example the delays is considered to belong to S 1 ). The proof is based on setting the appropriate matrices close to zero.

The following condition along the trajectories of ( 11)

Vj (t, x t , ẋt ) + 2α j V j (t, x t , ẋt ) ≤ 0 (13)
implies some decay rate on the LKF

V j (t, x t , ẋt ) ≤ e -2α j (t-t 0 ) V j (t, x t 0 , ẋt 0 ) ∀t 0 ∈ R. ( 14
)
The latter implies exponential stability of (11) since

x T (t)P j x(t) ≤ V j (t, x t , ẋt ) ≤ e -2α j (t-t 0 ) V j (t, x t 0 , ẋt 0 ) ≤ e -2α j (t-t 0 ) V j (t, x t 0 , ẋt 0 ) |α j =0 .

The following theorem gives conditions guaranteeing some decay rate in each mode separately.

It DOES NOT ensure stability of the system (11) for arbitrary switches.

Theorem 1: Given α j > 0, j = 1, 2, if there exist n × n-matrices P j > 0, R i j > 0, S i j > 0, i = 0, 1, 2, P 2 j , P 3 j , Y 1 j and Y 2 j such that the LMIs ( 15), ( 16) with ( 17) are feasible, then each mode j = 1, 2 of switched delay system ( 11) is exponentially stable with the rate α j for all fast-varying

delays τ ∈ [h j , h j+1 ]. Φ| χ=1 =          Φ 111 Φ 121 R01 + P T 21 A 1 -Y T 11 Y T 11 0 [Y T 11 -P T 21 A 1 ] Y T 11 * Φ 221 P T 31 A 1 -Y T 21 Y T 21 0 [Y T 21 -P T 31 A 1 ] Y T 21 * * Ŝ11 -( S01 + R01 ) 0 0 0 0 * * * Ŝ21 -( S11 + R21 ) R21 0 0 * * * * -( S21 + R21 ) 0 0 * * * * * -R11 0 * * * * * * -R11          < 0 ( 15
)
Φ| χ=0 =          Φ 112 Φ 122 R02 P T 22 A 2 -Y T 12 Y T 12 [Y T 12 -P T 22 A 2 ] Y T 12 * Φ 222 0 P T 32 A 2 -Y T 22 Y T 22 [Y T 22 -P T 32 A 2 ] Y T 22 * * Ŝ12 -( S02 + R02 + R12 ) R12 0 0 0 * * * Ŝ22 -( S12 + R12 ) 0 0 0 * * * * -S22 0 0 * * * * * -R22 0 * * * * * * -R22          < 0 (16)
holds, where

Ri j = e -2α j (h i+1 ) R i j , Si j = e -2α j h i+1 S i j , Ŝi j = e -2α j h i S i j , Φ 11 j = A T P 2 j + P T 2 j A + S 0 j -e -2α j h 1 R 0 j + 2α j P j , Φ 12 j = P j -P T 2 j + A T P 3 j , Φ 22 j = -P 3 j -P T 3 j + ∑ 2 i=0 (h i+1 -h i ) 2 R i j . ( 17
)
Proof: Computing (13) with the LKFs [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF] gives:

Vj (t, x t , ẋt ) + 2α j V j (t, x t , ẋt ) ≤ 2x T (t)P j ẋ(t) + 2α j x T (t)P j x(t) + ẋT (t)[∑ 2 i=0 (h i+1 -h i ) 2 R i j ] ẋ(t) -∑ 2 i=0 (h i+1 -h i )e -2α j (h i+1 ) t-h i t-h i+1 ẋT (s)R i j ẋ(s)ds+ ∑ 2 i=0 x(t -h i ) T e -2α j h i S i j x(t -h i ) -∑ 2 i=0 e -2α j h i+1 x T (t -h i+1 )S i j x(t -h i+1 ). ( 18
)
1) We start with the case of χ = 1, i.e. τ ∈ [h 1 , h 2 ], i = 0, 2, j = 1. Note that:

t-h 1 t-h 2 ẋT (s)R 11 ẋ(s)ds = t-τ(t) t-h 2 ẋT (s)R 11 ẋ(s)ds + t-h 1 t-τ(t) ẋT (s)R 11 ẋ(s)ds ,
Applying the Jensen's inequality [START_REF] Gu | Stability of time-delay systems[END_REF] for i = 0, 2 gives:

t-τ(t) t-h 2 ẋT (s)R 11 ẋ(s)ds ≥ 1 h 2 -h 1 t-τ(t) t-h 2 ẋT (s)dsR 11 t-τ(t) t-h 2 ẋ(s)ds, t-h 1 t-τ(t) ẋT (s)R 11 ẋ(s)ds ≥ 1 h 2 -h 1 t-h 1 t-τ(t) ẋT (s)dsR 11 t-h 1 t-τ(t) ẋ(s)ds t-h i t-h i+1 ẋT (s)R i1 ẋ(s)ds ≥ 1 h i+1 -h i t-h i t-h i+1 ẋT (s)dsR i1 t-h i t-h i+1 ẋ(s)ds, (19)
Then, denoting

v 11 = t-h 1 t-τ(t) ẋ(s)ds, v 12 = t-τ(t)
t-h 2 ẋ(s)ds, we obtain:

V1 (t, x t , ẋt ) + 2α 1 V 1 (t, x t , ẋt ) ≤ 2x T (t)P 1 ẋ(t) + 2α 1 x T (t)P 1 x(t) + ẋT (t) ∑ 2 i=0 (h i+1 -h i ) 2 R i1 ) ẋ(t) + ∑ 2 i=0 e -2α 1 h i x T (t -h i )S i1 x(t -h i ) -∑ 2 i=0 e -2α 1 h i+1 x T (t -h i+1 )S i1 x(t -h i+1 ) -[x(t) -x(t -h 1 )] T e -2α 1 h 1 R 01 [x(t) -x(t -h 1 )] -[x(t -h 2 ) -x(t -h 3 )] T e -2α 1 h 3 R 21 [x(t -h 2 ) -x(t -h 3 )] -v T 11 e -2α 1 h 2 R 11 v 11 -v T 12 e -2α 1 h 2 R 11 v 12 . (20)
We use further the descriptor method [START_REF] Fridman | New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems[END_REF]. The following quantity:

0 = 2[x T (t)P T 21 + ẋT (t)P T 31 ][Ax(t) + A 1 x(t -h 1 ) -A 1 v 11 -ẋ(t)],
with some matrices P 21 , P 31 of the appropriate size, is added into the right-hand side of [START_REF] Niemeyer | Towards force-reflecting teleoperation over the internet[END_REF].

Following [START_REF] He | Parameter-dependent lyapunov functional for stability of time-delay systems with polytopic-type uncertainties[END_REF], we also add free weighting matrices:

0 = 2[x T (t)Y T 11 + ẋT (t)Y T 21 ][-x(t -h 1 ) + x(t -h 2 ) + v 11 + v 12 ]. Setting η j (t) = col{x(t), ẋ(t), x(t -h 1 ), x(t -h 2 ), x(t -h 3 ), v 11 , v 12 }
, we obtain that the inequality:

V1 (t, x t , ẋt ) + 2α 1 V 1 (t, x t , ẋt ) ≤ η T 1 (t)Φη 1 (t) < 0, (21)
is satisfied if the LMI (15) holds.

2) Continuing with the case χ = 0, i.e. τ ∈ [h 2 , h 3 ], by applying the same method with j = 2 and i = 0, 1, we obtain:

V2 (t, x t , ẋt ) + 2α 2 V 2 (t, x t , ẋt ) ≤ 2x T (t)P 2 ẋ(t) + 2α 2 x T (t)P 2 x(t) + ẋT (t) ∑ 2 i=0 (h i+1 -h i ) 2 R i2 ) ẋ(t) + ∑ 2 i=0 x T (t -h i ) Ŝi2 x(t -h i ) -∑ 2 i=0 x T (t -h i+1 ) Si2 x(t -h i+1 ) -[x(t) -x(t -h 1 )] T R02 [x(t) -x(t -h 1 )] -[x(t -h 2 ) -x(t -h 1 )] T R21 [x(t -h 2 ) -x(t -h 1 )] -v T j1 R22 v j1 -v T j2 R22 v j2 . (22)
The equation [START_REF] Nilsson | Real-time control systems with delays[END_REF] with j = 2 is then satisfied if the LMI ( 16) holds.

This concludes the proof of Theorem 1.

Remark 4: Note that the Theorem 1 can also be easily applied to arbitrary switches by using the same LKF for both the switching modes, i.e., fixing α 1 = α 2 = α,

P 1 = P 2 = P, S i1 = S i2 = S i , R i1 = R i2 = R i , P 21 = P 22 = P 2 , P 31 = P 32 = P 3 , (i = 0, 1, 2) .
Form this remark, the following results can be obtained.

Corollary 1: Given α > 0, if there exist n × n-matrices P > 0, R i > 0, S i > 0, i = 0, 1,

B. Strategy for the output stabilization

The results presented here are direct consequence of theorem 1. They provide a design procedure to guarantee prescribed performances in each mode. As it was the case for theorem 1 it DOES NOT ensure the stability when arbitrary switches occurs. That is why it is assumed that in (9), x(t) = x(t), and in [START_REF] He | Parameter-dependent lyapunov functional for stability of time-delay systems with polytopic-type uncertainties[END_REF], δ i con (t) = δ i con (t) (separation principle holds). The first lemma is used for the state feedback design procedure and the second for the observer design procedure.

Lemma 1: Given some scalars α j > 0 and ε j > 0, j = 1, 2 if there exist some matrices P 1 j > 0, R i j > 0, S i j > 0, P 2 j , P 3 j , Y i j , and M j such that the LMIs ( 23) and ( 24) hold, then each mode j of the closed loop (9) with x(t) = x(t) is exponentially stabilized with the rate α j . The control gains are given by K j = M j P -1 2 j .

Φ con1 =          Φ 111 Φ 121 [ R01 -BM 1 -Y T 11 ] Y T 11 [Y T 11 + BM 1 ] Y T 11 0 * Φ 221 [-ε 1 BM 1 -Y T 21 ] Y T 21 [Y T 21 + ε 1 BM 1 ] Y T 21 0 * * Ŝ11 -S01 -R01 0 0 0 0 * * * Ŝ21 -S11 -R21 0 0 R21 * * * * -R11 0 0 * * * * * -R11 0 * * * * * * -S21 -R21          < 0, (23)
Φ con2 =          Φ 112 Φ 122 [-BM 2 -Y T 12 ] Y T 12 [Y T 12 + BM 2 ] Y T 12 R02 * Φ 222 [-ε 2 BM 2 -Y T 22 ] Y T 22 [Y T 22 + ε 2 BM 2 ] Y T 22 0 * * Ŝ22 -S12 -R12 0 0 0 R12 * * * -S22 0 0 0 * * * * -R22 0 0 * * * * * -R22 0 * * * * * * Ŝ12 -S02 -R02 -R12          < 0, (24)
with: Ri j = e -2α j h i+1 R i j ,

Si j = e -2α j h i+1 S i j ,

Ŝi j = e -2α j h i S i j , Φ 11 j = AP 2 j + P T 2 j A T + S 0 je -2α j h 1 R 0 j + 2α j P 1 j , Φ 12 j = P 1 j -P 2 j + ε j P T 2 j A,

Φ 22 j = -ε j P 2 j -ε j P T 2 j + ∑ 2 i=0 (h i+1 -h i ) 2 R i j . ( 25
)
Lemma 2: Given some scalars α j > 0 and ε j > 0, j = 1, 2, if there exist some matrices P 1 j > 0, R i j > 0, S i j > 0, P 2 j , P 3 j , Y i j , and M j such that the following LMIs ( 26) and ( 27) hold, then each mode of the observation error equation given in [START_REF] He | Parameter-dependent lyapunov functional for stability of time-delay systems with polytopic-type uncertainties[END_REF] with δ i con (t) = δ i con (t) is exponentially stabilized with a decay rate α j :

Φ obs1 =          Φ 111 Φ 121 [ R01 -W 1 C -Y T 11 ] Y T 11 [Y T 11 +W 1 C] Y T 11 0 * Φ 221 [-ε 1 W 1 C -Y T 21 ] Y T 21 [Y T 21 + ε 1 W 1 C] Y T 21 0 * * Ŝ11 -S01 -R01 0 0 0 0 * * * Ŝ21 -S11 -R21 0 0 R21 * * * * -R11 0 0 * * * * * -R11 0 * * * * * * -S21 -R21          < 0, (26)
Φ obs2 =          Φ 112 Φ 122 [-W 2 C -Y T 12 ] Y T 12 [Y T 12 +W 2 C] Y T 12 R02 * Φ 222 [-ε 2 W 2 C -Y T 22 ] Y T 22 [Y T 22 + ε 2 W 2 C] Y T 22 0 * * Ŝ22 -S12 -R12 0 0 0 R12 * * * -S22 0 0 0 * * * * -R22 0 0 * * * * * -R22 0 * * * * * * Ŝ12 -S02 -R02 -R12          < 0, (27)
with: Ri j = e -2α j h i+1 R i j ,

Si j = e -2α j h i+1 S i j ,

Ŝi j = e -2α j h i S i j , Φ 11 j = A T P 2 j + P T 2 j A + S 0 je -2αh 1 R 0 j + 2α j P 1 j , Φ 12 j = P 1 j -P T 2 j + ε j A T P 2 j ,

Φ 22 j = -εP 2 j -εP T 2 j + ∑ 2 i=0 (h i+1 -h i ) 2 R i j . (28)
Proof: Both proofs remain the same as for theorem 1. Only some matrix matrix manipulation and changes of variables are added. For more details, similar proof are in [START_REF] Jiang | A gain scheduling strategy for the control and estimation of a remote robot via internet[END_REF], [START_REF] Sun | Stability of switched systems with time-varying delays: delay-dependent common lyapunov functional approach[END_REF].

IV. GLOBAL STABILIZATION OF THE NETWORKED CONTROL SYSTEM WITH MINIMUM DWELL TIME

The previous part was devoted to design the state feedback and the observer to ensure some decay rate while the system does not switch. The present part gives material to design the switching rule of the controller guaranteeing stability with/o performances. The main idea is to exploit the ability of the controller to enlarge the delay by using buffer and avoid some switching occurrences. This feature permit to respect some dwell time requirement for the global stability with switches.

A. Networked control system Model

This subsection gives the global model of the networked controlled system. This model takes into account the interactions between the delay detection problem, the switching buffer size and the observer. In this part, there are no separation principle and the overall models has to be taken into account. It assumed that the controller gains K i and L i are designed using lemmas 1 and 2

on the non switching model for decay rates α i .

Let recall our networked control system.

   ẋ(t) = Ax(t) + Bu(t -δ con (t)), y(t) = Cx(t), (29)
With its controller:

         ẋ(t) = A x(t) + Bu(t -δ i con (t)) + L i (y(t -δ i obs (t)) -ŷ(t -δ i obs (t))), ŷ(t) = C x(t), u(t) = K i x(t) + ky c (t), (30)
Remind that the controller switches as soon as a bigger delay is detected but one can choose when it switch if the delay becomes smaller by increasing artificially the delay. The delay is measured with some lag (takes at maximum h max if measured on the measurement channel).Because of this lag and because of the buffer size problem, four modes arise including two additional modes when the estimated delay does not equals the real one: SS 1 mode, t ∈ [t 0 ,t 1 ]: The delays are in S 1 and are correctly detected. In this case, the small buffer is used and the controller is in the mode i = 1. In that case, δcon (t) = δ con . The controller gains, K 1 and L 1 , are designed for exponential stability with a decay rate α 1 . The equations of SS 1 with ζ (t) = col{x(t), e(t)} are given by:

ζ (t) = Ā0 ζ (t) + Ā11 ζ (t -δ 1 con (t)) + Ā31 ζ (t -δ 1 obs (t)) δ 1 con (t), δ 1 obs (t) ∈ [h 1 , h 2 ] ( 31
) with Ā0 = A 0 0 A , Ā11 = BK 1 -BK 1 0 0 , Ā31 = 0 0 0 -L 1 C
.

SU 1 mode, t ∈ [t 1 ,t 2 ]:
The system was in SS 1 then the delay grows and belongs now in S 2 .

The value is not detected yet by the communication structure implying that the controller stays in mode 1. This mode may be unstable since the controller gains K 1 and L 1 are not designed for. Its decay rate is denoted by α 3 and may be negative. The second effect is that the control buffer is smaller than the communication delay, the observed control input and the plant control input are no more synchronized during this period because δcon (t) = δ con (t). This mode has a maximum dwell time since the delay detection takes less than the maximum communication delay. The equation of SU 1 are:

ζ (t) = Ā0 ζ (t) + Ā13 ζ (t -δ 1 con (t)) + Ā23 ζ (t -δ 2 con (t)) + Ā33 ζ (t -δ 2 obs (t)). δ 1 con ∈ [h 1 , h 2 ], δ 2 con (t), δ 2 obs (t) ∈ [h 2 , h 3 ] (32)
for Ā0 =

A 0 0 A , Ā13 = 0 0 -BK 1 BK 1 , Ā23 = BK 1 -BK 1 BK 1 -BK 1 , Ā33 = 0 0 0 -L 1 C
. SS 2 mode, t ∈ [t 3 ,t 4 ]: The system was in mode SU 1 and finally detect the correct value of the delay. The controller switches to the mode 2. In that case, the buffer works again properly, i.e. δcon (t) = δ con (t). The controller gains, K 2 and L 2 , are designed for exponential stability with a decay rate α 2 . The equations are:

ζ (t) = Ā0 ζ (t) + Ā12 ζ (t -δ 2 con (t)) + Ā32 ζ (t -δ 2 obs (t)) δ 2 con (t), δ 2 obs (t) ∈ [h 2 , h 3 ] (33)
with Ā0 =

A 0 0 A , Ā12 = BK 2 -BK 2 0 0 , Ā32 = 0 0 0 -L 2 C
. SU 2 mode, t ∈ [t 3 ,t s ]: The delay becomes smaller and belongs to S 1 but the controller do not switch to mode i = 1. It waits to meet the dwell time condition. The control gains K 2 are still design for this mode since the delay is artificially kept in S 2 . And the observer gains L 2 keep the stability since if the conditions of theorem 1 hold for a given delay, it does also for a smaller one. The global decay rate is noted α 4 .

ζ (t) = Ā0 ζ (t) + Ā14 ζ (t -δ 2 con (t)) + Ā34 ζ (t -δ 1 obs (t)). δ 2 con (t) ∈ [h 2 , h 3 ], δ 1 obs (t) ∈ [h 1 , h 2 ] ( 34
) for Ā0 = A 0 0 A , Ā14 = BK 2 -BK 2 0 0 , Ā34 = 0 0 0 -L 2 C
.

The four modes is summarized in the following figure where the arrows represent the possible transitions, the only controller transition is SU 2 → SS1: To summarize:

SS 1 : t ∈ [t 0 ,t 1 ] i = 1, Bu f f er = h 2 -T δcon (t) = δ con (t) δ con (t) ∈ S 1 , δ obs (t) ∈ S 1 SU 1 : t ∈ [t 1 ,t 2 ] i = 1, Bu f f er = h 2 -T δcon (t) = δ con (t) δ con (t) ∈ S 2 , δ obs (t) ∈ S 2 SS 2 : t ∈ [t 2 ,t 3 ] i = 2, Bu f f er = h 3 -T δcon (t) = δ con (t) δ con (t) ∈ S 2 , δ obs (t) ∈ S 2 SU 2 : t ∈ [t 3 ,t 4 ] i = 2, Bu f f er = h 3 -T δcon (t) = δ con (t) δ con (t) ∈ S 2 , δ obs (t) ∈ S 1

B. Global stability conditions

In order to avoid unstable switching sequences, a minimum dwell time between the modes is computed here by using the exponential stability property. The global stability of the system will be studied by using the method of [START_REF] Liberzon | Switching in Systems and Control[END_REF]. The LKF V 1 , V 2 , V 3 and V 4 are respectively associated to the modes SS 1 , SS 2 , SU 1 and SU 2 . A common structure is chosen:

V j (t, x t , ẋt ) = x T (t)P j x(t) + ∑ 2 i=0 t-h i t-h i+1 e 2α j (s-t) x T (s)S i j x(s)ds + ∑ 3 k=1 ∑ 2 i=0 (h i+1 -h i ) -h i -h i+1 t t+θ e 2α j (s-t) ẋT (s)R k i j ẋ(s)dsdθ , (35)
LKF are used to compute the decay rate α j of each modes and a scale parameter µ i j between the modes:

V j (t, x t , ẋt ) ≤ µ i j V i (t, x t , ẋt ) ∀i, j = 1, . . ., 4 V j (t, x t , ẋt ) ≤ e -2α j (t-t 0 ) V j (t 0 , x t 0 , ẋt 0 ) ∀ j = 1, . . . , 4 (36)
The following lemma provides conditions ensuring V j (t, x t , ẋt ) ≤ µ i j V i (t, x t , ẋt ) for the consid- ered sequence.

Fig. 3. Minimum dwell time: Small delay from t 0 to t 1 , then big delay from t1 to t 3 , switches at t 2 and t s .

Lemma 3: The property V j (t, x t , ẋt ) ≤ µ i j V i (t, x t , ẋt ), (i, j) ∈ {(1, 4), (4, 2), (2, 3), (3, 1)}} is guaranteed if the following conditions are satisfied :

∀n ∈ {0, 1, 2}, ∀h ∈ {h n , h n+1 }, ∀k ∈ {1, 2, 3} P i ≤ µ i j P j ∀(i, j) ∈ {(1, 4), (4, 2), (2, 3), (3, 1)}, , e -2α i h S ni ≤ µ i j e -2α j h S n j , e -2α i h R k ni ≤ µ i j e -2α j h R k n j . (37)
The computation of the parameters α i and µ i j is done using the following lemma which is a specific version of theorem 1.

Lemma 4: Given some scalars α j and µ i j , (i, j) ∈ {(1, 4), (4, 2), (2, 3), (3, 1)}, if there exist scalars µ i j , i, j = 1, . . ., 4, matrices P j > 0, R k i j > 0 and S i j > 0(i = 0, 1, 2 and k = 1, 2, 3), P 2 j , P 3 j , Y 1k j and Y 2k j ( j = 1, 2, 3, 4 and k = 1, 2, 3) with proper dimensions, such that (38), (39) and [START_REF] Zampieri | Trends in networked control systems[END_REF] conditions hold then the system (IV-A) has positive functionals (35) fulfilling 36.

Φ| SS 1 = Ψ 11 Ψ 21 * Ψ 31 < 0, Φ| SS 2 = Ψ 12 Ψ 22 * Ψ 32 < 0, (38)
Φ| SU 1 = Ψ 13 Ψ 23 * Ψ 33 < 0, Φ| SU 2 = Ψ 14 Ψ 24 * Ψ 34 < 0, (39)
where:

Ψ 11 =      Φ 111 Φ 121 P T 21 A 11 + P T 21 A 31 + ∑ 3 k=1 ( Rk 01 -Y T 1k1 ) ∑ 3 k=1 Y T 1k1 0 * Φ 221 P T 31 A 11 + P T 31 A 31 -∑ 3 k=1 Y T 2k1 ∑ 3 k=1 Y T 2k1 0 * * Ŝ11 -S01 -∑ 3 k=1 Rk 01 0 0 * * * Ŝ21 -S11 -∑ 3 k=1 Rk 21 ∑ 3 k=1 Rk 21 * * * * -S21 -∑ 3 k=1 Rk 21      , (40)
Ψ 21 =      Y T 111 -P T 21 A 11 Y T 111 Y T 121 Y T 121 Y T 131 -P T 21 A 31 Y T 131 Y T 211 -P T 31 A 11 Y T 211 Y T 221 Y T 221 Y T 231 -P T 31 A 31 Y T 231 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      , (41)
Ψ 31 =        -R1 11 0 0 0 0 0 * -R1 11 0 0 0 0 * * -R2 11 0 0 0 * * * -R2 11 0 0 * * * * -R3 11 0 * * * * * -R3 11        , (42)
Ψ 12 =      Φ 112 Φ 122 ∑ 3 k=1 Rk 02 P T 22 A 12 + P T 21 A 31 -∑ 3 k=1 Y T 1k2 ∑ 3 k=1 Y T 1k2 * Φ 222 0 P T 32 A 12 + P T 32 A 32 -∑ 3 k=1 (Y T 2k2 ) ∑ 3 k=1 Y T 2k2 * * Ŝ12 -S02 -∑ 3 k=1 ( Rk 02 + Rk 12 ) ∑ 3 k=1 Rk 12 0 * * * Ŝ22 -S12 -∑ 3 k=1 Rk 12 0 * * * * -S22      , (43)
Ψ 22 =      Y T 112 -P T 22 A 12 Y T 112 Y T 122 Y T 122 Y T 132 -P T 22 A 32 Y T 132 Y T 212 -P T 32 A 12 Y T 212 Y T 222 Y T 222 Y T 232 -P T 32 A 32 Y T 232 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      , (44)
Ψ 32 =        -R1 22 0 0 0 0 0 * -R1 22 0 0 0 0 * * -R2 22 0 0 0 * * * -R2 22 0 0 * * * * -R3 22 0 * * * * * -R3 22        , (45)
Ψ 13 =      Φ 113 Φ 123 ∑ 3 k=1 Rk 03 + P T 23 Ā13 -Y T 113 P T 23 ( Ā23 + Ā33 ) +Y T 113 -Y T 123 -Y T 133 Y T 123 +Y T 133 * Φ 223 P T 33 Ā13 -Y T 213 P T 33 Ā23 + P T 33 Ā33 +Y T 213 -Y T 223 -Y T 233 Y T 223 +Y T 233 * * Ŝ13 -S03 -∑ 3 k=1 Rk 03 -R2 13 -R3 13 R2 13 + R3 13 0 * * * Ŝ23 -S13 -R1 23 -R2 13 -R3 13 R1 23 0 * * * * -S23 -R1 23      , (46)
Ψ 23 =      Y T 113 -P T 23 Ā13 Y T 113 Y T 123 -P T 23 Ā23 Y T 123 Y T 133 -P T 23 Ā33 Y T 133 Y T 213 -P T 33 Ā13 Y T 213 Y T 223 -P T 33 Ā23 Y T 223 Y T 233 -P T 33 Ā33 Y T 233 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      , (47)
Ψ 14 =      Φ 114 Φ 124 ∑ (48)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      , (50)
Ψ 34 =        -R1
and where

Ri j = e -2α j (h i+1 ) R i j ,

Si j = e -2α j h i+1 S i j ,

Ŝi j = e -2α j h i S i j , Φ 11 j = A T P 2 j + P T 2 j A + S 0 je -2α j h 1 R 0 j + 2α j P j , Φ 12 j = P j -P T 2 j + A T P 3 j , Φ 22 j = -P 3 j -P T

3 j + ∑ 2 i=0 (h i+1 -h i ) 2 R i j . (52)
Since the NCS global model switches are in a predetermined order, the global performance/stability is achieved if a one of the functional V j is decreasing each cycle. This gives the following condition to ensure a decay rate α g over a complete cycle:

V 1 (t s , x t s , ẋt s ) ≤ e -2α j (t s -t 0 ) V 1 (t 0 , x t 0 , ẋt 0 ).
This condition is illustrated on figure 3. The following theorem gives the stability conditions for a 4 sequenced mode switching system. system (IV-A) has positive functionals (35) fulfilling 36.

Theorem 2: Consider a 4 sequenced mode switching model described in (IV-A). Given some scalars α j and µ i j , (i, j) ∈ {(1, 4), (4, 2), (2, 3), (3, 1)}, if there exist scalars µ i j , i, j = 1, . . . , 4, matrices P j > 0, R k i j > 0 and S i j > 0(i = 0, 1, 2 and k = 1, 2, 3), P 2 j , P 3 j , Y 1k j and Y 2k j ( j = 1, 2, 3, 4 and k = 1, 2, 3) with proper dimensions, such that (37), [START_REF] Zhang | Modelling and control of networked control systems with both network-induced delay and packet-dropout[END_REF], (39) and condition (53) hold for some α g > 0 then the model (IV-A) is exponentially stable with the decay rate α g : µ 14 µ 42 µ 23 µ 31 e -2α 4 (t s -t 3 )-2α 2 (t 3 -t 2 )-2α 3 (t 2 -t 1 )-2α 1 (t 1 -t 0 ) ≤ e α g (t s -t 0 ) (53)

Proof: Using the properties [START_REF] Yu | An LMI approach to network control systems with data packet dropout and transmission delays[END_REF], it comes:

V 1 (t s , x t s , ẋt s ) ≤ µ 14 V 4 (t s , x t s , ẋt s )

≤ µ 14 e -2α 4 (t s -t 3 ) V 4 (t 3 , x t 3 , ẋt 3 )

≤ µ 14 µ 42 e -2α 4 (t s -t 3 ) V 2 (t 3 , x t 3 , ẋt 3 )

≤ µ 14 µ 42 e -2α 2 (t s -t 2 )-2α 2 (t 3 -t 2 ) V 2 (t 2 , x t 2 , ẋt 2 )
...

≤ µ 14 µ 42 µ 23 µ 31 e -2α 4 (t s -t 3 )-2α 2 (t 3 -t 2 )-2α 3 (t 2 -t 1 )-2α 1 (t 1 -t 0 ) V 1 (t 0 , x t 0 , ẋt 0 ). (54)
then V 1 (t s , x t s , ẋt s ) ≤ e -2α j (t s -t 0 ) V 1 (t 0 , x t 0 , ẋt 0 ) if µ 14 µ 42 µ 23 µ 31 e -2α 4 (t s -t 3 )-2α 2 (t 3 -t 2 )-2α 3 (t 2 -t 1 )-2α 1 (t 1 -t 0 ) ≤ e α g (t s -t 0 ) (55)

Remark 5: There is always a solution to the problem of finding the switch time t s in the two following cases:

1) α g < α 4 even in the case of fugitive modes,

2) α g < α 1 if the dwell time in mode SS 1 is sufficient.

This result could be used to provide recommendation, in the form of dwell time in SS1 mode to choose the network type needed to achieved given performances.

Note that the conditions given in corollary 4 and in theorem 2 are not strictly LMI since the α j and the µ i j must be fixed. A successive optimization of the parameters can be achieved to get a compromise between a high values of α j and low values µ i j . This optimization is possible since the conditions are harder to satisfy when α j grows / µ i j decreases.

The global approach for the switching controller design is summarized in the following algorithm:

Algorithm 1: Controller design steps: 1) use lemmas 1 and 2 to compute the gains with given α performances for modes 1 and 2.

2) use corollary 4 to compute a compromise between the minimum µ i j and the maximum α i .

3) implement the controller (7) that uses the following switching law:

i = 1, if δ obs (t)) ∈ S 1 and t > t s 2, otherwise. ( 56
)
t s is extracted by applying the logarithm function on the condition of theorem 2.

Remark 6: These result does not take into account the packet dropout case for sake of clarity.

Since the presented controller does not detect packet dropouts, it is treated using [START_REF] Seuret | Control of a remote system over network including delays and packet dropout[END_REF] to compute de worst case exponential decay rate for all modes. Remark also that these results can be extended to more zones and to the case where the control delay range is different from the observer delay range.

V. EXPERIMENTAL RESULTS

The experiments are done on two computers separated from about 40 kilometers away (Fig. 4). The Master program runs on the remote computer with an advanced computing capability, the Slave one on the local one which also communicates with a light-inertia robot Miabot of the company Merlin by Bluetooth.

A. The structure of the Master

In order to implement the model for the remote control system, a four-threads program is designed to fulfill the functions of Controller and Observer in Fig. 4, while the explication of all the parameters refers to [START_REF] Jiang | A gain scheduling strategy for the control and estimation of a remote robot via internet[END_REF]. we need to apply the control data at the time given by the master, the buffer used is a list list_Y where the packet are sorted by order of appliance.

u(t m,k ),t m,k y(t s,k ′ ),t s,k ′ -d 1 y(t s,k ′ ),t s,k ′ -d 1 x(t 0,q ) x(t 0,q ) u(t m,k ),t m,k

C. Experimental study

The model of the Miabot is derived experimentally. In order to avoid saturation and nonlinear behaviors, the speed has been limited to 2m/sec. The obtained model is given by (57). The state value is composed of its position and speed, the control is the value sent by Master.

Once the gain are computed, the dwell time parameters α i and µ i j are computed following theses steps:

1) The values of α 3 and α 4 are maximized for fixed µ i j = 100, α 1 and α 2 , satisfying corollary 4 conditions.

2) µ i j are minimized using the criterion µ 14 + µ 42 + µ 23 + µ 31 for the obtained α i and LKF matrices, satisfying corollary 4 conditions.

These steps lead to the following solutions: α 1 = 2, α 2 = 0.6, α 3 = -4, α 4 = 0.6. The global performance α g chosen is chosen to only keep stability: α g = 0.

D. Results of remote experiment

The result is shown in Fig. 7, in which the blue curve represents the set values; the green and red represent respectively the robot's estimated position and speed; the black corresponds to the real position of the Miabot. On Fig. 7, one can notice three kinds of step responses. The first one corresponds to the case when the time-delay is greater than 80ms, only the second subsystem is active. In this case, a decay rate α 2 is guaranteed. During the second step, only the first mode is active because delays are small. The performances are better: the response time is smaller since a decay rate of α 1 is achieved. In the last kind of response, where a some switches occur during the transient response. In that case, only the global stability is guaranteed. At the moment t ∈ [42ms 45ms],

the time-delay becomes small, but the switching strategy () do not permit to switch back to mode 1 in order to guarantee the global stability.

VI. CONCLUSION

A QoS-dependent controller has been presented to enhance the performances of a NCS system based on switching gains. This controller uses the time-stamps of packets to measure the communication delay and to synchronize the observer with the plant. The resulting controller switches according to the delay and a dwell time strategy ensuring the global stability.

For the sake of clarity, only two intervals for the delay range are considered to quantify the QoS but the conditions can be easily extended to the multiple intervals.

An experimental platform has been presented and illustrates the results of the network-based control theory, showing the effectiveness of the dwell time strategy.

Fig. 2 .

 2 Fig. 2. Switch among four modes.

Fig. 4 .

 4 Fig. 4. Structure of the global system

Fig. 5 .

 5 Fig. 5. Structure of the Master

Fig. 8

 8 illustrate the corresponding switched control signals from Master to Slave. The red curve is the real control while the green and the black ones are the controls calculated respectively for the two subsystems. We can see the switch points according to the values of time-delay. Fig.9depicts the variable global time-delays on the control communication channel and the corresponding switching signal.

Remark 8 :Fig. 8 .Fig. 9 .

 889 Fig. 7. Results of remote experiment

and α 2 Fig. 10 .

 210 Fig. 10. Performances analysis

  2, P 2 , P 3 , Y j1 and Y j2 , j = 1, 2 such that the LMIs (15),[START_REF] Lelevé | Telerobotics over IP networks: Towards a low-level real-time architecture[END_REF] with notations given in[START_REF] Liberzon | Switching in Systems and Control[END_REF] are feasible, then the arbitrary switching delay system[START_REF] Hespanha | stability of switched systems with average dwell-time[END_REF] is exponentially stable with the rate α for all fast-varying delays τ ∈ [h j , h j+1 ], j = 1, 2.

July 23, 2010

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Remark 7: Note that speed related component of the control gain K 2 has a positive value.

This just mean that to keep the system stability when the delay is high, it is needed slightly degrade the performances of the system stable open-loop pole.


L Bensaddek

M L Villarreal

M A Fliniaux

Induction and growth of hairy roots for the production of medicinal compounds

The development of genetically transformed plant tissue cultures and mainly of roots transformed by Agrobacterium rhizogenes (hairy roots), is a key step in the use of in vitro cultures for the production of secondary metabolites. Hairy roots are able to grow fast without phytohormones, and to produce the metabolites of the mother plant. The conditions of transformation (nature and age of the explants, bacterial strain, bacterial density, and the protocol of infection) deeply influence the frequency of the transformation events as well as the growth and productivity of the hairy roots. Then optimization of the culture parameters (medium constituents, elicitation by biotic or abiotic stress) may enhance the capability of the hairy roots to grow fast and to produce valuable compounds.

Introduction

The use of plants as medicines is a very ancient story and a traditional medical practice in all the passed civilisations [START_REF] Samuelsson | Drugs of Natural Origin: a Textbook of Pharmacognosy[END_REF]. Natural products and in particular plant metabolites are still extensively used for therapeutic applications, and it was evaluated that between 1981 and 2002, 28% of the 868 new chemical entities were natural products or derived from natural products, with another 24% created around a pharmacophore from a natural product [START_REF] Raskin | Plants and human health in the twenty-first century[END_REF][START_REF] Newman | Natural products as sources of new drugs over the period 1981-2002[END_REF][START_REF] Balunas | Drug discovery from medicinal plants[END_REF]. In the early years of the twenty-first century, plants are economically important pharmaceuticals and essential for human health. Examples of important drugs obtained from plants are, morphine and codeine from Papaver somniferum, vincristrine and vinblastine from Catharanthus roseus, digoxin from Digitalis lanata [START_REF] Hollman | Drugs and atrial fibrillation. Digoxin comes from Digitalis lanata[END_REF], and quinine and quinidine from Cinchona spp. Natural products have played a major role in lead discovery, mainly in the following areas: oncology, cardiovascular and metabolic diseases, and immuno suppression (Butler, 2004). It has been estimated that between 1981 and 2002, 60% of anticancer drugs and 75% of anti-infectious drugs already on the market or under clinical trial were of natural origin [START_REF] Cragg | Natural products in drug discovery and development[END_REF][START_REF] Shu | Recent natural products based drug development: a pharmaceutical industry perspective[END_REF][START_REF] Newman | Natural products as sources of new drugs over the period 1981-2002[END_REF][START_REF] Lam | New aspects of natural products in drug discovery[END_REF]. The antimalarial artemisinin, isolated from Artemisia annua L. is effective against multidrug resistant strains of Plasmodium, and a lead compound for the discovery of new antimalarial drugs [START_REF] Van Agtmael | Artemisinin drugs in the treatment of malaria: from medicinal herb to registered medication[END_REF]. Several clinically useful anti-cancer agents are plant products or their close derivatives: vinblastine, irinotecan, topotecan, etoposide, and paclitaxel [START_REF] Cragg | Plants as sources of anticancer agents[END_REF]. Huperzine A and galantamine (galanthamine) acting as acetylcholinesterase inhibitors have been approved for the treatment of Alzheimer's disease and other neurodegenerative pathologies [START_REF] Raves | 3D structure of acetylcholinesterase complexed with the nootropic alkaloid, (-)-huperzine A[END_REF][START_REF] Scott | Galantamine: a review of its use in Alzheimer's disease[END_REF]. The obtention of medicinal compounds from extraction of wild or cultivated plants can be limited by various problems: plants difficult to cultivate, risk of extinction for over exploited plants, and geopolitical problems, among other causes [START_REF] Verpoorte | Biotechnology for the production of plant secondary metabolites[END_REF]. To try to overcome these problems, many attempts were made during the last decades to evaluate the possibility of producing medicinal compounds by in vitro plant cell and organ cultures [START_REF] Berlin | Secondary products from plant cell cultures[END_REF][START_REF] Alfermann | Natural product formation by plant cell biotechnology -Results and perspectives[END_REF]. However, in most cases, the compounds were undetectable or were accumulated at low levels in the cultures. Several strategies such as screening and selection of high producing cell lines, cell immobilization, elicitation, and culture of differentiated tissues were developed. In each case problems were encountered and results did not allow the development of an economically valuable commercialization of the biotechnologically produced compounds [START_REF] Verpoorte | Biotechnology for the production of plant secondary metabolites[END_REF]. The in vitro transformation of plant material with Agrobacterium rhizogenes strains allowed to overcome some of the huge difficulties of in vitro plant organ cultures, and led to the obtention of fast growing organs, exhibiting extensive branching, and capable of producing the main metabolites of the mother plant or even new metabolites undetected in the mother plant nor in other kinds of in vitro cultures [START_REF] Nader | Production of triterpenoids in liquid-cultivated hairy roots of Galphimia glauca[END_REF]. The so called hairy roots offered a promising technology for secondary metabolite production [START_REF] Hamill | New routes to plant secondary products[END_REF] such as tropane alkaloids [START_REF] Flores | Metabolic relationships of putrescine, GABA and alkaloids in cell and root cultures of Solanaceae[END_REF][START_REF] Oksman-Caldentey | Regulation of tropane alkaloid metabolism in plants and plant cell cultures[END_REF] and many other metabolites [START_REF] Giri | Transgenic hairy roots: recent trends and applications[END_REF]. At the present time, the more precise knowledge about A. rhizogenes transformation of plant material as well as about hairy roots and their biotechnological use for the production of pharmaceutical products offer new prospects (Guillon et al., 2006a and b;[START_REF] Kuzovkina | Genetically transformed root cultures-generation, properties and application in plant sciences[END_REF][START_REF] Georgiev | Hairy root type plant in vitro systems as sources of bioactive substances[END_REF][START_REF] Srivastava | Hairy root culture for mass-production of high-value secondary metabolites[END_REF].

Agrobacterium rhizogenes strains and the induction of hairy roots

In the hairy roots disease, the infectious process by A. rhizogenes wild strains is characterized by the following four steps: 1) chimiotactism induced movement of agrobacteria towards the plant cells; 2) binding of the bacteria to the surface components of the cell wall; 3) activation of the virulence (vir) genes, and 4) transfer and integration of the transfer-DNA (T-DNA) into the plant genome [START_REF] Zupan | The Agrobacterium DNA transfert complex[END_REF]. The genetic information allowing this infection process is mainly contained in the Ri plasmid (pRi) carried by the bacteria. In the pRi, the vir region concentrates 6 to 8 genes involved in the DNA transfer. The right and left T-DNA regions (T R -DNA and T L -DNA) of the pRi, which are delimited by their border sequences, are the regions that are transferred to the plant. Within the T R section, loci involved in auxin biosynthesis are transferred to the plant genome, thus increasing the auxin level of the transformants. Other genes of the T R section are responsible of the synthesis of opines which are unusual amino acid sugar derivatives used by the bacteria for their feeding [START_REF] Gartland | Agrobacterium virulence[END_REF]. The wild A. rhizogenes strains, many of which have been used to produce hairy roots from medicinal plants, can be classified by their opine type. Agropine strains (A4, 15834, 1855, LBA 9402) induce agropine, mannopine and agropinic acid production while the mannopine strains (8196) and the cucumopine [START_REF] Petit | Further extension of the opine concept: plasmids in Agrobacterium rhizogenes cooperate for opine degradation[END_REF] strains induce the production of one single opine. Agropine strains pRi transfer independently both the T L -DNA and T R -DNA to the plant genome, while mannopine strains only transfer the T L -DNA. This pRi region contains the four rol genes A, B, C and D [START_REF] Schmülling | Single genes from Agrobacterium rhizogenes influence plant development[END_REF][START_REF] Petersen | Structure and function of root-inducing (Ri) plasmids and their relation to tumor-inducing (Ti) plasmids[END_REF] which enhance the auxin and cytokinin [START_REF] Estruch | The plant oncogene rolC is responsible for the release of cytokinins from glucoside conjugates[END_REF] susceptibility of plant cells and are responsible for the formation of roots by transformed tissues (Bonhomme et al., 2000a;[START_REF] Hong | Terpenoid indole alkaloid production by Catharanthus roseus hairy roots induced by Agrobacterium tumefaciens harboring rol ABC genes[END_REF]. The hairy root phenotype is mainly due to the rol genes (A, B, C and D), and in particular the rol B gene [START_REF] Nilsson | Getting to the root: the role of the Agrobacterium rhizogenes rol genes in the formation of hairy roots[END_REF], though hairy roots could also be obtained after transformation of Atropa belladonna, with the rol C gene alone (Bonhomme et al., 2000b). The choice of a bacterial strain is very important since some plants are very resistant to infection (monocots are for example harder to transform with Agrobacterium than dicotyledonous plants). Moreover, bacterial strains are more or less virulent according to the plant species. The LBA 9402 strain is hypervirulent and has been used to successfully transform Hyoscyamus muticus [START_REF] Vanhala | Virulence of different Agrobacterium strains on hairy root formation of Hyoscyamus muticus[END_REF], Centaurium erythraea [START_REF] Piatczak | Genetic transformation of Centaurium erythraea Rafn by Agrobacterium rhizogenes and the production of secoiridoids[END_REF], Saponaria vaccaria [START_REF] Schmidt | High efficiency Agrobacterium rhizogenes-mediated transformation of Saponaria vaccaria L. (Caryophyllaceae) using fluorescence selection[END_REF], Gentiana macrophylla [START_REF] Tiwari | Genetic transformation of Gentiana macrophylla with Agrobacterium rhizogenes: growth and production of secoiridoid glucoside gentiopicroside in transformed hairy root cultures[END_REF] and Papaver somniferum album [START_REF] Flem-Bonhomme | Hairy root induction of Papaver somniferum var. album, a difficult-to-transform plant by A. rhizogenes LBA[END_REF]. Besides the use of wild strains, genetically engineered bacterial strains with modified pRi or disarmed Agrobacterium tumefaciens with a plasmid containing rol genes together or separately have been also employed for the transformation. Hairy roots may also be initiated, containing constitutive expression constructs. This was the case for Cinchona officinalis [START_REF] Geerlings | Alkaloid production by a Cinchona officinalis 'ledgeriana' hairy root culture containing constitutive expression constructs of tryptophan decarboxylase and strictosidine synthase cDNA from Catharanthus roseus[END_REF]. These authors developed a binary vector whose T-DNA contained constitutive-expression versions (CaMV35S promoter with double enhancer and nos terminator) of two genes encoding rate-limiting enzymes: tryptophan decarboxylaxe (tdc) and strictosidine synthase (str) from C. roseus, together with an intron-possessing β-glucuronidase (gus-int) reporter gene and a hygromycin phosphotransferase (hpt) selection marker gene. This binary vector construct was used in conjunction with A. rhizogenes strain LBA 9402 to obtain tdc and strgene-transformed hairy roots of C. officinalis. This technique opened a wide field of applications in the regulation of biosynthetic pathways and bioconversions; however, plant transgenesis is still a discussed subject which encounters a strong opposition of the public opinion, more specifically in European countries.

Infection conditions of the plant material

Several protocols have been used for the infection of plant material by A. rhizogenes. However, the success of the transformation depends on various parameters such as the species and the age of the plant tissue, with the younger ones being in general more sensitive to bacterial infection [START_REF] Sevon | Agrobacterium rhizogenes mediated transformation: root cultures as a source of alkaloids[END_REF]. The bacterial strain used and the density of the bacterial suspension are also influential [START_REF] Park | Agrobacterium rhizogenes-mediated transformation of opium poppy, Papaver somniferum L., and California poppy, Eschscholzia californica Cham., root cultures[END_REF]. The explants most commonly used for infection are young tissues of sterile plantlets, hypocotyl segments, cotyledons, petioles and young leaves. The contact between bacteria and plant cells can be induced by direct injection of the bacterial suspension into the plantlet or by immersion of the plant tissues in the bacterial suspension. This last procedure can be enhanced with vacuum infiltration [START_REF] Tomilov | Agrobacterium tumefaciens and Agrobacterium rhizogenes transformed roots of the parasitic plant Triphysaria versicolor retain parasitic competence[END_REF]. In these cases, the explants have to be wounded before they are inoculated. The use of excised tissues, leaf disks [START_REF] Wang | Improved artemisinin accumulation in hairy root cultures of Artemisia annua by (22S, 23S)-homobrassinolide[END_REF] or organ sections [START_REF] Komaraiah | Enhanced production of antimicrobial sesquiterpenes and lipoxygenase metabolites in elicitor-treated hairy root cultures of Solanum tuberosum[END_REF] increases the contact surface between the plant tissue and the bacteria. With hard to transform plants alternative procedures may be implemented. Among these procedures, micro wounding through electroporation [START_REF] Matsuki | Tissue-specific expression of the rolC promotor of the Ri plasmid in transgenic rice plants[END_REF] or sonication can be used (in a process called sonication assisted Agrobacteriummediated transformation or SAAT) [START_REF] Trick | SAAT: Sonication-Assisted Agrobacterium-mediated transformation[END_REF][START_REF] Flem-Bonhomme | Hairy root induction of Papaver somniferum var. album, a difficult-to-transform plant by A. rhizogenes LBA[END_REF]. For the transformation of P. somniferum album, several factors of the SAAT protocol were investigated for their influence on transient gus expression: pre-culture period, sonication and co-culture duration. The highest number of GUS -positive hypocotyls (91%) was obtained after 60 seconds of sonication and 2 days of co-culture. In several experiments acetosyringone was used to activate the virulence genes of Agrobacterium, and to enhance the transfer of foreign genes into the plant genome [START_REF] Stachel | Identification of the signal molecules produced by wounded plant cells which activate the T-DNA transfer process in Agrobacterium tumefaciens[END_REF][START_REF] Gelvin | Agrobacterium and plant genes involved in T-DNA transfer and integration[END_REF][START_REF] Tao | Genetic transformation of Torenia fournieri L. mediated by Agrobacterium rhizogenes[END_REF][START_REF] Kumar | Agrobacterium rhizogenes mediated genetic transformation resulting in hairy root formation is enhanced by ultrasonication and acetosyringone treatment[END_REF]. The optimal concentration of acetosyringone varies from one experiment to another. For the transformation of Torenia fournieri, low concentrations (10-30 µM) enhanced the transformation, but higher ones did not significantly increase the transformation frequency. For Nicotiana tabacum or P. somniferum the acetosyringone concentrations used varied between 50 and 150 µM. The duration of the plant -bacterium contact during the inoculation and the co-cultivation are parameters that can be optimized. The average co-cultivation duration is about two or three days. After that, the explants must be transferred to a solid medium containing an antibiotic to eliminate the bacteria. Cefotaxime (250-500 mg L -1 ) and Timentin (200-300 mg L -1 ) are often used to eliminate the bacteria. The explants are then transferred onto a solid hormonefree medium in the dark at 20-25°C, and the first roots appear after a few weeks (usually 1 to 4). The roots are then transferred to erlenmeyer flasks containing liquid phytohormones -free medium. The typical transformed root phenotype is a highly branched root covered with a mass of tiny root hairs and these cultures do not require phytohormones. Concerning the growth rate, the average doubling time of hairy root lines is around 2-3 days. Species-related anatomic, morphologic and cytologic changes have been reported [START_REF] Webb | Characterization of transgenic root cultures of Trifolium repens, Trifolium pratense and Lotus corniculatus and transgenic plants of Lotus corniculatus[END_REF]. The putatively transformed roots are usually analyzed to check for T-DNA integration. Opine analysis using paper electrophoresis on root extracts is one of the techniques used to confirm the transgenic nature of the roots [START_REF] Giri | Transgenic hairy roots: recent trends and applications[END_REF][START_REF] Han | Hairy root induction and plant regeneration of crownvetch (Coronilla varia L.) transformed by Agrobacterium rhizogenes[END_REF]. Another alternative is the use of reporter genes. In this area, the use of binary vectors has proven useful for assessing the gene transfer to the plant genome, and following the long term stability of this transfer using selection or marker genes. Selection genes can be antibioticresistance genes such as nptII or hpt coding for neomycin phosphotransferase or hygromycin -phosphotransferase, respectively, expressed only in the transformed tissues. A study on Astragalus sinicus and Glycine max using feedback-insensitive anthranilate synthase (ASA2) cDNA isolated from a 5-methyltryptophan (5MT)-resistant tobacco cell line showed that hairy roots transformed with a 35s-asa2 construct could be directly selected using 20-75 µM 5MT. GUS staining or fluorescence microscopy following transformation with binary vectors containing GUS, GFP, dsRED or EYFP are useful for the identification of stable transformed roots [START_REF] Tomilov | Agrobacterium tumefaciens and Agrobacterium rhizogenes transformed roots of the parasitic plant Triphysaria versicolor retain parasitic competence[END_REF]. PCR and Southern-blotting of rol genes is also another way to confirm T-DNA integration into the plant genome [START_REF] Lorence | Camptothecin and 10hydroxycamptothecin from Camptotheca acuminata hairy roots[END_REF][START_REF] Tiwari | Genetic transformation of Gentiana macrophylla with Agrobacterium rhizogenes: growth and production of secoiridoid glucoside gentiopicroside in transformed hairy root cultures[END_REF]. Additional PCR and Southern-blotting of vir C gene is sometimes performed to check for total elimination of the agrobacteria [START_REF] Shi | Induction of cucumber hairy roots and effect of cytokinin 6-BA on its growth and morphology Sheng Wu Gong[END_REF]. The infection conditions are of capital importance and the choice of the Agrobacterium strain is a first -rate parameter. The strain virulence has strong repercussions on the transformed material properties (morphology, growth rate, and metabolite level). The A. rhizogenes strain LBA 9402 has showed stronger infective ability on Rheum palmatum while A. rhizogenes strain R1601 generated a faster growing clone. In this paper, like often reported, the secondary metabolite content and composition varied significantly between clones [START_REF] Yang | Induction of hairy roots and anthraquinone production in Rheum palmatum[END_REF]. The T-DNA integration into the plant genome which can be linked to the bacterial strain and the number of transferred copies has consequences on the growth and secondary metabolism of the transformed roots. In Whitania somnifera, T L -DNA and T R -DNA integration frequency was linked to the bacterial strain and had an effect on the transformed material morphology. Typical transformed roots, transformed rooty calluses and transformed calluses were obtained where the whitasteroids level was more related to the material morphology than to the inoculated bacterial strain [START_REF] Bandyopadhyay | Changes in morphological phenotypes and withanolide composition of Ri-transformed roots of Withania somnifera[END_REF]. In Gentiana macrophylla, transformed roots, T L -DNA and T R -DNA integration had an effect on the root specific secoiridoid glucoside gentiopicroside accumulation [START_REF] Tiwari | Genetic transformation of Gentiana macrophylla with Agrobacterium rhizogenes: growth and production of secoiridoid glucoside gentiopicroside in transformed hairy root cultures[END_REF].

Effect of medium components on growth and metabolite accumulation

The hairy roots growth rate is generally high, but great variations exist from one line to another. Mean doubling time after inoculation ranges from 24 to 90 h [START_REF] Payne | Plant cell and tissue culture in liquid systems[END_REF], but sometimes it is much longer. As an example, the doubling time of Galphimia glauca hairy roots was 6 days [START_REF] Nader | Production of triterpenoids in liquid-cultivated hairy roots of Galphimia glauca[END_REF], and even 15 days in the case of Cinchona hairy roots [START_REF] Geerlings | Alkaloid production by a Cinchona officinalis 'ledgeriana' hairy root culture containing constitutive expression constructs of tryptophan decarboxylase and strictosidine synthase cDNA from Catharanthus roseus[END_REF]. Optimization of the medium composition may sometimes increase the growth rate of the roots and/or the yield of accumulated metabolites. The use of modified culture media is generally required. These modifications involve changes in sugars, nitrogen, and phosphorous sources. The effect of nitrate and ammonium concentrations on growth and alkaloid accumulation of A. belladonna hairy roots was studied [START_REF] Bensaddek | The effect of nitrate and ammonium concentrations on growth and alkaloid accumulation of Atropa belladonna hairy roots[END_REF]). An increase of ammonium concentration in the culture medium resulted in lowering the growth rate while an increase of the nitrate concentration had a deleterious effect on the alkaloid biosynthesis and accumulation. The highest biomass and alkaloid yields were obtained with reduced levels of both nitrogen sources. The results obtained by Sivakumar and collaborators with ginseng hairy roots suggest that mineral elements are an important regulatory factor of growth and biomass [START_REF] Sivakumar | Optimization of organic nutrients for ginseng hairy roots production in large-scale bioreactors[END_REF]. In vitro culture of plants cells usually require the presence in the medium of plant growth regulators, mainly auxins and cytokinins. In the case of hairy roots, one characteristic of their phenotype is the fast hormone-independent growth. The result is that in media used for the culture of hairy roots hormones are generally lacking. Even more, it has been demonstrated that in transformed roots of Datura stramonium, treatment of the cultures with 2.0 mg L -1 naphtalene acetic acid (NAA) and 0.2 mg L -1 kinetin induced a de-differentiation of the root tissue and a redirection of primary nitrogen metabolism [START_REF] Ford | Phytohormone-induced GABA production in transformed root cultures of Datura stramonium: an in vivo 15 N NMR study[END_REF]. In several experiments this de-differentiation was accompanied by a significant decrease or even a cessation of alkaloid production [START_REF] Robins | Studies on the biosynthesis of tropane alkaloids by Datura stramonium L. transformed root cultures. 3. The relationship between morphological integrity and alkaloid biosynthesis[END_REF]. However, it was demonstrated more recently that when testing systematically the effect of different types of phytohormones upon root growth and secondary metabolite production, some of them could enhance either growth or metabolites production. In the case of A. annua hairy roots (Weathers et al., 2005), the response of cultures to five types of hormones: auxins, cytokinins, ethylene, gibberellins (GA) and abscissic acid (ABA) was evaluated. The highest biomass was obtained when 1-5 mg L -1 ABA was supplied in the medium, while 0.5-1 mg L -1 2-isopentenyladenine inhibited root growth but stimulated the production of artemisinin more than 2-fold. In other experiments (Yu et al., 2006), Polygonum multiflorum hairy root cultures were supplemented with 2,4-D, NAA and 6-BA at various concentrations. The results showed that 0.1 mg L -1 2-4 D had a deleterious effect on the root cultures; in contrast, NAA and 6-BA in certain conditions could stimulate the growth (0.3-0.4 mg L -1 BA; or 0.4 mg L -1 NAA) and the production of anthraquinones (0.4 mg L -1 BA). With the combined treatment of P. ginseng hairy roots with both 25 µM indole-3-butyric acid (IBA) and 100 µM MeJA the productivity of ginsenoside went to 10 mg L -1 d -1 , instead of 7.3 mg L -1 d -1 with MeJA alone (Kim et al., 2007).

Elicitation

The use of biotic or abiotic stress on tissue cultures has been shown to have an effect on the secondary metabolite accumulation. The elicitation procedure consists in treating the cultures with a physical or a chemical agent that will cause phytoalexin production leading to defence mechanisms in the plant cells. The eliciting agents are classified in two large categories: abiotic elicitors (physical, mineral and chemical factors), and biotic elicitors which are factors of plant or pathogen origin [START_REF] Yoshikawa | Diverse modes of action of biotic and abiotic phytoalexin elicitors[END_REF]. As the secondary metabolites are generally produced in nature as a defence mechanism against pathogenic and insect attack, elicitation is often used to enhance their in vitro accumulation levels. Elicitation is mainly used when the hairy root cultures have reached its stationary phase, usually around 2-3 weeks after inoculation. There are many recent examples combining hairy root culture and elicitation treatments (Table I), some of which involving the production of pharmacologically-active terpene-derived compounds. Abiotic elicitors such as NiSO 4 (20 µM), selenium (0.5 mM), and NaCl (0.1%) supplemented in transformed root cultures of P. ginseng, increased the saponin content 1.15-1.33 times compared to controls [START_REF] Jeong | Enhanced secondary metabolite biosynthesis by elicitation in transformed plant root system: effect of abiotic elicitors[END_REF]. Sodium acetate (10.2 mM), added for 24 h to the culture medium of Arachis hypogaea (peanut) hairy roots, lead to a 60-fold induction and secretion of resveratrol and trans-resveratrol into the culture medium [START_REF] Medina-Bolivar | Production and secretion of resveratrol in hairy root cultures of peanut[END_REF]. Sorbitol added as an osmoticum had a dramatic effect on tanshinone yield in Salvia miltiorrhiza Bunge hairy roots: that yield was increased 4.5-fold as compared to the control [START_REF] Shi | Enhancement of tanshinone production in Salvia miltiorrhiza Bunge (red or Chinese sage) hairy-root culture by hyperosmotic stress and yeast elicitor[END_REF]. The effect of biotic elicitors used at higher concentrations (5-400 mg L -1 ) seems to be clearly efficient. In transformed roots of P. ginseng, plant-derived oligosaccharides from Paris polyphylla var. yunanensis increased the saponin content by more than 3 times [START_REF] Zhou | Stimulation of saponin production in Panax ginseng hairy roots by two oligosaccharides from Paris polyphylla var. yunnanensis[END_REF]. Fungus-derived oligosacchrides (from the fungal endophyte Colletotrichum gloeosporoides), yeast elicitor (polysaccharide fraction of the yeast extract), and chitosan increased artemisinin (anti-malarial sesquiterpene endoperoxide) production in A. annua 1.5, 3 and 6-fold, respectively [START_REF] Wang | The preparation of an elicitor from a fungal endophyte to enhance artemisinin production in hairy root cultures of Artemisia annua L. Sheng Wu Gong[END_REF][START_REF] Putalun | Improvement of artemisinin production by chitosan in hairy root cultures of Artemisia annua L[END_REF]. Signal compounds such as salicylic acid and methyl jasmonate (MeJA) can be used as elicitors to enhance the accumulation of secondary metabolites already present in the cultures. In the case of Azadirachta indica hairy roots, addition of 100 mM jasmonic acid and salicylic acid showed a 6-9 fold enhancement of azadirachtin, a tetranortriterpenoid with pesticidal activity, as compared to control cultures (Satvide et al., 2007). These elicitors stimulate biosynthetic pathways. In transgenic Hyoscyamus niger hairy root cultures overexpressing putrescine N-methyltransferase, MeJA treatment enhanced both polyamine and tropane alkaloid biosynthesis [START_REF] Zhang | Tropane alkaloids production in transgenic Hyoscyamus niger hairy root cultures over-expressing putrescine N-methyltransferase is methyl jasmonate-dependent[END_REF]. Moreover, added to the medium of Centella asiatica at a concentration of 0.1 µM, MeJA triggered de novo accumulation of asiaticoside, an anti-inflammatory triterpene saponin which was not initially accumulated in the hairy roots (Kim et al., 2007). This accumulation followed a linear increase for 2 weeks and could be maintained at its top level (7.12 mg g -1 ) for one additional week. The same authors also state that the expression of CabAS, a putative beta-amyrin synthase, was higher than controls 12 hours after MeJA addition and during the 2 following weeks.

Conclusion and perspectives

At the present time, a constantly increasing number of species have been transformed for the establishment of hairy root cultures: 29 species in 1987, 116 in 1990, and 185 plants from 41 families in 2004 [START_REF] Kuzovkina | Genetically transformed root cultures-generation, properties and application in plant sciences[END_REF]. Hairy root cultures offer many advantages among which we can highlight the high and continuous yields of a wide range of metabolites and a high growth potential (Grzegorczyc et al., 2006). The large number of initiated clones offers a screening opportunity (Yu et al., 2006). Large scale culture feasibility and long term stability make this biotechnological approach not only a reliable source of secondary metabolites [START_REF] Peebles | Long-Term Maintenance of a Transgenic Catharanthus roseus Hairy Root Line[END_REF], but also an effective tool to study the biosynthetic pathways of complex plant products [START_REF] Robins | The application of root cultures to problems of biological chemistry[END_REF]. However, scaling up hairy roots to industrial levels poses a great challenge at the moment. The efficiency of the scaling up systems still needs optimization before industrial exploitation becomes valuable. Considering their high efficiency at extremely low concentrations, the use of MeJA and other signalling compounds for transformed root cultures elicitation is opening new ways for a possibly profitable in vitro secondary metabolite production. Genetic engineering [START_REF] Li | Overexpressio of the Saussurea medusa chalcone isomerase gene in S. involucrata hairy root cultures enhances their biosynthesis of apigenin[END_REF], nutritional modelling [START_REF] Cloutier | Predictive nutritional model for plant cells and hairy roots[END_REF], and cross-species co-culture systems involving hairy roots, are opening exciting future prospects in the field of enhanced production and bioconversion [START_REF] Lin | Production of podophyllotoxin using cross-species coculture of Linum flavum hairy roots and Podophyllum hexandrum cell suspensions[END_REF]. Kim et al., 2007 Table I : Elicitation of hairy root cultures accumulating pharmacologically-active compounds. a ratio as compared to the content of not elicited hairy roots.

Acknowledgments : the authors would like to thank ECOS-Nord and ANUIES for supporting the cooperation between the french and mexican laboratories on the biotechnological approach of promotion of mexican medicinal plants.


M Assafin

Vieira Martins

J I B Camargo

Da Silva Neto

PRAIA -Platform for Reduction of Astronomical Images Automatically

Introduction

PRAIA performs high precision differential photometry and astrometry on digitized images (CCD frames, Schmidt plate surveys, etc). The package main characteristics are automation, accuracy and processing speed. Written in FORTRAN 77, it can run in scripts and interact with any visualization and analysis software. PRAIA is in cope with the ever growing amount of observational data available from private and public sources, including data mining and next generation fast telescope all sky surveys, like SDSS, Pan-STARRS and others. PRAIA was officially assigned as the astrometric supporting tool for participants in the GAIA-FUN-SSO activities and will be freely available for the astronomical community.

PRAIA astrometric structure

Figure 1 displays the astrometric scheme of PRAIA package. The photometric part of the platform is out of scope in this text and will not be presented here.

1.1

Information extraction, parameter configuration. Results and information archiving.

PRAIA is composed of separate independent programs. Each program runs in automatic fashion without interaction with the user. Each program has an input data file which must be filled in. Due to the high degree of development of the algorithms used in the image and data treatments, there are quite a few critical parameters which cannot be internally computed and must be furnished by the user. Most of the information in the input data files regards to input and output file names. There are also parameters indicating reduction models and other auxiliary information. In some cases, there are also other input data files that must be present in order to run a task. Usually, those regard to target and to image header information. Image header extraction furnishes critical FOV information like field size, (α,δ) central coordinates and exposure mid-instants. Other auxiliary information is also extracted, as exposure time, filters, object name, image file name. Note that PRAIA reads only FITS format images, but recognizes many FITS header "free" styles. One can always edit and change the extracted information. Target information files contain: (α,δ), JD, standard object name. PRAIA includes programs which can automatically generate target files for solar system bodies by ephemeris extraction, using web services (IMCCE/Skybot or JPL/Horizons) and NAIF/SPICE package. More than one target, with or without changing coordinates with time, can be included in a single target file.

Results are archived on two types of output files with redundancy: (α,δ), exposures mid-times, magnitudes (psf-based and from catalogue), seeing, proper motions (computed, from catalogue), number of reference stars, position errors (mean error, standard error, (O-C)s, …), (x,y), Gaussian psf parameters and errors, magnitude errors, filters, object name, fits file, etc. One file type regards to individual image output and includes information for all objects measured in the FOV. The other type refers to the list of targets and gives in one single file the results from all treated images, only for the targets identified in the FOVs. All the files are formatted and written in standard ASCII, making it very easy for the user to analyze data using their own graphic and statistic tools. For consistency, only "fixed" objects like stars should be used in the procedure. Entries in the target list can be excluded, like solar system moving objects. Table 1 presents a sample of results which show PRAIA astrometry and data handling performance according to object type (star, quasar, solar system object), field size, reference star number, field star number, brightness and epoch.

Table 1 -PRAIA: astrometry and data handling.

Conclusion

PRAIA was officially assigned as the astrometric supporting tool for participants in the GAIA-FUN-SSO activities. No interaction is needed to run the programs of the package. PRAIA automatically identifies reference stars and targets, performs photometric and astrometric measurements, and computes positions and errors. All polynomial models, including radial distortions, magnitude and color terms are available. Default reference catalogues are the UCAC2 and the 2MASS. Others may be used, including secondary catalogues generated (or not) from former PRAIA runs. It also performs astrometry over overlapping frames, from single observations or from CCD mosaics for instance. All resultsincluding those for pre-selected targets -plus complete observational and reduction information are archived in one or few output files with redundancy. PRAIA also allows for a fast visual inspection of the results by screen plots of graphics, tables and statistics using your preferred tools. PRAIA astrometric and photometric performance is certified by a number of publications given in the references. They are examples of high precision photometry and astrometry of CCD observations of asteroids, occultation candidate stars, TNOs, ICRF radio sources and natural satellites. The photometric precision is compatible with DAOPHOT performance to 0.001mag or better. Depending on the reference catalogue, instrument, field size and exposition, positional errors range between 30mas and 70mas and astrometric accuracy (repeatability) can be as good as 20mas to 10mas.

Fig. 1 -

 1 Fig. 1 -Astrometric scheme of PRAIA package.


Anatoliy Ivantsov

Rustam Gumerov

Irek Khamitov

Zeki Aslan

William Thuillot

Gennadiy Pinigin

Daniel Hestroffer

Serge Mouret

Analysis of Astrometry and Photometry Observations of Asteroids at the RTT150

Introduction

The space astrometric mission Gaia, a cornerstone of the European Space Agency, will be launched in 2012 with the objective to make a 3D precise map of our Galaxy. The Gaia will furnish positions, distances and motions of a billion stars with unprecedented precision. Beside stars, the Gaia will observe asteroids with unprecedented precision from 0.5 to 3 mas, allowing the extremely fine orbit determinations [START_REF] Tanga | Gaia, an unprecedented observatory for Solar System dynamics[END_REF]. This precision has great significance for the determination of small effects influencing the dynamics (relativistic, gravitational, non-gravitational, etc.) of Solar system bodies. The determination of masses of a hundred asteroids with a relative precision better than 50% is expected in 5 years of Gaia operation (Mouret et al, 2007).

Considering the time length of the Gaia mission, there will be encounters between asteroids occurring either at the beginning or the end of the mission, so the maximum of deflection angle pertained to the perturbation maximum will not be observed. The precision of mass determinations based solely on the Gaia observations will deteriorate in such cases [START_REF] Hestroffer | Ground-based observations of solar system bodies in complement to Gaia[END_REF]. A possible way out consists of acquiring ground-based observations of high accuracy of selected asteroids and organizing a dedicated network [START_REF] Thuillot | A Ground-Based Network of Observers for Gaia Follow-Up[END_REF]. The RTT150 telescope is one of the professional telescopes, which has already shown its possibilities for researching orbital dynamics of asteroids [START_REF] Aslan | Observational Programs and First Results of Selected Asteroid Observations at RTT150 within International Cooperation[END_REF].

Astrometrical Results

The original list of Institut de Mécanique Céleste et de Calcul des Éphémérides (IMCCE) for astrometry measurements of asteroids consists of 279 perturbed asteroids to be observed during 2008-2010. If astrometrical accuracy of their positions is high enough, then they can be used for complementary science with the Gaia results, thus improving determination of masses for 27 asteroids, including Ceres and Vesta. The idea consists of recovering the orbits of the perturbed asteroids through accurate astrometrical ground-based observations spanned in time before the Gaia launch.

There are two difficulties in accomplishing this task: limited visibility of selected asteroids caused by changing their configuration with respect to the Sun and Earth, as the three year period is comparable to the synodic periods of main belt asteroids, and the limited allocated time for astrometrical observations of asteroids at the professional telescopes, allowing both aperture for imaging faint asteroids, appropriate field of view and scale for making astrometrical measurements at the accuracy level of 0.1″ with contemporary astrometric catalogues.

The selection of asteroids for observations at the RTT150 telescope was made of those perturbed asteroids which can make possible of mass determination for as many as possible asteroids, and those ones perturbed by Ceres and Vesta, which have observed effect greater than 50 mas. The first group consisted of 48 asteroids and the second made up of 22 ones, 70 perturbed asteroids in total. There was calculated visibility for each selected asteroid for the time period of 2008-2010, consisted of apparent visual magnitude, zenith distance at the meridian, solar elongation. The final observational programme was made of those asteroids, whose visibilities were limited by magnitude less than 18, zenith distance at the meridian less than 70°, solar elongation greater than 90° for attaining the best achievable conditions for astrometrical observations.

In the given allocated time of 3 years, there were observed 45 perturbed asteroids at the RTT150, making up a catalogue of 2437 astrometry positions. The astrometric reduction was made with the UCAC2 and UCAC3 catalogues. For making analysis of astrometrical observations of asteroids, there were calculated mean differences (O-C) for each series of observations using the HORIZONS system. The resulting distribution is given in Figure 1.

Fig. 1 -Distribution of (O-C) in position measurements of asteroids

Certainly, several points may belong to the measurements of one asteroid though in different nights.

The weighted errors of a single measurement are 0.16″ in right ascension and 0.13″ in declination. The standard error was calculated as a standard deviation of a single position from the mean one in the series of observation positions in one night. The weights for individual dispersions were assigned proportional to the number of positions in respective series.

Considering the above listed values of errors, the length of series greater than ten positions, one can expect to find significant discrepancies in (O-C) at the level 0.1″, assuming a normal distribution. As one can easily discover in Figure 1, the majority of (O-C) in both right ascension and declination described here has great "Student's ratio", and thus, the associated positions can be used for improving orbital elements of observed asteroids even now.

Photometrical Results

The differential photometry was made for all images where the reference stars of SDSS7 catalogue were present. The stellar magnitudes of SDSS7 were transformed to the BVR Johnson-Cousins-Bessel system using the adopted equations. Thus, there were photometrically reduced 1842 images. The weighted errors are 0.14 mag in B-band, 0.09 mag in V-band, 0.14 mag in R-band, which are greater than the best errors about 0.01 mag of the respective transformations, given by R. Lupton, on the site of SDSS Data Release 7 (http://www.sdss.org/dr7/algorithms/sdssUBVRITransform.html).

The light curve of one of the observed asteroids ( 35107

Conclusion

The telescope RTT150 is used for observations of perturbed asteroids under the observational programme made of the IMCCE list. There were observed 45 asteroids in 3 years run. The allocated time is a principal factor which limits the number of observed asteroids.

The achieved position precision is 0.16″ in right ascension and 0.13″ in declination and is limited by small effects of atmosphere and optics. There are possibilities to reduce precision less than 0.1″.

There was made differential photometrical reduction for 1842 images (82% of positions) in the BVR Johnson-Cousins-Bessel system. The weighted errors are 0.14 mag in B-band, 0.09 mag in V-band, 0.14 mag in R-band.

  ) 1991 VH belonging to Apollo group is given below, Figure 2. The photometrical data represents observations made on June 28, 2008. The light curve shows changes of brightness with the period of variability about 0.1 day and amplitude about 0.2 mag. The error bars indicate standard errors, resulting from the signal-to-noise registered.

Fig. 2 -

 2 Fig. 2 -Light curve of asteroid (35107) 1991 VH

Acknowledgements

A.I. and G.P. express their gratitude to the Organizing Committee for the partial support of their stay during the Gaia-FUN-SSO Workshop. R.G. is partially supported by Russian Foundation for Basic Research, grant 11-02-00262-a.