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Brief Paper

Stabilization of coupled linear heterodirectional hyperbolic PDE–ODE systems✩ Florent Di Meglio a , Federico Bribiesca Argomedo b , Long Hu c , Miroslav Krstic d a

CAS — Centre automatique et systèmes, MINES ParisTech, PSL Research University, 60 bd St Michel, 75006 Paris, France Université de Lyon - Laboratoire Ampère (CNRS UMR5005) - INSA de Lyon, 69621 Villeurbanne CEDEX, France c School of Mathematics, Shandong University, Jinan, Shandong 250100, China d Department of Mechanical & Aero. Eng., University of California, San Diego, La Jolla, CA 92093-0411, United States b

article

info

Article history: Received 5 October 2016 Received in revised form 9 May 2017 Accepted 30 August 2017 Available online 6 November 2017

a b s t r a c t We solve the problem of stabilizing a linear ODE having a system of a linearly coupled hyperbolic PDEs in the actuating path. The control design is based on a backstepping approach and yields exponential closedloop stability of the zero equilibrium. The existence of a Volterra transformation relies on a relatively general well-posedness result for hyperbolic backstepping kernel equations. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Stabilization Distributed parameter systems Predictor feedback Hyperbolic systems

1. Introduction The interest for coupled Ordinary Differential Equations–Partial Differential Equations (ODE–PDE) systems has first emerged when considering delays in the actuating and sensing paths of ODE. Delays can be seen as first-order hyperbolic PDEs. There are many approaches to deal with input or measurement delays, usually divided into two categories: memoryless controllers, which extend standard control techniques without explicitly accounting for the delay in the control design (Fridman & Shaked, 2002; Michiels & Niculescu, 2007; Yue & Han, 2005); and prediction-based controllers aiming at explicitly compensating the delay (BekiarisLiberis & Krstic, 2013; Bresch-Pietri, 2012; Smith, 1959). The use of Lyapunov and backstepping methods enabled dealing with more involved PDEs in the actuating and sensing paths. In Krstic (2009), an output feedback control law is derived for an ODE having a heat equation in the actuating and sensing paths. The coupled PDE–ODE system is stabilized using an observer–controller structure relying on a backstepping approach. The same approach has been used to deal with ODEs coupled ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Jamal Daafouz under the direction of Editor Richard Middleton. E-mail addresses: [email protected] (F. Di Meglio), [email protected] (F.B. Argomedo), [email protected] (L. Hu), [email protected] (M. Krstic). https://doi.org/10.1016/j.automatica.2017.09.027 0005-1098/© 2017 Elsevier Ltd. All rights reserved.

(rather than cascaded) with parabolic PDEs (Tang & Xie, 2011), uncertain parabolic PDEs (Li & Liu, 2012), orODE—Schrödinger cascades (Ren, Wang, & Krstic, 2013). Lyapunov methods enable the design of static output feedback controllers for nonlinear ODE– parabolic PDE cascades, as in Wu and Wang (2013). The first application of the backstepping approach to deal with hyperbolic PDE–ODE couplings is (Krstic & Smyshlyaev, 2008) where actuator and sensor delays are explicitly compensated. While this problem had already been tackled by, e.g., the Smith predictor (Smith, 1959), the reformulation of the delay as a linear ODE enabled numerous related problem to be tackled, most notably non-constant and uncertain delays (Bekiaris-Liberis & Krstic, 2013; Bresch-Pietri, 2012). In Tsubakino and Oliveira (2016), the problem of stabilizing a multi-input ODE with distinct delays is tackled using a backstepping approach. In Castillo Buenaventura, Witrant, Prieur, and Dugard (2012), an observer is designed for an ODE having a homodirectional1 hyperbolic PDE in the sensing path, relying on a Lyapunov analysis requiring to solve Linear Matrix Inequalities (LMI). As will appear, the systems of Tsubakino and Oliveira (2016) are particular cases of the system considered here. Here, we solve the problem of stabilizing an ODE with a system of first-order linear hyperbolic PDEs in the actuating path, i.e. we

1 i.e. where all the states transport in the same direction.

282

F. Di Meglio et al. / Automatica 87 (2018) 281–289

Here, we solve these problems within the general setting of Eqs. (1)–(5). The system is mapped to an exponentially stable target system using a Volterra transformation. The target system has a cascade structure ensuring its convergence to the zero equilibrium. The design is based on a recent result on heterodirectional systems of hyperbolic PDEs (Hu, Di Meglio, Vazquez, & Krstic, 2016). The paper is organized as follows. In Section 2 we present the backstepping control design. In Section 3 we present a general wellposedness result for a class of hyperbolic PDEs on a triangular domain. In Section 4 we apply these results to the considered problem and state the main result. We conclude in Section 5 with perspectives for future work.

Fig. 1. Schematic view of the ODE–PDE coupling structure.

consider the following system X˙ (t) = AX (t) + Bv (t , 0) ut (t , x) = −Λ ux (t , x) + Σ

(1)

v (t , x) vt (t , x) = Λ− vx (t , x) + Σ −+ u(t , x) + Σ −− v (t , x) u(t , 0) = Q0 v (t , 0) + CX (t) v (t , 1) = R1 u(t , 1) + U(t) +

++

u(t , x) + Σ

+−

(2) (3) (4) (5)

where t > 0 and x ∈ [0, 1] are respectively the time and space variables, X ∈ Rp is the ODE state, u(t , x) ∈ Rn and v (t , x) ∈ Rm are the PDE states and U(t) is the control input. The matrices Λ+ and Λ− are such that

Λ+ = diag (λ1 , . . . , λn ),

Λ− = diag (µ1 , . . . , µm )

(6)

with −µ1 < · · · < −µm < 0 < λ1 < · · · < λn . The system naturally features several feedback loops or couplings that can be sources of instabilities:

• Inside the ODE itself (the A matrix in Eq. (1)) • Coupling between hyperbolic states inside the spatial domain (the Σ ·· matrices in (2), (3)) • Coupling between hyperbolic states at the boundary (the Q0 and R1 matrices in (4), (5)) • Coupling between the PDE and the ODE (the B and C matrices in (1), (4)) • A combination of all the above. This structure is schematically depicted in Fig. 1. The (A, B) pair is assumed stabilizable. This problem is motivated by applications in the drilling industry, more precisely the suppression of mechanical vibrations. Drilling systems are composed of long flexible strings subject to axial and torsional vibrations that propagate upwards and downwards. At the bottom end, the so-called drill bit crushes rock to create the borehole and is subject to friction and cutting forces. The ODE state X then corresponds to the drill bit axial and torsional positions while the PDE states represent the propagation of torsional and axial waves from and to the drill bit. For more details on these models, the interested reader is referred e.g. to Di Meglio and Aarsnes (in press) and Germay, Denoël, and Detournay (2009). When damping of the vibrations along the drill string is neglected and the axial and torsional vibrations are coupled, the PDE reduces to two delay equations. Several contributions have taken advantage of this simplification and designed stabilizing feedback laws, e.g. relying on neutral system approaches (Saldivar, Mondie, Loiseau, & Rasvan, 2011), flatness approaches (Sagert, Di Meglio, Krstic, & Rouchon, 2013) or predictor-based approaches (BekiarisLiberis & Krstic, 2014; Bresch-Pietri & Krstic, 2014). However, no existing solution simultaneously allows stabilization

• taking into account damping inside the PDE domain (matrices Σ ±± in (2), (3)); • for a model of both axial and torsional vibrations, yielding four coupled PDE states rather than two delay equations. Details of such models can be found in Di Meglio and Aarsnes (in press) and Germay et al. (2009), where the corresponding matrices Λ, Σ , etc. are given.

2. Control design The control design is based on a Volterra transformation mapping the state (X , u, v ) to a target system (X , α, β ) with desirable properties. The target system equations are described in the next section. 2.1. Target system We design the target system as follows X˙ (t) =(A + BK)X (t) + Bβ (t , 0)

(7)

αt (t , x) = − Λ αx (t , x) + Σ α (t , x) + Σ β (t , x) + D(x)X (t) ∫ x ∫ x + + C (x, y)α (t , y)dy + C − (x, y)β (t , y)dy (8) +

++

+−

0

0

¯ β (t , x) βt (t , x) =Λ− βx (t , x) + G(x)β (t , 0) + Σ α (t , 0) =Q0 β (t , 0) + C0 X (t) β (t , 1) =0

(9) (10) (11)

¯ are where C0 , C + , C − and D have yet to be defined and G(·) and Σ defined as

⎛

0

⎜ ⎜ g2,1 (x) G(x) = ⎜ ⎜ . ⎝ .. gm,1 (x)

⎛ −− Σ11 ⎜ ⎜ ¯ =⎜ 0 Σ ⎜ . ⎝ .. 0

··· .. . .. . ··· 0

.. ..

.

. ···

⎞

0

··· .. . .. .

.. ⎟ .⎟ ⎟ .. ⎟ , .⎠

gm,m−1

0

0

⎞

··· .. . .. . ···

.. . .. .

Σm−− ,m

⎟ ⎟ ⎟ ⎟ ⎠

(12)

¯ is the diagonal matrix consisting of the diagothat is to say, Σ nal elements of Σ −− . The target system has a cascade structure schematically depicted in Fig. 2. Its stability properties are assessed in the following lemma. Lemma 1. Denote T the following closed triangular domain T = {0 ≤ y ≤ x ≤ 1} .

(13)

Under the following assumptions (i) A + BK is Hurwitz (ii) C + , C − ∈ L∞ (T ) (iii) G, D ∈ L∞ ([0, 1]) the zero equilibrium of System (7)–(11) is exponentially stable in the L2 sense. Notice that the existence of a matrix K such that Assumption (i) holds is guaranteed by the stabilizability of the (A, B) pair.

F. Di Meglio et al. / Automatica 87 (2018) 281–289

−

m ∑

µk Lik (x, 0)κkj

k=1

k

283 n ∑

⎡

⎤

m

∑ ⎢ ⎥ λk ⎣ckj + qkl κlj ⎦ K ik (x, 0)

k=1

(22)

l=1

l

with initial conditions

∀j , Fig. 2. Schematic view of the cascade structure of the target system. Here, ISS refers to Input-to-State Stability considering β as the (vanishing) input.

Proof. The proof follows from considering the following Lyapunov functional

∫

1

( )−1 V (t) = e−δ x α (t , x)⊤ Λ+ α (t , x)dx 0 ∫ 1 ( )−1 Rβ (t , x)dx + X (t)⊤ PX (t) + eδ x β (t , x)⊤ Λ−

(14)

0

where the symmetric positive definite matrix P, the diagonal matrix R = diag(r1 , . . . , rm ) and the parameter δ > 0 are design parameters, and using Young’s and Poincaré’s inequalities.

To map the original system (1)–(5) to the target system (7)–(11), we use the following Volterra transformation

α (t , x) ≡ u(t , x)

(15)

(16)

0

where the kernels K , L and γ have yet to be defined. Differentiating (16) w.r.t. space and time yields the following kernel equations for 1 ≤ i ≤ m, 1 ≤ j ≤ n

µi ∂x Kij (x, y) −λj ∂y Kij (x, y) = −σii−− Kij (x, y) n m ∑ ∑ + σkj++ Kik (x, y) + σkj−+ Lik (x, y)

Lij (1, x) = lij (x).

(24)

These are degrees of freedom in the control design. However, their effect on the closed-loop performances are still unclear, thus, to study well-posedness, which we do in the next section, we only impose lij ∈ L∞ ([0, 1]). Besides, provided the K and L kernels are well-posed, the coefficients of G for j < i and the matrices C0 C + , C − and D are given by gij (x) = µj Lij (x, 0) −

n ∑

λk Kik (x, 0)qk,j −

k=1

C − (x, y) = Σ +− L(x, y) + D(x) = Σ +− γ (x) +

bkj γik (x)

(25)

k=1 x

∫

∫

p ∑

C − (x, s)K (s, y)ds

(26)

C − (x, s)L(s, y)ds

(27)

y x

y x

∫

C − (x, y)γ (y)dy

(28)

k=1

k=1

(18)

ϵi (x)∂x Fi (x, y) + νi (y)∂y Fi (x, y) = Σi (x, y)F (x, y) (30) ( )T where F = F1 · · · Fn and the functions ϵi , νi are locally Lipschitz on [0, 1]. Each unknown Fi satisfies boundary conditions on a subset Ωi ⊂ ∂ T of the following form, for i = 1, . . . , n

along with the following set of boundary conditions, for i = 1, . . . , m

µi + λj σij−−

(19) ∆

= lij µi − µj p n ∑ ∑ ∀j ≥ i µj Lij (x, 0) = λk Kik (x, 0)qk,j + bkj γik (x) k=1

(20)

(21)

k=1

where the qk,j in (21) are the elements of Q0 . Besides, γ satisfies the following ODE for all i, j

⎡ ⎤ p m ∑ ∑ ⎢ ⎥ µi γij′ (x) = bkl κlj ⎦ γik (x) − σii−− γij (x) ⎣akj + l=1

l

3. A general class of kernel equations

We consider the following class of equations on the triangular domain T defined by (13)

k=1

σij−+

We prove well-posedness of the kernel equations over the next two sections. First, we study a relatively general class of hyperbolic PDEs on a triangular domain.

(17)

k=1

µi ∂x Lij (x, y) +µj ∂y Lij (x, y) = −σii−− Lij (x, y) m n ∑ ∑ + σkj−− Lik (x, y) + σkj+− Kik (x, y)

(29)

3.1. Problem setup

for 1 ≤ i ≤ m, 1 ≤ j ≤ m

k=1

∀j < i ,

C0 = C + Q0 K.

K (x, y)u(t , y)dy β (t , x) = v (t , x) − 0 ∫ x L(x, y)v (t , y)dy − γ (x)X (t) −

∀j ̸= i, Lij (x, x) = −

where the κij are the entries of the control matrix gain K. To ensure well-posedness of the system, we add the following arbitrary boundary conditions

0

x

∀j, Kij (x, x) = −

(23)

C + (x, y) = Σ +− K (x, y) +

2.2. Volterra transform

∫

γij (0) = κij

∀(x, y) ∈ Ωi Fi (x, y) = fi (x, y) +

n ∑

Γij (x, y)Fj (x, y)

(31)

j=1

where fi , Γij ∈ L∞ (Ωi ). Although we do not impose a priori restrictions on the nature of the Ωi , they typically are edges of the triangular domain, as illustrated in Eqs. (19)–(21). Moreover, Assumption (ii) in Theorem 3.2 imposes certain relations between the Ωi and the transport velocities ϵi and νi for the system to be well-posed. The functions Γij , defined on the boundaries of the triangular domain T , are boundary couplings between the different kernels Fi . The well-posedness of (30), (31) depends on the sparsity of the matrix Γ = (Γij ). More precisely, consider the following definition. Definition 3.1. Let G be the directed graph whose vertices are the Fi and whose edges are defined by the matrix whose are ) ( coefficients the infinity-norm of the coefficients of Γ , i.e. Γij ∞ . In other words, there is an edge between nodes i and j if and only if there

284

F. Di Meglio et al. / Automatica 87 (2018) 281–289

Fig. 3. Examples of characteristic lines that satisfy Assumption (ii) of Theorem 3.2.

exists (x, y) ∈ Ωi such that Γij (x, y) ̸ = 0. A path of length p in the graph is a p–uplet a = (a1 , . . . , ap ) such that p−1 ∏ Γa

k ,ak+1

∞

̸= 0.

(32)

k=1

By convention, a path (a1 ) of length p = 1 is the single node Fa1 . The following theorem gives a sufficient condition on the structure of G for the system to be well-posed. Theorem 3.2. Consider system (30) with boundary conditions (31). Assume (i) that the uncoupled system, obtained by taking Σ (x, y) ≡ 0 in (30) and Γij = 0, ∀i, j in (31), is well-posed; (ii) that there exists α > 1 such that, for all i = 1, . . . , n, the following inequality holds

∀(x, y) ∈ T

αϵi (x) − νi (y) > δ > 0

(33)

(iii) the graph G is acyclic, i.e. is does not contain any cycles. ∞

Then there is a unique solution F ∈ L (T ). Remark 1. A necessary and sufficient condition for Assumption (i) to be satisfied is that, for every i = 1, . . . , n the characteristics defined by the ϵi , νi uniquely connect each point of T to Ωi . Remark 2. Assumption (ii) is a simple geometric condition for the well-posedness of the system: the tangent vector (ϵi (x), µi (y)) to all the characteristics, at all points (x, y) ∈ T must lie in the half-space such that the scalar product with (α, −1)T is positive. In other words, the characteristics leaving the boundaries where (31) are defined must always ‘‘point away’’ from a certain line y = α x, with α > 1. Examples of such characteristics are pictured in Fig. 3. The proof of Theorem 3.2 is quite involved and spans over the next few sections. It relies on the transformation of (30), (31) into integral equations. For this, we define in the next section the characteristic curves. 3.2. Transformation into integral equations Assumption (i) of Theorem 3.2 yields the existence and uniqueness of characteristic curves, defined as follows

Fig. 4. Example of characteristics to be integrated along, corresponding to the Γ matrix given by (37). Each colored line corresponds to the characteristics of a given kernel Fi , i = 1, 2, 3, 4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Definition 3.3. For each i = 1, . . . , n and any (x, y) ∈ T there exists (χi0 (x, y), ξi0 (x, y)) ∈ Ωi and sFi (x, y) ∈ R+ such that

⎧ ⎧ dξi (s; x, y) dχi (s; x, y) ⎪ ⎪ ⎪ = ϵi (χi (s; x, y)) ⎪ = νi (ξi (s; x, y)) ⎨ ⎨ ds ds χi (0; x, y) = χi0 (x, y) ∈ Ωi ⎪ ξi (0; x, y) = ξi0 (x, y) ∈ Ωi ⎪ ⎪ ⎪ ⎩ F ⎩ χi (sFi (x, y); x, y) = x ξi (si (x, y); x, y) = y.

(34)

The curves (χi (s), ξi (s)) are the characteristic curves associated with Fi . For any two points (M1 , M2 ) ∈ T , we denote Ci (M1 , M2 ) the characteristic curve associated with Fi starting in M1 = (x1 , y1 ) and ending in M2 = (x2 , y2 ), when such a curve exists, i.e. when

χi (sF (x1 , y1 ); x1 , y1 ) = x2 ,

ξi (sF (x1 , y1 ); x1 , y1 ) = y2 .

(35)

Notice that this curve is then unique by application of Cauchy– Lipschitz’s Theorem. In the absence of the boundary couplings Γij , the proof of wellposedness would consist in integrating (30) along (34) and using a method of successive approximations. Here, this yields Fi (x, y) = fi (χi0 (x, y), ξi0 (x, y))

+

n ∑

Γij (χi0 (x, y), ξi0 (x, y))Fj (χi0 (x, y), ξi0 (x, y))

j=1 sFi (x,y)

∫ +

Σi (χi (s; x, y), ξi (s; x, y))F (χi (s; x, y),

0

× ξi (s; x, y))ds.

(36)

The second term still contains unknowns, and the method of successive approximations does not straightforwardly apply. Rather, the second term must, again, be integrated along the characteristics of the Fj ’s for which Γij is non-zero. This situation is depicted in Fig. 4 for an example corresponding to the following matrix

Γij (x, y) =

{

1 0

if (i, j) = (1, 2), (1, 3) or (2, 3) otherwise.

(37)

To avoid this situation repeating infinitely (infinitely many ‘‘rebounds’’), we use the following basic results from graph theory.

F. Di Meglio et al. / Automatica 87 (2018) 281–289

285

3.3. Basic results from graph theory

Eq. (41) can be rewritten as

The following definitions and lemmas are classical results, see e.g. West et al. (2001).

Fi (x, y) =

(a) There exist terminal nodes, i.e. there exists a set TN ⊂ {1, . . . , n} such that ∀i ∈ TN , ∀k ∈ {1, . . . , n} Γik = 0. (b) All the valid paths are of finite length. In other words, any valid path can be completed with a valid path that leads to a terminal node and has a uniformly bounded length. This allows us to add the following two definitions. Definition 3.4. For any node Fi , we define its depth di as the length of the longest valid path to a terminal node. We also define dmax as the maximum length of any path dmax = maxi=1,...,n di . Definition 3.5. Let a = (a1 , . . . , ap ) be a (not necessarily valid) path. Then, we recursively define the sequence of points Mka (x, y) ∈ T , k = 0, . . . , p such that M0a (x, y) = (x, y)

(38)

Mka (x, y) = χa0k (Mka−1 ), ξa0k (Mka−1 ) ∈ Ωak

)

(39)

where χa0k (·) and ξa0k (·) are defined by (34). In other words, Mka is the point on the boundary of T such that the characteristic curve Cak (Mka , Mka−1 ) exists.

= fi (M1a (x, y)) +

(

(

))

= Mp(b+,a) q (x, y)

p=1 a=(a1 ,...,ap ) a1 =i

(i)

(i)

+

·

Remark 3. The sum

(

× Mka

(

∑

(i) M1

fap (Mpa

Fi (x, y) =

+

(i)

(i)

Γij (M1 (x, y))

(

))

(i) M1

p−1 ∏

Σi F

Γak ,ak+1

k=1

)

)

)

∫ )+

(

(

(i)

Cap Mpa M1

(i) fi (M1 (x

)

)) ( (i) ,Mpa−1 M1

, y)) +

∫ (

d n ∑ ∑

∑

( fap+1 (Mpa+1

(i)

(i)

Ci (M1 (x,y),M0 (x,y))

p=1 j=1 a=(a1 ,...,ap ,ap+1 ) a2 =j, a1 =i

·

)

∫

(x, y)) +

Cap (Mpa (x,y),Mpa−1 (x,y))

Σa p F .

(i)

(41)

Fi (x, y) = fi (M1 (x, y)) +

+

denotes the sum over all (possibly

We now prove Eq. (41) by recursion on the depth d. Case d = 1. Consider a node Fi such that di = 1, i.e. Fi is a terminal node. Assuming by convention that, the empty product is equal to 1, i.e.

k=1

(i)

Ci (M1 (x,y),M0 (x,y))

Σap F

(45)

i.e.

p ∏

Σi F

) Γak ,ak+1

Mka

(

) (x, y)

k=1

)

∫ (

)

Cap+1 Mpa+1 (x,y),Mpa (x,y)

(42)

d ∑

∫ (i)

(i)

Ci (M1 (x,y),M0 (x,y))

( p ∏

∑

p=1 a=(a1 ,...,ap+1 ) a1 =i

invalid) paths of length p starting from the node Fi . However, a large number of the terms of this sum is zero due to the product of Γak ,ak+1 inside this sum.

Γak ,ak+1 ≡ 1.

∫

(

d n ∑ ∑

a=(a1 ,...,ap ) a1 =i

0 ∏

(44)

Σap+1 F

(46)

i.e.

fap Mpa (x, y) +

∑

Σi F .

p=1 j=1 a=(a1 ,...,ap ) a1 =j

( ) Γak ,ak+1 Mka (x, y)

)

(i)

Fi (x, y) = fi (M1 (x, y)) +

)

( (

(i)

Notice that all the Fj for which Γij ̸ = 0 are of depth dj ≤ d. Applying Eq. (41) to them and changing the order of summation yields

(

k=1

(43)

(i)

Ci (M1 (x,y),M0 (x,y))

3.4.1. Transformation into integral equations The proof relies on the following transformation of (30), (31) into integral equations. For any i = 1, . . . , n, any M = (x, y) ∈ T , one has

∑

Σi F

Γij (M1 (x, y))Fj (M1 (x, y))

+

·

∑

Σa 1 F

j=1

3.4. Proof of Theorem 3.2

Fi (x, y) =

n ∑

∫

(40)

Proof. Classically, the proof consists in transforming the PDEs into integral equations and using a method of successive approximations.

Ci (M1a (x,y),M0a (x,y))

(i)

We are now ready to prove Theorem 3.2.

(p−1 ∏

Ca1 (M1a (x,y),M0a (x,y))

∫

Fi (x, y) = fi (M1 (x, y)) +

where (b, a) denotes the concatenation of the two paths.

di

∫

which exactly corresponds to integrating (30) along the characteristics associated to Fi since for all j = 1, . . . , n, Γij = 0 for a terminal node (see (36) with Γij = 0). Recursion d → d + 1. Assume now that (41) is true for all nodes of depth less or equal to d, for some d ∈ {1, . . . , dmax }. Consider now Fi of depth d + 1. Integrating (30) along the characteristics and plugging in the boundary conditions (31) yield

Property 3.6. For any two paths a = (a1 , . . . , ap ) and b = (b1 , . . . , bq ) one has Mpa Mqb x, y

fa1 (M1a (x, y)) +

p=1 a=(a1 ) a1 =i

Lemma 2. If G is acyclic, then the following holds

(

1 ∑ ∑

Σi F

Γak ,ak+1

Mka+1

(

) ) (x, y)

k=1

( ·

fap+1 (Mpa+1 (x, y))

)

∫ + (

)

Cap+1 Mpa+1 (x,y),Mpa (x,y)

Σap+1 F

i.e. Fi (x, y) =

(i) fi (M1 (x

, y)) +

∫ (i)

(i)

Ci (M1 (x,y),M0 (x,y))

Σi F

(47)

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F. Di Meglio et al. / Automatica 87 (2018) 281–289

+

d+1 ∑

(p−1 ∏

∑

p=2 a=(a1 ,...,ap ) a1 =i

·

Γak ,ak+1

Mka+1

) (x, y)

k=1

( fap (Mpa

Corollary 3.7. For any path a = (a1 , . . . , ap ) of length p ≥ 0 and any k = 0, . . . , p, denote Mka (x, y) = (xk , yk ). Then one has

) (

(x, y)) +

α xk − yk ≤ α x − y.

)

∫ (

)

Cap Mpa (x,y),Mpa−1 (x,y)

Σa p F

(48)

i.e., using (42) Fi (x, y) =

di ∑

(p−1 ∏

∑

p=1 a=(a1 ,...,ap ) a1 =i

Γak ,ak+1

·

) Mka+1 (x, y)

fap (Mpa (x, y)) +

(58)

Assume now that (57) is satisfied for some k = 1, . . . , p − 1, then, by definition of the Mka (·, ·), one has

k=1

(

Proof. We prove the result by induction. For k = 0, given the definition of the Mka (Eq. (38)), one has

α x0 − y0 = α x − y.

) (

(57)

)

∫

) Σap F

(

(49)

Cap Mpa (x,y),Mpa−1 (x,y)

α xk+1 − yk+1 = αχa0k (xk , yk ) − ξa01 (xk , yk ).

(59)

Using Lemma 3 for i = ak+1 , this yields

which concludes the proof by induction since di = d + 1.

α xk+1 − yk+1 < α xk − yk .

3.4.2. Method of successive approximations The end of the proof follows the classical successive approximations method, applied( to (41). More) precisely, we define the ⊤ following operators Φ = Φ1 · · · Φn

Then, using the induction assumption, this concludes the proof.

Φi [F ](x, y) = (p−1 di ∏ ∑ ∑ p=1 a=(a1 ,...,ap ) a1 =i

Γak ,ak+1

Mka (x

(

, y)

)

Σap F

(50)

as well as the following vector φ = φ1

(

φn

···

∫

) Γak ,ak+1 Mka (x, y)

(

)

fap Mpa (x, y) .

(

)

δ

(61)

q+1

k=1

(62)

[αχi (s; x, y) − ξi (s; x, y)]q ds =

α x−y

αχi0 (x,y)−ξi0 (x,y)

(51)

τ p dτ αϵi (χ¯ i (τ ; x, y)) − νi (ξ¯i (τ ; x, y))

(63)

i −1 where we have abusively denoted χ¯ i (τ ; x, y) = χi ((ψ(x ,y) ) (τ ); x, y) i −1 ¯ (resp. ξi (τ ; x, y) = ξi ((ψ(x,y) ) (τ ); x, y)). Using (33) this yields sFi (x,y)

∫

(52)

F q+1 (x, y) = φ (x, y) + Φ [F q ](x, y)

(53)

[αχi (s; x, y) − ξi (s; x, y)]q ds

0

1 [α x − y]

q+1

( )⊤ = φ1 (x, y) · · · φn (x, y) + ( )⊤ Φ1 [F q ](x, y) · · · Φn [F q ](x, y) .

< (54)

∆F q = F q − F q−1 .

(55)

∑+∞

Provided the limit exists, then F = limq↦→+∞ F = q=1 ∆F is a solution to (41). To prove that the series is convergent, we rely on the following lemmas q

δ

[ ]q+1 − αχi0 (x, y) − ξi0 (x, y) . q+1

(64)

Since (χi0 (x, y), ξi0 (x, y)) ∈ T and α > 1, one has αχi0 (x, y) − ξi0 (x, y) > 0 which yields the result.

Finally, define the following sequence for q ≥ 1

q

Lemma 3. Assume inequality (33) holds. Then for all i = 1, . . . , n, (x, y) ∈ T , the following function

ψ(xi ,y) : s ∈ [0, sFi (x, y)] ↦→ αχi (s; x, y) − ξi (s; x, y)

Lemma 5. For any path a = (a1 , . . . , ap ) of length p ≥ 0 and any k = 1, . . . , p, one has

∫

[ Cak (Mka (x,y),Mka−1 (x,y))

∫

(56)

ψ(xi ,y) (sFi (x, y)) = α x − y > αχi0 (x, y) − ξi0 (x, y) = ψ(xi ,y) (0). i Thus, ψ(x of [0, sFi (x, y)] onto its im[ 0 ,y) defines0 a diffeomorphism ] age αχi (x, y) − ξi (x, y), α x − y .

Proof. The proof is trivial since for i = 1, . . . , n and (x, y) ∈ T , one (s) = αϵi (χi (s; x, y)) − νi (ξi (s; x, y)) and recalling (33).

αχak (·) − ξak (·)

]q

(

≤

)q+1

1 αx − y

δ

q+1

.

(65)

Proof. Denoting Mka (x, y) = (xk , yk ), one has from Lemma 4 Cak (Mka (x,y),Mka−1 (x,y))

is strictly increasing. In particular, the following inequality holds

ds

sFi (x,y)

∫

F 0 (x, y) = 0

has

[αχi (s; x, y) − ξi (s; x, y)] ds ≤

i where ψ(x ,y) (·) is defined by (56). It yields

)⊤

Define now the following sequence for q ∈ N

i dψ(x ,y)

(

τ = ψ(xi ,y) (s)

0

(p−1 ∏

)q+1

1 αx − y

q

Proof. Consider the following change of variables

k=1

Cap (Mpa (x,y),Mpa−1 (x,y))

p=1 a=(a1 ,...,ap ) a1 =i

sFi (x,y)

∫

where α and δ are defined by (33).

)

∫

=

Lemma 4. For any i = 1, . . . , n and any (x, y) ∈ T , one has

0

×

φi (x, y) di ∑ ∑

(60)

≤

[ ]q αχak (·) − ξak (·)

1 [α xk−1 − yk−1 ]q+1

δ

q+1

.

(66)

Applying Corollary 3.7 yields the results. Lemma 6. Define

φ¯ = max ∥φi (·, ·)∥L∞ (T ) , i=1,...,n

Γ¯ =

max

i,j∈{1,2,...,n}

∥Γij ∥∞

(67)

F. Di Meglio et al. / Automatica 87 (2018) 281–289

¯ = Σ

max

(x,y)∈T , i=1,...,n

M=

(dmax ∏

dmax

δ

|||Σi (x, y)||| )

(68)

max −1

(n − k) Γ¯ d

¯. Σ

(69)

k=0

Assume that for some q ≥ 1, one has, for all (x, y) ∈ T q q ⏐ q ⏐ ⏐∆F (x, y)⏐ ≤ φ¯ M (α x − y) . i q!

∀i = 1, . . . , n

287

for 1 ≤ j ≤ m −− µ1 ∂x L1j (x, y) + µj ∂ξ L1j (x, y) = −σ11 L1j (x, y) m n ∑ ∑ + σpj−− L1p (x, y) + σkj+− K1k (x, y) p=1

(70)

∀j, K1j (x, x) = −

Then, one has

⏐ ⏐ M q+1 (α x − y)q+1 ⏐ q+1 ⏐ . (71) ⏐∆Fi (x, y)⏐ ≤ φ¯ (q + 1)! Assume that (70) holds for some fixed q ≥ 1. Then, one has, for all i = 1, . . . , n ⏐ ⏐ ⏐ q+1 ⏐ (72) ⏐∆Fi (x, y)⏐ ⏐ ⏐ q ⏐ ⏐ = Φi [∆F ](x, y) (73) ∫ di ∑ ∑ ⏐ ⏐ ⏐Σap ∆F q ⏐ . (74) ≤ Γ¯ p−1 ∀i = 1 , . . . , n

Cap (Mpa (x,y),Mpa−1 (x,y))

p=1 a=(a1 ,...,ap ) a1 =i

∑

∑

¯ Γ¯ p−1 Σ

p=1 a=(a1 ,...,ap ) a1 =i

( )q M αχap (·) − ξap (·) φ¯ × . q! Cap (Mpa (x,y),Mpa−1 (x,y)) q

(75)

≤

¯ φ¯ Γ¯ p−1 Σ

M q (α x − y)q+1 (q + 1)!

p=1 a=(a1 ,...,ap ) a1 =i

≤

dmax i

dmax i

∏ ⎝

∑

∀j, µ1 γ1j′ (x) =

(81)

(82)

−− akj γ1k (x) − σ11 γ1j (x)

(83)

λk ckj K 1k (x, 0)

(84)

k=1 n

+

∑ k=1

{

1 0

(85)

if y = 0 otherwise

.

⎞ max −1

(n − k)⎠ Γ¯ di

¯ φ¯ Σ

k=1 (n

M q (α x − y)q+1 (q + 1)!

− k) paths of

4. Well-posedness of (17)–(24) and control law In this section, we apply the results of Section 3 to prove the well-posedness of the kernel equations. The following theorem assesses the well-posedness of the kernel equations. Theorem 4.1. System (17)–(24) has a unique solution K , L ∈ L∞ (T ). Proof. We prove the result by induction on i = 1, . . . , n. i = 1. For i = 1, the equations are rewritten as follows for 1 ≤ j ≤ n

p=1

−− akj γ˜k (x, y) − σ11 γ˜j (x, y)

n ∑

] K (x, y)λk ckj 1k

(87)

k=1

with boundary conditions

γ˜j (x, x) = 1{y=0} (x, x)κ1j . (77)

which, in turn, yields the result given the definition of M (Eq. (69)). Finally, Lemma 6 ensures that the series (55) is uniformly convergent, thus the kernel equations (30) with boundary conditions (31) are well-posed (see, e.g. Di Meglio, Vazquez, and Krstic, 2013 for a detailed proof). In the next section, we apply Theorem 3.2 to prove well-posedness of (17)–(24).

−− µ1 ∂x K1j (x, y) − λj ∂ξ K1j (x, y) = −σ11 K1j (x, y) n m ∑ ∑ + σkj++ K1k (x, y) + σpj−+ L1p (x, y)

n ∑ k=1

(76)

∏p

(86)

one can define γ˜j such that ∀(x, y) ∈ T γ˜j (x, y) = 1{y=0} (x, y)γ1j (x) or, equivalently, the γ˜j satisfy the following PDEs of the form (30)

+

k=1

k=1

∆

k=1

µ1 ∂x γ˜j (x, y) = 1{y=0} (x, y)

Noticing that there cannot be more than length p from a given node i, this yields

⎛

(80)

k=1 p

[

Using Lemma 5 yields

∑

∆

= k1j

= l1j µ1 − µj p n ∑ ∑ ∀j ≥ 1 µj L1j (x, 0) = λk K1k (x, 0)qk,j + bkj γ1k (x)

1{y=0} (x, y) =

∫

di ∑

µ1 + λj σ1j−−

One can readily check that (78)–(79) are of the form (30). Besides, the ODE (84) can also be put under the form (30) by ‘‘embedding’’ it into T . More precisely, denoting

di

≤

∀j ̸= 1, L1j (x, x) = −

σ1j−+

∀j, γ1j (0) = κ1j .

Using (70) yields

(79)

k=1

(88)

Besides, boundary conditions (80)–(82), (88) are of the form (31), with the boundary coupling coefficients Γij being zero for every kernel except the Lij on the y = 0 boundary. Therefore, the graph defined by Γij is acyclic, and Theorem 3.2 applies to (78)–(85) which is well-posed, i.e. has a unique solution with K1j , L1j ∈ L∞ (T ) and γ1j ∈ L∞ ([0, 1]). {1, . . . , i − 1} → i. Let i ∈ {2, . . . , m} be fixed and assume that for k = 1, . . . , i − 1 there exist Kkj , Lkj ∈ L∞ (T ) and γkj ∈ L∞ ([0, 1]), for all j. Then, Eqs. (17)–(24) are of the form (30), (31) with coefficients in L∞ since they are linear in the Kij , Lij and γij variables with coefficients that depend on the Kkj , Lkj and γkj for k < j. Thus, Theorem 3.2 applies again and the equations are well-posed. This yields the main result of the paper, stated in the following theorem. Theorem 4.2. Consider System (1)–(5) where (A, B) is assumed stabilizable with the following control law U(t) = −R1 u(t , 1) +

1

∫

K (1, y)u(t , y)dy 0 1

∫

L(1, y)v (t , y)dy + γ (1)X (t)

+

(89)

0

(78)

where K , L and γ are defined by (17)–(24). Then, the zero equilibrium is exponentially stable in the L2 sense.

288

F. Di Meglio et al. / Automatica 87 (2018) 281–289

Proof. Theorem 4.1 ensures the existence of K , L ∈ L∞ (T ), γ ∈ L∞ ([0, 1]) such that (15), (16) hold and (α, β ) satisfies (7), (11). Lemma 1 and the invertibility of the Volterra transformation yields the result. Remark 4. Interestingly, this result encompasses the result of Tsubakino and Oliveira (2016). More precisely, the systems considered in Tsubakino and Oliveira (2016) have the structure (1)–(5) with the matrices Σ ·· , C , R1 , Q0 taken equal to zero. With these parameters, the target system (7)–(11) is identical to the one in Tsubakino and Oliveira (2016) and both approaches use a Volterra transformation to perform the change of coordinates from (u, X ) to (α, X ) (the β state is identically zero in this case). Since Theorem 3.2 gives uniqueness of the transformation kernels, the two approaches necessarily yield the same controller. 5. Conclusion and perspectives We have presented a control design for ODEs with a system of hyperbolic PDEs in the actuating path. The design results in a full-state feedback law needing measurements of the distributed actuator states along the spatial domain. This is not realistic in practice and future contributions will focus on the design of an observer solely relying on (collocated) boundary measurements. Besides, the result opens the door to control design for other systems involving cascaded hyperbolic PDEs. In particular, networks of systems of hyperbolic balance laws are instrumental in modeling, e.g. oil production systems, networks of open channels (Bastin, Coron, & d’Andréa Novel, 2009) or power transmission lines (Günther, 2001). References Bastin, G., Coron, J.-M., & d’Andréa Novel, B. (2009). On lyapunov stability of linearised saint-venant equations for a sloping channel. Networks and Heterogeneous Media, 4(2), 177–187. Bekiaris-Liberis, N., & Krstic, M. (2013). Nonlinear control under nonconstant delays. SIAM. Bekiaris-Liberis, N., & Krstic, M. (2014). Compensation of wave actuator dynamics for nonlinear systems. IEEE Transactions on Automatic Control, 59(6), 1555– 1570. Bresch-Pietri, D. (2012). Robust control of variable time delay systems. theoretical contributions and applications to engine control. MINES ParisTech, (Ph.D. thesis). Bresch-Pietri, D., & Krstic, M. (2014). Output-feedback adaptive control of a wave PDE with boundary anti-damping. Automatica, 50(5), 1407–1415. Castillo Buenaventura, F., Witrant, E., Prieur, C., & Dugard, L. (2012). Dynamic boundary stabilization of hyperbolic systems. In Proceedings of CDC, page n/c, Maui, Hawaï, États-Unis, December. Di Meglio, F., & Aarsnes, U. J. F. (2015). A distributed parameter systems view of control problems in drilling In (in press) in 2nd IFAC workshop on automatic control in offshore oil and gas production, Florianópolis, Brazil. Di Meglio, F., Vazquez, R., & Krstic, M. (2013). Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input. IEEE Transactions on Automatic Control, 58(12), 3097–3111. Fridman, E., & Shaked, U. (2002). H–∞ control of linear state-delay descriptor systems: an LMI approach. Linear Algebra and its Applications, 352, 271–302 Fourth Special Issue on Linear Systems and Control. Germay, C., Denoël, V., & Detournay, E. (2009). Multiple mode analysis of the selfexcited vibrations of rotary drilling systems. Journal of Sound and Vibration, 325(12), 362–381. Günther, M. (2001). A Pdae model for interconnected linear rlc networks. Mathematical and Computer Modelling of Dynamical Systems, 7(2), 189–203. Hu, L., Di Meglio, F., Vazquez, R., & Krstic, M. (2016). Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes. IEEE Transactions on Automatic Control, 61(99), 3301–3314. Krstic, M. (2009). Compensating actuator and sensor dynamics governed by diffusion Pdes. Systems & Control Letters, 58(5), 372–377. Krstic, M., & Smyshlyaev, A. (2008). Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Systems & Control Letters, 57(9), 750–758. Li, J., & Liu, Y. (2012). Adaptive control of the ode systems with uncertain diffusiondominated actuator dynamics. International Journal of Control, 85(7), 868–879.

Michiels, W., & Niculescu, S. (2007). Stability and stabilization of time-delay systems. An eigenvalue based approach. Philadelphia: SIAM Publications. Ren, B., Wang, J.-M., & Krstic, M. (2013). Stabilization of an ode–schrödinger cascade. Systems & Control Letters, 62(6), 503–510. Sagert, C., Di Meglio, F., Krstic, M., & Rouchon, P. (2013). Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling. In IFAC sympositium on system structure and control, (pp. 779–784). Saldivar, M. B., Mondie, S., Loiseau, J.-J., & Rasvan, V. (2011). Stick-slip oscillations in oillwell drillstrings: Distributed parameter and neutral type retarded model approaches. In Proceedings of the 18th IFAC World congress, Vol. 18, (pp. 284– 289). Smith, O. (1959). A controller to overcome dead time. ISA Journal, 6, 28–33. Tang, S., & Xie, C. (2011). State and output feedback boundary control for a coupled pde–ode system. Systems & Control Letters, 60(8), 540–545. Tsubakino, M., Krstic, D., & Oliveira, T. R. (2016). Exact predictor feedbacks for multiinput {LTI} systems with distinct input delays. Automatica, 71, 143–150. West, D. B., et al. (2001). Introduction to graph theory, Vol. 2. Upper Saddle River: Prentice hall. Wu, H.-N., & Wang, J.-W. (2013). Observer design and output feedback stabilization for nonlinear multivariable systems with diffusion PDE-governed sensor dynamics. Nonlinear Dynamics, 72(3), 615–628. Yue, D., & Han, Q.-L. (2005). Delayed feedback control of uncertain systems with time-varying input delay. Automatica, 41, 233–240.

Florent Di Meglio is tenured professor at the Centre Automatique et Systémes of MINES ParisTech, part of PSL Research University. He received his Ph.D. from the same university in Mathematics and Control in 2011, and was a Postdoctoral Researcher at UC San Diego from 2011 to 2012. His current research interests include control and estimation design for hyperbolic PDEs, with application to process control, most notably multiphase flow control and oil drilling.

Federico Bribiesca Argomedo was born in Zamora, Mexico, in 1987. He received the B.Sc. degree in Mechatronics Engineering from the Tecnológico de Monterrey, Monterrey, Mexico, in 2009, the M.Sc. degree in Control Systems from Grenoble INP, Grenoble, France, in 2009, and the Ph.D. degree in Control Systems from the GIPSA-Lab, Grenoble University, Grenoble. He held a post-doctoral position with the Department of Mechanical and Aerospace Engineering, University of California at San Diego, San Diego, CA, USA. He is currently an Assistant Professor with the Department of Mechanical Engineering, Institut national des sciences appliquèes de Lyon, Lyon, France, attached to Ampère Laboratory. In particular, he has applied his research techniques to tokamak safety factor control and batteries. His current research interests include control of partial differential equations and nonlinear control theory.

Long Hu received the Ph.D. degrees in Applied Mathematics from Fudan University, Shanghai, China, in June 2015 and Université Pierre et Marie Curie, Paris, France, in September 2015. He is currently an Assistant Professor at Shandong University, Jinan, China. His research interests include control and stabilization theory for linear and nonlinear hyperbolic systems.

Miroslav Krstic is Distinguished Professor of Mechanical and Aerospace Engineering, holds the Alspach endowed chair, and is the founding director of the Cymer Center for Control Systems and Dynamics at UC San Diego. He also serves as Associate Vice Chancellor for Research at UCSD. As a graduate student, Krstic won the UC Santa Barbara best dissertation award and student best paper awards at CDC and ACC. Krstic is Fellow of IEEE, IFAC, ASME, SIAM, and IET (UK), Associate Fellow of AIAA, and foreign member of the Academy of Engineering of Serbia. He has received ASME Oldenburger Medal, ASME Nyquist Lecture Prize, ASME Paynter Outstanding Investigator Award, the PECASE, NSF

F. Di Meglio et al. / Automatica 87 (2018) 281–289 Career, and ONR Young Investigator awards, the Axelby and Schuck paper prizes, the Chestnut textbook prize, and the first UCSD Research Award given to an engineer. Krstic has also been awarded the Springer Visiting Professorship at UC Berkeley, the Distinguished Visiting Fellowship of the Royal Academy of Engineering, the Invitation Fellowship of the Japan Society for the Promotion of Science, and honorary professorships from four universities in China. He serves as Senior Editor in

289

IEEE Transactions on Automatic Control and Automatica, as editor of two Springer book series, and has served as Vice President for Technical Activities of the IEEE Control Systems Society and as chair of the IEEE CSS Fellow Committee. Krstic has coauthored twelve books on adaptive, nonlinear, and stochastic control, extremum seeking, control of PDE systems including turbulent flows, and control of delay systems.