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Supervisory Switching Control for Linear Hyperbolic Systems
Pierre-Olivier Lamare
To cite this version:
Pierre-Olivier Lamare. Supervisory Switching Control for Linear Hyperbolic Systems. Automatica,
Elsevier, 2019, �10.1016/j.automatica.2019.01.034�. �hal-02398607�
Supervisory Switching Control for Linear Hyperbolic Systems
Pierre-Olivier Lamare
a,1aCentre Automatique et Syst`emes, MINES ParisTech, PSL University, 60 bd St Michel 75006 Paris France.
Abstract
We consider the stabilization problem for linear systems of balance laws subject to uncertain gains affecting the boundary conditions and the reaction term. The stabilization is achieved using thesupervisory control, a well established approach in finite dimension. In the context where a controller exists for each estimator, the supervisory control with a performance signal built with the boundary output effectively steers the system to the origin. This novel approach is illustrated with a traffic flow control problem modelled by the Aw-Rascle-Zhang equations.
Key words: Hyperbolic systems; Supervisory control; Uncertainty.
1 Introduction
Hyperbolic systems are suitable to model a large num- ber of physical systems, such as traffic networks [1] or oil well drilling [2]. It shows the great potential of industrial applications linked to these models. Thus, a large num- ber of results showed up these last two decades: adap- tive observers [2,3], adaptive control [4], stability analy- sis and controller synthesis based on Linear Matrix In- equalities (LMI) [5], control of switched hyperbolic sys- tems [6,7], etc.
A challenging issue for this type of systems remains their control in presence of large parameter uncertainties. In this context, the so-called adaptive control is used. It consists of the unknown parameter evaluation by some update laws. Using the certainty equivalence principle, which invariably considers the estimated parameters as the true ones, a candidate controller designed for the current estimate of the system is applied. This method has been successfully used in several cases. For instance, Krstic and Smyshlyaev investigated the adaptive control for parabolic equations (see the book [8]). Other exam- ples are the use of the adaptive control for the Burgers’
equation with unknown viscosity in [9], for a non-local
? This paper was not presented at any IFAC meeting.
Email address:
[email protected] (Pierre-Olivier Lamare).
1 This work was partially performed while affiliated with Inria Sophia-Antipolis, BIOCORE project team.
hyperbolic PDE in [10], for an anti-stable wave equation in [11] or for a hyperbolic PDE with an unknown reac- tion coefficient in [4]. Recently, adaptive observers have been developed for hyperbolic systems in [2,3].
In finite dimension, the use of the conventional adaptive control faces some issues, exhibiting at best bad perfor- mances [12]. In order to overcome these issues, thesu- pervisory control , which is an adaptive control follow- ing a switching-based principle has been developed. This procedure, which usually embeds ahysteresis switching policy, has been meticulously analyzed in [13]. More pre- cisely, the supervisory control consists of several com- ponents: a multi-estimator based on the available mea- surements, the candidate controllers, a performance sig- nal, and a switching policy. The difference between this latter method and the conventional adaptative frame- work is the abrupt change in the update law due to the switching policy. To the best of our knowledge, supervi- sory control is up to now exclusive to finite-dimensional systems.
In this article, we aim at introducing supervisory control in order to stabilize linear hyperbolic systems of balance laws subject to uncertain gains affecting the boundary conditions and the reaction term (Section 2). We show that the proposed supervisory control is effective to steer the system to the origin (Section 3). We illustrate our ap- proach with a problem of traffic flow control modelled by the Aw-Rascle-Zhang (ARZ) equations [1] (Section 4).
Notations.R+ is the set of nonnegative real numbers.
N is the set of natural numbers. Rn×n is the set of square real matrices of dimension n. Given a matrix A, the transpose of the matrix A is denoted by A>. For a symmetric matrix A ∈ Rn×n, A being positive definite is denoted by A > 0, while A being positive semi-definite is denoted byA≥0. The usual Euclidian norm in Rn is denoted by k·k. The spectral radius of a matrix A ∈ Rn×n is denoted by ρ(A) and its norm induced by the usual Euclidian norm in Rn is denoted bykAk. The smallest eigenvalue of a matrixA∈Rn×n is denoted byλmin(A) while its largest eigenvalue is de- noted byλmax(A). The derivative of a matrixA(x) with respect to x is denoted by A0(x). The identity matrix of dimension n is denoted by In. The set of functions y : [a, b] → Rn such thatkyk2L2(a,b)=Rb
a kyk2dx <∞, is denoted by L2(a, b). Furthermore, we denote the set L2(0,1) byE. The set of functionsy∈Esuch that there exists a functiong∈Esuch thatR1
0 yϕ0dx=−R1 0 gϕdx for all ϕ ∈ Cc1 =
h∈C1([0,1])|supp(h)⊆(0,1) where supp(h) = {x∈(0,1)|h(x)6= 0}, is denoted by H1(0,1).
2 Problem Statement
We consider the following system
∂tu+ Λ+∂xu= Γ1u+ Γ2v , (1)
∂tv−Λ−∂xv= Γ3u+ Γ4v , (2) where t ∈ R+ is the time variable, x ∈ (0,1) is the spatial variable,u: (0,1)×R+→Rm,v: (0,1)×R+→ R(n−m). The matrices Γ1, Γ2, Γ3, and Γ4are inRm×m, Rm×(n−m),R(n−m)×m, andR(n−m)×(n−m)respectively.
Matrices Λ+ and Λ− are diagonal positive definite. To obtain an unique and well-defined solution to system (1) and (2) we consider the following boundary conditions
"
u(0, t) v(1, t)
#
=G
"
u(1, t) v(0, t)
#
+LU(t), (3)
together with the following initial conditions
u(x,0) =u0(x), x∈(0,1), (4) v(x,0) =v0(x), x∈(0,1) , (5) whereGandLare matrices inRn×n andRn×p respec- tively, for some integer psatisfying n > p. The vector U ∈Rpis the control input and u0, v0
∈E×E. The output of the system is given by
y(t) =C
"
u(1, t) v(0, t)
#
, (6)
whereC is inRq×n. Let us introduce the following no- tations
|Λ|=diag
Λ+,Λ−
, (7)
Γ =
"
Γ1 Γ2 Γ3 Γ3
#
. (8)
The matrices G, L, C, and Γ are supposed to be un- certain. Let us denote byG∗, L∗,C∗, and Γ∗ the true matrices. These real matrices belong to a (known) set of N >0 admissible candidates. These candidates are de- notedωi= (Gi, Li, Ci,Γi),i= 1, . . . , N. This assertion is formally stated in the following assumption.
Assumption 1 Matrices G∗, L∗, C∗, and Γ∗ belong to a known discrete set Ω. More precisely, there ex- ists N such that I = {1, . . . , N} and Ω = ∪i∈Iωi =
∪i∈I(Gi, Li, Ci,Γi). Let j∗ ∈ I be the mode which cor- responds to the true matricesG∗,L∗,C∗, andΓ∗. Assumption 1 requires that one element in Ω matches the true parameters. This condition may be strong from an applicative point of view. In a more realistic situation, it may be supposed that only a compact set is known for the unknown parameters. Then, dividing this set in sev- eral subsets and selecting a representative element per subset yields a finite collection of possible elements for the true parameters. This procedure has been done for the finite dimensional case in [14]. The stabilization of the uncertain system is achieved under the hypotheses that the discretization is thin enough and that the con- trollers are robust to small uncertainties. However, a thin division entails a large number of modesN, which could be expensive from a computational point of view. This drawback can be avoid using a state-sharing technique, see e.g. [12]. This technique is suitable for our system, but only in special cases. Nonetheless, since the purpose of the paper is to introduce the supervisory switching control for a PDE system, we supposed that one mode matches the true parameters.
Remark 2 Here, it is assumed that matrices Λ+, Λ−, andΓare constant. However, the results may be gener- alized to spatially dependent matrices.
We aim at using the outputyin (6) to identify a stabi- lizingsupervisory controlfor system (1)–(5).
Let us recall the definitions ofGlobal Exponential Sta- bilityin theL2-norm for system (1)–(5).
Definition 3 System (1)–(5)is said to be Globally Ex- ponentially Stable (GES) in the L2-norm if there exist ν > 0andM >0 such that, for every initial condition
u0, v0
∈E, the solution to system(1)–(5)satisfies, for
2
allt≥0, h
u> v>
i>
(·, t) E
≤M e−νt h
u0> v0>
i>
E
. (9)
In the proof of the main result of the paper we shall con- sider system such as (1), (2) with the boundary condi- tion
"
u(0, t) v(1, t)
#
=A
"
u(1, t) v(0, t)
#
+δ(t), (10) whereδ: [0,∞)→Rn is aL2-function andA∈Rn×n. Definition 4 Let consider system (1), (2) along with the boundary condition (10) and the initial condi- tions (4) and (5). It is said to be Input-to-State-Stable (ISS) in theL2-norm with respect to inputδinL2(0,∞) if there exist ν > 0,C > 0, and aK-function θ, such that, for every initial condition u0, v0
∈ E, the solu- tion satisfies, for allt≥0,
h
u> v>
i>
(·, t) E
≤Ce−νt h
u0> v0>
i>
E +θ
kδkL2(0,t)
. (11)
3 Supervisory Control Strategy
In this section, the supervisory control for system (1)–(5) is introduced. We first present the control method and then demonstrate its ability to stabilize the system. This construction follows the conventional approach as stated by Hespanha, Morse, and Liberzon in the late nineties, see e.g. [15] and [13]. A schematic representation of the closed-loop is given by Figure 1 at the end of the section.
3.1 Multi-Estimator Structure
The multi-estimator is defined as the set of estimator systems
∂tui+ Λ+∂xui= Γ1,iui+ Γ2,ivi, (12)
∂tvi−Λ−∂xvi= Γ3,iui+ Γ4,ivi, (13) with boundary conditions
"
ui(0, t) vi(1, t)
#
= (Gi+QiCi)
"
ui(1, t) vi(0, t)
#
+LiU(t)
−Qiy(t), (14) and initial conditions
ui(x,0) =u0i(x), x∈(0,1), (15) vi(x,0) =v0i(x), x∈(0,1), (16)
whereh
u0i> vi0>
i∈E,Qi∈Rn×q,i∈ I. Let us denote the error variables with respect to the estimatoriby
˜
ui(x, t) =ui(x, t)−u(x, t), (17)
˜
vi(x, t) =vi(x, t)−v(x, t). (18) Then, let us denote by ei(t), i ∈ I, the measurement deviation with respect to the output of system (1)–(5)
ei(t) =Ci
"
ui(1, t) vi(0, t)
#
−y(t). (19)
As mentioned in Section 2, the analysis of supervisory switching control requires an ISS result with respect to the injection error ei for estimator system (12), (13), and (14). It is given by the following lemma
Proposition 5 Let us consider PDEs (12), (13) with boundary condition
"
ui(0, t) vi(1, t)
#
=Gi
"
ui(1, t) vi(0, t)
#
+LiUi(t) +δ(t), (20)
whereδis a perturbation inL2(0,∞). If the controllerUi stabilizes the unperturbed system (δ≡0), then it is ISS with respect to perturbationδinL2(0,∞).
PROOF. System (12), (13), and (20) is linear, which implies by Lemma 1 and Proposition 3 in [16] that it is ISS with respect to δ. This concludes the proof of Proposition 5.
Let us write the error system forj∗. It satisfies
∂t˜uj∗+ Λ+∂xu˜j∗= Γ1,j∗u˜j∗+ Γ2,j∗˜vj∗, (21)
∂t˜vj∗−Λ−∂xv˜j∗= Γ3,j∗u˜j∗+ Γ4,j∗v˜j∗, (22) with boundary conditions
"
˜ uj∗(0, t)
˜ vj∗(1, t)
#
= (Gj∗+Qj∗Cj∗)
"
˜ uj∗(1, t)
˜ vj∗(0, t)
#
. (23)
The matricesQi,i ∈ I, have to be designed such that every systems (21), (22), and (23) with j∗ replace by i are exponentially stable, which is guaranteed by the following assumption.
Assumption 6 (cf conditions of Theorem 3.1. in [5]) For everyi∈ I, there existνi>0,θiinR, matricesFiin
Rn×q, diagonal positive definite matricesSi+inRm×m, andSi−inR(n−m)×(n−m)such that forSi(x)defined by
Si(x) =diag
e2θixSi+, e−2θixSi−
, (24)
the following conditions hold, for allx∈[0,1],
"
diag
e−2θiSi+, Si−
(Si(0)Gi+FiCi)>
Si(0)Gi+FiCi diag
S+i , e−2θiS−i
#
>0, (25)
−2θiSi(x) +Si(x)Γ>i Λ−1
+ Λ−1ΓiSi(x)≤ −2νiSi(x)Λ−1. (26) Using Assumption 6, the sought matrices Qi,i∈ I are given by
Qi=Si−1(0)Fi. (27) Using Assumption 6, Proposition 2.1. in [5] may be applied and one can conclude that system (21), (22), and (23) is exponentially stable.
Remark 7 Assumption 6 comes from a Lyapunov anal- ysis. More precisely, the candidate Lyapunov function takes the following form
Zi(y) = Z 1
0
y>(x)Λ−1Pi(x)y(x)dx , (28) wherePi=Si−1. More details may be found in [5].
Remark 8 Condition(26)involves the spatial variable x∈[0,1], meaning that the number of conditions to ver- ify is infinite. In order to overcome this issue, over- approximation techniques are proposed in [5], which con- sists of reducing the problem to a finite number of LMI using polytopic embeddings.
3.2 Multi-Controller
In this paper, the candidate controllers for sys- tem (1)–(5) are stated in an abstract way and only depend on the states of the multi-estimator
Ui(t) =Ki[ui, vi] (t). (29) The controllerUi is designed such that it stabilizes the system (12), (13) with the boundary condition
"
ui(0, t) vi(1, t)
#
=Gi
"
ui(1, t) vi(0, t)
#
+LiUi(t). (30)
Controller (29) may be static as in [5] or derived by backstepping (see e.g. [17]). The only requirement is that it exists a controller for each mode, which is state in
Assumption 9 For everyi∈ I, there exists a controller Ui, which stabilizes system (12),(13), and (30).
3.3 Switching Logic
Let us introduce the performance signal. It is designed with the available measurementy(t). It satisfies
˙
µi(t) =−ξµi(t) +kei(t)k2 , (31) µi(0) =εi>0, (32) whereeiis defined in (19),εi,i= 1, . . . , N, are the corre- sponding initial conditions for each performance signal and 0< ξ <2νwith
ν= minνi, (33)
is a tuning parameter. The parameters νi, i ∈ I, are those used in the construction of the gainsQiin Assump- tion 6. It corresponds to the decay rate of the Lyapunov function of the error system ifiwere the true index. Let us explain the boundξ <2ν. Foremost, we need the fol- lowing lemma
Lemma 10 Let Assumption 6 holds. Then, for every 0≤t1< t2the following holds
kej∗k2L2(t
1,t2)≤M1 max
t∈[t1,t2]
h
˜ u>j∗ ˜vj>∗
i>
(·, t)
2
E
+M2 h
˜ u>j∗ ˜vj>∗
i>
(·, t1)
2
E
, (34)
whereM1, M2>0.
The proof of Lemma 10 is postponed in Appendix A page 8.
The solution to (31) and (32) is
µi(t) =e−ξtεi+ Z t
0
e−ξ(t−s)kei(s)k2ds . (35)
Let us define the time instanttnby
tn=nτ , (36)
with
τ =ρ(|Λ|). (37)
Using the exponential stability of the dynamics of
4
h
˜ u>j∗ ˜v>j∗
i>
and Lemma 10 withtnandtn+1we get
kej∗k2L2(tn,tn+1)≤M1e−2νj∗tn h
˜ u0j∗> v˜j0∗>
i>
2
E
+M2e−2νj∗tn h
˜ u0j∗> v˜j0∗>
i>
2
E
=M e−2νj∗tn h
˜ u0j∗> v˜j0∗>
i>
2
E
, (38) whereM =M1+M2. Besides, we have
kej∗(t)k2≤ kej∗k2L2(tn,tn+1) , (39) for almost everyt∈[tn, tn+1]. Thus, using (38) and (39) we get
kej∗(t)k2≤M e−2νj∗tn h
˜ u0j>∗ v˜j0∗>
i>
2
E
, (40)
for almost every t ∈ [tn, tn+1]. Let us rewrite (35) for i=j∗as
µj∗(t) =e−ξtεj∗+
N−1
X
n=0
Z tn+1 tn
e−ξ(t−s)kej∗(s)k2ds +
Z t tN
e−ξ(t−s)kej∗(s)k2ds . (41)
Using (40) in (41) we obtain µj∗(t)≤e−ξtεj∗+M
hu˜0j>∗,˜v0j∗>
i>
2
E
×
N−1
X
n=0
e−2νj∗tn Z tn+1
tn
e−ξ(t−s)ds
+M h
˜ u0j>∗ ˜v0j∗>
i>
2
E
×e−2νj∗tN Z t
tN
e−ξ(t−s)ds . (42)
In the following analysis we will need the boundedness of the signal µj∗. Therefore, using inequality (42), we get that ξhas to satisfyξ <2νj∗. Since the true index parameter j∗ is unknown the condition ξ < 2ν must hold. This property should be understood as follows:
the “learning rate” ξ of the performance signal must be slower than the “convergence rate” of the estimator systems.
Remark 11 Let us notice that we have decomposed the time interval [0, T] in several sub-intervals whose the length is given by the minimum of time needed for a par- ticle to travel trough the whole space domain. Otherwise,
Process y(t) hu
i(1,t) vi(0,t)
i
ei(t) Controller
Uσ(t) Multi- Estimator
y(t)
Monitoring Signal Generator Switching
Logic σ
µi, i∈ I
+ -
Figure 1. Closed-loop representation of the supervisory con- trol for system (1)–(5).
we would haveT in the right-hand side of inequality(42) and we could not conclude on the boundedness ofµj∗. Finally, let us introduce the procedure to switch from one mode to the other. This procedure satisfies a scale- independent hysteresis switching, such as in [15]:
σ(t) =
σ(t−) if∀i∈ I,
(1 +h)µi ≥µσ(t−), arg min
i∈I µi(t) elseif.
(43)
σ(0) =
arg min
i∈I εi ifεi6=εj,
∀i, j∈ I, pick randomlyεk,
k∈ I s. t.εk= min
i∈I εi elseif.
(44)
Let us stateσ(t−) = lims→t,s<tσ(s). Roughly speaking, t−is the value ofσ“just beforet”. The parameterhis a strictly positive constant, calledhysteresis constant. The switching signalσ constructed in this way is piecewise constant and continuous from the right.
In summary, the closed-loop of the supervisory switching control used in this paper is illustrated by Figure 1. It is based on the representations given in [12].
3.4 Stabilization under Supervisory Control Let us state the main result of the paper.
Theorem 12 Under Assumptions 1, 6, and 9 sig- nals defined by the supervisory control system (1)–(5), (12)–(16), (29), (31)–(33),(43), and (44)exist for all t ∈ R+ and remain bounded for any initial conditions h
u0> v0>
i∈E. Moreover, there exists a time T∗ > 0 such thatσ(t) =i∗∈ I for allt≥T∗ (i.e. the switching
stops in finite time) and
t→∞lim h
u> v>
i>
(·, t) E
= 0. (45)
PROOF. Two points need to be proved:
• first, the switching phenomena stops;
• second, the state converges to 0∈E.
As a first step, let us assume that [0, T) is the largest interval of time for which a solution exists to (1)–(5), (12)–(16), (29), (31)–(33), (43), and (44).
Let us assume that the true index parameter isj∗∈ I.
The finiteness of the switching borrows the same lines as in the finite dimensional case, see e.g. [13] and [12].
The nature of the dynamical system is not crucial. We may change the performance signals (31) by
µi=eξtµi(t), (46) where i ∈ I. As described in [12], the scaling by the functionθ(t) =eξt does not affect the switching signal generated by (43) and (44). Let us notice that the signal µi is monotonically non decreasing because of
µ˙i=ξµi+eξtµ˙i=eξtkei(t)k2≥0. (47) Introducing the scaled performance signals (46) allows to get the switching finiteness by applying Theorem 1 in [13].
The application of Theorem 1 in [13] requires: a finite number of elements inI (Assumption 1); the bounded- ness of the signalµj∗(proved in Subsection 3.3); the fact that all the signals µi, with i ∈ I, are monotonically nondecreasing (see (47)), and the positiveness of all the initial conditions of the signalsµi(see (32)). These three latter conditions combined with the switching proce- dure (43) and (44) guarantee that some index will never be used over the time. This latter observation in addi- tion to a reasoning based on the continuity of the signals µi are both the basis of the proof of Theorem 1 in [13].
Thus, if [0, T) is the largest time interval on which sys- tem (1)–(5), (12)–(16), (29), (31)–(33), (43), and (44) is defined, there exists a time instant T∗ in [0, T) such that σ(t) = i∗ for all t ≥ T∗ and for which µi∗ is bounded on [0, T). Let us remark that the parameters ωi∗ = (Gi∗, Li∗, Ci∗,Γi∗) may be the true parameters ωj∗ or not. Moreover,µi∗ is bounded on [0, T). Indeed, if we assume that it is not the case, then the switching logic would be violated becauseµj∗ is bounded by (42).
Because the switching stops and using again the latter reasoning one gets that every switching signal is bounded on [0, T).
As system (1)–(5) is linear, the growth rate of the norm of the solution is controlled, see e.g. Theo- rem A.4. page 251 in [17]. Therefore, an accumula- tion of switches at a given time is the only reason for a local (in time) existence of a solution to sys- tem (1)–(5), (12)–(16), (29), (31)–(33), (43), and (44).
However, we just proved that this condition cannot arise. Therefore, the global (T = ∞) existence and uniqueness of a solution to (1)–(5), (12)–(16), (29), (31)–
(33), (43), and (44) is given by Proposition 3.1 in [7] in the context of weak solutions. The existence of a solution (globally in time) for the estimator systems (12)–(16) holds by the same arguments. Thus, there always exists a solution to system (12)–(16) for each i ∈ I, whose norm may grow, potentially, to the infinity. This latter statement together with the boundedness of every per- formance signal prove the first assertion of Theorem 12.
Now let us prove the convergence of the system to 0, that is let us prove relationship (45). Let us denote byX1(t) the vector h
u>(0, t) v>(1, t) u>i∗(0, t) vi>∗(1, t) i>
and X2(t) the vector h
u>(1, t) v>(0, t) u>i∗(1, t) vi>∗(0, t) i>
. One has
X1(t) =
"
Gj∗ 0
−Qi∗Cj∗ Gi∗+Qi∗Ci∗
# X2(t)
+
"
Lj∗Ki∗[ui∗, vi∗] (t) Li∗Ki∗[ui∗, vi∗] (t)
#
=
"
Gj∗+Qj∗Cj∗ −Qj∗Ci∗
0 Gi∗
# X2(t)
+
"
Lj∗Ki∗[ui∗, vi∗] (t) Li∗Ki∗[ui∗, vi∗] (t)
# +
"
Qj∗ Qi∗
#
ei∗(t). (48)
Sinceei∗goes to zero (µi∗is bounded) andUi∗stabilizes the system (12), (13) with boundary condition (30) by Assumption 9, it follows from (48) thatui∗ and vi∗ go to zero inL2-norm by Proposition 5. Since the system satisified byu,v,ui∗, andvi∗is a cascade system,uand v also converges to zero, it follows (45). This concludes the proof of Theorem 12.
Remark 13 The main difference of the supervisory con- trol analysis for system (1)–(5) compared to the finite dimension consists of ensuring the stability of the dis- tributed state with a performance signal built with the boundary measurements. The “hidden” regularity prop- erty of hyperbolic systems has been used to prove it.
4 Application to the Traffic Flow Control In this section, the switching supervisory control is illus- trated with the Aw-Rascle-Zhang model (see e.g. [1]),
6
which represents the density z(x, t) and the velocity w(x, t) at time t∈ R+ and space-location x∈ (0,1) of vehicles on a road. The dynamics is given by
∂tz+∂x(zw) = 0, (49)
∂t(w−V (z)) +w∂x(w−V(z)) = (V(z)−w)
τ . (50)
The function V, thedesired velocity function or equi- librium velocity function, establishes a functional rela- tionship between a density and a velocityw=V(z). By linearizing around a steady state (z?, w?) and using Rie- mann coordinates, equations (49) and (50) become
∂tu1+w?∂xu1=−1
τu1, (51)
∂tu2+ (w?+z?V0(z?))∂xu2=−1
τu1, (52) respectively. Let us assume thatw?+V0(z?) is positive, which corresponds to a traffic free-flow mode. Therefore, it yieldsn= 2 andm= 2. Let us suppose the possibility to control the inflowi.e.the controller takes the following form
W(t) =w(0, t)z(0, t). (53) In Riemann coordinates, boundary control (53) is equiv- alent to
"
u1(0, t) u2(0, t)
#
=
"
g1 g2
g3 g4
# "
u1(1, t) u2(1, t)
# +
"
`1
`2
#
U(t), (54)
for a suitable choice of the control inputW(t), W(t) =− 1
V0(z?)(g3u1(1, t) +g4u2(1, t) +`2U(t) +w∗) ((g1−g3)u1(1, t) + (g2−g4)u2(1, t)
+ (`1−`2)U(t)−V0(z?)z?). (55) We assume that the density z at x = 1 is the output of the system. Moreover, this output is not perfectly measured. The sensor returns a fraction of the correct value. In Riemann coordinates it reads
y(t) = f? V0(ρ?)
h
−1 1i
"
u1(1, t) u2(1, t)
#
, (56)
wheref?corresponds to the level of confidence. In such a situation p= 1 and q = 1. Now, let us assume that the relaxation term τ, the parameters gi, i = 1, . . . ,4,
`1,`2, andf?are uncertain. Let us choose the following equilibrium velocity function
V(z) =−90
802z2+ 90. (57)
The following steady-state are considered (w?, z?) = 77.3438 km.h−1,30 veh.km−1
. (58) The road is 5 km long. The set Ω is composed by four elementsωi= (Gi, Li, Ci, τi) given by
ω1=
"
1 1
−1 0.5
# ,
"
1
−0.9
# ,
"
0.0038
−0.0038
#>
,40
, (59)
ω2=
"
0 1
−1 0.5
# ,
"
1 0.2
# ,
"
0.0034
−0.0034
#>
,45
, (60)
ω3=
"
0.5 0.25
−1 0.5
# ,
"
1 0.2
# ,
"
0.0030
−0.0030
#>
,50
, (61)
ω4=
"
0.5 0.25 1 −0.5
# ,
"
−0.5 0.6
# ,
"
0.0047
−0.0047
#>
,35
.(62)
We decide to use a static controller in (29), i.e.
Ui(t) = Kih
u>i (1, t) vi>(0, t) i>
. The matrices Ki are derived with the method developped in [5] and are given by
K1=h
−0.9911 1i
, K2=h
−0.0083−1i
, (63)
K3=h
−0.5080 −0.25i
, K4=h
0.9886 0.5016 i
. (64) The following matrices has been found for the multi- estimator
Q1=
"
−129.9559 260.0338
# , Q2=
"
−292.6952 292.5401
#
, (65)
Q3=
"
0.7208 334.3375
# , Q4=
"
−106.2027
−212.7604
#
. (66)
The true parameters are those corresponding to the third index The initial conditions for eachµi,i∈ I, are arbi- trary chosen as
µ01, µ02, µ03, µ04
=
0.3,0.3,0.4,0.2
. (67) The signalσ is initialized such thatσ(0) = 4. The pa- rameters for the performance signal (31) and switching strategy (43) are
(ξ, h) = 10−2,0.5
. (68)
0 200 400 600 800 1000 1200 0
0.5 1 1.5
1 2 3 4
Figure 2.L2-norm of the state
[u1u2]>(·, t)
Eand switch- ing signalσ(t) generated by (43).
The initial conditions (4), (5), (15), and (16) for the simulation are
h
u01(x) u02(x) i>
=h
1.3851 1.3889 i>
(69) h
u01i(x) u02i(x) i>
=h
u01(x) u02(x) i>
, i∈ I. (70)
The evolution of h
u1 u2
i>
(·, t) E
andσ(t) are plotted on Figure 2. As expected by Theorem 12 the L2-norm of the state
h
u1 u2
i>
(·, t) E
goes to 0.
5 Conclusions
In this paper, a supervisory control framework has been introduced for stabilizing linear systems of balance laws subject to large modeling uncertainties affecting the boundary conditions and the reaction term. This adaptive method has been illustrated by a traffic flow control. This paper is the first application of supervisory control to hyperbolic PDE. Several questions remain open. One prospect could be the analysis of the multi- estimator initialization using the approach developed in [18] for finite dimensional problems. Besides, a more advanced supervisory control could be considered when every controller fails to stabilize the system while the stabilization is achieved by switching among them [19].
Finally, the stabilization of the system whenever none of the possible parameters match the true parame- ter should be investigated in the future. In particular, special effort should be put to estimate the minimum division to ensure stabilization.
Acknowledgments
The author is deeply grateful to Antoine Girard and Christophe Prieur for many constructive suggestions.
A Proof of Lemma 10
PROOF. Let us consider system (21)–(23), we assume that h
˜ u>j∗ v˜>j∗
i>
(·, t1) ∈ H1(0,1). Then, h
˜ u>j∗ v˜>j∗
i>
∈ C1([t1, t2] ;E)∩C [t1, t2] ;H1(0,1)
. Let us define the following functions
g1(x) =e−2θj∗x Λ+−1
S+j∗
−1
˜
uj∗(x, t) (A.1) g2(x) =e2θj∗x Λ−−1
Sj−∗
−1
˜
vj∗(x, t), (A.2) whereSj+∗andSj−∗are defined in Assumption 6. By first multiplying (21) and (22) byg1(x) andg2(x) respectively then integrating over ∆ = (t1, t2)×(0,1), we obtain
Z 1 0
"
˜
uj∗(x, t2)
˜
vj∗(x, t2)
#>
Sj−1∗ (x)
"
˜
uj∗(x, t2)
˜
vj∗(x, t2)
# dx
− Z 1
0
"
˜
uj∗(x, t1)
˜
vj∗(x, t1)
#>
Sj−1∗(x)
"
˜
uj∗(x, t1)
˜
vj∗(x, t1)
# dx
+ Z t2
t1
"
˜ uj∗(1, t)
˜ vj∗(0, t)
#>
G
"
˜ uj∗(1, t)
˜ vj∗(0, t)
# dt
= 2 Z
∆
"
˜ uj∗(x, t)
˜ vj∗(x, t)
#>
P(x)
"
˜ uj∗(x, t)
˜ vj∗(x, t)
#>
dxdt , (A.3)
where
G=h Im 0m,n−m
Y−+
j∗ Y−−
j∗
i>
Ien,mSj−1∗ (1)h Im 0m,n−m
Y−+
j∗ Y−−
j∗
i
−h Y++
j∗ Yj+−∗ 0n−m,mIn−m
i>
Ien,mSj−1∗ (0)h Y++
j∗ Yj+−∗ 0n−m,mIn−m
i (A.4) Yj∗=Gj∗+Qj∗Cj∗=
Y++
j∗ Y+−
j∗
Y−+
j∗ Y−−
j∗
(A.5) P(x) = Γ>j∗Sj−1∗ (x)
+
−θj∗e−2θj∗x(S+)−1 0n,n−m
0n−m,n θj∗e2θj∗x(S−)−1
, (A.6) withYj++∗ ,Yj+−∗ ,Yj−+∗ , andYj−−∗ inRm×m,Rm×(n−m), R(n−m)×m, and R(n−m)×(n−m) respectively. Using As- sumption 6 it can be proved that the matrixGin (A.4) is positive definite (see the proof of Proposition 5). Thus,
8
from (A.3) we get λmin(G)
h
˜
u>j∗(1, t) ˜vj>∗(0, t) i>
2
L2(t1,t2)
≤m1 max
t∈[t1,t2]
h
˜ u>j∗ ˜v>j∗
i>
(t)
2
E
+m2 h
˜ u>j∗ v˜>j∗
i>
(t1)
2
E
, (A.7)
where
m1= 2 (t2−t1) max
x∈[0,1]λmax(P(x)), (A.8) m2= max
x∈[0,1]λmax Sj−1∗(x)
. (A.9)
Inequality (34) readily follows from (A.7) with Mk= kCj∗k2mk
λmin(G) , k= 1,2. (A.10) By density, inequality (34) also holds for the function h
˜ u>j∗ ˜v>j∗
i>
(·, t1) only inE. This concludes the proof of Lemma 10.
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