I Zafeiratou

I Prodan

F Boem
email: f.boem@ucl.ac.uk

L Lefèvre

Handling power losses in a DC microgrid through constrained optimization

Keywords: DC microgrid, Power dissipation, Load balancing, Differential flatness, MPC

This paper extends a hierarchical control approach for power balancing in a meshed DC microgrid while minimizing the power losses in the central transmission network. The control strategy is divided into three layers in hierarchical framework: i) the high level solves a continuous-time optimization problem which minimizes the DC-bus power loss and the electricity cost from the external grid power purchase through the combined use of differential flatness with B-splines parametrization; ii) the middle level employs a tracking Model Predictive Control (MPC) method which mitigates the discrepancies among the optimal and the actual profiles; iii) the low level controller handles the switching activity of the converters. The proposed approach is validated in simulation for a specific meshed DC microgrid system.

INTRODUCTION

Energy management and power distribution in microgrids play a major role and influence the power system operation. Power losses in the DC-bus network can significantly affect the power quality during transmission. Hence, their mitigation constitutes a very important factor for the power transmission improvement.

Different approaches have been proposed over the years for the power loss reduction, concentrating either on the components connected to the microrgid or on the central transmission network. Some works focus on topological issues or optimal scheduling of energy storage (ES) systems or renewable sources, as in [START_REF] Iovine | Nonlinear control of a dc microgrid for the integration of photovoltaic panels[END_REF]. [START_REF] Nahata | Hierarchical control in islanded dc microgrids with flexible structures[END_REF] propose a three-layer hierarchical control approach to solve the energy management problem in islanded microgrids. A two-level plug and play hierarchical controller was proposed in [START_REF] Vazquez | A fully decentralized adaptive droop optimization strategy for power loss minimization in microgrids with pv-bess[END_REF].

This paper extends the hierarchical control scheme for a meshed DC microgrid in [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF] and concentrates on the minimization of power losses in the central transmission network (Fig. 1). The meshed DC microgrid and its components were already presented in [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF] (see also Fig. 1). In the high level, an optimization problem under constraints for power balancing and cost minimization was developed neglecting the power losses of the central transmission network. In here we concentrate on the high and middle levels proposing the following contributions: i) minimization of the power dissipation of the central transmission network is included in the optimization problem to optimize the power flow routing while minimizing the electricity cost; ii) detailed model description of the central power dissipation to represent the transmission lines including voltage drops Institute of Engineering Univ. Grenoble Alpes among the connecting nodes 1 . Supplementary constraints are considered to maintain the voltage in the DC network close to 400 V ; iii) validation of the approach through simulations considering different scenarios in the power losses among the sources and the loads.

The cost function and the constraints will be written, through differential flatness 2 , in function of the B-splines to ensure continuous-time constraints validation.

MESHED DC MICROGRID ARCHITECTURE

This section will briefly present the ES model and will focus on the dynamics of the central transmission network. The hereinafter dynamics emerge from the port-1 A connecting node is the point where a source or a load is linked to the central network as in Fig. 1 for the nodes 1, 2, 3, 4. 2 Differential flatness is a structural property of a class of nonlinear dynamical systems, denoting that all system variables (the states and the control inputs) can be written in terms of a set of specific variables, the so-called flat outputs (equal in number to the number of inputs), and their derivatives [START_REF] Levine | Analysis and control of nonlinear systems: A flatness-based approach[END_REF].

Hamiltonian (PH) modeling method which is detailed in [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF].

ES dynamical model: The ES system (Fig. 2) contains a lead acid battery connected to the Split-Pi converter. The state-space representation of the ES system is given in equations ( 4)-( 7) in [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF] and will not be detailed here due to space limitation. Note that, from the ES circuit in Fig. 2, the battery's current is equal to i b (t) = i R 1b (t) and the battery's voltage is

equal to v b (t) = q 1b (t) C 1b , where i R 1b is the current of R 1b
resistor, q 1b is the charge of C 1b capacitor and C 1b is its corresponding capacitance. These two relations will be used later in the control part. and Sw 4sc ). The high-voltage DC-bus of the microgrid constrains the Split-Pi converter to always operate in down-conversion (towards the sources) and up-conversion (towards the central network) [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF]. Consequently, d 1sc (t) will be always 0 and d 2sc (t) ∈ (0, 1). Hence, let us denote in the following:

α(t) = 1 -d 2sc (t),
(1) which describes the relation between the two duty cycles in the Split-Pi converter. Central transmission network: The transmission lines in this work are considered as resistors and, through the PH formalism, the network's model is described providing explicitly the relations of the voltages and currents on the connecting nodes3 (see also Fig1):

• from node 1:

i loads = i R1 +i R2 = v es -v loads R 1 + v ug -v loads R 2 , (2a)
• from node 2:

i es = i R3 -i R1 = v pv -v es R 3 - v es -v loads R 1 , (2b)
• from node 3:

i ug = i R4 -i R2 = v pv -v ug R 4 - v ug -v loads R 2 , (2c)
• from node 4:

i pv = i R3 -i R4 = v pv (t) -v es (t) R 3 - v ug (t) -v pv (t) R 4 . (2d)
Because of the transmission-line power losses, there are deviations in the voltages on the connecting nodes. Hence, boundaries must be defined, which will maintain the voltages, v ug (t), v pv (t), v es (t), v loads (t), close to 400 V , as follows:

v min,h DC ≤ v ug (t), v pv (t), v es (t), v loads (t) ≤ v max,h DC .
(2) Next, using (2a-2d), we express the power variables of the DC network as follows:

P ug = v ug i ug = v ug [i R4 -i R2 ] (3a) = v ug v ug -v pv R 4 - v loads -v ug R 2 , P es = v es i es = v es [i R3 -i R1 ] (3b) = v es v pv -v es R 3 - v es -v loads R 1 , P pv = v pv i pv = v pv [i R3 -i R4 ] (3c) = v pv v pv -v es R 3 - v ug -v pv R 4 , P loads = v loads i loads = v loads [i R1 -i R2 ] (3d) = v loads v es -v loads R 1 - v loads -v ug R 2 ,
where P ug , P es , P pv , P loads are the electrical powers produced/consumed by the UG, ES, PV and loads components, respectively. The variables v es (t) and i es (t), v pv (t) and i pv (t), v ug (t) and i ug (t), v loads (t) and i loads (t) denote the input voltage and current on the connecting nodes of ES, PV, UG and loads components, respectively, as in Fig. 1. Next, the electrical power of the dissipative elements are also introduced:

P R1 = [v es -v loads ] 2 R 1 , (3e)
P R3 = [v pv -v es ] 2 R 3 , (3f)
P R2 = [v loads -v ug ] 2 R 2 , (3g)
P R4 = [v ug -v pv ] 2 R 4 , (3h)
where, P R1 , P R2 , P R3 , P R4 correspond to the power losses within the DC bus. In addition, the power conservation equation is presented below: P ug (t)+P pv (t) -P es (t) -P loads (t) -P R1 (t)-

-P R2 (t) -P R3 (t) -P R4 (t) = 0. ( 4
)
Flat representation of the ES: As mentioned in the Introduction we use differential flatness [START_REF] Levine | Analysis and control of nonlinear systems: A flatness-based approach[END_REF] since it allows us to make the connection between the nonlinear dynamics of the ES [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF]) and the forthcoming continuous-time constrained optimization problem for power balancing and power loss minimization. Therefore, the set of flat outputs together with the flat representation found for the ES system in Zafeiratou et al.

(2020) are considered. Next, the B-spline parametrization is employed. Due to its properties (convexity, smoothness and differentiability), the continuous-time constraints validation can be verified (see also [START_REF] Prodan | Necessary and sufficient LMI conditions for constraints satisfaction within a B-Spline framework[END_REF] and the references therein).

HIERARCHICAL CONTROL APPROACH

This section presents the extension of the three-level hierarchical control problem of the meshed DC microgrid including the power dissipation. Priority is given to the high and the middle level whose structure must be adapted appropriately to the dynamics of the central transmission network contrary to [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF]. The three supervision levels consist of (see also the control scheme in Fig. 3): i) the high level, which generates optimal profiles not only for the power balancing, the current and the voltage of the battery, but also for the voltage on the connecting nodes and the duty cycle of the Split-Pi converter; ii) the middle level, where a tube-MPC contoller will track the optimal profiles obtained at the high level to minimize the discrepancies among the reference and the real profiles; iii) the low level, where the switching activity in the converter is validated [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF].

Fig. 3. Control scheme for the meshed DC microgrid.

High level problem formulation

Primary, the high level will be thoroughly analyzed. The objective function considered in [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF] was: min

Pug(t)
t f t0 e(t)P ug (t)dt, where only the electricity cost was penalized neglecting power losses. Enhancing the previous work, we consider here the complete power conservation equation ( 4), where the P ug is replaced by: P ug (t) =P es (t) + P loads (t) -P pv (t) + P R1 (t)+ +P R2 (t) + P R3 (t) + P R4 (t), (5) including in this way the overall dynamics of the system. Consequently, the general objective function is4 :

min i b ,v b ,α,ves,vpv,vug t f t0 e(t) Q cost P es (t) i b (t)v b (t) (t)+P loads (t)- -P pv (t) + Q loss P R (t) dt,
(6a) subject to : system dynamics , (6b) power conservation (4), (6c) constraints f or v b , i b and q 2b , (6d)

constraints f or v ug , v pv , v es , v loads (2), (6e)
P min,h ug -P loads (t) + P pv (t) -P R (t) ≤P es (t)

P es (t) ≤ P max,h ug + P loads (t) -P pv (t)-P R (t), (6f)
where P R (t) = P R1 (t) + P R2 (t) + P R3 (t) + P R4 (t) is the total power dissipation. In the following, we continue with the integration of the central dynamics in the optimization problem (objective function and constraints).

Note first that the converter is considered as an ideal element (power dissipation in the switches is neglected). Therefore, the input power, P sc in (t), is equal to the output power, P sc out (t) (Fig. 2): P sc in (t) =P sc out (t) (7a) v sc in (t)i sc in (t) =v sc out (t)i sc out (t).

(7b) Additionally, from the Ohm's law, we have:

v sc in (t) = v es (t) -R 1sc i es (t) (8a) v sc out (t) = v b (t) + i b (t)R 1b
(8b) Combining the above equations ( 7) and ( 8) and considering that i sc in (t) = i es (t) and i sc out (t) = i b (t) lead to:

v es (t) = v b (t) + i b (t)R 1b α(t) + R 1sc α(t)i b (t), (9a)
i es (t) = α(t)i b (t), (9b) where v b and i b are written in function of the flat outputs as presented in [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF]. Therefore, from (9a) and (9b) the ES power and from (3e)-(3h) the P R are deduced as follows:

P es (t) =v es (t)i es (t) = [v b (t) + i b (t)R 1b ]i b (t)+ + R 1sc [α(t)i b (t)] 2 , ( 10a
) P R (t) =( [v es (t) -v loads (t)] 2 R 1 + [v loads (t) -v ug (t)] 2 R 2 + + [v ug (t) -v pv (t)] 2 R 4 + [v pv (t) -v es (t)] 2 R 3 , (10b)
where v es (t), i es (t) are defined by ( 9a) and (9b) respectively. Additionally, some further restrictions are considered for the voltage of the loads, v loads , the power generated by the PV, P pv , and the power consumption, P loads , as shown below (note that the voltage of the ES, v es , is in function of the battery's dynamics as given above in (9a)):

f rom (2a) : v loads (t) = R 1 i es (t) + (1 + R 1 R 3 )• (11a) • v es (t) - R 1 R 3 v pv (t), f rom (3c) : P pv (t) = v pv (t)i pv (t) = v pv (t)• (11b) • v pv (t) -v es (t) R 3 - v ug (t) -v pv (t) R 4 , f rom (3d) : (11c) P loads (t) -loads ≤ v loads (t) v es (t) -v loads (t) R 1 - - v loads (t) -v ug (t) R 2 ≤ P loads (t) + loads ,
where loads ∈ R makes (11c) a soft constraint, hence relaxing the load's demand to ensure the feasibility of the optimization problem in (6a)-(6f).

Two constraints remain for α and P ug (t) in (6f):

1 α(t) > 1, (12)
P min,h ug ≤P ug (t) ≤ P max,h ug , (13) To ensure constraint validation in continuous time, we consider ( 12) and ( 13) in function of the B-splines. In [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF] the flat outputs found for the battery are written in function of the B-splines.

Similarly to the flat outputs of the battery, we parametrize the α factor from (12) ( 1 α(t)

∈ (1, +∞)) using a set of Bsplines, b i of order d α as follows:

1 α(t) = Nα j=1 p α j b i,d α (t) (14)
where N α is the number of control points p α j , for which (according to B-splines properties) we deduce:

p α j > 1, ∀i = 1, . . . , N α . (15)
Next, for the constraint (13), we take into account (2c) and via (11a), ( 9a) and (9b), P ug is rewritten as:

P ug (t) =v 2 ug (t)( 1 R 4 + 1 R 2 ) + v ug (t)v pv (t)( R 1 R 2 R 3 - 1 R 4 ) - 1 R 2 v ug (t) (1 + R 1 R 3 ) v b (t) + R 1b i b (t) α(t) + + (R 1 + R 1sc + R 1 R 1sc R 3 )α(t)i b (t) .
(16) Therefore, the constraint ( 13) is defined as:

P max,h ug ≥ v 2 ug (t)( 1 R 4 + 1 R 2 ) + v ug (t)v pv (t)( R 1 R 2 R 3 - 1 R 4 ) - 1 R 2 v ug (t) (1 + R 1 R 3 ) v b (t) + R 1b i b (t) α(t) - -(R 1 + R 1sc + R 1 R 1sc R 3 )|i b (t)| , (17a)
P min,h ug ≤ v 2 ug (t)( 1 R 4 + 1 R 2 ) + v ug (t)v pv (t)( R 1 R 2 R 3 - 1 R 4 ) - 1 R 2 v ug (t) (1 + R 1 R 3 ) v b (t) + R 1b i b (t) α(t) + + (R 1 + R 1sc + R 1 R 1sc R 3 )|i b (t)| , (17b
) which can be deduced also in function of the B-splines via v b (t) and i b (t). Hence, the optimization problem (6a)-( 6f) is finally rewritten in function of the B-splines considering the constraints of the central transmission network as in ( 2), ( 11a)-( 11c), ( 12

Middle level

At this level, a tube-MPC controller tracks under perturbation the reference profiles obtained at the high level, minimizing the deviations among the reference and the real profiles. The discretized dynamical model of the battery will be considered [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF]. When considering the power losses, the MPC tracking problem is reformulated as follows:

min ũ(k) k+Np-1 i=k (ỹ(i) -ỹref (i)) Q ỹ (ỹ(i) -ỹref (i))+ +(ũ(i) -ũref (i)) R ũ ũ(i) -ũref (i) ( 18a
-Ploads (k) + Ppv (k) -PR1 (k) -PR2 (k)- -PR3 (k) -PR4 (k) ≤ Pes (k), ( 19a
) P max,m ug -Ploads (k) + Ppv (k) -PR1 (k)- -PR2 (k) -PR3 (k) -PR4 (k) ≥ Pes (k), ( 19b
)
where Pes (k) = ĩes (k)ṽ es (k). The ĩes (k) and ṽes (k) are calculated with respect to the α factor reference profile α ref , obtained at the high level and the equations ( 9a) and (9b) in discrete time.

SIMULATION AND COMPARISON RESULTS

In Table 1, the parameters of the DC microgrid are illustrated. Table 2 5 depicts the parameters and constraints of the high and middle level controllers. The simulations are implemented in MATLAB 2015a. Furthermore, we use the YALMIP optimization toolbox for both high and middle level. This allows the use of IPOPT solver capable to handle nonlinear optimization problems.

Table 1. Parameters of the system.

Variable Values Units R 1sc , R 1b , R 2b 1, 0.025, 0.088 [Ω] I 1sc , I 2sc 0.25, 0.25 [H] C 1sc , C 2sc , C 3sc 0.0008, 0.0008, 0.0008 [F ] C 1b , C 2b 86400, 21600 [F ] R 1 , R 2 , R 3 , R 4 1 [Ω]
High level: In Fig. 4 the PV power and load profile are illustrated. The UG power and the ES power optimal profiles are generated through B-spline parametrization, with N = 27 and N α = 18 control points. The simulation is based on a constrained dynamics implementation in continuous-time over a horizon of 24 hours. In the figures, the red lines at the upper and lower level of each simulation represent the corresponding constraints (Table 2).

According to Fig. 4, the consumers' demand increases in the afternoon, while during the day until 3p.m. is more or less stable. Therefore, from 12p.m. to 12a.m., the U G and the P V charge the batteries. There is, also, a surpass of energy generated from the PV and it is sold to the UG (approximately 13% of the total power consumed). Furthermore, in Fig. 5, the battery's reference profiles for the current, voltage and charge are depicted as well as the α factor of the Split-Pi converter with the constraint verification. The electricity cost is equal to 2.713 euros. Afterwards, the power losses are presented in Fig. 6 together with the constraint validation for the voltage on the four connecting nodes (Fig. 7) where the sources and the loads stand. The power losses in lines R 3 and R 4 are caused by the PV purchase towards the UG or the ES system. The total loss of R 1 and R 2 , about 0.33%, exists due to the load demand that increases after 4p.m.. The total calculation time of the simulation is around 12 m.

v min,h b , v max,h b 12, 13 [V ] i min,h b , i max,h b -9, 9 [A] q min,h 2b , q max,h
Middle level: As aforementioned, in the middle level, we use MPC for reference tracking with a prediction horizon N p equal to 5 and a sampling time T s equal to 300 s.

In Fig. 8, we observe the tracking profiles of the P ower Balancing, and the control input, v sc out (the output voltage of the Split-Pi converter), which is in function of the current, i b , and the voltage, v b of the battery as in (8b). From the figures Fig. 8 and Fig. 9, we observe that the optimal profiles obtained at the high level are very closely followed.

Comparison with MPC for optimal profile generation: As a next step, we compare the reference trajectories generation at the high level (for R 1 =R 2 =R 3 =R 4 =1 Ω) obtained through differential flatness and B-spline parametrization with MPC method, used for instance in [START_REF] Velarde | On the comparison of stochastic model predictive control strategies applied to a hydrogen-based microgrid[END_REF]. Hence, we present the simulation results for the reference profiles obtained with the profiles obtained using MPC in Fig. 10 with a prediction horizon equal to 24 and a sampling time equal to 1200. Similar trajectories are observed with a slight difference in the cost as it is depicted also in Table 4 of about 0.3 -3%. Furthermore, we observe that because of the dynamics discretization in MPC, the value of the sampling time influences the simulation's performance (note in particular the variation in the electricity cost).

CONCLUSION

The paper extends the method presented in [START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF] for the multilevel supervision of a meshed DC microgrid. A constrained optimization-based control approach (via differential flatness, B-spline parametrization and Model Predictive Control) was presented for solving the power balancing problem minimizing, at the same time, the cost and the power dissipation. Further improvements are related to the analysis of the control scheme robustness under unexpected events, such as continuity of the system operation in case of faulted transmission lines, will be investigated.

Fig. 1 .

 1 Fig. 1. Meshed DC microgrid architecture.

Fig. 2 .

 2 Fig. 2. Electrical circuit of ES.

  )-(13). Next, the middle level is adjusted to the new considered dynamics with power losses included in the central transmission network. The reference profiles from the high level are: the battery current (i ref b ), battery voltage (v ref b ), input voltage of the ES (v ref es ), output voltage (v ref sc out ) and α factor (α ref ) of the Split-Pi converter.

  ) subject to : the battery s discretized dynamics , es (k) and ũref (k) = ṽref sc out (k), taken at T s sampling time. The last constraint P ug (t) is replaced by: P min,m ug

Fig. 4 .

 4 Fig. 4. Optimal power flow of the power sources.

Fig. 5 .Fig. 8 .

 58 Fig. 5. Reference profiles of the high level.

  Fig. 9. Tracking profiles.

Fig. 10 .

 10 Fig.10. Comparisons of P es and P ug optimal profiles obtained by flatness approach and MPC.

  The duty cycles, d 1sc (t) and d 2sc (t), characterize the operation of the converter and are the control variables of the ES system, (d 1sc (t) for switches Sw 1sc and Sw 2sc and d 2sc (t) for switches Sw 3sc

Table 2 .

 2 Variables and constraints for the high and the middle level.

	Level	Variable	Values	Units
		N	27
		na as in (14)	18
	High level	d=da as in (14) Qcost as in (6a)	4 1
		Q loss as in (6a)	1
	Constraints

Table 3 .

 3 Power percentage with respect to the total power produced or consumed (high level).

	Power	Power produced [%]	Power consumed [%]
	Pug	42.14%	13% sold to the UG
	Pes	6.58%	6.7% for ES charging
	Ppv	51.28%	-
	P loads	-	79.66% for load usage
			Total: 0.64%
			R 1 : 0.12%
	P loss	-	R 2 : 0.21%
			R 3 : 0.13%
			R 4 : 0.18%

Table 4 .

 4 Simulation results obtained for optimal profiles with MPC.

	Prediction	Sampling	Electricity	Power
	horizon N p	time T s [s]	cost [euros]	loss [%]
	24	1200	2.801	1.63%
	15	1800	2.747	1.65%
	10	1800	2.788	1.64%

For compactness, whenever it is clear in the context, we discard the time dependence.

Note that (6b) and (6d) correspond to equations (4)-(7) and (15c)-(15e) respectively given in[START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF].

Note that N and d correspond to equations (17a)-(17b) given in[START_REF] Zafeiratou | Meshed dc microgrid hierarchical control: A differential flatness approach[END_REF].

ACKNOWLEDGEMENTS

This work is funded by the French National Research Agency within the framework of the project ANR-15-CE05-004-02 C 3 µ (Components, Control and Communication). We also acknowledge the support of the PHC Alliance project No.43234RH. Finally, we thank BUI Ngoc Thanh Hung for helping with the code implementation.


Zhiqi Tang
email: zhiqitang@tecnico.ulisboa.pt

Rita Cunha

Tarek Hamel
email: thamel@i3s.unice.fr

Carlos Silvestre
email: csilvestre@umac.mo

Bearing Leader-Follower Formation Control under Persistence of Excitation

Keywords: Multi-agent systems, Decentralized control, Nonlinear cooperative control, Distributed nonlinear control, Lyapunov methods, Stability of nonlinear systems

This paper addresses the problem of formation control with a leader-follower structure in three dimensional space by exploring persistence of excitation (PE) of the desired bearing reference, which is provided by a possibly moving or time-varying desired formation. Using only bearing measurements, control laws are proposed for a group of agents with single-integrator dynamics. By defining a desired formation such that the corresponding inter-vehicle bearing measurements are persistently exciting, relaxed conditions on the interaction topology (which do not require bearing rigidity nor constraint consistence) can be used to derive distributed control laws that guarantee exponential stabilization of the desired formation in terms of relative position. A key outcome of this approach is that even if there is only one connection originated from each follower, exponential stability of the formation can be achieved as long as the excitation conditions are met on the desired formation. The approach generalizes stability results provided in prior work for leader-first follower (LFF) structure, based on bearing rigidity and constraint consistence that required at least two connections for each follower except for the first one. Simulations results are provided to illustrate the performance of the proposed control method.

INTRODUCTION

The problem of formation control has been extensively studied over the last decades both by the robotics and the control communities. The main categories of solutions can be classified as [START_REF] Oh | A survey of multiagent formation control[END_REF]: 1) position-based formation control, [START_REF] Ren | Distributed multi-vehicle coordinated control via local information exchange[END_REF], 2) displacement-based formation control, [START_REF] Ren | Coordination variables and consensus building in multiple vehicle systems[END_REF], 3) distance-based formation control, [START_REF] Anderson | Control of directed formations with a leader-first follower structure[END_REF] and more recently 4) bearing-based formation control, [START_REF] Basiri | Distributed control of triangular formations with angle-only constraints[END_REF]. This latter control category has received growing attention due to its minimal requirements on the sensing ability of each agent. Early works on bearing-based formation control were mainly about controlling the subtended bearing angles which are measured in each agent's local coordinate frame and was limited to the planar formations only, [START_REF] Basiri | Distributed control of triangular formations with angle-only constraints[END_REF]; [START_REF] Bishop | A very relaxed control law for bearingonly triangular formation control[END_REF]. The main body of work however builds on concepts from bearing rigidity theory, which investigates the conditions for which a static geometric pattern of a framework is uniquely determined by the corresponding constant bearing measurements. Bearing rigidity theory in two- dimensional space (also termed parallel rigidity) is explored in [START_REF] Eren | Sensor and network topologies of formations with direction, bearing, and angle information between agents[END_REF]; [START_REF] Servatius | Constraining plane configurations in computer-aided design: Combinatorics of directions and lengths[END_REF] and more recently it has been extended to arbitrary dimensions in [START_REF] Zhao | Bearing rigidity and almost global bearing-only formation stabilization[END_REF]. More specifically, a formation control solution based on bearing measurements is proposed in [START_REF] Zhao | Bearing rigidity and almost global bearing-only formation stabilization[END_REF], under the assumption that the graph is undirected and infinitesimally rigid. The resulting bearing controller guarantees convergence to a target formation that is centroid invariant and scale invariant with respect to the initial conditions of the formation.

In the more challenging context of directed graphs, achieving stabilization of a formation requires not only bearing rigidity, as in the case of undirected graphs, but also the constraint consistence (also termed bearing persistence, in [START_REF] Zhao | Bearing-based formation stabilization with directed interaction topologies[END_REF]), which is the ability to maintain consistence between constraints induced by the desired bearing measurements. In [START_REF] Eren | Using angle of arrival (bearing) information for localization in robot networks[END_REF], the conditions for directed bearing rigidity of a digraph in two-dimensional space are stated and a bearing control law for nonholomonic agents is proposed. The authors in [START_REF] Schiano | A rigidity-based decentralized bearing formation controller for groups of quadrotor uavs[END_REF] proposed a control strategy that relies on an extension of the rigidity theory to directed bearing frameworks defined in R 3 × S 1 while at least one distance between two agents is required to correct the scale of the formation. In [START_REF] Trinh | Bearing-based formation control of a group of agents with leader-first follower structure[END_REF], bearing control laws were proposed to stabilize formations that satisfy a leader-first follower (LFF) interaction topology in an arbitrary dimensional space, obtained from a bearing Henneberg construction. LFF formations are In (a), agent 4 is collinear with its two neighbors agent 2 and 3 and thus the framework is not bearing rigid. The framework in (b) forms a square with only one connection originating from each follower, and obviously it is not bearing rigid. The framework in (c) is bearing rigid but not constraint consistent. The stability of these three formations can not be guaranteed by the bearing controller relying only on bearing rigidity theory and constraint consistence [START_REF] Zhao | Bearing-based formation stabilization with directed interaction topologies[END_REF] but can be guaranteed by the proposed control law under the PE condition.

uniquely determined up to a translation (the leader's position) and a scaling factor, meaning that knowledge of the distance between the leader and the first follower is required to control the formation scale.

In this paper, we consider the problem of controlling a multiagent system with single-integrator dynamics to stabilize the actual formation to a reference one using bearing measurements only. We propose an approach that draws inspiration from the work in [START_REF] Trinh | Bearing-based formation control of a group of agents with leader-first follower structure[END_REF], which presents a first-order bearing formation control law, considering an LFF graph topology. A distinctive feature of the present work is the shift of focus from static formations to moving and time-varying formations. In this context, persistence of excitation (PE) can be explored to significantly relax the conditions on the graph topology that are needed to guarantee stabilization of the formations. As a result, we are able to define general PE criteria that incorporate and generalize convergence criteria used in prior work.

In particular, we will show that under the PE condition 1) the formation stabilization is achieved for a leader-follower structure that is not necessarily bearing rigid nor constraint consistent (i.e. a directed acyclic graph which has a spanning tree, as shown in Fig. 1); 2) the desired formations may correspond to fixed geometric patterns that translate and rotate, as long as that the PE condition is provided by rotational motion; 3) scale ambiguity, which is a characteristic of bearing rigidity, can be removed and convergence of the desired formation in terms of scale can be guaranteed, without the need to measure the distance between any two agents. In general, a desired formation that is persistently exciting can be time-varying (in scale, translation, rotation, and even in shape), similar to those typically considered in position-based formation control Brinón-Arranz et al. ( 2014); [START_REF] Dong | Time-varying formation control for unmanned aerial vehicles: Theories and applications[END_REF].

We also show that static formations under the LFF structure [START_REF] Trinh | Bearing-based formation control of a group of agents with leader-first follower structure[END_REF] can be treated as a special case, for which the proposed bearing control law guarantees formation stabilization up to a scale factor.

The body of the paper is organized as follows. Section II presents mathematical background on graph theory and introduces the leader-follower graph structure together with the bearing PE condition exploited in the paper. Section III presents the bearing formation control laws along with stability analysis.

Section IV shows the performance of the proposed control strategy. The paper concludes with some final comments in Section V.

PERSISTENCE OF EXCITATION AND LEADER-FOLLOWER INTERACTION TOPOLOGY

Persistence of excitation

Let S 2 := {y ∈ R 3 : y = 1} denote the 2-Sphere and . the euclidean norm. The operator [.] × yields the skew-symmetric matrix associated to its vector argument and λ max (.)(λ min (.)) represents the maximum (minimum) eigenvalue of its matrix argument.

For any y ∈ S 2 , we can define the projection operator π y

π y := I -yy ≥ 0, (1)
which is such that, for any vector x ∈ R 3 , π y x provides the projection of x on the plane orthogonal to y. Lemma 2. Let Q := l i=1 π yi . Then the matrix is persistently exciting, if one of the following conditions is satisfied:

Definition 1. A matrix Σ(t) ∈ R n×n ,
(1) there is at least one of the directions y i that is persistently exciting, (2) there are at least two direction y i and y j , i, j ∈ {1, ..., l}, i = j such that they are uniformly noncollinear. That is, for all t ≥ 0 there exists an 1 > 0 such that |y i (t)

y j (t)| ≤ 1 -1 .
Proof. The proof of this lemma is given in (Le Bras et al., 2017, Lemma 3).

Preliminaries on graph theory

Consider a system of n connected agents. The underlying interaction topology can be modelled as a digraph (directed graph) G := (V, E), where V = {1, 2, . . . , n} is the set of vertices and E ⊆ V × V is the set of directed edges. In this work, the graph is interpreted as sensing graph, meaning that if the ordered pair (i, j) ∈ E then agent i can access or sense information about agent j, which is called a neighbor of agent i. Note that the information flow is in the opposite direction. The set of neighbors of agent i is denoted by The green, blue and black dashed lines represent possible neighbors of agent 2, 3 and 4, respectively.

N i := {j ∈ V|(i, j) ∈ E}. Define m i = |N i |, where |.| denotes the cardinality of a set. A directed path is a finite sequence of distinct vertices ν 1 , ν 2 , . . . , ν k-1 , ν k , such that (ν i-1 , ν i ), 2 ≤ i ≤ k belongs to
E. A directed cycle is a directed path with the same start and end vertices, i.e.

ν 1 = ν k . A digraph G is called an acyclic digraph if it has no directed cycle.
The digraph G is called a directed tree with a root vertex i, i ∈ V, if for any vertex j = i, j ∈ V, there exists only one directed path connecting j to i. Note that a directed tree is acyclic. We say that G has a directed spanning tree, if there exists a subgraph of G that is a directed tree and contains all the vertices of G.

Leader-follower structure

In this section, we will define a leader-follower structure that is not necessarily bearing rigid nor constraint consistent. Definition 3. A digraph G is called a leader-follower structure if it is acyclic and has a spanning tree.

Definition 4. A digraph G = (V, E) is called a minimal leader- follower structure if it is a directed tree.
In this setting of a leader-follower structure, the leader is the root vertex which has no neighbors and each of the other followers has at least one neighbor. Without loss of generality, the agents are numbered (or can be renumbered) such that agent 1 is the leader, i.e. N 1 = ∅, and for each agent i ≥ 2 the set of neighbors satisfies N i ⊆ {1, 2, . . . , i -1}. An example of a possible 5-agent leader-follower graph is shown in the Figure 2.

The first follower (agent 2) has only one neighbor which is the leader (agent 1). The second follower (agent 3) has two possible neighbors: agent 1 and 2, and so forth.

Remark 1. Note that the leader-follower structure defined above is more general than the leader-first follower structure (LFF) considered in [START_REF] Trinh | Bearing-based formation control of a group of agents with leader-first follower structure[END_REF], for which each follower has two neighbors except the first follower which is only connected to the leader.

Given the digraph G as defined in Definition 3, let each agent i ∈ V be modeled as a point p i ∈ R 3 in a common inertial frame. Then, the stacked vector p = [p 1 , ..., p n ] ∈ R 3n is called a configuration of G. The digraph G and the configuration p together define a formation G(p) in the 3-dimensional space.

Defining the relative position vectors p ij := p j -p i , i, j ∈ V, i = j (3) then as long as p ij = 0, the bearing of agent j relative to agent i is given by the unit vector

g ij := p ij / p ij ∈ S 2 . ( 4
ṗ * i = v * i -K j∈Ni π g * ij (p * i -p * j ), ∀i ≥ 2, (5)
with arbitrary initial conditions and K a positive definite matrix. Assume the desired formation satisfies the PE condition. Then p * i converges uniformly globally exponentially (UGE) to the unique p * i . Proof. Consider the error variables p * i := p * i -p * i defined for i = 2, . . . , n and the corresponding dynamics obtained from (5). For i = 2, we have N 2 = {1} and it is straightforward to verify that the dynamics of p * 2 is given by ṗ * 2 = -Kπ g * 21 p * 2 (6) and that p * 2 = 0 is UGE stable under the PE condition (by direct application of (Le Bras et al., 2017, Lemma 4)). For i = 3 and N 3 = {1}, the proof is exactly the same as for agent 2. For N 3 = {2} or {1, 2}, the dynamics of p * 3 can be written as

ṗ * 3 = -K j∈N3 π g * 3j p * 3 + Kπ g * 32 p * 2 (7)
which together with (6) forms a cascaded system with p * 2 as input to (7). Using the fact that p * 2 = 0 is UGE stable and system ( 7) is continuously differentiable and globally Lipschitz in (p * 3 , p * 2 ), it follows (by direct application of (Le Bras et al., 2017, Proposition 1)) that p * 3 = 0 is also UGE stable. In the general case, we can write

ṗ * i = -K j∈Ni π g * ij p * i + K j∈Ni\{1} π g * ij p * j , (8)
for i = 2, . . . , n and the proof of that p * i = 0 is UGE stable can be obtained in a similar way. Remark 2. For the static case where v * i -v * j = 0, ∀(i, j) ∈ E, we obviously conclude that g * 21 is not persistently exciting. However, if each agent i (i ≥ 3) has two neighbors 1 ≤ j = k < i with g * ij = ±g * ik , the desired formation becomes exactly the same as the desired LFF formation described in [START_REF] Trinh | Bearing-based formation control of a group of agents with leader-first follower structure[END_REF] and uniqueness of the target formation can still be guaranteed if the distance d * 21 = p * 1 -p * 2 is provided. Under the proposed controller, which will be defined in the next section, the formation will converge to the desired shape up to a scaling factor as discussed in [START_REF] Trinh | Bearing-based formation control of a group of agents with leader-first follower structure[END_REF].

Note that under the condition of Theorem 1, the shape and the size of the desired formation may be time-varying. There are some situations in which a rigid motion of the desired formation ensures the PE condition. More specifically, the rigid motion should include a rotational motion as stated in the following Corollary. Corollary 1. Consider a minimal leader-follower formation subjected to a continuous rigid motion such that:

g * ij (t) = R(t) g * ij (0), ∀(i, j) ∈ E
where R(t) ∈ SO(3) is the orientation matrix of a virtual frame attached to the formation (at some point) with respect to the common inertial frame. Let Ω(t) be the orientation velocity of the formation expressed in the virtual frame, such that Ṙ(t) = R(t)[Ω(t)] × . Assume that Ω(t) is a uniformly continuous vector. If there exists (T, ) > 0 and τ ∈ [t, t + T ], ∀t such that Ω(t) and g * ij (t) are uniformly non-collinear: Ω(τ ) × g * ij (τ ) ≥ , ∀(i, j) ∈ E, then the desired formation satisfies the PE condition.

Proof. Recalling that in the case of a minimal leader-follower structure, |N i | = 1 (∀i ≥ 2), it then follows that the desired formation satisfies the PE condition if and only if g * ij (t), ∀(i, j) ∈ E satisfies the PE condition of Definition 2. Using the fact that ġ * ij (t) = [Ω(t)] × g * ij (t) along with Lemma 1, it is straightforward to verify that the PE condition is satisfied as long as Ω(t) is uniformly non-collinear to g * ij (t), ∀(i, j) ∈ E.

BEARING FORMATION CONTROL

Problem formulation

Consider the framework G(p), where each agent i ∈ V is modelled as a single integrator with the following dynamics: ṗi = v i (9) where p i ∈ R 3 is the position and v i ∈ R 3 the velocity input, all expressed in a common inertial frame.

We assume that the n-agent system satisfies the following assumptions. Assumption 1. The desired velocity v * i (t) are bounded for all t. The desired positions p * i (t) are such that the resulting desired bearings g * ij (t) are well-defined for all t and the resulting desired formation satisfies the PE condition. Assumption 2. The sensing topology of the group is described by a digraph G(V, E) that satisfies the leader-follower structure defined in Definition 3. Each agent i ≥ 2 can measure the relative bearing vectors g ij to its neighbors j ∈ N i . Assumption 3. As the formation evolves in time, no inter-agent collisions and occlusions occur. In particular, we assume that the bearing information g ij (t), (i, j) ∈ E is all the time welldefined.

With all these ingredients, we can define the bearing formation control problem as follows. Problem 1. Consider the system (9) and the underlying framework G(p). Under Assumptions 1-3, design stabilizing distributed control laws based on only bearing measurements that guarantee convergence to the desired formation.

Exponential stabilization of the formations

For distinct agents i, j ∈ V, define the desired relative position vectors p * ij according to (3) and the relative position error pij := p ij -p * ij . In addition, for i ∈ V and j ∈ N i , define the dynamics of the error:

ṗij = v j -v * j -(v i -v * i ). (10)
The relative error pij is defined for all i, j ∈ V (with i = j), which may include pairs for which (i, j) / ∈ E. However, the dynamics of the error is only defined for pij (i, j) ∈ E.

The following control law is proposed for each agent i ∈ V

v i = - j∈Ni k i π gij p * ij + v * i , (11)
where k i is positive gains. Remark 3. Assume for now the results that will be shown in the following sections hold, i.e., assume that the equilibrium point pij = 0, (i, j) ∈ E of the system (10) is Exponentially Stable (ES). To express this result in terms of the formation and the positions of the agents p i , note that under the given assumption it also holds that pi1 = 0, 2 ≤ i ≤ n is ES. Then, we can conclude that ∀2 ≤ i ≤ n, p i exponentially converges to p * i up to a translation given by p 1 -p * 1 . The latter can be driven to zero provided that agent 1 can sense its position and velocity.

Stability and convergence of the first follower Lemma 3. For i = 2 consider the dynamics of the error (10) along with the control law (11). If the Assumptions 1-3 are satisfied, then the equilibrium point p21 = 0 is exponentially stable (ES).

Proof. The control law (11) for agent 1 is v 1 = v * 1 and for agent 2 is v 2 = -π g21 p21 + v * 2 . Recalling (10), the closed-loop system for the state p21 is expressed as ṗ21 = -k 2 π g21 p21 .

(12) Consider the following Lyapunov function candidate:

L 21 = 1 2 p21 2 (13)
Taking its time-derivative, it yields 21 along with a direct application of (Lorıa and Panteley, 2002, Lemma 5) one can conclude that p21 = 0 is ES. Remark 4. Note that in the above lemma, assumption 3 relies on the evolution of state variables. This assumption serves here to show that if there is no collision or occlusion, the bearings are well-defined and the proposed control design yields the desired convergence properties (Lemma 3 and even in the following results: Lemma 4 and Theorem 2). Trying to more specifically characterize the set of initial conditions for which the system's solutions avoid collision and occlusion is out of the scope of the paper.

L 21 = -k 2 p 21 π g21 p21 , (14)
Stability and convergence of the second follower Lemma 4. For i = 3 consider the dynamics of the error (10) along with the control law (11). If the Assumptions 1-3 are satisfied and Lemma 3 is valid, then the equilibrium point p3j = 0, ∀j ∈ N 3 is ES.

Proof. According to the leader-follower structure described in subsection 2.2, the second follower (agent 3) can have three possible sets of neighbors: N 3 = {1}, N 3 = {2} and N 3 = {1, 2}.

CONCLUSION

This paper studies bearing formation control using persistence of excitation and an underlying interaction graph with a leaderfollower structure. The desired formation is determined by the desired relative positions between agents, which are chosen such that the corresponding desired bearing vectors satisfy the persistence of excitation condition. Under the proposed control laws, the desired formation is achieved with exponential rate of convergence, that is, the relative positions of the group of agents converge exponentially to the desired values, without relying on bearing rigidity nor on estimation of range. Simulation results are provided to validate the control laws. Future work will be dedicated to the incorporation of collision avoidance in the bearing control laws to bypass assumption 3 in order to ensure at least semi-global exponential stability.

  This work was partially supported by the Project MYRG2015-00126-FST of the University of Macau; by the Macao Science and Technology, Development Fund under Grant FDCT/026/2017/A; by Fundac ¸ão para a Ciência e a Tecnologia (FCT) through Project UID/EEA/50009/2019 and Project PTDC/EEI-AUT/5048/2014; and by the EQUIPEX project Robotex. The work of Z. Tang was supported by FCT through Ph.D. Fellowship PD/BD/114431/2016 under the FCT-IST NetSys Doctoral Program. Carlos Silvestre in on leave from Instituto Superior Técnico, Universidade de Lisboa, Portugal.

Fig. 1 .

 1 Fig. 1. Examples of frameworks with leader-follower structure.In (a), agent 4 is collinear with its two neighbors agent 2 and 3 and thus the framework is not bearing rigid. The framework in (b) forms a square with only one connection originating from each follower, and obviously it is not bearing rigid. The framework in (c) is bearing rigid but not constraint consistent. The stability of these three formations can not be guaranteed by the bearing controller relying only on bearing rigidity theory and constraint consistence[START_REF] Zhao | Bearing-based formation stabilization with directed interaction topologies[END_REF] but can be guaranteed by the proposed control law under the PE condition.

Fig. 2 .

 2 Fig. 2. Possible connections of a leader-follower structure when n = 5. The black solid line represents the only neighbor of the first follower (agent 2) which is the leader (agent 1).The green, blue and black dashed lines represent possible neighbors of agent 2, 3 and 4, respectively.

Fig. 3 Fig. 4 .

 34 Fig. 3. 3-D Trajectory described by a formation under a single directed path topology. Colored solid lines represent the agents' trajectories and dashed lines represent the desired trajectories. The black solid lines represent the connections between agents.

  Definition 5. We say that a desired formation G(p * (t)) satisfies the PE condition if for all agent i ≥ 2, the matrices j∈Ni π g * ij (t) satisfy the PE condition as defined in Lemma 2.The following lemma will show the uniqueness of the desired formation of a leader-follower structure under PE condition.

	Theorem 1. Consider a leader-follower formation. Assume that
	the leader's position p * 1 (t), the velocity vectors {v * i (t)} i∈{1,...,n} and the bearing vectors {g * ij (t)} (i,j)∈E are well-defined, known,
	and bounded. Let p * 1	p * 1 and p * i denote the estimate of p * i , for
	i = 2, . . . , n with the following dynamics:

) Let v i := ṗi denote the velocity of agent i. Let p * (t) = [p * 1 (t), ..., p * n (t)] ∈ R 3n be the desired configuration. Let {v * i (t)} i∈{1,...,n} and {g * ij (t)} i,j∈E be the set of all velocity vectors and the set of all bearing vectors, respectively, of the desired configuration p * (t).

Case i): N 3 = {1}, the proof is identical to the proof of Lemma 3.

Case ii): N 3 = {2} or N 3 = {1, 2}. The closed-loop system for the state p3j , j ∈ N 3 is expressed as ṗ3j = -

Since v 1 = v * 1 and v 2 is a function of variables p21 , p31 = p32 + p21 , we can interpret (16) as a cascaded system that has p21 as input to the unforced system ṗ3j = -k 3 l∈N3 π g 3l p3j .

(17)

Now the proof becomes analogue to the proof of Lemma 3. Consider the following Lyapunov function candidate:

and its time-derivative is given by L3j

which is negative-semidefinite. Thus state p3j is bounded. Due to the fact that p3j = p3k + pkj , k = j, k, j ∈ {1, 2} and p21 = 0 in the unforced system (17), one has p3j = p3k . It is straightforward to verify that

with γ 3 = min l∈N j p * 3l 2 ( p3j (0)+max l∈N j p * 3l ) 2 . Using the PE condition along with direct application of (Lorıa and Panteley, 2002, Lemma 5), we can conclude that the equilibrium point p3j = 0, j ∈ N 3 of the unforced system (17) is ES. This in turn implies that the equilibrium point p3j = 0, j ∈ N 3 is ES for the system (16).

The n-agents system Theorem 2. For all i ∈ V\{1} and ∀j ∈ N i , consider the system (10) in closed-loop with the proposed control law (11). If the Assumptions 1-3 are satisfied, then the equilibrium point pij = 0, i = 2, . . . , n, ∀j ∈ N i is ES.

Proof. We will prove the convergence of pij = 0 by mathematical induction. Firstly, for k = 2 we have p21 = 0 is ES based on Lemma 3. Thus Theorem 2 is true for k = 2. It is also true for k = 3 from the conclusion of Lemma 4.

Secondly, we suppose Theorem 2 is true for 4 ≤ k ≤ i -1, that is pkj = 0, ∀j ∈ N k is ES for all 4 ≤ k ≤ i -1 and we will prove that it is also true for k = i. Recall (10), the closed-loop system for the states pij , j

where v j is a function of variables pjm , m ∈ N j , piq = pij + pjq . Note that since the graph is connected, pjq can be represented by the error variables pkm , 2 ≤ k ≤ i -1, m ∈ N k . System (21) can then be considered as a cascaded system with pkm , 2 ≤ k ≤ i -1, m ∈ N k , being inputs of the unforced system analogously to system (17). Using a similar argument as shown in Lemma 4, one can conclude that the equilibrium point pij = 0, ∀j ∈ N i of the unforced system is ES. Because Theorem 2 is true for 2 < k ≤ i -1, we can conclude that the equilibrium point pij = 0, ∀j ∈ N i for system ( 21) is also ES. This in turn implies that Theorem 2 is true for k = i.

Then, by mathematical induction, if follows that the claim is true for all k ∈ V\{1}.

SIMULATION RESULTS

In this section, we consider a four-agent system defined in R 3 , V = {1, 2, 3, 4}, with digraphs that satisfy a minimal leaderfollower graph formed by a single directed path, that is, each follower has only one neighbor such that N i = {i -1}, i ∈ V\{1}. According to Assumption 1, the desired formation is chosen such that the four agents form a squared shape in R 2 that rotates about agent 1. Note that the desired configuration is not bearing rigid. For the sake of simplicity, the leader (agent 1) is static at position p 1 = [0 0 0] . The desired trajectories for the followers are p

. The gains used are k i = 3. Fig. 3 depicts the trajectories of the four agents during the time evolution of the formation and we can see that the four agents converge to the desired trajectories. Fig. 4 shows the time evolution of the error state p . We can conclude that the proposed control law stabilizes the formation efficiently without requiring of bearing rigidity.


Pierre-Alain Toupance
email: pierre-alain.toupance@lcis.grenoble-inp.fr

Bastien Chopard
email: bastien.chopard@unige.ch

Laurent Lefevre
email: laurent.lefevre@lcis.grenoble-inp.fr

Identification of complex network topologies through delayed mutual information

Keywords: complex systems, information theory, probabilistic models, interaction topology, adjacency matrix identification, community reconstruction, voter model

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

In [START_REF] Conant | Laws of information which govern systems[END_REF], the author already pointed out the interest of information theoretic approaches for the analysis of complex dynamical systems. He specifically highlighted their additive complexity which make them attractive for large scale systems. These approaches are based on the computation and analysis of information flows (and losses) between subsystems and components of the complex system. Therefore it seems quite logical to use them for the analysis of control systems topologies, with applications in mind to coarse graining or partitioning (decomposition) problems, for instance. Delayed mutual information was introduced in Schreiber (2000) -with appropriate conditioning of transition probabilities -to distinguish driving and responding elements through the analysis of the correlation between two stochastic signals. Introducing the time delay and detecting asymmetry in the interaction of subsystems allows one to distinguish information that is actually exchanged between two subsystems from shared information due to common history and input signals.

The structural analysis of complex systems dynamics and input-output properties has a long history. Approaches have been developed which make use of interconnection graph or "inference diagram", describing the existing (analytical) relations between a priori given input, state and output variables [START_REF] Lin | Structural controllability. Automatic Control[END_REF]; [START_REF] Siljak | Decentralized control of complex systems[END_REF]. Such approaches give structural results on controllability and observability, together with efficient graph algorithms. Some recent results on structural controllability and observability using this approach are presented in [START_REF] Liu | Controllability of complex networks[END_REF][START_REF] Liu | Observability of complex systems[END_REF]. They require a priori knowledge of the system interconnection topology. When trying to isolate the most influential or measurable nodes in some complex system, this information is often missing or incomplete. Besides, these results on controllability and observability only conclude to some existing causal relation between the considered sets of variables, but not how much the dynamics of a node may be measured or controlled from another node. Therefore we proposed in Toupance (2019) the use of delayed mutual and multi-informations to analyze the most influential components in a complex system with no a priori knowledge on the interconnection topology. This approach is non-intrusive in the sense that it may be performed by simply sampling the state dynamics, even if the underlying dynamics is unknown. We proved -on the example of the so-called voter model -that the nodes may be ranked according to their influence (the impact on the average opinion of the entire group) by monitoring the time-delayed multi-information and that this ranking closely relates to controllability/observability grammians singular values.

In this paper we investigate how this delayed mutual information approach may be used to reconstruct the interconnection topology (for instance the incidence matrix when the dynamics is defined on a graph). Our goal is to develop an approach which is only based on the sampling of the state dynamics and could be further applied online, for instance with a moving time frame for the computation of the mutual information.

In many practical situations, the topological structure of a complex network is unknown or uncertain and the topological structure of a complex network plays a pivotal role in it dynamic, and control. Therefore, exploring the underlying topology of a complex network is of great importance. This problem has therefore been studied in many papers and some articles propose an information theoretic approach. In neuroscience this approach seems to be preferred. For example, in the papers [START_REF] Becq | Détection dynamique de graphes appliquée à l'analyse de signaux électroencéphalographiques issus du sommeil de rat[END_REF] and [START_REF] Amblard | The relation between granger causality and directed information theory: A review[END_REF], the authors were interested in constructing the graph of interactions between the areas of the brain applied to encephalogram analysis. They identified the topology of this complex system using the transfer entropy [START_REF] Schreiber | Measuring information transfer[END_REF]) and the Granger causality concept [START_REF] Granger | Some recent development in a concept of causality[END_REF]). [START_REF] Wilmer | Timedelayed mutual information of the phase as a measure of functional connectivity[END_REF] used a time-delayed mutual information for detecting nonlinear synchronization in electrophysiological data.

Ideally the approach should be effective when changes occur in the topology, making us able to detect theses changes and reconstruct quickly the new system topology. We will consider again -as an illustration example -a probabilistic voter model where the vote dynamics is defined on a scale-free graph representing somehow the influence between agents in a social network. The quantitative nature of the mutual information and multi-information suggests a new way to detect groups or communities in the complex system, based on the information flux between these communities. We will show how this partitioning algorithm works on the example of the voter model and will suggest that it could be an alternative approach either for coarse graining or partitioning, or for estimation, control or diagnosis purpose (see for instance [START_REF] Ocampo-Martínez | Partitioning approach oriented to the decentralised predictive control of large-scale systems[END_REF]).

The paper is organized as follows : section 2 introduces the voter model which will be used throughout the paper (subsection 2.1) and the metrics from the theory of information that we will use (subsection 2.2), together with a summary of results previously obtained when using these metrics to measure the relative influence of the agents in the voter model. Our main contributions are presented in section 3 where it is shown that time-delayed multi-information and mutual information allow us to reconstruct the interconnection topology of a complex system (subsection 3.1) and partition the graph into communities which are defined from the information flows (subsection 3.2).

THE VOTER MODEL

Description of the model

Simple models that abstracts the process of opinion formation have been proposed by many researchers [START_REF] Castellano | Statistical physics of social dynamics[END_REF]; [START_REF] Galam | Competing species dynamics: Qualitative advantage versus geography[END_REF]. The version we consider here is an agent-based model defined on a graph of arbitrary topology, whether directed or not.

A binary agent occupies each node of the network. The dynamics is specified by assuming that each agent i looks at every other agent in its neighborhood, and counts the percentage ρ i of those which are in the state +1 (in case an agent is linked to itself, it obviously belongs to its own neighborhood). A function f is specified such that 0 ≤ f (ρ i ) ≤ 1 gives the probability for agent i to be in state +1 at the next iteration. For instance, if f would be chosen as f (ρ) = ρ, an agent for which all neighbors are in state +1 will turn into state +1 with certainty. The update is performed synchronously over all n agents. Formally, the dynamics of the voter model can be express as

s i (t + 1) = 1 with probability f (ρ i (t)) 0 with probability 1 -f (ρ i (t)) (1)
where s i (t) ∈ {0, 1} is the state of agent i at iteration t, and

ρ i (t) = 1 |N i | j∈Ni s j (t).
(2)

The set N i is the set of agents j that are neighbors of agent i, as specified by the network topology.

The global density of all n agents with opinion 1 is obviously obtained as

ρ(t) = 1 n n i=1 s i (t) (3)
In what follows, we will use a particular function f ,

f (ρ) = (1 -)ρ + (1 -ρ) = (1 -2 )ρ + (4)
The quantity 0 ≤ ≤ 1/2 is called the noise. It reflects the probability to take a decision different from that of the neighborhood.

To illustrate the behavior of this model, we consider a random scale-free graph G, as simple instance of a social network [START_REF] Barabási | Scalefree characteristics of random networks: the topology of the world-wide web[END_REF]). We use the algorithm of Béla Bollobás [START_REF] Bollobás | Handbook of graphs and networks: from the genome to the internet[END_REF]) to generate this graph.

Figure 1 shows the corresponding density of agents with opinion 1, as a function of time. We can see that there is a lot of fluctuations due to the fact that states "all 0's" or "all 1's" are no longer absorbing states when = 0.

Delayed Mutual and multi-information

Let us consider a set of random variables X i (t) associated with each agent i, taking their values in a set A. For instance, X i (t) = s i (t) would be the opinion of agent i at iteration t.

Since we want to assess the temporal causality, we measure the influence of the vote of agent i at time t on the vote of agent j at time t + τ , we define the τ -delayed mutual information w i,j as

ω i,j (t, τ ) = I(X i (t), X j (t + τ )) (5) = (x,y)∈A 2 p xy log p xy p x p y (6)
with p xy = P(X i (t) = x, X j (t + τ ) = y) and p x = P(X i (t) = x) and p y = P(X j (t + τ ) = y)

We also define the τ -delayed multi-information w i as a measure of the influence of agent i on all the others

ω i (t, τ ) = I(X i (t), Y i (t + τ )) (7)
Y i (t + τ ) = k =i X k (t + τ ) (8)
These information metrics can be computed by sampling.

In the sequel we will consider N = 10 5 instances of the system in order to perform an ensemble average. At the begining of each simulation, the probabilities at time zeros are reset, and the random seed is changed. For example to compute an approximation of the joint probability of (X i (t), X j (t + τ )), we count among these N simulations the number of couple (k, l) ∈ {0, 1} 2 that we obtained.

According to the central limit theorem, we know that, with this number of instances, we obtain a precision of 3 × 10 -2 with a risk of 5% for the approximate values of the probabilities that we compute.

The τ -delayed multi-information can be used as a measure of the influence of opinion of each node i on the vote of the other agents. For instance, Fig. 2 shows ω i (τ = 2) in a steady state, where the origin of time is arbitrary. We observe that some agents i exhibit a more pronounced peak of multi-information towards the rest of the system, suggesting that the opinion of these agents may affect the global opinion of all agents. Note that this results is obtained only by probing the systems, without modifying any of its components. For this reason, we describe this approach as "non-intrusive".

Controllability and information theoretical

In the paper "Controllability of the Voter Model : an information theoretic approach" (Toupance ( 2019)), we showed that the delayed multi-information is indeed a good metrics to identify the influential agents of the system. To illustrate this result we compared w i (τ ) with the impact of forcing the vote of agent i to 1 at all time.

As a result of this forcing, the density of vote

ρ(t) = 1 n n j=1 s j (t) (9)
oscillates around a value different from that observed when agent i is free. This variation of the average value of ρ is defined as the influence of agent i on the whole system. This measure of influence is "intrusive" as it is the result of an action on the system.

The two metrics (intrusive and non-intrusive) are shown in Fig. 3, with a color representations, for the given scalefree graph. It shows that the multi-information (right Fig. 2. τ -delayed multi-information w i ) as a function of for a test graph G, similar but not identical to the graph shown in Fig. 3. The case show here has n = 50 agents and a noise level = 0.001 and τ -delayed multiinformation is computed at initial state. panel) detects correctly the influence of the nodes panel) since the variation of gray levels are similar in both cases. In this way we can then identify, by non-intrusive observations, which agents are those whose control will be the most influential to the system when their vote is forced.

The strong correlation that exists between influence and the delayed multi-information can be proven rigorously in the case of a 1D unidirectional voting model. In [START_REF] Toupance | Controllability of the voter model: an information theoretic approach[END_REF] it is shown that the probability π i that agent i votes 1 when the system is stationary, is

π i = 1 2 + 1 2 exp - i c ( 10
)
where c is defined as

c = 1 ln 1+2 1-2 (11)
We have also shown that delayed mutual information decreases exponentially with the distance between two agents and the noise. By simulation, we obtained that the multi-information between agents i and j (where

j > i) with a delay τ = j -i is ω i,j (j -i) = α i exp (-λ i (j -i))
where λ i depends on the noise level, . Figure 2.3 shows that λ i is proportional to 1/ c , confirming the strong link between our concept of influence and that of time-delayed multi-information.

TOPOLOGY OF THE SYSTEM

1-delayed mutual information and adjacency matrix

After the results presented in the previous section about the link between controllability and information theory, we are interested here in the identification of topology of a system. The aim is to us our information metrics to construct the interaction topology of the unknown graph underlying the dynamics of a complex system. Fig. 3. Scale free graph colored as a function of the values of the influence (left) and the τ -delayed multi-information (right), for τ = 4. The value of τ is chosen so as to match the diameter of the graph. In this case, the multiinformation is computed in the transient regime that follows the initial state.

Figure 4 shows the values of 1-delayed mutual information ω i,j (1) between agent i and all the others in the system. These values were calculated by sampling, when the system has reached its steady state. In this case, the highest values of ω i,j (1) are obtained for the neighbors of agent i.

For i = 42 we observed a peak for agents 15, 36, 42, 44 (the agent is neighbor of itself).

Thus, we can use the 1-Delayed Mutual information to get the edges of a graph. For each agent i, we fixed a threshold T i for ω i,j (1) that indicates that j is a neighbor of i. This threshold is defined as T i = µ i + a i σ i where µ i is the mean of the values of the 1-delayed mutual information between agent i and the others agents, and σ i is the standard deviation of these values.

There are two possible values for a i according to the following criteria where w i (t, τ ) is agent i's τ -delayed multi-information at time t, as defined in eq. ( 7). The value of t is chosen to be in the initial regime and τ is taken large enough to capture the influence over the rest of the system. The values α and β are respectively the average and the standard deviation of w i (t, τ ) over i. In the first case (a i = 0.2), the agent i is considered as very influential.

a i = 0.2 if w i (t, τ ) > α + 1 2 β 0.7 otherwise (12)
The estimation of the adjacency matrix, denoted M = (m i,j ) 1 i,j N , is defined by :

m i,j = 1 if ω i,j (1) > T i or ω j,i (1) > T j 0 otherwise
In other words, when ω i,j (1) > T i or ω j,i (1) > T j , it is assumed that agents i and j are neighbors.

The values of a i were determined by an empirical method.

We have tested all possible values of a from 0 to 1 with Fig. 5. Graph built with 1-delayed mutual information calculated when the system is in a steady state. The dashed red edges are the ones that have not been found, and the solid pink edges are those that were wrongly added.

a step of 0.1 over several scale free graphs, and we have chosen the values in order to minimize the error rate r between this matrix and the actual adjacency matrix A.

The error rate is defined by r = ∆(M,A)

n 2
, where ∆(M, A) is the Hamming distance, namely the number of values that differ between M and A. n is the number of agents in the graph.

For example, Fig. 5, shows the graph that is reconstructed by this procedure, and compares it to the original graph. In this case, the error rate is r = 1.3%. An even better result is obtained when 1-delayed mutual information is computed during the initial transient regime (see Fig. 6). The error rate is now r = 0.24%. In the transient regime the results are probably better because it really probes the direct influences. In order to provoke such a transient regime, one may disrupt the system by temporarily increasing the noise, while calculating the mutual information. This method has been tested by randomly generating 20 scale free graphs. The average error rate obtained is 0.9% and the standard deviation is 0.0026.

Community detection

The existence and structures of communities in a graph is an important concept analysis of social networks. Communities are sub-graph with dense connections and sparse connections between these In our case, partitioning the agents in communities should allow us to better control the network. In [START_REF] Papadopoulos | Community detection in social media[END_REF]) the author presents different method to determine communities. In our case, we will rather use the delayed mutual information. In a scale free graph, we expect a seed-centric approaches for community detection (see [START_REF] Kanawati | Seed-centric approaches for community detection in complex networks[END_REF]). In the voter model these seeds are the most influential nodes, each of them characterizing a different community. In the first part of this paper, we showed that the delayed multi-information can determine Fig. 6. Graph built with the 1-delayed mutual information computed when the system evolves from its initial state. The red edges are the ones that have not been found.

these seeds : if we want to partition into k communities, to find the seeds, we determine the k largest values of multiinformation.

Then, to build the corresponding community, we define the proximity p(i, j) between two agents i and j as

p(i, j) = 1 r r τ =1 ω i,j (τ ) (13)
where r is an approximation of radius of the graph. [START_REF] Bollobás | The diameter of a scale-free random graph[END_REF] proved that the diameter of a scale-free random graph is asymptotically log(n) log(log(n)) where n is the size of the graph. We can choose

r = 1 2 log(n) log(log(n))
this value is approximately the radius of the graph. Each agent j is associated with the community of the seed agent i, where i is obtained as

i = arg max i∈I p(i, j) ( 14
)
where I is the set of seed agents.

The communities obtained with this algorithm are shown in Figs. 7.

To evaluate the quality of the partitioning, we compute their modularity Q (see [START_REF] Newman | Modularity and community structure in networks[END_REF]) defined by :

Q = 1 2m i,j A i,j - q i q j 2m δ(c i , c j ) ( 15
)
where m is the number of edges of the graph, A i,j is the coefficient (i, j) of the adjacency matrix, q i is the degree of agent i, c i is the community of agent i, and δ is the Kronecker symbol, we have δ(c i , c j ) = 1 if i and j are in the same community 0 otherwise For our example, the coefficients of modularity are shown in table 3.2. This indicates a good assortativity of the proposed communities.

CONCLUSION

In this paper we showed that information theory is related to control theory and further that is offers efficient tools to determine the unknown interaction topology of a complex dynamical system. The results are obtained using a sampling approach on a voter model defined on a scale free graph. The key information-theoretic quantities introduced in this work is the τ -delayed mutual information and multi-information. In addition to reconstructing the graph topology, it can be used to determine communities that partition the graph.

In a future work we plan to extend the present results in three directions: (1) to detect possible changes in the graph topology over time. This will be obtained by computing the information metrics over sliding time windows; (2) To used the detected communities to apply different control strategies;

(3) to use community as a way to reduce the complexity of the full system (model reduction).

Fig. 1 .

 1 Fig. 1. Time of of opinion 1 with noise = 0, 001 and n = 200 agents connected through a scale-free network.

Fig. 4 . 1

 41 Fig. 4. 1 Delayed mutual information between agent = 42and the rest of the system. Peaks are visible for the neighbors of i.

Fig. 7 .

 7 Fig. 7. Partition of the graph in 3, 4 and 6 communities.

Table 1 .

 1 Modularity for different partitioning.

	Number of communities	3	4	6
	Q	0.3607 0.4208 0.4382


Yassine Boukal
email: y.boukal@gmail.com

Mohamed Zerrougui
email: mohamed.zerrougui@univ-amu.fr

Ali Zemouche

Rachid Outbib

Nonlinear Observer design for Systems with Sampled Measurements: An LPV Approach

Keywords: Observers design, Lipschitz nonlinarity, Sampled measurements, LMI Approach, LPV system

The aim of this work is to propose a design methodology of observers for a class of Lipschitz nonlinear dynamical systems with sampled measurements by using the differential mean value theorem (DMVT) which allows us to transform the nonlinear part of the estimation error dynamics into a linear parameter varying (LPV) system. The designed observer must ensure the stability of the estimation error subject to a sampled measurements. An LMI-based minimization problem is provided to ensure the stability and the existence of the observer using Lyapunov theory. Thus, the measurements sampling period is included in the LMI as a decision parameter. Indeed, this allows to widen the sampling period as much as possible, which helps optimization of energy consumption while guaranteeing the convergence of the observer. Finally, to illustrate the performance of the proposed methodology, a numerical example is presented.

INTRODUCTION

In real applications, most systems are naturally linear time-varying or nonlinear. Controlling this kind of system is not an easy task, because it requires the development of efficient control boxes with sufficient computation frequency to satisfy the process specifications. Generally, the high number of operation performed, implies a significant energy consumption. This presents a disadvantage in the case of embedded systems in some applications like autonomous mobile systems.

Nonlinear observer design has been a topic of great research in the last decades [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF], [START_REF] Simon | A game theory approach to costrained minmax state estimation[END_REF], [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF], [START_REF] Rajamani | Observers for Lipschitz nonlinear systems[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF], [START_REF] Arcak | Observer-based control of systems with slope-restricted nonlinearities[END_REF], [START_REF] Kravaris | Nonlinear observer design for state and disturbance estimation[END_REF], those ones are required especially when we don't have access to all the system states. The estimated states obtained by the observer are used for different purposes such as the design of control laws for guarantee some desired performance. For example, in the field of autonomous vehicles, measurement of some variables, such as longitudinal distances, velocities and accelerations of other nearby vehicles, requires significant expense in terms of energy. Some of sensors, such as slip angle and roll angle, can be extremely expensive to measure, requiring sensors that cost thousands of dollars [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], [START_REF] Rajamani | Demonstration of integrated longitudinal and lateral control for the operation of automated vehicles in platoons[END_REF]. In addition, several important tasks cannot be performed due to purely and simply unavailability of sensors.

Designing sampled-data observers has attracted much attention in recent years due to the growth of the number of applications where the measurements are sampled, see [START_REF] Bouraoui | Observer design for a class of uncertain nonlinear systems with sampled outputs-application to the estimation of kinetic rates in bioreactors[END_REF] and references therein. One of the major difficulties in the design of sampled-data observers is how to enlarge the sampling intervals and ensure the convergence of the estimations. Some results on the output feedback sampled-data control of nonlinear systems using high-gain observers are proposed in [START_REF] Dabroom | Output feedback sampled-data control of nonlinear systems using high-gain observers[END_REF]. In [START_REF] Ahmed-Ali | Global exponential sampled-data observers for nonlinear systems with delayed measurements[END_REF], the problem of global exponential sampled-data observers design for nonlinear systems with delayed measurements is studied. The exact and Euler approximate models are used by the authors of [START_REF] Abbaszadeh | Robust H ∞ observer design for sampled-data lipschitz nonlinear systems with exact and euler approximate models[END_REF] to design Robust H ∞ observer for sampled-data Lipschitz nonlinear systems.

The aim of this note is to analyze and to design observers for a class of nonlinear systems with sampled measurements by using the LPV approach. Firstly, the nonlinear part of the considered system is analyzed. This is done by using the differential mean value theorem (DMVT) which allows us to transform the nonlinear part of the estimation error dynamics into a linear parameter varying (LPV) system (see [START_REF] Zemouche | On LMI conditions to design observers for Lipschitz nonlinear systems[END_REF]). Moreover, the existence and stability conditions of such observer are given as an LMI-based minimization problem taking into account the sampling period of the measure. In fact, maximize the sampling time enable to reduce the access frequency to the energy source which quantifies the energy consumption. Finally, a simulation example is given to illustrate the effectiveness of the proposed approach.

Notation: In the sequel R n and R n×m will denote the n dimensional Euclidean space and the set of all n × m real matrices, respectively. A T denotes the transpose of matrix A. A is symmetric positive definite matrix if and only if A T = A and A > 0. Sym{X} is used to denote X T + X.

PROBLEM FORMULATION AND PRELIMINARY RESULTS

Preliminaries and useful Lemmas

In this section we introduce some definitions and preliminaries which will be of crucial use in the proposed approach Definition 1. [START_REF] Zemouche | On LMI conditions to design observers for Lipschitz nonlinear systems[END_REF]) Consider two vectors

X =    x 1 . . . x n    ∈ R n and Z =    z 1 . . . z n    ∈ R n .
For all i = 0, ..., n, we define an auxiliary vector X Zi ∈ R n corresponding to X and Z as follows:

                     X Zi =          z 1 . . . z i x i+1 . . . x n          for i = 1, ..., n X Z0 = X (1)
Lemma 1. [START_REF] Zemouche | On LMI conditions to design observers for Lipschitz nonlinear systems[END_REF]). Consider a continuous function Ψ : R n -→ R. Then, for all

X =    x 1 . . . x n    ∈ R n and Z =    z 1 . . . z n    ∈ R n there exist functions ψ j : R n × R n -→ R, j = 1, ..., n such that Ψ(X) -Ψ(Z) = j=n j=1 ψ j X Zj-1 , X Zj e n (j) X -Z (2)
where e n (j) is the j th vector of the canonical basis of R n .

Lemma 2. [START_REF] Zemouche | On LMI conditions to design observers for Lipschitz nonlinear systems[END_REF]). Consider a function Ψ : R n -→ R n . Then, the two following items are equivalent:

• Ψ is γ Ψ -Lipschitz with respect to its argument, i.e.:

Ψ(X) -Ψ(Z) ≤ γ Ψ X -Y , ∀ X, Z ∈ R n (3)
• for all i, j = 1, ..., n, there exist functions

ψ ij : R n × R n -→ R and constants γ ψij ≤ 0, γψij ≥ 0, such that ∀ X, Z ∈ R n , Ψ(X) -Ψ(Z) = i=n i=1 j=n j=1 ψ ij H ij X -Z (4) and -γ Ψ ≤ γ ψij ≤ ψ ij ≤ γψij ≤ γ Ψ (5)
where

ψ ij ψ ij X Zj-1 , X Zj and H ij = e n (i)e n (j)
Lemma 3. Let X and Y two given matrices of appropriate dimensions. the following inequality is true:

X T Y + Y T X ≤ µX T X + 1 µ Y T Y. (6)
Lemma 4. [START_REF] Gu | An integral inequality in the stability problem of time-delay systems[END_REF]). For any constant symmetric positive matrix M ∈ R m×m , scalars t 1 , t 2 and vector function v : [t 1 , t 2 ]→R m then the following inequality holds (Jensen Inequality):

t 2 t 1 v T (β)dβ M t 2 t 1 v(β)dβ ≤ (t 2 -t 1 ) t 2 t 1 v T (β)M v(β)dβ (7)
Remark 1. Lemma 4 can be viewed as a direct application of a variant of Halanay's inequality [START_REF] Hien | New generalized halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems[END_REF] or the well-known Razumikhin's Theorem.

Finally, we introduce the following main lemma, which will be used to guarantee exponential convergence of the estimation error. Lemma 5. If there exist a Lyapunov function V (t, x(t)) and positive scalars α > 0, β > 0, β < α and ϕ = (α -

β)e -ατmax such that d dt V (x(t)) ≤ -αV (x(t)) + βV (x(t k )) (8) ∀t ∈ [t k , t k+1 [, k ≥ 0 (9)
Then, the function V (x(t)) converges exponentially to zero. i.e.

V (x(t)) ≤ e -ϕ(t-t k ) V (x(0)) (10)
Proof. By multiplying the inequality ( 8) by e αt and integrating from t k to t, leads to

t t k e αt d dt V (x(t))dt ≤ t t k -αe αt V (x(t))dt + t t k βe αt V (x(t k ))dt (11)
∀t ∈ [t k , t k+1 [, k ≥ 0 Applying the integration by parts formula to the left hand side of the inequality (11), we get the following inequality

e αt V (x(t)) -e αt k V (x(t k )) - t t k αe αt V (x(t))dt ≤ t t k -αe αt V (x(t))dt + t t k βe αt V (x(t k ))dt (12) ∀t ∈ [t k , t k+1 [, k ≥ 0 which can be simplified as e αt V (x(t)) ≤ e αt k V (x(t k ))+ β α e αt V (x(t k )) - β α e αt k V (x(t k )) (13) ∀t ∈ [t k , t k+1 [, k ≥ 0 or V (x(t)) ≤ e -α(t-t k ) + β α - β α e -α(t-t k ) V (x(t k )) ∀t ∈ [t k , t k+1 [, k ≥ 0. ( 14
)
Let ϕ = (α -β)e -ατmax since β < α, then it can be easily shown that

e -α(t-t k ) + β α - β α e -α(t-t k ) ≤ e -ϕ(t-t k ) (15)
from ( 14) and ( 15) and the fact that V (x(t)) ∈ R + is continuous we obtain the inequality (10). This ends the proof of Lemma 5.

System Description

For simplicity of the presentation and to explain well what we propose in this note, we consider the following triangular form of nonlinear systems as in [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]:

                 ẋ =        ẋ1 ẋ2 . . . ẋn-1 ẋn        =        x 2 x 3 . . . xn f (x)        y = x 1 (t k ) ( 16
)
with f : R n → R satisfies the Lipschitz property formulated under the flowing form:

f (x 1 + ∆ 1 , . . . , x n + ∆ n ) -f (x 1 , . . . , x n ) ≤ γ f n j=1 |∆ j | . (17)
and x ∈ R n denotes the state vector, y ∈ R the measurement output vector is sampled at instants

t k satisfy 0 ≤ t 0 ≤ • • • ≤ t k ≤ t k+1 , the sampling interval τ k = t k+1 - t k is a positive constant satisfy τ k < τ max ∀k ≥ 0,
where τ max is the largest value of the sampling period τ k . System (16) can be written as

ẋ(t) = Ax(t) + Bf (x) y(t k ) = Cx(t k ), ( 18
) with B = [0 . . . 0 1] T , C = [1 0 . . . 0]
where the matrix A is defined as

(A) i,j = 1 if j = i + 1 0 if j = i + 1 .
To estimate the state of the system (18) subject to sampled measurements y(t k ), we consider the following Luenberger observer ẋ(t) = Ax(t)

+ Bf (x) + L y(t k ) -C x(t k ) ( 19
)
The dynamics of the estimation error e(t) = x(t) -x(t) is then given by:

ė(t) = A -LC e(t) + B f (x) -f (x) + LC e(t) -e(t k ) (20)
The aim consists in finding a matrix L that ensures exponential convergence of the estimation error e(t) despite the availability of measures only at instants t k .

MAIN RESULTS

LPV/LMI-Based Approach

In this section, the problem of designing an observer for a non-linear systems is considered, where the non-linear function satisfies the Lipschitz property defined as (17). Then, by using the Lemma 2, this allows us to transform the non-linear part in the dynamic of the estimation error as LPV term.

Since f (.) is γ f -Lipschitz, then following Lemma 2 there are functions ψ j : R n × R n -→ R and constants γ ψj and γψj , such that

f (x) -f (x) = j=n j=1 ψ j e n (j) e (21)
and

γ ψj ≤ ψ j ≤ γψj ( 22
)
where

ψ j ψ j x xj-1 k , x xj
is defined as in Lemma 2. For the sake of brevity, we use only ψ j instead of ψ j x xj-1 , x xj .

Now, define the matrix function

A Ψ = A + B j=n j=1 ψ j e n (j), ∀Ψ ∈ R n (23)
Consequently, the dynamics (20) can be rewritten as

ė(t) = A Ψ -LC e(t) + LC e(t) -e(t k ) (24)
According to ( 22), the vector parameter Ψ belongs to a bounded convex set H n for which the set of vertices is defined by:

V Hn = Φ ∈ R n : Φ j ∈ γ ψj , γψj . (25)
In the following theorem, the problem of designing an observer of Lipschitz systems will be formulated as solving a LMI subject to some constraints. Theorem 1. The observer ( 19) is asymptotically convergent if there exist a symmetric positive definite matrix P and a matrix Y of appropriate dimension such that the following LMI conditions hold:

A(Ψ) T P -C T Y T + PA(Ψ) -YC + YY T + R ≤ -µI, ∀ Φ ∈ V Hn (26) where R = 2τ 2 max δR T R 1 < δ, R = [0, 1, 0, . . . , 0] Hence (t -t k ) < τ max ∀t ∈ [t k , t k+1 [ 0 < β = 2τ 2 max λ max (P )L 2 1 0 < α 1 = µλ min (P ) 0 < α 2 = 2τ max (δ -1))
β < min{α 1 , α 2 } and, the observer gain is given by

L = P -1 Y.
Proof. To analyse the stability of the estimation error, we introduce the following candidate Lyapunov function

V (t) = V 1 (t) + δV 2 (t) (27a) where V 1 (t) = e T (t)P e(t) (27b) V 2 (t) = t t k t ς e 2 (s) 2 dsdς (27c)
The derivative of V 1 (t) along the solution of ( 24) is given as follows

d dt V 1 (t) = e T (t) Sym A(Ψ) -LC T P e(t)
+ Sym e T (t)P LC e(t) -e(t k )

Υ1

(28)

Applying Lemma 3 on Υ 1 , where µ = 1, leads to the following inequality

Υ 1 ≤ e(t) -e(t k ) T C T C e(t) -e(t k ) + e T (t)P LL T P T e(t) (29)
From ( 28) and ( 29), we obtain the following inequality

d dt V 1 (t)≤ e T (t) Sym A(Ψ) -LC T P +P LL T P T e(t) + e(t) -e(t k ) T C T C e(t) -e(t k ) Υ2 (30) Since Υ 2 = e 1 (t) -e 1 (t k ) T e 1 (t) -e 1 (t k ) = e 1 (t) -e 1 (t k ) 2 = t t k ė1 (s)ds 2 ≤ (t -t k ) t t k ė1 (s) 2 ds ≤ (t -t k ) t t k e 2 (s) -L 1 e 1 (t k ) 2 ds ≤ 2τ max t t k e 2 (s) 2 + L 1 e 1 (t k ) 2 ds (31) this leads to d dt V 1 (t)≤ e T (t) Sym A(Ψ) -LC T P + P LL T P T )e(t) + 2τ max t t k e 2 (s) 2 + L 1 e 1 (t k ) 2 ds (32a) which is equivalent to d dt V 1 (t)≤ e T (t) Sym A(Ψ) -LC) T P + P LL T P T )e(t) + 2τ 2 max L 1 e 1 (t k ) 2 + 2τ max t t k e 2 (s) 2 ds (32b)
The dynamic of the Lyapunov function V 2 (t) along the solution trajectories of ( 24) is

d dt V 2 (t) = τ max e 2 (t) 2 - t t k e 2 (s) 2 ds ≤ 2τ 2 max e 2 (t) 2 -2τ max t t k e 2 (s) 2 ds (33)
then, by substituting ( 28) and ( 33) in (27a) we obtain the derivative of V (t) defined in (27a) along the solution trajectories of ( 24) is given as

d dt V (t)≤ e T (t) Sym A(Ψ) -LC) T P + P LL T P T )e(t) + 2τ 2 max δ e 2 (t) 2 + 2τ 2 max L 1 e 1 (t k ) 2 + 2τ max t t k e 2 (s) 2 ds -2τ max δ t t k e 2 (s) 2 ds
From the above inequality, we obtain

d dt V (t) ≤ e T (t) Sym A(Ψ) -LC T P + P LL T P T + 2τ 2 max δR T R e(t) + 2τ 2 max L 1 e 1 (t k ) 2 -2τ max (δ -1) t t k e 2 (s) 2 ds (34)
Let 1 < δ, from (27c), we have

V 2 (t) = t t k t ς e 2 (s) 2 dsdς ≤ τ max t t k e 2 (s) 2 ds (35) and,
-2τ max (δ -1)

t t k e 2 (s) 2 ds ≤ - 3 2 (δ -1)V 2 (t) (36) Now, let Sym A(Ψ) -LC T P + P LL T P T + 2τ 2 max δR T R ≤ -µI (37) then, d dt V (t) ≤ -µe T (t)e(t) -2τ max (δ -1) t t k e 2 (s) 2 ds + 2τ 2 max L 1 e 1 (t k ) 2 d dt V (t) ≤ -α 1 V 1 (t) -α 2 V 2 (t) + βV (t k ) where 0 < β = 2τ 2 max λ max (P )L 2 1 0 < α 1 = µλ min (P ) 0 < α 2 = 2τ max (δ -1) d dt V (t) ≤ -µV (t) + βV (t k ) where α = max{-α 1 , -α 2 }
Then, according to Lemma 5, we have

V (x(t)) ≤ e -ϕ(t-t k ) V (x(t k )) ∀t ∈ [t k , t k+1 [, k ≥ 0 ( 38
)
where ϕ = (α -β)e -αT and β < α, then V (t) converge exponentially toward zero. This completes the theorem proof.

Remark 2. Let us distinguish between 2 cases. In the first case where α = α 1 , we deduce from the fact β < α = α 1 that

β < α 1 ⇒ τ max < T 1 = 2µλ min (P ) 3λ max (P )L 2 1 (39)
In the second case where α = α 2 , from the fact β < α = α 2 we deduce that

β < α 2 ⇒ τ max < T 2 = (δ -1) λ max (P )L 2 1 (40)
The obtained conditions (39) and ( 40) are summarised as τ max < min{T 1 , T 2 } (41)

NUMERICAL EXAMPLE

In this section, the performance of the proposed oberver ( 19) is tested using a numerical example. We consider a two-dimensional nonlinear system under the triangular form ( 16) given as follows

A = 0 1 0 0 , B = 0 1 , C = [1 0], (42a)
and the nonlinear function f (x) = [0 0.3 * sin(x 1 )] T satisfies the Lipschitz property (17).

However, applying the proposed approach, we obtain

A(α 1 ) = 0 1 -L f -L f , A(α 2 ) = 0 1 L f -L f , (43a)
A(α 3 ) = 0 1 -L f L f , A(α 4 ) = 0 1 L f L f , (43b)
Then, by solving the LMI-based minimization problem proposed in Theorem 1 leads to deduces the matrix gain of the observer ( 19) and the sampling time. This can be done by using any toolbox dedicated for solving this kind of optimization problem.

A feasible solution of the optimization problem defined as ( 26) is given by the following matrices

for τ max = 0.1s which ensures the stability of the estimation error.

To illustrate the effectiveness of the proposed method in this work, the evolution of the system states ant their estimates are plotted in figures 1 and 2. In addition, the sampled measurement and sampled output of the observer are also plotted in (3). In this work, a design methodology of observers for a class of Lipschitz nonlinear dynamical systems with sampled measurements is investigated. Firstly, the nonlinear part of the estimation error dynamics is reformulated by using the differential mean value theorem (DMVT). Then, to ensure the stability of the estimation error subject to sam-pled measurements, an LMI-based optimization problem is addressed such that, the measurements sampling period is taken as a decision parameter. This allows to reduce the energy consumption. Finally, a simulation result is presented to illustrate the performance of the proposed methodology.

As future work, we aim to extend the results of this paper to other interesting estimation problems, namely for systems with noises and uncertainties. We also plan to extend the main ideas of this paper to systems with time-varying sampling period and event-triggered measurements.

Fig. 1 .

 1 Fig.1. The evolution of x 1 (t) and its estimate x1 (t) Due to scale effect, the figures 1, 2 and 3 are zoomed in order to show the convergence of the estimated states to the real states.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 2. evolution of x 2 (t) and its estimate x2 (t)