Yaogong Wang

Natalya Rozhnova
email: natalya.rozhnova@lip6.fr

Ashok Narayanan
email: ashokn@cisco.com

David Oran
email: oran@cisco.com

Injong Rhee
email: rhee@ncsu.edu

An

Keywords: C, 2, 2 [Computer-Communication Networks]: Network Protocols Congestion Control, Information-Centric Networking

est destinée au dépôt

INTRODUCTION

The communication paradigm of NDN [START_REF] Zhang | Named Data Networking Project[END_REF] has two prominent features: 1) all traffic is receiver-driven; 2) content retrieved in response to an interest traverses exactly the same links in reverse order. These two unique characteristics make hop-by-hop interest shaping a better option for NDN congestion control than tra- Dropping interest packets early wastes fewer resources than dropping data packets late. More significantly, end-to-end congestion control is severely handicapped in NDN. Extensive content multihoming and caching make it very difficult to identify interests belonging to a single "flow" and sharing the same congestion path. Performing interest shaping in a hop-by-hop manner can significantly alleviate this problem, especially if the shaping scheme does not rely on flow identification. It also enables sophisticated forwarding strategies such as congestion-driven rerouting to higher-cost (but uncongested) paths. Hence, incorporating hop-by-hop interest shaping with a backpressure mechanism appears to be a more viable option for NDN congestion control.

A number of schemes have been proposed along this path (e.g., [START_REF] Rozhnova | An Effective Hop-by-hop Interest Shaping Mechanism for CCN Communications[END_REF][START_REF] Oueslati | Flow-aware Traffic Control for a Content-Centric Network[END_REF][START_REF] Carofiglio | Joint Hop-by-hop and Receiver-driven Interest Control Protocol for Content-Centric Networks[END_REF]) but all these interest shapers consider single or multiple unidirectional flows. As Figure 1(a) shows, if we assume that interests are sent in just one direction or the bandwidth consumption of the interests in the reverse direction is negligible, then the design of the interest shaping algorithm is quite straightforward: we simply pace the interests so that the contents they bring back will saturate the reverse link. However, in practice, a link will see interests and contents flowing in both directions simultaneously. Also content names in interests can be long since it includes transactional information in many applications (e.g., [START_REF] Jacobson | VoCCN: Voice over Content-Centric Networks[END_REF]). As each interest packet fetches exactly one data packet, interests may consume a non-negligible fraction of the link bandwidth. In light of these factors, the interest shaping algorithm in one direction can no longer assume that the entire reverse link bandwidth is available for returning contents. It needs to shape the interests properly so that enough room is left on the reverse link for interests in the reverse direction. The same logic applies to the interest shaper on the other side of a link and they have a recursive interdependence as shown in Figure 1(b). Since it is non-trivial to determine how much room should be left by the shaper, an imprudent algorithm may cause starvation or link under-utilization.

In this paper, we examine the interdependence between interests and contents in bidirectional flows, and study its impact on the design of hop-by-hop interest shapers in NDN. Section 2 formulates this issue as an optimization problem and obtains optimal solutions under various scenarios. Section 3 presents a practical shaping algorithm based on these solutions, to be executed at the output interface of NDN routers. Section 4 evaluates the performance of our algorithm in conjunction with simple AIMD clients via simulation. Finally, Section 5 discusses related work and Section 6 concludes the paper.

PROBLEM FORMULATION

As depicted in Figure 2, let i 1 and i 2 denote the interest arrival rate in each direction. They are shaped down to s 1 and s 2 respectively by the interest shapers at the routers. Let r 1 and r 2 denote the average size ratio between contents and interests in each direction and let c 1 and c 2 denote the link capacity in each direction. Note that both r 1 and r 2 must be greater than 1. We can then formulate the interest shaping problem under steady state as follow:

Objective:

max u(s 1 ) + u(s 2 ) (1)
Subject to: where u(•) is the utility function discussed later. The objective of this optimization is to maximize network utility (Eq. 1) subject to demand (Eq. 2 and 3) and bandwidth (Eq. 4 and 5) constraints. As Figure 3 shows, the feasible region of this optimization problem is convex. Hence, if the utility function constitutes a concave objective function, the formulation is a convex optimization problem which is mathematically tractable. Note that the utility function we pick has profound impacts on the solution we would obtain. The utility function must maximize data throughput while maintaining certain degree of fairness between the traffic in both directions. As mentioned above, reverse interests compete with forward data packets for bandwidth and the shaper needs to appropriately divide resources between them. It has been shown in [START_REF] Kelly | Charging and Rate Control for Elastic Traffic[END_REF] that logarithmic utility functions can achieve proportional fairness between competing flows. Hence, we present below a closed-form solution to the described optimization problem under logarithmic utility functions (equivalent to maximizing the product of s1 and s2).

0 ≤ s 1 ≤ i 1 (2) 0 ≤ s 2 ≤ i 2 (3)
s 1 + r 2 s 2 ≤ c 1 (4)
r 1 s 1 + s 2 ≤ c 2 (5)
s 1 s 2 s 1 = i 1 s 2 = i 2 i 2 i 1 c 1 c 1 /r 2
First, if we temporarily assume infinite loads in both directions, we can lift the constraints in Eq. 2 and 3. Second, from Figure 3 the optimal solution of the problem lies on the boundary of the feasible region, so we can convert the inequality constraints in Eq. 4 and 5 into equality constraints. Now let us begin by solving the optimization problem under each equality constraint independently. Solving for Eq. 4:

maximize s 1 s 2 subject to s 1 + r 2 s 2 = c 1
The optimal solution for this problem can be determined using Lagrange multipliers:

� s 1 = c1 2 s 2 = c1 2r2 (6)
This optimal point has been labeled by an "x" in Figure 3. Similarly, solving for Eq. 5:

maximize s 1 s 2 subject to r 1 s 1 + s 2 = c 2
The solution (also labeled by an "x" in Figure 3) is:

� s 1 = c2 2r1 s 2 = c2 2 (7)
Now we consider the solution to the original problem under infinite demand (i.e., the optimization problem consisting Eq. 1, 4 and 5). From Figure 3 we can see that the optimal solution primarily depends on how the two bandwidth constraint lines cross each other. If one of the optimal points is within the feasible region, then the optimal solution is just that point (Eq. 6 or 7). If neither optimal point lies within the feasible region (the case shown in Figure 3), the optimal solution is given by the intersection of the two lines (labeled by a circle):

� s 1 = r2c2-c1 r1r2-1 s 2 = r1c1-c2 r1r2-1 (8)
It is trivial to show that the only case where both optimal points lie within the feasible region occurs for the degenerate case of r 1 = r 2 = 1, in which case both points coincide and represent the optimal solution. Now we reintroduce the condition of finite load (Eq. 2 and 3). In practice, it is possible that the traffic load in one direction is inherently low (e.g., the first hop link from a client may have lots of outgoing interests but few incoming interests). Under such cases, the reduced load in one direction should result in increased shaping rate in the opposite direction. The limiting case is unidirectional demand in which case:

i 2 = 0 (9) s 1 = c 2 r 1 (10)
Hence, the instantaneous shaping rate should be variable. We seek proportional fairness between the twoway traffic only if both directions have excessive demand and are competing for the link capacity in both directions. If the load in one direction is inherently low, our scheme is work-conserving and will let the traffic in the other direction grab as much bandwidth as it can.

An adaptive algorithm that adjusts the shaping rate between Table 1 and Eq. 10 based on the fluctuating demand will be presented in the next section.

case s 1 s 2 c1 c2 < 2r2 r1r2+1 c1 2 c1 2r2 2r2 r1r2+1 ≤ c1 c2 ≤ r1r2+1 2r1 r2c2-c1 r1r2-1 r1c1-c2 r1r2-1 c1 c2 > r1r2+1 2r1 c2 2r1 c2 2
Table 1: Optimal solution under infinite loads in both directions

PRACTICAL ALGORITHM

As we have shown, obtaining the optimal interest shaping rate is mathematically tractable if the shaper has knowledge of the content/interest size ratio (r 1 and r 2 ), link capacity (c 1 and c 2 ) and demand (i 1 and i 2 ) in both directions. Due to the symmetric routing of interests and contents in NDN, the shapers can independently measure r 1 and r 2 by observing the interests and contents arriving at and leaving the interface. If we assume that link bandwidths c 1 and c 2 are static parameters, they can also be made known to the shapers easily. However, the offered demand (i 1 and i 2 ) is constantly varying and cannot be accurately measured. Further, the shaper at one end of a link cannot know the interest load on the other side without additional message exchange. Hence, we have designed an adaptive algorithm that does not require accurate knowledge of the offered demand on both sides, which we present below.

From Eq. 10, we can compute the maximum interest shaping rate max s 1 (which occurs when i 2 = 0). We can also determine the minimum interest shaping rate (min s 1 ) from Table 1. To determine the actual shaping rate, we measure the incoming interest rate and use it as an estimation of the load on the other side. Assuming a similar shaper is running on the other side of the link, if there is sufficient demand in the reverse direction, then the observed incoming interest rate (obs s 2 ) should be no less than the rate given by the s 2 column in Table 1 (we call it the expected minimum incoming interest expmin s 2 ). Hence, if obs s 2 ≥ expmin s 2 , the shaping rate is set to min s 1 . Otherwise, the shaping rate is calculated as follows:

min s 1 + (max s 1 -min s 1 )(1 - obs s 2 expmin s 2 ) 2 (11)
This equation adjusts the outgoing interest shaping rate between min s 1 and max s 1 . If obs s 2 = 0, the shaping rate become max s 1 . As we measure higher incoming interest rate, we reduce the shaped outgoing interest rate until it hits min s 1 . We observe that the quadratic control used here is more conservative and robust than a linear control. Figure 4 shows how the proposed interest shaper is implemented on each interface of an NDN router. It is important to note that hop-by-hop interest shaping alone is inadequate to solve the entire congestion control problem. If a client issues more interests than the network can handle, the excessive interests are discarded by the shaper. This interest loss must be signaled back to the client so that it can slow down its request rate. In this paper, we use the simplest droptail policy for the interest queue in the shaper and reject interests with negative acknowledgments (NACK). A similar NACK mechanism has been proposed in [START_REF] Yi | A Case for Stateful Forwarding Plane[END_REF]. Compared with explicit congestion notification in the current Internet, we believe that using NACK to signal congestion in NDN networks with hop-by-hop interest shaping has some unique advantages. When a neighboring router forwards an interest across a link, enough bandwidth has been accounted for in the reverse direction of this link to accommodate the expected returning content. If this interest is then rejected with a NACK towards the client, this NACK takes the place of the accounted-for content. As long as NACKs are smaller on average than contents, there should be enough bandwidth in all the downstream links for NACKs and they should never get lost due to congestion. Hence, using our shaping scheme, NACKs are a much more reliable and timely method of congestion notification than existing proposals (e.g., the timeout mechanism in [START_REF] Carofiglio | ICP: Design and Evaluation of an Interest Control Protocol for Content-Centric Networking[END_REF]).

Clients should react to congestion-triggered NACKs and throttle their outstanding interest rate. Intermediate routers can also react to this congestion signal and implement some sophisticated forwarding strategies such as dynamic rerouting to alternative paths, or midstream throttling of flows. In this paper, we demonstrate our shaping algorithm with a simple windowbased control on clients only. We defer the design of more sophisticated router reaction to our future work.

PERFORMANCE EVALUATION

We have implemented our proposal in ndnSIM and evaluated its performance across a number of different scenarios. In our current implementation, each shaper estimates r 1 and r 2 by monitoring the sizes of the interests and contents passing in both directions. The average size of interests (or contents) is calculated by smoothing out the observed samples similar to TCP round-trip time (RTT) estimation. Also we allow for 2% headroom in the shaping to accommodate traffic burstiness. The sizes of the shaper queues as well as the Layer 2 queues are all set to 60 packets. In-network caching is not enabled at this point. All simulations last for 70s and the session start times are randomly picked between 0s and 5s. Metric measurement ignores the first 10s to eliminate any transient effect during the warm-up phase. All test cases have been repeated 12 times and the confidence intervals are calculated.

In the baseline scenario shown in Figure 5, clients at the two ends of the network issue interests for the contents served at the other end. Content payload size is fixed at 1000B and the interest size is 24B (which slowly increases to 28B due to the increasing number of digits in the content name: /pref ix/1, /pref ix/2, ...). Table 2 shows the simulation results under this scenario as well as a few other scenario that will be discussed later. As can be seen from the table, the hop-by-hop interest shaper effectively controls congestion so that the packet loss rate at the bottleneck link (due to queue overflow) is zero. About 0.015% of interests issued by the clients are rejected by the shapers with NACK and they serve as congestion signals propagating back to the clients. The throughputs achieved by the clients on both sides of the bottleneck are close to 9.56Mbps. This illustrates the bandwidth consumed by interests and the motivation for our optimization: to achieve 9.56Mbps of data throughput in each direction, we need an additional 9.56 × 26 ÷ 1000 ≈ 0.24856Mbps of interest rate in each direction. Therefore, the total throughput of interest and content traffic is 9.80856Mbps. Taking into account the 2% headroom reserved by the shaper, our shaping algorithm achieves the optimal possible throughput. We now investigate a few variations of the baseline scenario. First, we evaluate the shaper under randomized packet size. Recall that the shaper estimates r 1 and r 2 by smoothing packet size samples. When the sizes vary over time, the shaper may have inaccurate estimation of r 1 and r 2 , leading to sub-optimal shaping behavior. However, as shown in Table 2, when the interest packet size is uniformly distributed between 27B and 62B, and the content payload size between 600B and 1400B, the shaper still achieves zero packet loss and we only experience minor throughput loss. Second, we evaluate the scenario where the content/interest size ratios in two directions are asymmetric (i.e., r 1 � = r 2 ). In this scenario, the content payload size on Client/Server2 is set to 500B while the other side still has 1000B payloads. Due to the reduced payload size, Client/Server1 needs to send interests at twice the original rate to achieve the same data throughput. Hence, we experience lower data throughput at Client/Server2 because more link bandwidth is allocated to the interests from Client/Server1. Finally, we simulate the asymmetric link bandwidth scenario (i.e., c 1 � = c 2 ), as is commonly found in today's residential access networks. Here the bottleneck link bandwidth from Router1 to Router2 is reduced to 1Mbps while the reverse link remains at 10Mbps. As simulation results show, our hop-by-hop interest shaper still manages to control congestion under this highly asymmetric scenario. Client/Server2 achieves approximately 0.72Mbps data throughput out of the 1Mbps link, leaving the rest to the interests from Client/Server1 that bring back 9.77Mbps of contents. This also outlines the importance of the logarithmic utility function which avoids starvation of either side.

Next, we evaluate our scheme under the dumbbell topology shown in Figure 6. Three different scenarios have been simulated. First, we launch two clients on the left side of the bottleneck link. They retrieve contents from the two servers on the right side respectively. 3 along with other scenarios under the same topology. As Figure 7 shows, the two flows roughly converge to fair bandwidth sharing despite the randomized session start time. Note that, in this scenario, there are no interests coming from the right side of the bottleneck link, and therefore all this bandwidth can be used for the returning contents. Our shaping algorithm has correctly adapted to this situation, as evidenced by the sum of the throughputs of these two flows being around 9.8Mbps.

We reran this test but set the link latency between Router2 and Client/Server4 to 20ms so that the two flows now have heterogeneous RTTs. The results are also shown in Table 3. Figure 8 compares the evolution of the bottleneck queue length over time between the homogeneous RTT case and the heterogeneous RTT case. As the figures show, we have minimum queues (oscillating between zero and one packet) under homogeneous RTTs since the interests going out from Router1 to Router2 are perfectly paced by the shaper. With heterogeneous RTTs, we see more fluctuations in the queue length due to the burstiness of the traffic incurred by RTT heterogeneity. However, even in this case the queue is well controlled (maximum of 17 packets). Note that our shaper is an open-loop controller that aims at long-term fairness and stability of the system. An alternative is a closed-loop controller that adjusts the shaping rate based on the instantaneous queue length on the other side. However, such a solution may be much more complicated and costly since the queue states need to be communicated using some messaging or piggybacking technique. We believe that with proper parameter tuning, our open-loop controller is able to make the system converge to the optimal steady state. 3 show that our hop-by-hop interest shaper works perfectly in this scenario, too.

RELATED WORK

A number of hop-by-hop interest shaping algorithms have been proposed in the literature. [START_REF] Yi | A Case for Stateful Forwarding Plane[END_REF] proposed the simplest algorithm where the interest rate is shaped so that the returning data rate matches the reverse link capacity multiplied by a pre-determined coefficient between 0 and 1. A more sophisticated shaping algorithm was proposed in [START_REF] Rozhnova | An Effective Hop-by-hop Interest Shaping Mechanism for CCN Communications[END_REF] where the shaping rate is dynamically adjusted based on the instantaneous queue length and available capacity to send data packets at time t. A window-based mechanism described in [START_REF] Carofiglio | ICP: Design and Evaluation of an Interest Control Protocol for Content-Centric Networking[END_REF] controls the interest sending rate of a client in TCP-like fashion. This mechanism was extended in [START_REF] Carofiglio | Joint Hop-by-hop and Receiver-driven Interest Control Protocol for Content-Centric Networks[END_REF] by adding a hop-by-hop interest shaping scheme where each flow is allocated a fair share of the link bandwidth and shaped accordingly. [START_REF] Oueslati | Flow-aware Traffic Control for a Content-Centric Network[END_REF] also presented a per-flow bandwidth fair sharing mechanism. Interests exceeding the fair share are discarded with NACKs and simple AIMD schemes are used to adjust the client's interest rate.

Our scheme differs from these existing efforts in a few ways. First, our scheme works well in bidirectional traffic scenarios, where we believe that fair allocation of bandwidth between interests and contents requires the consideration of their interdependence. This is missing in all prior proposals. Second, we note that flow identification at intermediate routers in NDN is difficult due to the absence of any client identifying information in interest packets. Our scheme does not require any awareness of flows by the shaper. Third, the proposed shaping algorithm is well-suited to handle highly asymmetric link bandwidth and highly varying packet sizes, maximizing utilization in both directions while preventing starvation in either. Fourth, NACKs generated by our shaper offer a timely congestion signal to clients. Our shaping scheme ensures that NACKs have a very low probability of getting dropped, thus avoiding the traditional problem with congestion-generated NACKs exacerbating the congestion problem. These NACKs can also be used by intermediate routers to enforce client shaping. Finally, our scheme detects congestion downstream of the bottleneck link. This enables congestionaware rerouting of interests at the correct point without any additional messaging.

CONCLUSION

In conclusion, we have presented a hop-by-hop interest shaping algorithm for NDN to avoid network congestion and achieve optimal network resource utilization. Our proposed solution accounts for the interdependence between interests and contents in opposite directions and is capable of optimally sharing link bandwidth without extra message exchange. We evaluated the performance of our mechanism when combined with simple AIMD clients using ndnSIM, and studied its behavior under varying conditions of bandwidth, load, and packet size ratios. Our future work includes the design of more sophisticated backpressure mechanisms as well as congestion-aware multipath routing.

Figure 1 :

 1 Figure 1: Interdependence between interests and contents in reverse directions and its impact on hop-by-hop interest shaping ditional TCP-like mechanisms. TCP congestion control reacts to congestion after data packets are lost. By contrast, interest shaping proactively prevents data packet loss by regulating the interest rate in the first place.Dropping interest packets early wastes fewer resources than dropping data packets late. More significantly, end-to-end congestion control is severely handicapped in NDN. Extensive content multihoming and caching make it very difficult to identify interests belonging to a single "flow" and sharing the same congestion path. Performing interest shaping in a hop-by-hop manner can significantly alleviate this problem, especially if the shaping scheme does not rely on flow identification. It also enables sophisticated forwarding strategies such as congestion-driven rerouting to higher-cost (but uncongested) paths. Hence, incorporating hop-by-hop interest shaping with a backpressure mechanism appears to be a more viable option for NDN congestion control.A number of schemes have been proposed along this path (e.g.,[START_REF] Rozhnova | An Effective Hop-by-hop Interest Shaping Mechanism for CCN Communications[END_REF][START_REF] Oueslati | Flow-aware Traffic Control for a Content-Centric Network[END_REF][START_REF] Carofiglio | Joint Hop-by-hop and Receiver-driven Interest Control Protocol for Content-Centric Networks[END_REF]) but all these interest shapers consider single or multiple unidirectional flows. As Figure1(a)shows, if we assume that interests are sent in just one direction or the bandwidth consumption of the interests in the reverse direction is negligible, then

Figure 2 :

 2 Figure 2: Modeling hop-by-hop interest shaping

Figure 3 :

 3 Figure 3: The feasible region of the optimization problem is convex.

Figure 4 :

 4 Figure 4: Shaper implementation

Figure 5 :

 5 Figure 5: Baseline topology

Figure 6 :

 6 Figure 6: Dumbbell topology

Figure 7 :

 7 Figure 7: Client window size evolution under homogeneous RTTHere the two flows have homogeneous RTTs of 60ms and their performance is shown in Table3along with other scenarios under the same topology. As Figure7shows, the two flows roughly converge to fair bandwidth sharing despite the randomized session start time. Note that, in this scenario, there are no interests coming from the right side of the bottleneck link, and therefore all this bandwidth can be used for the returning contents. Our shaping algorithm has correctly adapted to this situation, as evidenced by the sum of the throughputs of these two flows being around 9.8Mbps.We reran this test but set the link latency between Router2 and Client/Server4 to 20ms so that the two flows now have heterogeneous RTTs. The results are also shown in Table3. Figure8compares the evolution of the bottleneck queue length over time between the homogeneous RTT case and the heterogeneous RTT case. As the figures show, we have minimum queues (oscillating between zero and one packet) under homogeneous RTTs since the interests going out from Router1 to Router2 are perfectly paced by the shaper. With heterogeneous RTTs, we see more fluctuations in the queue length due to the burstiness of the traffic incurred by RTT heterogeneity. However, even in this case the queue is well controlled (maximum of 17 packets). Note that our shaper is an open-loop controller that aims at long-term fairness and stability of the system. An alternative is a closed-loop controller that adjusts the shaping rate based on the instantaneous queue length on the other side. However, such a solution may be much more complicated and costly since the queue states need to be communicated using some messaging or piggybacking technique. We believe that with proper parameter tuning, our open-loop controller is able to make the system converge to the optimal steady state.

Figure 8 :

 8 Figure 8: Comparison of bottleneck queues Finally, we simulate the following heterogeneous RTT scenario: one client resides on Client/Server1 and retrieves contents from Client/Server3. Another client resides on Client/Server4 and retrieves contents from Client/Server2. Since the link latency between Router2 and Client/Server4 is 20ms, these two flows sharing the bottleneck link in reverse directions have different RTTs. The results in Table3show that our hop-by-hop interest shaper works perfectly in this scenario, too.

  The outgoing packets are first classified into interests and contents. Content packets are passed directly to the link output queue without shaping. Interest packets

	Scenario	Data Throughput (Mbps)	Packet Loss Rate (%)	Interest Rejection Rate (%)
		Client/Server1	Client/Server2	Router1	Router2	Client/Server1	Client/Server2
	Baseline	9.558421 ± 0.001261	9.559624 ± 0.001550	0	0	0.0150 ± 0.0006	0.0153 ± 0.0011
	Randomized Packet Size	9.432117 ± 0.005931	9.434337 ± 0.007859	0	0	0.0180 ± 0.0014	0.0167 ± 0.0015
	Asymmetric Size Ratio	9.373692 ± 0.014214	9.326215 ± 0.000921	0	0	0.0074 ± 0.0006	0.0155 ± 0.0006
	Asymmetric Link Bandwidth	9.774441 ± 0.001723	0.719525 ± 0.000139	0	0	0.0119 ± 0.0005	0.0576 ± 0.0000

Table 2 :

 2 Simulation results over baseline topology

	Interests	Shaper
	Outgoing packets
	Contents	L2 Queue

Table 3 :

 3 Simulation results over dumbbell topology

	Client/Server1 100Mbps 10ms		Client/Server3 100Mbps 10ms
		10Mbps
	Router1	10ms	Router2
	100Mbps 10ms		100Mbps 10ms
	Client/Server2		Client/Server4


Swann Marx
email: swann.marx@ens-cachan.fr.

Vincent Andrieu
email: vincent.andrieu@gmail.com.

Christophe Prieur
email: christophe.prieur@gipsa-lab.fr.

Using a high-gain observer for a hybrid output feedback: finite-time and asymptotic cases for SISO affine systems

This article suggests a design method of hybrid output feedbacks for affine systems under observability and stabilizability assumptions. Our aim is to use the separation principle on systems controlled by hybrid feedback laws. We investigate two constructive methods for the high-gain observer: the first one is based on a finite-time convergence of the observation error, the second one is based on an asymptotic convergence of the observation error. We illustrate one of our main results on a well-known example: integrators chain.

I. INTRODUCTION

In recent decades, many methods have been introduced for designing output feedback laws that asymptotically stabilizes the origin of a nonlinear system (see e.g [START_REF] Teel | Global stabilizability implies semi-global stabilizability by output feedback[END_REF], [START_REF] Andrieu | Global asymptotic stabilization for nonminimum phase nonlinear systems admitting a strict normal form[END_REF], [START_REF] Andrieu | A unifying point of view on output feedback designs for global asymptotic stabilization[END_REF], [START_REF] Astolfi | Output feedback stabilization with an observer in the original coordinates for nonlinear systems[END_REF]). More recently, thanks to the hybrid formalism described in [START_REF] Goebel | Hybrid Dynamical Systems[END_REF], new methods have been introduced to design asymptotically stabilizing hybrid state feedbacks laws. This allows to consider a larger class of system (for instance the Brockett integrator [START_REF] Brockett | Differential Geometric Control Theory, chapter Asymptotic stability and feedback stabilization[END_REF]). Moreover, hybrid state feedbacks laws may increase performances (see for instance [START_REF] Prieur | Uniting local and global controllers with robustness to vanishing noise[END_REF]).

The design of output feedback controllers may be obtained from an observer and a state feedback design. Note however that for nonlinear continuous systems, designing separately each of these tools leads only to local result. Following [START_REF] Teel | Global stabilizability implies semi-global stabilizability by output feedback[END_REF], when the observer is tuned based on the robustness property of the continuous state feedback, a semi-global result may be achieved. In this paper, we extend this approach to the case in which the state feedback is hybrid.

This paper is organized as follows. In Section II, by considering stabilizability and observability assumptions, a hybrid output feedback law is designed by considering a high-gain observer which converges in finite-time. Such observers are based on the homogeneity notion (see e.g. [START_REF] Andrieu | High-gain observer with uniform in the initial condition finite time convergence[END_REF]). This type of design may imply numerical problems. In Section III, by considering different stabilizability and observability assumptions that are stronger, we etablish a second theorem that deals with a more classical high-gain observer, because it converges asymptotically. This result is illustrated by the design of an output feedback controller for an integrators chain. Section IV collects some concluding remarks. Finally the appendix collects the proofs of the main results.

Due to space limitation, some poofs are omitted.

II. FIRST SET OF ASSUMPTIONS: THE FINITE-TIME CASE

Let us consider the single-input single-output system:

ẋp = f p (x p ) + g p (x p )u y = h p (x p , u), (1)
where

x p ∈ R np , y ∈ R, u ∈ R, f p : R np → R np , g p :
R np → R np and h p : R np × R → R are locally Lipschitz functions. We assume that the origin is an equilibrium point for [START_REF] Andrieu | Global asymptotic stabilization for nonminimum phase nonlinear systems admitting a strict normal form[END_REF].

A. Stabilizability assumption

Consider the following nonlinear hybrid system H := (F, F, J , G):

H ẋ = F (x), x ∈ F x + = G(x), x ∈ J (2)
where F ⊂ R n and J ⊂ R n are called respectively the flow and jump sets associated to the continuous and discrete dynamics given respectively by F : R n → R n and G : R n → R n . Given a closed set A and denoting

satisfies |x(t, j)| A ≤ α(|x(0, 0)| A ), for all (t, j) ∈ dom(x);
• the set A is uniformly globally attractive (UGA) for ( 2) if for each ε > 0 and r > 0, there exists T > 0 such that for any solution x to [START_REF] Andrieu | A unifying point of view on output feedback designs for global asymptotic stabilization[END_REF]

such that |x(0, 0)| A ≤ r is complete and satisfies |x(t, j)| A ≤ ε, ∀(t, j) ∈ dom(x), t + j ≥ T ;
• the set A is uniformly globally asymptotically stable (UGAS) for [START_REF] Andrieu | A unifying point of view on output feedback designs for global asymptotic stabilization[END_REF] if it is both uniformly globally stable and uniformly globally attractive;

• given a set Γ, the A is uniformly semi-globally asymptotically stable with respect to Γ for (2) if A is uniformly globally asymptotically stable, by considering only initial conditions in Γ.

Inspired by [START_REF] Sontag | Stability and stabilization: discontinuities and the effect of disturbances[END_REF] and [START_REF] Prieur | Robust optimal stabilization of the Brockett integrator via a hybrid feedback[END_REF], we assume that the origin of (1) can be stabilized by a hybrid state feedback law.

Assumption 1 (Stabilizability).

There exists a hybrid controller defined by (F c , J c , f c , g c , θ c ), where F c and J c are closed sets, F c ∪ J c = R np+nc , g c : R np+nc → R nc , f c : R np+nc → R nc and θ c : R np+nc → R are continuous functions, such that the origin of the system:

ẋp = f p (x p ) + g p (x p )θ c (x p , x c ) ẋc = f c (x p , x c ) (x p , x c ) ∈ F c (3a) x + p = x p x + c = g c (x p , x c ) (x p , x c ) ∈ J c (3b)
is uniformly globally asymptotically stable.

B. Observability assumption

Following [START_REF] Astolfi | Output feedback stabilization with an observer in the original coordinates for nonlinear systems[END_REF], we define recursively the following functions for all

(x p , ṽ) ∈ R np × R np , ϕ 0 (x p , v 0 ) = h p (x p , v 0 ), ϕ i (x p , v 0 , . . . , v i ) = ∂ϕi-1 ∂xp f p (x p , v 0 ) + i-1 k=0 ∂ϕi-1 ∂v k v k+1
. for all i = 1, . . . , n p -1, where the notation ṽ = (v 0 , . . . , v np-1 ) has been used. We consider also the function

φ c : R np × R np → R np defined as φ c (x p , ṽ) =    ϕ 0 (x p , v 0 ) . . . ϕ np-1 (x p , v 0 , . . . , v np-1 )   
Given a smooth function u : [0, ∞) → R, we denote, for all x p ∈ R np and t ≥ 0,

φ(x p , t; u) = φ c (x p , u(t), . . . , u (np-1) (t)) .
where, for each k ∈ N, u (k) denotes the k-th derivative of the function u. We can now state the observability assumption employed in the first main result:

Assumption 2 (Observability for a suitable controller).

There exists a smooth controller ū : [0, ∞) → R, such that: (i) For all t ≥ 0, the function x → φ(x, t; ū) is injective on R np ; (ii) For all t ≥ 0 and for all x p ∈ R np , the matrix ∂φ(xp,t;ū) ∂xp is invertible.

Remark 1. With this property, and given a compact set of initial condition, it is possible to design a finite time highgain observer. Indeed, if (1) satisfies Assumption 2, then for each t ≥ 0, the function φ(•, t; ū) is a diffeomorphism from R np to R np . Inspired by [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF], by setting u = ū(t) and Z = φ(x p , t; ū), the system (1) can be rewritten as follows:

Ż = SZ + Bδ(Z, t) (4)
where

Z =      z p1 z p2 . . . z pn p      , S =         0 1 0 . . . 0 0 0 1 . . . 0 . . . . . . . . . . . . . . . 0 0 . . . . . . 1 0 . . . . . . . . . 0         , B = 0, • • • , 0, 1 T and δ : R np × [0, ∞) → R is a nonlinear
continuously differentiable function. Following [START_REF] Andrieu | High-gain observer with uniform in the initial condition finite time convergence[END_REF], for all compact set of initial conditions and for all T > 0 it is possible to design an observer which converges in time T .

Under Assumptions 1 and 2, we are interested in the design of a hybrid output feedback law that makes the origin of the system (1) semi-globally asymptotically stable by coupling the state feedback considered in Assumption 1 and a finite-time high-gain observer that will be obtained from Assumption 2.

Coupling Assumptions 1 and 2 together with a temporal timer and a high-gain strategy as in [START_REF] Gauthier | Deterministic Observation Theory and Applications[END_REF] yields the following hybrid system:

       ẋp = f p (x p ) + g p (x p )U(x p , x c , σ) ẋc = f c (x p , x c ) ẋp = ψ p (x p , x c , σ, U, y) σ = s(σ) (x p , x c , σ) ∈ F c × [0, T ] (5a)
         x + p = x p x + c = g c (x p , x c ) x+ p = xp σ + = σ (x p , x c , σ) ∈ J c × [T, +∞) (5b) where U = U(x p , x c , σ) is such that U = ū(σ) if σ ≤ T and u = θ c (x p , x c ) if σ > T ,
ψ p is a continuous vector field to be designed, σ stands for the timer state and s

: [0, ∞) → [0, ∞) is a smooth function such that s(σ) = 1 for σ ≤ T and s(2T ) = 0.
C. First main result Theorem 1. (Attractivity for appropriate initial timer states) Under Assumptions 1 and 2, for all compact sets Γ ⊂ R np , there exist a positive real number T and a function ψ p such that by focusing on solutions satisfying σ(0) = 0, the origin of ( 5) is attractive with a basin of attraction containing Γ × {0}. More precisely, for all x ♯ ∈ Γ, the solutions of (5) starting from (x p (0), x c (0), xp (0), σ(0)) = (x ♯ , 0, 0, 0)

converge to {0} × [0, 2T ].
The proof of this result is omitted due to space limitation. Remark 2. Let us emphasize that the property written in Theorem 1 is not the uniform semi-global asymptotic stability since only solutions of ( 5) with initial conditions satisfying σ(0) = 0 are considered, and since we were not able to state the stability property. Remark 3. By exploiting the high-gain observer and the timer, a convergence of the error observation is obtained in finite time. Roughly speaking, the designed output feedback controller first observes the state and then stabilizes it. Therefore this approach allows to exclude the Zeno solutions, and impose that the solutions follow a continuous time dynamics during T units of time, so that the observer is able to converge.

III. SECOND SET OF ASSUMPTIONS: THE ASYMPTOTIC CASE

A. Stabilizability Assumption

In this section, we consider an approach similar to the one of [START_REF] Teel | Global stabilizability implies semi-global stabilizability by output feedback[END_REF]. Indeed, based on some hybrid stabilizability assumption and observability assumption, we obtain semiglobal asymptotic stabilizability of the origin. Note however that due to some particular effects of hybrid dynamics (for instance Zeno solutions) we require a persistent flow condition on the stabilizing state feedback.

Assumption 3 (Persistent flow stabilizability).

There exist a hybrid controller (F c , J c , f c , g c , θ c ), a real λ in (0, 1] such that the set {0}×[0, 1] in R np+nc ×[0, 1] is uniformly globally asymptotically stable for the following system

     ẋp = f p (x p ) + g p (x p )θ c (x p , x c ) ẋc = f c (x p , x c ) σ = 1 -σ (x p , x c , σ) ∈ F c × [0, +∞), (6)
     x + p = x p x + c = g c (x p , x c ) σ + = 0 (x p , x c , σ) ∈ J c × [λ, +∞). (7)
Remark 4. This assumption is related to the notion of persistent flow, which charaterizes that a small dwell time λ > 0 should exist between two successive jumps.

B. Observability Assumption

Moreover, in this context, we need an observability assumption uniform in the control input.

Assumption 4 (Complete Uniform Observability). System (1) is completely uniformly observable, that is (i) For all ṽ in R np , the function x p → φ c (x p , ṽ) is injective on R np ; (ii) The matrix ∂φc(xp,ṽ) ∂xp

is invertible for all (x p , ṽ) ∈ R np × R np .

C. Second main result

With the previous assumptions, it holds the following: Theorem 2 (Semi-global asymptotic stability). Assume there exists a function γ such that, for all (x p , x c ) in J c , we have:

|g c (x p , x c )| ≤ γ(|x c |) . ( 8
)
Under Asumptions 3 and 4, the origin of system ( 1) is uniformly semi-globally asymptotically stabilizable by a hybrid dynamic output feedback. More precisely, for all compact sets Γ contained in R np , there exist a C 1 function Ψ p : R np × R × R → R and a positive real number c x such that the set {0} × [0, 1] in R 2np+nc × [0, 1] is uniformly asymptotically stable for the system

               ẋp = f p (x p ) + g p (x p )u ẋp = Ψ p (x p , y, u) ẋc = f c (x p , x c ) σ = 1 -σ y = h p (x p , u) , u = θ c (x p , x c ) (9)
(x p , x c , σ) ∈ F c × [0, +∞)          x + p = x p x+ p = xp x + c = g c (x p , x c ) σ + = 0 (x p , x c , σ) ∈ J c × [λ, +∞). (10)
where 1 xp = Sat cx (x p ) , [START_REF] Liu | Eigenstructure Assignment for Control System Design[END_REF] with basin of attraction containing

Γ × {0} × R + (which is a subset of R 2np+nc × R + ).
Remark 5. Note that the timer avoids Zeno solutions. See e.g [START_REF] Johansson | On the regularization of Zeno hybrid automata[END_REF] and [START_REF] Goebel | Solutions to hybrid inclusions via set and graphical convergence with stability theory applications[END_REF].

The main steps of the proof can be found in Appendix A.

D. Example: integrators chain

We want to illustrate the Theorem 2 by applying it on the system ẋp1 = x p2 , ẋp2 = u + x 2 p2 and y p = x p1 , with the set Γ = {x p ∈ R 2 : x p1 ≤ 50, x p2 ≤ 50}. Because this system has the same structure as (4), Assumption 4 holds.

We design a global controller:

u g = -x p1 -x 2 p2 -k 1 x p2 - k 2 (x p2 + k 1 x p1 ),
where k 1 = 1 and k 2 = 2. We focus on the linearization around the origin and design a more efficient controller (u l = -Kx p , where K is a matrix with appropriate dimensions and computed with a pole placement method, as described in e.g. [START_REF] Liu | Eigenstructure Assignment for Control System Design[END_REF]). By uniting these two controllers thanks to a discrete variable x c ∈ [0, 1] (see e.g [START_REF] Prieur | Uniting local and global controllers with robustness to vanishing noise[END_REF]) and setting λ = 0.01, we get a hybrid controller that satisfies Assumption 3. Let us numerically compute the solutions whose initial conditions are x # p = (20, 10) and x # c = 0. The phase portrait of the plant state and the time-evolution of x c -variable are respectively given in Figures 1 and2. The stabilization is illustrated by Fig. 1. It is checked on Fig. 2 that there is no Zeno solution.

IV. CONCLUSION

In this paper, two techniques are proposed to design a hybrid output feedback controller. Both methods combine an observer and a hybrid stabilizing state feedback law. The first case considers a finite-time converging observer, whereas the second method suggests to use a high-gain controller asymptotically converging to the state. The asymptotic design is illustrated on a nonlinear control system. This work lets different questions open. The first one is the relaxation of the persistent flow condition considered in Assumption 3 used in the second main result. This may be done by applying robustness arguments. Moreover the applications of these results to other nonlinear control systems are under actual investigation.

APPENDIX

A. Proof of Theorem 2 1) Lower bounded existence time in a compact set:

This subsection is devoted to the tunning of the saturating parameter c x and to the construction of some sets. Indeed the construction of the observer is based on the construction of a specific set that needs to be selected in an appropriate way taking into account jumps that may occur due to the hybrid dynamics. Along this subsection, we consider the system defined by, for all (x p , x c , σ) in

(F c ×R + )∪(J c ×[λ, +∞)),      ẋp = f p (x p ) + g p (x p )θ c (ω, x c ) ẋc = f c (ω, x c ) σ = 1 -σ (ω, x c , σ) ∈ F c × [0, +∞), (12)
     x + p = x p x + c = g c (ω, x c ) σ + = 0 (ω, x c , σ) ∈ J c × [λ, +∞), (13)
where ω is an external time function in L ∞ loc ([0, +∞); R np ). Note that in the particular case where ω = xp , which is defined in [START_REF] Liu | Eigenstructure Assignment for Control System Design[END_REF], the solution to system (12)-( 13) (without the dynamics xp ) is also solution to system (9)- [START_REF] Johansson | On the regularization of Zeno hybrid automata[END_REF].

To define the set of interest, consider V : R np+nc+1 → R + a continuously differentiable positive definite and proper Lyapunov function associated to the hybrid stabilizing state feedback of Assumption 3. Hence a real α in (0, 1) exists such that V verifies:

V (x p , x c , σ) = 0 ⇒ x p = 0 , x c = 0 , σ ∈ [0, +∞) and ∂V ∂x (x)F (x) ≤ -V (x), ∀x ∈ F c × [0, +∞) V (G(x)) -V (x) ≤ -αV (x), ∀x ∈ J c × [λ, +∞) (14)
where we used the compact notation x = (x p , x c , σ) and

F (x) =   f p (x p ) + g p (x p )θ c (x p , x c ) f c (x p , x c ) 1 -σ   , G(x) =   x p g c (x p , x c ) 0   .
Let m = max xp∈Γ,σ∈[0,λ] V (x p , 0, σ) and consider the following compact sets:

D m := {x ∈ R np+nc+1 , V (x) ≤ m}, D l1 := {x ∈ R np+nc+1 , V (x) ≤ l 1 }, (15)
where l 1 > m. Consider also the set

D + l1 = {(x p , x c , 0) ∈ R np+nc+1 , ∃(x - c , σ -) ∈ R nc+1 , |x c | ≤ γ(|x - c |), (x p , x - c , σ -) ∈ D l1 }. We finally define the two last sets D l2 = {x ∈ R np+nc+1 , V (x) ≤ l 2 } and D n := {x ∈ R np+nc+1 , V (x) ≤ n} where l 2 > l 1 is such that D + l1 ⊂ D l2
and where n > l 2 . Let us now establish the following property for solutions of system ( 12)-( 13) initiated from D m . Lemma 1. (Lower bounded existence time of solution in D n ) Let c x = max (xp,xc,σ)∈Dn {|x p |}. There exists T min > 0 such that for all ω in L ∞ loc ([0, +∞); R np ) with ω(t) ≤ c x for all t in [0, T min ], and all x # := (x # p , x # c , σ # ) in D m , all solutions x(•, •) of ( 12)-( 13) with x(0, 0) = x # and all (t, j) in dom(x) then x(t, j) ∈ D n for all t ≤ T min .

Proof. Let V the positive real numbers defined by

V = max x∈Dn,|ω|≤cx ∂V ∂x (x)F ω (x, ω)
where

F ω (x, ω) =   f p (x p ) + g p (x p )θ c (ω, x c ) f c (ω, x c ) 1 -σ   .
In the remaining part of the proof, we show that Lemma 1 holds with T min chosen as any positive real number satisfying

T min < inf -ln(1 -λ), l 1 -m V , l 2 -n V .
Let x # be in D m and let x be a solution of system ( 12)-( 13) whose initial conditions are x # . For all (t, j) in dom(x), we denote V(t, j) = V (x(t, j)).

Let (t, j) in dom(x) such that t ≤ T min . To prove the lemma, we need to show that x(t, j) is in D n . First of all, we show that j ≤ 1. Indeed, assume j ≥ 2. This implies that there exist t 0 and t 1 such that 0 ≤ t 0 < t 1 ≤ t such that (t 0 , 0), (t 0 , 1), (t 1 , 1), (t 1 , 2) are in dom(x). Note that σ(t 0 , 1) = 0 and σ(t 1 , 1) = λ. Moreover, for all s in [t 0 , t 1 ], (s, 1) is in dom(x) and that ∂σ ∂t (s, 1) = 1σ(s, 1). Hence, t ≥ t 1t 0 = -ln(1λ) ≥ T min . Hence this is impossible, and therefore j ≤ 1.

So two cases may be distinguished. j = 0 This implies that s ∈ [0, t] → x(s, 0) is a continuous mapping with x(0, 0) in D n ⊂ D l1

∂V ∂t (s, 0) = ∂V ∂x (x(s, 0))F ω (x(s, 0), ω(s)) ≤ V .
This gives

V(t * , 0) ≤ V t * + V(0, 0) ≤ V T min + m < l 1 .
Hence x(t * , 0) is in the interior of D l1 . It yields that there exists ε > 0 such that x(t * + ε, 0) is in the interior of D l1 which contradicts the fact that t * is an extremum. j = 1 This implies that there exists t 0 in [0, t] such that (t 0 , 0) and (t 0 , 1) is in dom(x). Following the first case study, it is possible to show that x(t 0 , 0) is in D l1 . Moreover, we have x p (t 0 , 1) = x p (t 0 , 0) and, due to [START_REF] Goebel | Hybrid Dynamical Systems[END_REF] x c (t 0 , 1) = g(x c (t 0 , 0), w(t 0 )) ≤ γ(x c (t 0 , 0)) .

This implies that x(t 0 , 1) ∈ D + l1 . Note that [t 0 , t] → x(s, 1) is a continuous mapping with x(t 0 , 1) in D l2 ⊂ D n . As in the previous case, we define t * , the largest time in [t 0 , t] such that x(s, 1) is in D n (i.e. t * = max s∈[t0,t],x(ℓ,1)∈Dn,∀ℓ∈[t0,s] {s}). Note that if t * = t then this implies that x(t, 1) is in D n , hence the result. Assume t * < t. This implies that, for all s in [t

0 , t * ], it holds ∂V ∂t (s, 1) = ∂V ∂x (x(s, 1))F ω (x(s, 1), ω(s)) ≤ V .
This implies

V(t * , 1) ≤ V t * + V(t 0 , 1) ≤ V T min + l 2 < n .
Hence x(t * , 1) is in the interior of D n and following the previous case, we get a contradiction. This concludes the proof of Lemma 1.

2) Robustness margin in the compact set:

Lemma 2. (Robustness margin) Under Assumption 3, let V be a function which satisfies [START_REF] Sontag | Stability and stabilization: discontinuities and the effect of disturbances[END_REF]. Let D n be defined in the previous section. There exist a class K function ρ and a positive real number ε r such that, for all e ∈ R np , with |e| ≤ ε r and all (x p , x c , σ) in D n the following relations hold. This concludes the proof of Lemma 2.

• If (x p + e, x c , σ) ∈ F c × [0, +∞) ∂V ∂x (x)   f p (x p ) f c (x p + e, x c ) 1 -σ   + ∂V ∂x (
3) Tuning the high-gain observer: In this subsection we design a high-gain observer for the system (1) in the form ẋp = Ψ p (x p , y, u) .

(18)

The function Ψ p is selected to estimate the state x p for initial conditions in the set Moreover, the observer has to be designed such that the estimation error is smaller than the stability margin of the

D np = {x p ∈ R np , ∃(x c , σ) ∈ R nc × R + , (x p , x c ) ∈ D n } .

  F c and J c are computed from the local version of the basin of attraction. Note that we can find a function γ such that |g c (x p , x c )| := |1x c | ≤ γ(|x c |) We tune the observer xp = f p (x p ) + g p (x p )θ c (x p , x c ) + k(y p , ŷp ) where: k(y p , ŷp ) := -280 -2 * 280 2 (y p -ŷp ).

Fig. 1 .

 1 Fig. 1. Phase portrait of the plant state xp. The two main jumps of the xc-variable are marked by a circle 1 Given a positive real number c, Satc : R n → R n is the saturating vector function defined by Satc(0) = 0 and Satc(x) := x min n 1, c |x| o , ∀x = 0.

Fig. 2 .

 2 Fig. 2. Time-evolution of the xc variable. The two main jumps are marked by a circle.

2 V 2 V 2

 222 x p + e, x c ) ≤ -1 (x p , x c ) + ρ(|e|) (16)• If (x p + e, x c , σ) ∈ J c × [λ, +∞) V (x p , g c (x p + e, x c ), 0) -V (x p , x c , σ) p , x c , σ) + ρ(|e|)Proof. Employing (14), the set D n being compact and the functions (F, G, V ) being continuous, there exists ε r such that for all |e| ≤ ε r and (x p , x c , σ) in D n then:• If (x p + e, x c , σ) in F c × [0, +∞), ∂V ∂x (x)F (x) ≤ -1 (x) , • If (x p + e, x c , σ) in J c × [λ, +∞) V (G(x)) -V (x) ≤ -1 2 αV (x), ∀x ∈ J c × [0, λ] .Consider now the strictly increasing function ρ : [0, ε r ) → [0, +∞) as the function ρ(s) ≥ max max (xp+e,xc,σ)∈Dn∩Fc×R+,|e|≤s ν 1 (x, e) , max (xp+e,xc,σ)∈Dn∩Jc×R+,|e|≤sν 2 (x, e) x p ) f c (x p + e, x c ) c (x p + e, x c )θ c (x p , x c )) (x, e) = V (x p , g c (x p + e, x c ), 0) -V (x p , g c (x p , x c ), 0).

  and the control input u is a measurable function such that |u(t)| ≤ ū with ū = max xp∈Dnp,(xc,σ)∈Dnc θ c (x p , x c ) whereD nc = {(x c , σ) ∈ R nc+1 , ∃x p ∈ R np , (x p , x c ) ∈ D n }.

  . Hence we can define t * , the largest time in [0, t] such that x(s, 0)is in D l1 (i.e. t * = max s∈[0,t],x(ℓ,0)∈D l 1 ,∀ℓ∈[0,s] {s}).Note that if t * = t then this implies that x(t, 0) is in D l1 , hence the result. Assume t * < t. This implies that for all s in [0, t * ] we have

This work has been partially supported by the ANR project LimICoS contract number 12 BS03 005 01.

This work has been partially supported by CONICYT grant ACT-1106.

The proof of this result is omitted due to space limitation. 4) Proof of Theorem 2: Consider the positive real number c x obtained in Lemma 1 and the function k obtained in Lemma 3. In the first part of the proof, we show attractivity of the set {0}×[0, 1] in R 2np+nc ×[0, 1] along the solutions to system (9)-( 10) whose initial conditions are in Γ×{0}×R + .

Let

and defined on its time domain denoted dom # . Note that the system ( 9)-( 10) can be rewritten as the hybrid system ( 12)-( 13) with ω = Sat cx (x p ) and xp is given with the observer (18).

With Lemma 1 and with the σ dynamics (persistent flow), we know that there exists j 0 such that (T min , j 0 ) is in dom # and for all (t, j) in dom # with t ≤ T min then (x p (t, j), x c (t, j), σ(t, j)) is in D n . Note moreover that this implies that for all (t, j) in dom # with t ≤ T min then the control input satisfies |u(t, j)| ≤ ū. With Lemma 3, we get that the observer state is well defined for all (t, j) in dom # with t ≤ T min . Moreover, for all (t, j) in dom # with t ≥ T min , |x p (t, j) -xp (t, j)| ≤ c e .

We can now show that for all (t, j) in dom # , x(t, j) is in D n . We will argue by contradiction to prove this assertion. By assuming that it is false, two cases may occur.

1) The solution escapes D n when flowing. Hence, there exists (t 0 , j 0 ) in dom # such that (x p , x c , σ)(t 0 , j 0 ) is in D n and for all ε > 0 there exists δ < ε such that (t 0 + δ, j 0 ) is in dom # and (x p , x c , σ)(t 0 + δ, j 0 ) is not in D n . Note that this implies that (x p , x c )(t 0 , j 0 ) is at the boundary of D n . Consequently, this implies V(t 0 , j 0 ) = n. Note moreover, that keeping in mind that |x p (t 0 , j 0 )-xp (t 0 , j 0 )| ≤ c e ≤ ε r we get employing Lemma 2 ∂V ∂t (t 0 , j 0 ) ≤ -1 2 V(t 0 , j 0 )+ρ(c e ) ≤ -n 6 . This implies that the function s → V(t 0 + s, j 0 ) is strictly decreasing. It contradicts the existence of small ε.

2) The solution escapes D n when jumping. Hence, there exists (t 0 , j 0 ) in dom # such that (x p , x c , σ)(t 0 , j 0 ) is in D n and (x p , x c , σ)(t 0 , j 0 + 1) is not in

)n < n. This is a contradiction with the escape of the solution from D n . Consequently, for all (t, j) in dom # , we have x(t, j) is in D n . Note that the timer forces the t component of the time domain to be unbounded. Hence, thanks to the Lemma 3, lim t+j→+∞ |x p (t, j)x p (t, j)| = 0. We get the result employing the triangular structure of the system with the ISS property in D n (i.e. Lemma 2).

To conclude the proof, let us prove the stability property. From the same argument, the set

is invariant along solutions for sufficiently small v 0 , is an open set and contains {0} × [0, 1] in R 2np+nc × [0, 1]. Note moreover that for all neighborhoods of {0} × [0, 1] in R 2np+nc × [0, 1] there exists v 0 sufficiently small such that S v0 is included in it. This allows to conclude stability of the set R 2np+nc × [0, 1], and the proof of Theorem 2.


Najla Lassoued

Mohamed Naceur Khelil

Saloua Rejeb

Essaïd Bilal
email: bilal@emse.fr

Mohamed Chaouachi

Mohamed Nejib Rejeb

Issam Guénole-Bilal

Essaid Bilal

Mohamed Néjib Rejeb

Issam Guenolé-Bilal

Transfert heavy metal sewage sludge as fertilizer from soil to ray grass

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Introduction

The contamination of soil by heavy metals is a major environmental problem for two reasons. Firstly, it presents an ecotoxicology risk for food chains because of the strongly toxic properties of these elements for all the human beings. Secondly, this contamination can have very long-term effects because elements in tracks present generally a strong chemical and physical affinity for the solid matrix of soil and have a residence time in soil very long (Echevarria and More, 2006). The assessment of risks associated to the soil pollution requires, not only knowledge of the total content in metal, but also the content in metal in the various constituent compartments of the soil (Jeans, 2007). The heavy metals are unwanted constituents of the urban sewage sludge. As some of them are potentially toxic and present no agronomic interest, their presences generate a certain concern, perfectly understandable when it is about spread this waste on soils intended to produce food for the Man or the animals (Kiss, 2006). The agricultural use of the residual sludge allows the recycling of precious components such as the organic matter and many nourishing elements of the plant [START_REF] Logan | Physical characteristics of alkaline stabilized sewage sludge (Nvitro soil) and their effects on soil properties[END_REF]. It aims at arranging natural resources and at avoiding any wasting of organic matter due to the incineration or to the burying in the discharges [START_REF] Lambkin | The importance of precision in sampling sludges, biowastes and treated soils[END_REF], residual sludge can replace or reduce the use of expensive fertilizers. Sludge is a product susceptible to supply in the cultures of the nourishing elements useful for their growth and for their development. Furthermore, certain sewage sludge (composted or limed) can play the role of amendment by allowing maintaining, to improve the structure of the soil, its biological activity or still control soil its acidity. Nevertheless, from an agronomic point of view, and if the farmer respects the agronomic advice, the economy on the purchases of fertilizers can be considerable. The most popular elements are mainly the phosphor and the nitrogen. Savings are easily estimable for the elements, which are fixing by the soil (phosphor or lime). For the nitrogen, savings are more difficult to amount because the availability depends on weather conditions, which determine the mineralization (passage of the organic shape towards the mineral shape) [START_REF] Atlan | Les boues d'épuration et leurs perspectives de gestion en île de France. In : agriculture, environnement, ruralité[END_REF]. The factors bound to the physiological behavior of plants also play an important role in the bioavailability of heavy metals. The influence of the organ of the plant (heavy metals are essentially absorbed in roots and much less in edible parts), the species, the variety (plants with fast growth are more accumulate) and the phenomena of antagonism and synergy between the various mineral elements of sludge [START_REF] Juste | Les micro-polluants métalliques dans les boues résiduaires des stations d'épuration urbaines[END_REF].

Material and methods

The soil of the try is sandy-clayey, an alkaline pH (7.50 -8.31), an electric conductivity from 0.42 to 0.89 ms/cm what indicates a low of salinity bound to the sandy nature of the soil and a low content in total limestone (5 %) and in organic matter (0.39 % to 1.98 %). Besides, we note a better composition in organic matter, N and P in the superficial horizon (0-20 cm) compared with the other horizons (20-40) and (40-60) (Table 1). We used sewage sludge of the KORBA water-treatment plant. The sludge stemming from this station have sudden a stabilization in aerobe followed by a drying on beds. The analysis of both chaps of sludge (liquid and dry) shows that they are very different by their chemical compositions. The pH is acid and their electric conductivities translate an important salinity.

Concerning heavy metals, the Fe and Zn are the most represented. The average contents found in dry sludge are organizing according to the following sequence:

Fe >>Mn> Zn > Cu >Pb> Ni > Cd > Co.
The average contents of heavy metals brought by sludge are representing on the Table 2. Provide sufficient detail to allow the work to be reproducing, with details of supplier and catalogue number when appropriate. Methods already published should be indicate by a reference: only relevant modifications should be described.

The plant material used in our experiment is the English Ray Grass (Lolium perenne). In this study, we made five successive cuttings at the bolting stage, at 3 cm of the surface of the soil, at the interval of one month.

Five treatments are realizing for this experiment. Besides the treatment control soil, four treatments including two types of sludge, a dry sludge and a liquid sludge (thickened sludge) brought each there two doses (5 and 10t/ha) were applied in four repetitions:

Treatment T: Control soil without contributions of sludge. Treatment 5 BS: Contribution of dry sludge 5 t/ha, equivalent of 0.5kg/m ² . Treatment 10 BS: Contribution of dry sludge 10 t/ha, equivalent of 1 kg/m ² . Treatment 10 BL: Contribution of liquid sludge 10 t/ha, equivalent of 1 l/m ² . Treatment 5 BL: Contribution of liquid sludge 5 t/ha, equivalent of 0.5 l/m ² .

Results and discussion

For the same applied dose, dry matter production was higher with dry mud (Figure 1). Slurries with a leaching loss of an amount of nitrogen provided by the sludge are likely. The contribution of 5t/ha liquid sludge's had no significant effect on the production of dry material, while the dose 10 t/ha is accompanied by a significant increase of the production of the dry material compared with the control soil without contribution of sludge. This increase of the production of material dry registered with treatments sludge is probably the result of the contribution by sludge of essential nourishing elements for the growth of the plant, such as the nitrogen and the phosphor. For the same applied dose, the production of dry material is more raising with the dry sludge. With liquid sludge, a loss by lixiviation of a quantity of nitrogen brought by sludge is probable. The comparison between both types of sludge shows that the contribution of 10t/ha liquid sludge is equivalent to the contribution of 5t/ha dry sludge from point of view effect on dry matter production. The contribution of liquid sludge has not positive effect on the production of roots (Table 3). However, the weight of roots is increase with dry sludge. We also note a significant effect of the dose of dry sludge on the weight roots. The concentration of diverse heavy metals measured in the air parts of the Ray Grass expressed this evolution. We note absence of effect measures of sludge on the content in Zn of the plant (Figure 2). Besides, the comparison between cuttings shows a significant increase of the content in Zn obtained with cuttings four and five as well for the dry sludge as the liquid sludge. This increase is linking to a competition between Zn and Cd. The Cd contents of the dose of sludge have not any effect of the air part of the Ray Grass. For dry sludge, the contents tend to increase with the contribution of sludge 10 t/ha dose for the first three cuttings. We observed no effect of the dose or the type of sludge on the content in Cu of the air part of the Ray Grass and of the dose or the type of sludge on the content in Cu of the air part of the Ray Grass.

The average Pb contents oscillate around 8 µg/g. These values situated below threshold of toxicity of vegetables (30 µg/g). The sampling of roots realized further to the last cutting of the ray fat. The contribution of sludge increases significantly the content in Zn, Cd and Pb in roots (Table 4) compared with the control soil sample without any contribution. However, the Pb and Cd a significant effect of the dose indicated in roots as well for liquid sludge as for dry sludge. Accumulation of heavy metals in the soil: The analysis of the soil realized after the last cutting of the Ray Grass (Figure 3), shows a significant decrease of the content of the Zn in the superficial horizon of the soil (0-20 cm) for both doses and both types of brought sludge. The analysis of the soil realized after the last cutting of the Ray Grass (Figure 3), shows a significant decrease of the content of the Zn in the superficial horizon of the soil (0-20 cm) for both doses and both types of brought sludge. The analysis of the soil realized after the last cutting of the Ray Grass (Figure 3), shows a significant decrease of the content of the Zn in the superficial horizon of the soil (0-20 cm) for both doses and both types of brought sludge.

As for the deeper horizon (20-40 and 40-60 cm), no significant effect of the dose of dry sludge was noticed on the content of Zn in the soil between the beginning of the try and after the last cutting. However, for the liquid sludge in particular for the dose 10t/ha the content of Zn tends to decrease between the beginning and the end of the try. For the Pb, we note a significant decrease of the content in Pb of the soil between the beginning and the end of the try in the superficial horizon. This decrease noticed also in the deeper soil horizon (20-40 and 40-60 cm). Accumulation of heavy metals in the water of drainage -In this part, we use the analysis of the water leaving the lysimeter boxes for the different treatments in order to explore the possibility of migration of tracks elements to the water. The analysis of drainage water showed for various sludge treatments (Table 4) that the levels are not significantly different from control soil treatment. This suggests that the Cd, Zn, Cu and Pb retained rather in the soil.

The spreading of liquid and dry sludge resulted in an increase in the production of biomass Ray Grass. This increase is a function of increasing sludge doses applied. This the positive effect of sludge on yield found in other cultures. Indeed, many authors [MOREL and GUCKERT, 1979;[START_REF] Larry | Interaction chimique pour l'élévation de la biodisponibilité des métaux en traces du sol[END_REF][START_REF] Rejeb | Incidence de l'apport de boues résiduaires urbaines sur la composition minérales et la productivité de quelques espèces cultivées en Tunisie[END_REF][START_REF] Rejeb | Effet des boues urbaines sur la production dela pomme de terre[END_REF]] reported a significant increase in yields of various crops such as sorghum, corn, chili, potato. Our tests also showed a better performance with dry and liquid sludge with the year of application. This could also be the result of a share of a probable loss of nitrate made in slurries and other wealth of the sludge brought dry matter. In fact, the dry matter content from the sludge is variable about 5% and 67% in the slurry and dry sludge respectively. Moreover, the contribution of sludge caused heavy metals enrichment soil and more particularly to the horizon (0-20 cm). However, this increase is not significant except for Cd content, which has indeed increased, significantly in the presence of sludge. In the lower horizon soil (20-40 and 40-60 cm), the Pb, Zn, and Cu contents remained equivalent to the control soil. Regarding the accumulation of heavy metals in the plant, our results showed that the contents of Cd, Pb, Zn, and Cu in the aerial part of the plant does not vary with the addition of 5 and 10 t/ha of dry or liquid sludge. These results are in agreement with those of [START_REF] Berthet | Etude des voles de transfert du plomb del'environnement aux végétaux cultivées;application a l'utilisation agricole de boues desstations d'épuration[END_REF] who studied the evolution of the levels of Pb in tissues of Ray Grass, subject to three types of treatment (control soil without mineral fertilizer and sludge amendment). They showed that there is no significant difference between the mean levels of Pb in different crops Ray Grass, from different treatments of the same culture series, performed on the same date. They found the same effect for Zn [START_REF] Berthet | Etude expérimentale de la relation entrela forme physicochimique du Zn et sa disponibilité à l'égard de végétaux cultivés. Application à l'utilisation agricole de boues de stations d'épuration[END_REF]. By cons, other studies have shown that Zn absorbed easily by plants [START_REF] Juste | Effets d'applications massives de boues de stations d'épuration urbaines en monoculture de maïs : actions sur le rendement et la composition des plantes et sur quelques caractéristiques du sol[END_REF][START_REF] Chang | Accumulation of Cd and Zn in barley grown on sludge-treated soils: a longterm field study[END_REF]REJEB, 1999] and that this absorption could Pb to a significant increase in the content in the seed [START_REF] Juste | Les micro-polluants métalliques dans les boues résiduaires des stations d'épuration urbaines[END_REF]SOLDA, 1977, CHANG et al., 1983].

For Cu, no dose sludge effects observed in the Ray Grass. According to our experiments, the concentrations of Cu ranged between 4 and 15 mg/kg in the leaves of Ray Grass. These contents are by [REUTER and ROBINSON, 1997] for optimal growth. According to the same authors, the toxicity threshold reached in Cu from a Cu concentration of 15 to 30 mg/kg DM in the leaves of crops. However, the Cu content in the plant can reach a critical concentration variable, depending on the plant species from which appear symptoms of phytotoxicity [START_REF] Marschner | Mineral Nutrition of Higher Plants[END_REF]. A slight accumulation of Zn in the roots of Ray Grass observed for both types of sludge. The Pb content is increasing with 10t/ha doses of liquid and dry sludge. The Cd increased in roots with the addition of sludge and function of the applied dose.

Various studies have shown an accumulation of Cd into the roots of certain plants [START_REF] Carlson | The role of AVS in determining Cd bioavailability and toxicity in freshwater sediments[END_REF], or in the leaves of other species [START_REF] Kim | Relative concentrations of Cd and Zn in tissue selected food plants grown on sludge-treated soils[END_REF].

The flow heavy metals made by urban sludge are very low on the scale of a year and their accumulation does not represent a risk to the environment if the prescribed conditions are met [START_REF] Maisonnave | Impact des épandages de boues sur la teneur en éléments traces métalliques dans un sol et un végétal[END_REF]. However, the accumulation of heavy metals depends on the plant species. In addition, the response of plants to metals in sludge also varies depending on the elements. In early try, Pb testing shows that there is no significant difference in the surface horizon soil (0-20 cm) compared to control soil any dose given 5t/ha or 10 t/ha for both types of sludge. Yet at the end of the trial, the Pb contents are a significant decrease for all treatments with a more pronounced decrease Pb contents in horizon soil (20-40 cm) receiving liquid or dry sludge.

From application of sludge containing heavy metals results a very significant increase heavy metals contents in surface horizon soil [START_REF] Legret | Migration et spéciation des métaux lourds dans un sol soumis à des épandages de boues de station d'épuration à très forte charge en Cd et Ni[END_REF] and in particular the horizon (0-20 cm). The displacement of Pb in soil stopped at horizon (20-40cm). The Pb is associated with acid phases-soluble and reducible; it not moved into horizon soil 20-40 cm. In case of application of dry sludge and slurry, the Cu contents in the horizons (0-20 cm) and (20-40 cm) show a significant decrease in the soil at the end of Culture. However, the horizon soil (40-60 cm) has not the chemical effect. The mobility of Cu in soil is defining as the ability of this element to transfer to the horizon soil where it is less strongly retaining. The deep horizon soil is representing by the solution into the soil, which the Cu is immediately available to plants [START_REF] Juste | Les micro-polluants métalliques dans les boues résiduaires des stations d'épuration urbaines[END_REF]. The Cu added distributed in two forms, a liquid phase and a liquid phase. The Cu contents in the solution into the soil are generally very low. At the end of culture, with both types of sludge tested, we observed a significant decrease in the content of Cd in the surface layer (0-20 cm), increased levels of Cd in the layer (20-40 cm) and a slight increase in levels of the layer (40-60 cm).

In this work, analysis of drainage showed that the addition of increasing doses (5t/ha and 10t/ha) liquid and dry sludge has no significant effect on the levels of Zn, Cd, Pb and Cu in lixivia; these levels are negligible and similar to the witness.

Figure 1 :

 1 Figure 1: Evolution of the production accumulated by dry material (MS) of the air part according to the various doses of (BL) liquid and (BS) dry sludge.

Figure 2 :

 2 Figure 2: Variation according to the time of the accumulation of Cd, Zn, Cu and Pb (mg/kg) to the plants of Ray Grass cultivated in the presence of the increasing doses of liquid and dry sludge.

Figure 3 :

 3 Figure 3: Comparison of Pb, Cd, Zn and Cu contents (mg/kg) in the soil before and after the last cutting of the Ray Grass.

Figure 4 :

 4 Figure 4: Cd, Zn, Pb and Cu contents (mg/kg) in waters of drainage during the growth of the Ray Grass.

Table 1 :

 1 Characteristics of the various horizons of the soil

	Parameters	0-20 cm	20-40 cm	40-60 cm
	pH	8,03	8,18		8,06
	CE (ms/cm)	0,61	0,58		0,70
	MO (%)	1,58	1,07		0,65
	Total limestone (%)	1,12	2,12		2,28
	N (%)	0,05	0,04		0,03
	Heavy metals (mg/kg)
	Mn	136.50	125.00		107.40
	Pb	34.40	29.80		28.40
	Cd	2.06	1.72		1.23
	Co	3.47	3.72		2.77
	Cu	10.25	8.75		4.00
	Fe	5337.00	7000.00	4638.00
	Zn	22.50	21.70		17.90
	Ni	14.07	13.10		11.20
	Parameters	Dry sludge	Liquid sludge
	pH	6.46			6.30
	Humidity (%)	33.30			95.40
	CE (ms/cm)	6.95			6.17
	MO (% MS)	40.20			94,00
	N (% MS)	3.35			5.80
	C/N	8.56			7.94

Heavy metals (mg/kg of dries matter)

	Mn	349	108
	Pb	135	393
	Cd	22	28
	Cu	206	413
	Fe	6954	9456
	Zn	331	848
	Ni	30	347

Table 2 :

 2 The sewage sludge characteristic

Table 3 :

 3

Effect of liquid (BL)

and dry (BS) sludge on the production of root.

Table 4 :

 4

The variation of Pb, Cd, Cu and Zn Fe and Mn contents (mg/kg) in the roots linked with different dose of liquid (BL) and dry (BS) sludge treatment.

Acknowledgements

This work was supported by National Institute for Rural Engineering research, Water and Forestry, Tunis and Geosciences & Environment Department of Ecole Nationale Supérieure des Mines de Saint Etienne, France.