Event-based control of networks modeled by a class of ...researchers.lille.inria.fr/~nespitia/PhD_defenseNicolasESPITIAV2.pdfNetworks of conservation laws Event-based stabilization

Event-based control of networks modeled by a class ofinfinite-dimensional systems

Nicolas Espitia Hoyos

Supervisors: Antoine Girard, Nicolas Marchand and Christophe Prieur

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

Ph.D Defense

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

Networks of conservation laws Event-based stabilization EBC-Backstepping approach Conclusion

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 telecommunication networks[D’Apice, Manzo and Piccoli; 2006];

· · ·

[“Stability and Boundary Stabilization of 1-D Hyperbolic Systems”, Bastinand Coron; 2016].

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 time controllers

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,Tabuada, 2007, Girard, 2015, Bastin and Coron, J.-M., 2016]Event-based control of networks modeled by a class of infinite-dimensional systems/PhD Defense 2017

2 / 45


Outline

1 Networks of conservation laws (Chapter 1)Fluid-flow modelingISS stability

2 Event-based stabilization (Chapter 2)ISS static event-based stabilization

3 EBC-Backstepping approach (Chapter 4)

4 Conclusion and Perspectives

Event-based control of networks modeled by a class of infinite-dimensional systems/PhD Defense 2017

3 / 45


Fluid-flow modeling - communication networks

1

2

3

4d1 e4

q23

q24

q13

q12

q34

Figure: Example of a compartmental network.

1 In: set of the number of compartments, numbered from 1 to n.

2 Di ⊂ In: index set of downstream compartments connected directly tocompartment i (i.e. those compartments receiving flow from compartment i).

3 Ui ⊂ In: index set of upstream compartments connected directly tocompartment i (i.e. those compartments sending flow to compartment i).

Index sets involved in the example:

In = {1, 2, 3, 4}, U1 = ∅, U2 = {1}; U3 = {1, 2}, U4 = {2, 3}D1 = {2, 3}, D2 = {3, 4}, D3 = {4}, D4 = ∅.


4 / 45


Transmission linesTransmission 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 flow zoneand congested zone.

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


5 / 45


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

fij(ρij) = λij · ρijfor 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 / 45


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 / 45


The output flow ri(zi) (processing rate function):

ri(zi) =zi

θi(zi)

The residence time is the averaged time at which packets stay in the serverwhen being processed.

θi(zi) =1 + zi

ǫi

with ǫi > 0 as the maximal processing capacity of each server.

Processing rate function

ri(zi) =ǫizi

1 + zi


8 / 45


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.


9 / 45


The complete model for the network is:

{

∂tqij(t, x) + λij∂xqij(t, x) = 0

zi(t) = wi(t)di(t) +∑

k6=ik∈Ui

wi(t)qki(t, 1) − ri(zi(t))(1)

with dynamic boundary condition

qij(t, 0) = uij(t)ri(zi(t)), ri ≥ 0 (2)

output functionei(t) = ui(t)ri(zi(t)) (3)

and initial conditions{

qij(0, x) = q0ij(x), x ∈ [0, 1]

zi(0) = z0i .(4)


10 / 45


Free-flow steady-state

For a given constant input flow demand d∗i ,

the system (1)-(4) has infinitely many equilibrium points{q∗

ki, z∗i , u

∗ij , u

∗i , w

∗i , e

∗i }

w∗i d

∗i +

∑

k 6=ik∈Ui

w∗i q

∗ki − ri(z

∗i ) = 0

q∗ij = u∗ijri(z

∗i )

e∗i = u∗i ri(z

∗i ).

We assume that the system admits a free-flow steady-state;

Multi-objective optimization problem.

Maximizing the total output flow rate of the network;Minimizing the total mean travel time.


11 / 45


Linearized system around an optimal free-flowequilibrium 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 (B.C)

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

and initial condition (I.C)

y(0, x) = y0(x), x ∈ [0, 1]

Z(0) = Z0.

P


12 / 45


ISS properties for Hyperbolic systems

. [Prieur, Mazenc; 2012];

. [Prieur, Winkin, Bastin; 2008];

. [Tanwani, Prieur,Tarbouriech; CDC 2016].

Definition (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 (5)

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


13 / 45


Contributions on this framework:

Modeling of communication networks under fluid-flow and compartmentalrepresentation;

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


14 / 45



Modeling of communication networks under fluid-flow and compartmentalrepresentation;

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


14 / 45


Closed-loop setting objective:

Reduce the asymptotic gain.

Therefore, the linearized coupled PDE-ODE system in closed-loop with Cbecomes:

{

∂ty(t, x) + Λ∂xy(t, x) = 0

Z(t) = (A+BwKz)Z(t) + (Gy +BwKy)y(t, 1) +Dd(t)

B.C:

y(t, 0) = (Gz +BuLz)Z(t) +BuLyy(t, 1)

I.C:{

y(0, x) = y0(x)

Z(0) = Z0

P


15 / 45


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

uQΛBuLy;

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.


16 / 45


Optimization issues and control constraints

minimizeγ

2νe2µ

subject to Mc ≤ 0;

‖Kzi‖ ≤pδwiβz

; ‖Kyi‖ ≤(1− p)δwi

βy; ‖Lzi‖ ≤

δuij

βz


17 / 45


Numerical simulation

Running example

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.


18 / 45


Exogenous input flow demand

0 5 10 15 20 25 30 35 4098

99

100

101

102

103

104

Time[s]

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

In open loop (black line) - Asymptotic gain: 40.48

In closed loop (red dashed line) - Asymptotic gain: 3.3


19 / 45


Event-based control (Chapter 2)


20 / 45


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 (ETM);

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

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


21 / 45


There exists a causal operator ϕ

ϕ : Cpw(R+,Rn) → Cpw(R

+,Rm)

u = ϕ(y(·, 1))

y0 ∈ Cpw([0, 1],Rn), y(·, 1) ∈ Cpw(R

+,Rn)

There exists a unique solution to

the closed-loop system withcontroller u = ϕ(y(·, 1))

ϕ

Stability Analysis

?ASSUMPTION [A]

N. Espitia, A. Girard, N. Marchand, C. Prieur “Event-based control of linear hyperbolic systems of

conservation laws” Automatica; 2016.


22 / 45


ϕ1 : ISS static event-based stabilization

Let the control function be defined

u(t) = 0 ∀t ∈ [tu0 , tu1 )

u(t) = Ky(tuk , 1) ∀t ∈ [tuk , tuk+1), k ≥ 1

(6)

Let us rewrite the boundary condition:

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

= Hy(t, 1) +BKy(tuk , 1)

= (H +BK)y(t, 1) +BK(−y(t, 1) + y(tuk , 1))

= Gy(t, 1) + d(t)

d is seen as a deviation between the continuous controller u = Ky(t, 1) andthe event based controller (6).

Seek for ISS property with respect to d!


23 / 45


ϕ1 : ISS static event-based stabilization

Let the control function be defined

u(t) = 0 ∀t ∈ [tu0 , tu1 )

u(t) = Ky(tuk , 1) ∀t ∈ [tuk , tuk+1), k ≥ 1

(6)

Let us rewrite the boundary condition:

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

= Hy(t, 1) +BKy(tuk , 1)

= (H +BK)y(t, 1) +BK(−y(t, 1) + y(tuk , 1))

= Gy(t, 1) + d(t)

d is seen as a deviation between the continuous controller u = Ky(t, 1) andthe event based controller (6).

Seek for ISS property with respect to d!


23 / 45


Recall:

Let G = H + BK. The design of matrix K ∈ Rn×m such that the

closed-loop system is GES relies on : [Coron, J.-M. et al., 2008]

Sufficient condition for exponential stability[Coron, Bastin, d’Andrea-Novel; 2008]

ρ1(G) = Inf{

‖∆G∆−1‖; ∆ ∈ Dn,+

}

< 1

Dn,+: set of diagonal positive definite matrices.

Proposition [Diange, Bastin, Coron; 2012]

ρ1(G) < 1 if and only if there exist µ > 0, and a diagonal positive definitematrix Q ∈ R

n×n such that the following matrix inequality holds

GTQΛG < e

−2µQΛ. (7)


24 / 45


Let us consider the Lyapunov function

V (y) =

∫ 1

0

y(x)TQy(x)e−2µxdx (8)

Let us note that for all t > 1λwith λ = mini=1,..,n{λi},

V (y(t, ·)) =n∑

i=1

Qii

∫ 1

0

[

Hiy(t−xλi, 1) +Biu(t−

xλi)]2

e−2µx

dx (9)

Computing the right time-derivative of V , we obtain

D+V = y

T (·, 0)QΛy(·, 0) − yT (·, 1)e−2µ

QΛy(·, 1)

− 2µ

∫ 1

0

yT (Λe−2µx

Q)ydx


25 / 45


Under the sufficient condition for stability ρ1(G) < 1, we get strictLyapunov condition + ISS property

D+V ≤ −2νV + ρ‖d‖2 (10)

for suitable values of ν and ρ.

We rewrite

D+V ≤ −2ν(1− σ)V + (−2νσV + ρ‖d‖2), σ ∈ (0, 1) (11)

Therefore, we introduce the increasing sequence of time instants (tuk) beingdefined iteratively by tu0 = 0, tu1 = 1

λand for all k ≥ 1,

tuk+1 = inf{t ∈ R+|t > tuk ∧ ‖BK(−y(t, 1) + y(tuk), 1)‖

2 ≥ 2σνρV (t)}

(12)


26 / 45


Under the sufficient condition for stability ρ1(G) < 1, we get strictLyapunov condition + ISS property

D+V ≤ −2νV + ρ‖d‖2 (10)

for suitable values of ν and ρ.

We rewrite

D+V ≤ −2ν(1− σ)V + (−2νσV + ρ‖d‖2), σ ∈ (0, 1) (11)

Therefore, we introduce the increasing sequence of time instants (tuk) beingdefined iteratively by tu0 = 0, tu1 = 1

λand for all k ≥ 1,

tuk+1 = inf{t ∈ R+|t > tuk ∧ ‖BK(−y(t, 1) + y(tuk), 1)‖

2 ≥ 2σνρV (t)}

(12)


26 / 45


We define the event-based controller ϕ1

Definition

Let ϕ1 be the operator which maps y(·, 1) to u such that

tu0 = 0, tu1 = 1λ

and for all k ≥ 1,

tuk+1 = inf{t ∈ R

+|t > tuk∧ ‖BK(−y(t, 1) + y(tu

k, 1))‖2 ≥ 2σν

ρV (t) + ε1(t)}

(13)

u(t) = 0 ∀t ∈ [tu0 , tu1 )

u(t) = Ky(tuk, 1) ∀t ∈ [tu

k, tu

k+1), k ≥ 0 (14)

ε1(t) = ςV ( 1λ)e−ηt for all t ≥ 1

λ, is a decreasing function which is crucial to

prove that there is no Zeno phenomena;

Note that due to (9), the triggering condition (13) is a function of the outputonly.

Theorem (Espitia, Girard, Marchand, Prieur; 2016)

The system P with controller u = ϕ1(y(t, 1)) has a unique solution and isglobally exponentially stable. Moreover, it holds for all t ≥ 1

λ,

D+V (t) ≤ −2ν(1− σ)V (t) (15)

[Espitia et al., 2016]Event-based control of networks modeled by a class of infinite-dimensional systems/PhD Defense 2017

27 / 45



Running example.Network of conservation laws without ODE coupling

q3(t, x)

q4(t, x)

e4(t)

q1(t, x)

q2(t, x)

d1(t)

q5(t, x)

u24(t)q1(t, x)

u23(t)q1(t, x)u12(t)d1(t)

u13(t)d1(t)


28 / 45


Total output flow of the network

2 3 4 5 6 7 8

72

74

76

78

80

82

84

86

88

90

92

Time[s]

Continuous time controller ϕ0 (black line)

ISS-static event-based controller ϕ1 (red line)

D+-event-based controller ϕ2 (blue dashed-dot line)

ISS-dynamic event-based controller ϕ3 (green dashed line)


29 / 45


Stabilization of linear hyperbolic systems viaevent-triggered sampling and quantization (Chp 3).

ETMC

P

tk

∂∂ty(t, x) + Λ ∂

∂xy(t, x) = 0

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

u(t) = Kη(t)

ϕ

η(t) = −αη(t) + αq(y(tk, 1))

u(t)

Quantizer∆q

q(y(tk, 1))

y(tk, 1)

∆q


Event-triggered mechanisms (Lyapunov-based) to sample the output y(t, 1);

Well-posedness and ISS with respect to measurements errors (related toquantization) → Practical stability in H1 norm.


30 / 45


Event-based control via Backstepping

approach (Chapter 4)


31 / 45


EBC via Backstepping approach

Let us consider the unstable 2× 2 linear hyperbolic system

Original system

ut(t, x) + λ1ux(t, x) = c1v(t, x)

vt(t, x)− λ2vx(t, x) = c2u(t, x)

B.C u(t, 0) = qv(t, 0)

v(t, 1) = U(t)

Some assumptions on the system parameters (well-posedness issues underevent-based control);

Open loop unstable. Mainly due to coupling terms;

To stabilize it: Backstepping method [Vazquez, Krstic, Coron; 2011];

Full-state feedback.


32 / 45


Main idea: to transform the original “unstable” system into a stable one.

Target system: linear hyperbolic system of conservation laws.

αt(t, x) + λ1αx(t, x) = 0

βt(t, x)− λ2βx(t, x) = 0

B.C α(t, 0) = qβ(t, 0)

β(t, 1) = 0

Global exponentially stable;

Nice stability properties, e.g. finite time convergence to the origin;

We know how to deal with.


33 / 45


Backstepping Volterra transformation which maps the originalsystem into the target system:

α(t, x) = u(t, x)−

∫ x

0

Kuu(x, ξ)u(t, ξ)dξ −

∫ x

0

Kuv(x, ξ)v(t, ξ)dξ

β(t, x) = v(t, x)−

∫ x

0

Kvu(x, ξ)u(t, ξ)dξ −

∫ x

0

Kvv(x, ξ)v(t, ξ)dξ

Inverse transformation which maps the target system into theoriginal system:

u(t, x) = α(t, x) +

∫ x

0

Lαα(x, ξ)α(t, ξ)dξ +

∫ x

0

Lαβ(x, ξ)β(t, ξ)dξ

v(t, x) = β(t, x) +

∫ x

0

Lβα(x, ξ)α(t, ξ)dξ +

∫ x

0

Lββ(x, ξ)β(t, ξ)dξ

Kernels K and L are well-posed.


34 / 45


The backstepping transformation is used to get U(t) under the form([Vazquez, Krstic, Coron; 2011])

U(t) =

∫ 1

0

Kvu(1, ξ)u(t, ξ)dξ +

∫ 1

0

Kvv(1, ξ)v(t, ξ)dξ

or

U(t) =

∫ 1

0

Lβα(1, ξ)α(t, ξ)dξ +

∫ 1

0

Lββ(1, ξ)β(t, ξ)dξ (16)


35 / 45


Event-based control

Original system

ut(t, x) + λ1ux(t, x) = c1v(t, x)

vt(t, x)− λ2vx(t, x) = c2u(t, x)

B.C u(t, 0) = qv(t, 0)

v(t, 1) = Ud(t)

Target “perturbed” system

αt(t, x) + λ1αx(t, x) = 0

βt(t, x)− λ2βx(t, x) = 0

B.C α(t, 0) = qβ(t, 0)

β(t, 1) = d(t)

Ud(t) = U(t) + d(t);

d can be seen as a deviation between the continuous-time controller and theevent-based one.

d(t) =∫ 1

0Lβα(1, ξ)α(tk, ξ)dξ +

∫ 1

0Lββ(1, ξ)β(tk, ξ)dξ

−∫ 1

0Lβα(1, ξ)α(t, ξ)dξ −

∫ 1

0Lββ(1, ξ)β(t, ξ)dξ

(17)


36 / 45


Event-based controller ϕ

Definition

We define ϕ the operator from C0(R+;L2([0, 1];R2)) to Cpw(R+,R) that

maps (α, β)T to Ud as follows:

Let the increasing sequence of time instants (tk) be defined iteratively byt0 = 0 , and for all k ≥ 1,

tk+1 = inf{t ∈ R+|t > tk ∧ θBeµd2 ≥ θσυV (t) −m(t)} (18)

where m satisfies the ordinary differential equation,

m(t) = −ηm(t) +

(

Beµd2 − συV (t) − κ1α2(t, 1)− κ2β

2(t, 0)

)

(19)

Let the control function be defined by:

Ud(t) =

∫ 1

0Lβα(1, ξ)α(tk , ξ)dξ +

∫ 1

0Lββ(1, ξ)β(tk , ξ)dξ (20)

for all t ∈ [tk , tk+1).


37 / 45


On the existence of the minimal dwell-time

Inspired by [Tabuada; 2007] and [Girard; 2015].

Events are triggered to guarantee, for all t > 0,

θBeµd2(t) ≤ θσυV (t)−m(t)

1

tk tk+1t′k

0t

ψ

ψ(tk)

ψ(tk+1)

ψ =θBeµd2+

12m

θσυV −12m

ψ is continuous on[tk, tk+1];

m(t) ≤ 0.

∃t′

k > tk, ∀t ∈ [t′

k, tk+1] ψ(t) ∈ [0, 1]Event-based control of networks modeled by a class of infinite-dimensional systems/PhD Defense 2017

38 / 45


1

tk tk+1t′k

0t

ψ

ψ(tk+1) = 1

ψ(t) ≤ Ψ(t) ∀t ∈ [t′

k, tk+1]

ψ =2θBeµdd+ 1

2m

θσυV − 12m

−(θσυV − 1

2m)

θσυV − 12m

ψ

N. Espitia, A. Girard, N. Marchand, C. Prieur “Event-based boundary control of 2 × 2 linear hyperbolic

systems via Backstepping approach” IEEE TAC; 2017.


39 / 45


1

tk tk+1t′k

0t

ψ

ψ(tk+1) = 1

ψ(t) ≤ Ψ(t) ∀t ∈ [t′

k, tk+1]

ψ =2θBeµdd+ 1

2m

θσυV − 12m

−(θσυV − 1

2m)

θσυV − 12m

ψ

N. Espitia, A. Girard, N. Marchand, C. Prieur “Event-based boundary control of 2 × 2 linear hyperbolic

systems via Backstepping approach” IEEE TAC; 2017.


39 / 45


ψ ≤ a0 + a1ψ + a2ψ2

a0 = Beµε3συ

+ η + ε2 + 1

a1 = −συ + µmax{λ1, λ2}+ η + ε2 + 1 + 12θ

a2 = −συ + 12θ

Then, by the Comparison principle,

ψ(t) ≤ Ψ(t) where Ψ is the solution of

Ψ = a0 + a1Ψ+ a2Ψ2

It follows that the time needed by ψ (or Ψ) to go from ψ(t′

k) = 0 (or

Ψ(t′

k) = 0 ) to ψ(tk+1) = 1 (or Ψ(tk+1) = 1) is at least

τ =

∫ 1

0

1

a0 + a1s+ a2s2ds

Thus,

tk+1 − tk ≥ tk+1 − t′

k ≥ τ

τ a lower bound of the inter-execution times or minimal dwell-time.


40 / 45



Transport velocities: λ1 = 1, λ2 =√2,

Coupling terms: c1 = 1.5, c2 = 2 and q = 1/4.

Initial conditions: u0(x) = qv0(x) with v0(x) = 10(1 − x) for all x ∈ [0, 1].

Trajectories involved in the triggering condition.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time[s]

θσνV − m

θBeµd2

Execution times


41 / 45


Time-evolution of

Continuous-time control U (black dashed line)

Event-based control Ud (blue line with red circle marker)

0.5 1 1.5 2 2.5 3 3.5 4

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

Time[s]


42 / 45


Second components v(t, x) of the closed-loop system

under U(t) under Ud(t)


43 / 45


Conclusion and Perspectives

Modeling of communication networks: coupled PDE-ODEs.

Input-to-state stability of coupled PDE-ODEs and boundary controlsynthesis.

Extension of event-based controls developed for finite-dimensional systems tolinear hyperbolic systems by means of Lyapunov techniques;

Backstepping approach (full-state);

Existence of a minimal dwell-time;

Global exponential stability and well-posedness of the system under E.B.C.


44 / 45


Short term perspectives

To find a “period” using looped functionals in order to impose a dwell-time(In progress).

Periodic event-based control.

ETMC

P

tk

∂∂ty(t, x) + Λ ∂

∂xy(t, x) = 0

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

u(t) = Kη(τj)

η(t) = Aη(t) +Bηy(tk, 1)

u(t)

PTM

τj

∀t ∈ [τj , τj+1) withT = τj+1 − τj , theperiod;

Long term perspectives

Event-triggering conditions depending on observed states (for theBackstepping approach);

Event-based boundary control of parabolic equations for the modeling ofnetworks.


45 / 45

Thank you for your attention!

Optimization issues and control constraints

minimizeγ

2νe2µ

subject to Mc ≤ 0;

‖Kzi‖ ≤pδwiβz

; ‖Kyi‖ ≤(1− p)δwi

βy; ‖Lzi‖ ≤

δuij

βz

Change of variables:

X = P−1;

Q3 = (QΛ)−1;

YKz= KzX , YLz

= LzX, YKy= KyQ3 and YLy

= LyQ3

Mc is given by

XAT + AX + 2µλX + Y TKz

BTw + BwYKz GyQ3 + BwYKy D XGT

z + Y TLz

BTu

⋆ −e−2µQ3 0 Y TLy

BTu

⋆ ⋆ −γI 0⋆ ⋆ ⋆ −Q3

Combined with a line search on µ,

minimizeγ

2νe2µ

subject to Mc ≤ 0;(

ηHe(X)−η2I YKz

⋆

(

pδwiβz

)2

I

)

≥ 0;

(

ηHe(X)−η2I YLz

⋆

(

pδuijβz

)2

I

)

≥ 0

(

ηHe(Q3)−η2I YKy

⋆

(

(1−p)δwiβy

)2

I

)

≥ 0;

ϕ3: ISS dynamic event-based stabilization

This approach uses the static triggering condition previously introduced.

ISS static event-based stabilization

Events are triggered so that ‖d‖2 − κV − ε is always less than 0

We impose that the weighted averaged value of ‖d‖2 − κV − ε is lessthan 0.

Internal dynamic:

m(t) = e−ηt

∫ t

1λ

eηs(

−κV (s)− ε(s) + ‖d(s)‖2)

ds ∀t ≥ 1λ

Consider the following Lyapunov function candidate W ,

W (y,m, ε) = V (y) + ρ

η−2ν(1−σ)ε− ρm (21)

Computing the right time-derivative of W

D+W = D

+V − η ρ

η−2ν(1−σ)ε− ρ(−ηm− κV − ε+ ‖d‖2)

Recall: D+V ≤ −2νV + ρ‖d‖2.

⇒ D+W (t) ≤ −2ν(1− σ)W (t) + ρ(−2ν(1− σ) + η)m(t) (22)

We define the event-based controller ϕ3

Definition

Let ϕ3 be the operator which maps z to u such that

tu0 = 0, tu1 = 1λ

and for all k ≥ 1,

tuk+1 = inf{t ∈ R

+|t > tuk∧m(t) ≥ 0} (23)

u(t) = Ky(tuk , 1) ∀t ∈ [tuk , tuk+1), k ≥ 0 (24)

Theorem (Espitia, Girard, Marchand, Prieur; NOLCOS 2016)

The system P with controller u = ϕ3(y(t, 1)) has a unique solution and isglobally exponentially stable.

Vehicle traffic flow on a roundabout

∂

∂ty(t, x) + Λ

∂

∂xy(t, x) = 0

y(t, 0) = (H +BK)y(t, 1)

with

Λ =(

1 00

√2

)

, H =(

0 0.70.9 0

)

, B = I2, K =(

0 0.3−0.9 0

)

.

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.

Theorem

Let σ ∈ (0, 1), µ > 0, υ = µmin{λ1, λ2}, A = eµ, B = eµq2 + 1, ε1. Letη ≥ υ(1− σ) and 0 < θ ≤ min{ 1

2συ, 12Beµε1

}, κ1 and κ2 such that

max{2θBeµε1, 2θσυ} ≤ κ1 ≤ 1 (25)

2θσυ ≤ κ2 ≤ 1 (26)

holds. Then the system (17)-(17) with event-based controller Ud = ϕ has aunique solution and is globally exponentially stable.

W (α, β,m) =

∫ 1

0

(Aα2(x)e−µx

λ1+Bβ

2(x)eµx

λ2)dx−m (27)

W ≤ −υ(1− σ)W

Periodic event-based control.

ETMC

P

tk

∂∂ty(t, x) + Λ ∂

∂xy(t, x) = 0

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

u(t) = Kη(τj)

η(t) = Aη(t) +Bηy(tk, 1)

u(t)

PTM

τj

∀t ∈ [τj , τj+1) with T = τj+1 − τj , the period;

Can we obtain a sufficient condition for stability in terms of the period T ?

Looped functional

W (η, y, t− τj) = V (η, y) + V(η, t− τj)

V(η, t− τj) = (τj+1 − t)(η(t)− η(τj))TS1(η(t)− η(τj)) + (τj+1 − t)

∫ t

τjη(θ)Rη(θ)dθ

Bastin, G. and Coron, J.-M. (2016).Stability and Boundary Stabilization of 1-D Hyperbolic Systems.88. Birkhauser Basel, first edition.

Bastin, G., Coron, J.-M., and d’Andrea Novel, B. (2008).Using hyperbolic systems of balance laws for modeling, control and stabilityanalysis of physical networks.In Lecture notes for the Pre-Congress Workshop on Complex Embedded andNetworked 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.

Coron, J.-M., Bastin, G., and d’Andrea Novel, B. (2008).Dissipative boundary conditions for one-dimensional nonlinear hyperbolicsystems.SIAM Journal on Control and Optimization, 47(3):1460–1498.

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.

Espitia, N., Girard, A., Marchand, N., and Prieur, C. (2016).

Event-based control of linear hyperbolic systems of conservation laws.Automatica, 70:275–287.

Girard, A. (2015).Dynamic triggering mechanisms for event-triggered control.IEEE Transactions on Automatic Control, 60(7):1992–1997.

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.

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

Tabuada, P. (2007).Event-triggered real-time scheduling of stabilizing control tasks.IEEE Transactions on Automatic Control, 52(9):1680–1685.

Documents

Event-based control of networks modeled by a class of ...researchers.lille.inria.fr/~nespitia/PhD_defenseNicolasESPITIAV2.pdfNetworks of conservation laws Event-based stabilization