event-based boundary control of networks of conservation laws · n. espitia, a. girard, n....

Event-based boundary control of networks of conservationlaws

Nicolas Espitia Hoyos, Antoine Girard, Nicolas Marchand andChristophe Prieur

Universite de Grenoble, Gipsa-lab, Control Systems Department, Grenoble FranceLaboratoire des signaux et systemes, CentraleSupelec , Gif-sur-Yvette, France

This work has been partially supported by the LabEx PERSYVAL-Lab(ANR-11-LABX-0025-01)

Event-based boundary control of networks of conservation laws

1D-Hyperbolic partial differential equations

Modeling of physical networks

Hydraulic: Saint-Venant equations for open channels [Bastin,Coron, and d’Andrea-Novel; 2008];

Road traffic [Coclite, Garavello and Piccoli; 2005];

Data/communication: Packets flow on telecommunicationnetworks [D’Apice, Manzo and Piccoli; 2006].

Gas pipeline : Euler equations [Gugat, Dick and Leugering; 2011];

Electrical lines : Transmission and wave propagation[Magnusson,Weisshaar,Tripath and Alexander; 2000];

· · ·

Event-based boundary control of these applications

To propose a framework for event-based control of hyperbolic systems.A rigorous way to implement digitally continuous timecontrollers for hyperbolic systems.

To reduce control and communication constraints.

[Bastin et al., 2008, Coclite et al., 2005, Gugat et al., 2011,D’Andrea-Novel et al., 2010, D’Apice et al., 2006, Magnusson et al., 2000]


Outline

1 Networks of conservation lawsFluid-flow modelingISS stability

2 Event-based control of linear hyperbolic systemsEvent-based stabilization

3 Conclusion and Perspectives


Networks of conservation laws

Fluid-flow modeling

Fluid-flow modeling - communication networksCompartmental representation

1 2

34

5

d1

d3

d2

e5

e4

q45

q25

q32

q14

q12

q34

Figure: Example of a compartmental network.

1 In is the set of the number of compartments, numbered from 1 to n.2 Di ⊂ In is the index set of downstream compartments connected

directly to compartment i (i.e. those compartments receiving flow fromcompartment i).

3 Ui ⊂ In is the index set of upstream compartments connected directlyto compartment i (i.e. those compartments sending flow tocompartment i).

4 / 17



Fluid-flow modeling

Transmission lines

Transmission lines may be modeled by the following conservation laws[D’Apice, Manzo, Piccoli; 2008]:

∂tρij(t, x) + ∂xfij(ρij(t, x)) = 0

ρij(t, x) is the density of packets;

fij(ρij(t, x)) is the flow of packets, x ∈ [0, 1], t ∈ R+, i ∈ In, j ∈ Di.

σijρijρmax

ij

f(ρij)

Figure: Fundamental triangulardiagram of flow-density

fij(ρij) =

{

λijρij , if 0 ≤ ρij ≤ σij

λij(2σij − ρij), if σij ≤ ρij ≤ ρmaxij

σij is the critical density - free flowzone and congested zone.

λij is the average velocity of packetstraveling through the transmission line.

5 / 17



Fluid-flow modeling

We focus on the case in which the network operates in free-flow, i.e.

fij(ρij) = λij · ρij

for 0 ≤ ρij ≤ σij .

Let us denote the flow fij(ρij) := qij .

We rewrite the conservation laws as Kinematic wave equation [Bastin,Coron, d’Andrea-Novel; 2008]: [Bastin et al., 2008]

Linear hyperbolic equation of conservation laws.

∂tqij(t, x) + λij∂xqij(t, x) = 0

6 / 17



Fluid-flow modeling

Servers: Buffers and routers

zi(t)+

θi(zi)

vi(t) ri(zi)

(1− wi)vi

di(t)

Figure: Compartment: buffer.

Dynamics for each buffer i ∈ In(see e.g. Congestion control incompartmental network systems [Bastin, Guffens; 2006]): [?]

zi(t) = vi(t)− ri(zi(t))

vi is the sum of all input flows getting into the buffer;

ri is the output flow of the buffer (processing rate function) .

with vi(t) = di(t) +∑

k 6=ik∈Ui

qki(t, 1).7 / 17



Fluid-flow modeling

Control functions and dynamic boundary condition

Control functions

1 wi: To modulate the input flow vi and reject information.

2 uij : To split the flow through different lines.

zi(t) = wi(t)di(t) +∑

k 6=ik∈Ui

wi(t)qki(t, 1)− ri(zi(t)), wi(t) ∈ [0, 1]

Dynamic boundary condition

qij(t, 0) = uij(t)ri(zi(t))

Splitting control (routing control): uij(t) ∈ [0, 1], j ∈ Di, i ∈ In.

The output function for each output compartment i ∈ Iout is given by

ei(t) = ui(t)ri(zi(t))

with∑

i6=jj∈Di

uij(t) + ui(t) = 1.8 / 17



Fluid-flow modeling

Linearized system around an optimal free-flow equilibrium point

Coupled linear hyperbolic PDE-ODE.{

∂ty(t, x) + Λ∂xy(t, x) = 0

Z(t) = AZ(t) +Gyy(t, 1) +BwW (t) +Dd(t)

with dynamic boundary condition

y(t, 0) = GzZ(t) +BuU(t)

and initial condition

y(0, x) = y0(x), x ∈ [0, 1]

Z(0) = Z0.

P

9 / 17



ISS stability

Input-to-State stability ISS

The system P is Input-to-State Stable (ISS) with respect tod ∈ Cpw(R

+;Rn), if there exist ν > 0, C1 > 0 and C2 > 0 such that, forevery Z0 ∈ R

n, y0 ∈ L2([0, 1];Rm), the solution satisfies, for all t ∈ R+,

(‖Z(t)‖2 + ‖y(t, ·)‖2L2([0,1],Rm)) ≤

C1e−2νt(‖Z0‖2 + ‖y0‖2L2([0,1];Rm)) +C2 sup

0≤s≤t

‖d(s)‖2 (1)

C2 is called the asymptotic gain (A.g).

10 / 17



ISS stability

Contributions on this framework:

Modeling of communication networks under fluid-flow andcompartmental representation;

Characterization of suitable operating points for the network;

Open-loop analysis (Lyapunov-based):Sufficient condition for ISS - LMI formulation;Asymptotic gain estimation;

Closed-loop analysis (Lyapunov-based):Control synthesis to improve the performance of the network;LMI formulation;Control constraints.Minimization of the Asymptotic gain;

W (t) =[

Kz Ky

]

[

Z(t)y(t, 1)

]

U(t) =[

Lz Ly

]

[

Z(t)y(t, 1)

]

C

11 / 17



ISS stability


Modeling of communication networks under fluid-flow andcompartmental representation;

Characterization of suitable operating points for the network;

Open-loop analysis (Lyapunov-based):Sufficient condition for ISS - LMI formulation;Asymptotic gain estimation;

Closed-loop analysis (Lyapunov-based):Control synthesis to improve the performance of the network;LMI formulation;Control constraints.Minimization of the Asymptotic gain;

W (t) =[

Kz Ky

]

[

Z(t)y(t, 1)

]

U(t) =[

Lz Ly

]

[

Z(t)y(t, 1)

]

C

12 / 17



ISS stability

Numerical simulation

z3(t)

q23(t, x)

z4(t)

q24(t, x)

e4(t)

z1(t)

q12(t, x)

z2(t)

q13(t, x)

d1(t)

(1− w1(t))d1(t)

(1− w2(t))q12(t, 1)

(1− w3(t))(q13(t, 1) + q23(t, 1))

(1− w4(t))(q24(t, 1) + q34(t, 1))

q34(t, x)

u24(t)r2(z2(t))

u23(t)r2(z2(t))u12(t)r1(z1(t))

u13(t)r1(z1(t))

Figure: Network of compartments made up of 4 buffers and 5 transmission lines.

13 / 17



ISS stability

Exogenous input flow demand

0 5 10 15 20 25 30 35 4098

99

100

101

102

103

104

Time[s]

Asymptotic gain in open loop:40.48

Total output flow of the network

5 10 15 20 25 30 35 40

79

80

81

82

83

84

85

86

87

Time[s]

e4

Asymptotic gain in closed loop:3.3

14 / 17


EBC of Lin. hyperbolic sys

Event-based stabilization

EBC of Linear hyperbolic system of conservation laws

ETM

K

C

P

tk

∂∂ty(t, x) + Λ ∂

∂xy(t, x) = 0

y(t, 0) = Hy(t, 1) +Bu(t)

y(t, 1)u(t) = Ky(tk, 1)

ϕ


Event-triggered mechanisms (Lyapunov-based);

tk+1 = inf{t ∈ R+|t > tk ∧ some suitable triggering condition}

ISS static event-based stabilization ϕ1;D+V event-based stabilization ϕ2;ISS dynamic event-based stabilization ϕ3;

15 / 17


Conclusion

Conclusion and Perspectives

Modeling and dynamic boundary control of Coupled PDE-ODE.

Extension of event-based controls developed for finite-dimensionalsystems to linear hyperbolic systems by means of Lyapunov techniques;

New way of sampling in time in order to implement digitallycontinuous time controllers for linear hyperbolic systems;

Perspectives:

Self-triggered implementations;

Event-based boundary control of parabolic equations.

16 / 17



Fluid-flow modeling

About my P.h.D

Starting date: October 2014;

Expected defense date: September 2017.

Publications:Peer reviewed international journals

N. Espitia, A. Girard, N. Marchand, C. Prieur “Event-based control oflinear hyperbolic systems of conservation laws”. Automatica, Vol 70,pp.275-287, 2016.

Under review international journals

N. Espitia, A. Girard, N. Marchand, C. Prieur “Event-based boundarycontrol of 2× 2 linear hyperbolic systems via Backstepping approach”Under review as a technical note to IEEE Transactions on AutomaticControl.



Fluid-flow modeling

Peer reviewed international conferences

N. Espitia, A. Girard, N. Marchand, C. Prieur “Fluid-flow modelingand stability analysis of communication networks”. IFAC WorldCongress, Toulouse, France, 2017.

N. Espitia, A. Girard, N. Marchand, C. Prieur “Event-basedstabilization of linear systems of conservation laws using a dynamictriggering condition”. Proc. of the 10th IFAC Symposium on NonlinearControl Systems (NOLCOS), 2016.

Under review international conferences

N. Espitia, A. Girard, N. Marchand, C. Prieur “Dynamic boundarycontrol synthesis of coupled PDE-ODEs for communication networksunder fluid flow modeling”. Submitted to the 56th IEEE Conference onDecision and Control (CDC) 2017.

N. Espitia, A.Tanwani, S. Tarbouriech “Stabilization of boundarycontrolled hyperbolic PDEs via Lyapunov-based event triggeredsampling and quantization”. Submitted to the 56th IEEE Conferenceon Decision and Control (CDC) 2017.



Fluid-flow modeling

Thank you for your attention!



Fluid-flow modeling

Bastin, G., Coron, J.-M., and d’Andrea Novel, B. (2008).Using hyperbolic systems of balance laws for modeling, control andstability analysis of physical networks.In Lecture notes for the Pre-Congress Workshop on Complex Embeddedand Networked Control Systems, Seoul, Korea.

Coclite, G. M., Garavello, M., and Piccoli, B. (2005).Traffic flow on a road network.SIAM Journal on Mathematical Analysis, 36(6):1862–1886.

D’Andrea-Novel, B., Fabre, B., and Coron, J.-M. (2010).An acoustic model for automatic control of a slide flute.Acta Acustica united with Acustica, 96(4):713–721.

D’Apice, C., Manzo, R., and Piccoli, B. (2006).Packet flow on telecommunication networks.SIAM Journal on Mathematical Analysis, 38(3):717–740.

Gugat, M., Dick, M., and Leugering, G. (2011).Gas flow in fan-shaped networks: Classical solutions and feedbackstabilization.SIAM Journal on Control and Optimization, 49(5):2101–2117.



Fluid-flow modeling

Magnusson, P.-C., Weisshaar, A., Tripathi, V.-K., and Alexander,G.-C. (2000).Transmission lines and wave propagation.CRC Press.



Fluid-flow modeling

Theorem (Control synthesis)

Let λ = min{λij} i∈Inj∈Di

. Assume that there exist µ, γ > 0, a symmetric

positive definite matrix P ∈ Rn×n a diagonal positive matrix

Q ≥ I ∈ Rm×m, as well as control gains Kz, Ky, Lz and Ly of adequate

dimensions, such that the following matrix inequality holds:

Mc =

M1 M2 M3

⋆ M4 0⋆ ⋆ M5

≤ 0

with

M1 = (A+BwKz)TP+P (A+BwKz)+2µλP+(Gz+BuLz)TQΛ(Gz+BuLz);

M2 = P (Gy + BwKy) + (Gz + BuLz)TQΛBuLy;

M3 = PD;

M4 = −e−2µQΛ + LyTBT

u QΛBuLy;

M5 = −γI.

Then, the closed-loop system P is ISS with respect to d ∈ Cpw(R+;Rn), and

the asymptotic gain (A.g) satisfies

A.g ≤γ

2µλe2µ.



Fluid-flow modeling

Optimization issues and control constraints

minimizeγ

2νe2µ

subject to Mc ≤ 0;

‖Kzi‖ ≤pδwiβz

; ‖Kyi‖ ≤(1− p)δwi

βy

; ‖Lzi‖ ≤δuij

βz



Fluid-flow modeling

NT = 8000 with ∆t = 1× 10−3.

0

0.005

0.01

0.015

0.02

0.025

0.03

10−3 10−2.5 10−2 10−1.5 10−1 10−0.5 100 100.5 10Inter-execution times

Den

sity

ϕ1

Mean value of triggering times: 158.3events;

Mean value inter-execution times:0.0432.

0

0.005

0.01

0.015

0.02

0.025

10−2.5 10−2 10−1.5 10−1 10−0.5 100

Inter-execution times

Den

sity

ϕ3

Mean value of triggering times: 109.1events;

Mean value inter-execution times:0.0640.

event-based boundary control of networks of conservation laws · n. espitia, a. girard, n....

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