Output feedback control of general linear heterodirectional hyperbolic ODE–PDE–ODE systems

Output feedback control of general linear heterodirectional hyperbolic ODE–PDE–ODE systems

Automatica 95 (2018) 472–480 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper ...

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Automatica 95 (2018) 472–480

Contents lists available at ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Output feedback control of general linear heterodirectional hyperbolic ODE–PDE–ODE systems✩ Joachim Deutscher a, *, Nicole Gehring b , Richard Kern c a b c

Lehrstuhl für Regelungstechnik, Universität Erlangen-Nürnberg, Cauerstraße 7, D–91058 Erlangen, Germany Institut für Regelungstechnik und Prozessautomatisierung, Universität Linz, Altenberger Straße 69, 4040 Linz, Austria Lehrstuhl für Regelungstechnik, Technische Universität München, Boltzmannstraße 15, D-85748 Garching, Germany

article

info

Article history: Received 3 November 2017 Received in revised form 15 March 2018 Accepted 8 May 2018

Keywords: Distributed-parameter systems Hyperbolic systems Coupled ODE–PDE–ODE systems Backstepping Boundary control Observers

a b s t r a c t This paper considers the backstepping design of observer-based compensators for general linear heterodirectional hyperbolic ODE–PDE–ODE systems, where the ODEs are coupled to the PDEs at both boundaries and the input appears in an ODE. A state feedback controller is designed by mapping the closed-loop system into a stable ODE–PDE–ODE cascade. This is achieved by representing the ODE at the actuated boundary in Byrnes–Isidori normal form. The resulting state feedback is implemented by an observer for a collocated measurement of the PDE state, for which a systematic backstepping approach is presented. The exponential stability of the closed-loop system is verified in the ∞-norm. It is shown that all design equations can be traced back to kernel equations known from the literature, to simple Volterra integral equations of the second kind and to explicitly solvable boundary value problems. This leads to a systematic approach for the boundary stabilization of the considered class of ODE–PDE–ODE systems by output feedback control. The results of the paper are illustrated by a numerical example. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, the backstepping approach (see, e.g., Krstic & Smyshlyaev, 2008) was utilized to provide systematic solutions for the boundary stabilization of PDE–ODE systems. At first, PDE–ODE cascades were considered, where an ODE is coupled to a PDE or vice versa (see, e.g., Krstic, 2009). A more challenging problem is the stabilization of coupled PDE–ODE systems, which arise directly in the modelling if dynamic boundary conditions (BCs) have to be taken into account (see, e.g., Sagert, Di Meglio, Krstic, & Rouchon, 2013, Tang & Xie, 2011). Recently, the extension of the previous backstepping results to coupled PDE–ODE systems, where the PDEs describe a general heterodirectional hyperbolic system, attracted the attention of many researchers. The interest in this problem stems from applications including coupled string networks (see, e.g., Ch. 6 Luo, Guo, & Morgul, 1999), networks of open channels and transmission lines (see, e.g., Bastin & Coron, 2016). By making use of the results in Hu, Meglio, Vazquez, and Krstic (2016), a backstepping approach for this system class with constant coefficients ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Rafael Vazquez under the direction of Editor Miroslav Krstic. Corresponding author. E-mail addresses: [email protected] (J. Deutscher), [email protected] (N. Gehring), [email protected] (R. Kern).

*

https://doi.org/10.1016/j.automatica.2018.06.021 0005-1098/© 2018 Elsevier Ltd. All rights reserved.

was presented in Di Meglio, Argomedo, Hu, and Krstic (2018). Subsequently, the work (Deutscher, Gehring, & Kern, 2018) considered the case of spatially-varying coefficients by making use of the results in Hu, Vazquez, Meglio, and Krstic (2015). The approach in Di Meglio et al. (2018) only treats the state feedback controller design, while in Deutscher et al. (2018) also an observer for an anticollocated measurement was presented in order to design an observer-based compensator. For enlarging the applicability of the backstepping method, it is reasonable to consider ODE–PDE–ODE systems, where the actuated boundary is described by an ODE with an input. The latter describes the dynamics of the actuator and thus leads to a much more involved stabilization problem for the underlying distributed-parameter system (DPS). A first solution of this problem for a 2 × 2 hyperbolic system with a fully actuated ODE at z = 1 is presented in Bou Saba, Bribiesca-Argomedo, Di Loreto, and Eberard (2017). Therein, a single transformation is proposed to map the system into a cascade of an ODE and a coupled PDE–ODE system. The backstepping transformation follows from a new type of kernel equations, that consists of PDEs coupled with ODEs. Furthermore, exponential stability of the coupled target system is shown in the L2 sense. In this paper, the design of observer-based compensators for general linear heterodirectional ODE–PDE–ODE systems with spatially-varying coefficients is considered. The input of the system acts on the ODE appearing at z = 1. Furthermore, a collocated boundary measurement of the distributed state is assumed for

J. Deutscher et al. / Automatica 95 (2018) 472–480

the observer design. In order to solve the corresponding output feedback stabilization problem, the results in Deutscher et al. (2018) are generalized. Therein, a two-step approach is utilized to determine the controller for the coupled PDE–ODE system. In the first step, the DPS is mapped into backstepping coordinates. As the related target system is of much simpler structure, this significantly facilitates the decoupling into a stable PDE–ODE cascade in the second step. In the paper this method is applied to the design of the state feedback controller in order to map the closed-loop system into an ODE–PDE–ODE system, where the PDE subsystem is decoupled from the ODE at the unactuated boundary. The corresponding transformations can directly be obtained by solving the kernel equations in Hu et al. (2015) and simple Volterra integral equations of the second kind. As the input acts on the ODE, the last step for obtaining a stable ODE–PDE–ODE cascade requires the introduction of new coordinates to represent the ODE at z = 1 in its multivariable Byrnes–Isidori normal form (see, e.g., Isidori, 1995, Ch. 5.1). For this, a vector relative degree of one is assumed, which is a requirement often met in applications. Typical examples are hyperbolic flexible structures, where a rigid body is attached at the actuated boundary. This assumption also includes the full actuation considered in Bou Saba et al. (2017) as a special case. On the basis of the resulting Byrnes–Isidori normal form, the state feedback controller can easily be determined. For its implementation, a collocated observer is designed. Compared to an anticollocated observer, this is a much more challenging problem as the ODE at the unactuated boundary of the observer is, in this case, subject to a coupling with the PDEs from both boundaries. By making use of the two-step approach, it is shown that only the usual observer kernel equations for the PDE subsystem and simple Volterra integral equations of the second kind have to be solved for the observer design. The solutions of all other design equations are obtained explicitly. The separation principle is verified for the corresponding closed-loop system. This is possible by utilizing the simple structure of the target systems in order to calculate the closed-loop solution pointwise in space. On the basis of this result, the exponential decay of the distributed closed-loop states w.r.t. the ∞-norm, i.e., pointwise in space, is shown. This leads to the systematic design of observer-based compensators for a large class of coupled hyperbolic PDEs with dynamic BCs at both boundaries. The next section presents the formulation of the considered output feedback stabilization problem. In Section 3, the state feedback is designed. In order to implement this controller, Section 4 considers the observer design for a collocated measurement. Section 5 is devoted to the stability analysis of the closed-loop system with the observer-based compensator. The results of the paper are illustrated by means of a numerical example.

Consider the general linear hyperbolic ODE–PDE–ODE system

Σn1 (w1 )

that consists of n coupled transport PDEs (1a) with the distributed state x(z , t) = [x1 (z , t) . . . xn (z , t)]⊤ ∈ Rn and the ODEs (1d) and (1e) with the lumped states w0 (t) ∈ Rn0 and w1 (t) ∈ Rn1 . The input is u(t) ∈ Rp and the collocated measurement is y(t) ∈ Rm with p + m = n and p, m ≥ 1. Furthermore, Λ(z) in (1a) is given by

Λ(z) = diag(λ1 (z), . . ., λn (z)),

(2)

where λi ∈ C 1 [0, 1], i = 1, 2, . . . , n, and λ1 (z) > · · · > λp (z) > 0 > λp+1 (z) > · · · > λn (z), z ∈ [0, 1]. Moreover, the matrix A(z) = [Aij (z)] in (1a) satisfies Aij ∈ C 1 [0, 1], i, j = 1, 2, . . . , n and Aii (z) = 0, z ∈ [0, 1], i = 1, 2, . . . , n. Note, that the latter condition means no loss of generality (see, e.g., Hu et al., 2016). The initial conditions (ICs) of (1) are x(z , 0) = x0 (z) ∈ Rn , z ∈ [0, 1], w0 (0) = w0,0 ∈ Rn0 and w1 (0) = w1,0 ∈ Rn1 . With the matrices

[ ] E1 =

Ip 0

∈ Rn×p and E2 =

[ ] 0 Im

∈ Rn×m ,

⇔ z =1

Σ∞ (x)

⇔ z =0

Σn0 (w0 )

(3)

the transport in the negative direction of the spatial coordinate z is described by the states x1 (z , t) = E1⊤ x(z , t) ∈ Rp while the remaining states x2 (z , t) = E2⊤ x(z , t) ∈ Rm take the transport in the opposite direction into account. This gives rise to the state partitioning x(z , t) = col(x1 (z , t), x2 (z , t)) for the PDE subsystem (1a)–(1c). The PDEs for the states x1 are defined on (z , t) ∈ [0, 1) × R+ , while the PDEs for the states x2 evolve on (z , t) ∈ (0, 1]× R+ . Hence, the distributed-parameter subsystem (1a)–(1c) is a heterodirectional system (see Hu et al., 2016). The following assumptions are imposed: (A1) (A2) (A3) (A4)

(F0 , B0 ) is controllable, (C0 , F0 ) is observable, rank C1 = rank B = p, n1 ≥ p and det(C1 B) ̸ = 0.

The Assumptions (A1) and (A2) are needed for the stabilization of the ODE subsystems appearing in the state feedback and observer design. Moreover, Assumptions (A3) and (A4) are introduced so that a Byrnes–Isidori normal form exists for (C1 , F1 , B1 ) (see Section 3.2) and n1 ≥ p is assumed in order to avoid overactuation, i.e., more inputs than states. Finally, (A4) means that (C1 , F1 , B) has a vector relative degree equal to one. This paper is concerned with the backstepping design of an observer-based compensator, that stabilizes the system (1). 3. State feedback design In what follows the state feedback controller u(t) = K[w0 (t), w1 (t), x(t)]

2. Problem formulation

u→

473

(4)

with the formal feedback operator K is determined by mapping (1) into a stable ODE–PDE–ODE cascade. 3.1. Decoupling of the PDE subsystem

↓ y

It is shown in Deutscher et al. (2018) that the backstepping transformation and the transformation to decouple the PDE subsystem from the w0 -system at z = 0

described by

∂t x(z , t) = Λ(z)∂z x(z , t) + A(z)x(z , t) x2 (0, t) = Q0 x1 (0, t) + C0 w0 (t), x1 (1, t) = Q1 x2 (1, t) + C1 w1 (t), w ˙ 0 (t) = F0 w0 (t) + B0 x1 (0, t), w ˙ 1 (t) = F1 w1 (t) + B1 x2 (1, t) + Bu(t), y(t) = x2 (1, t),

(1a)

z



t>0

(1b)

x˜ (z , t) = x(z , t) −

K (z , ζ )x(ζ , t)dζ = T1 [x(t)](z)

t>0

(1c)

t>0

(1d)

x˜ (z , t) = T2−1 [ϑ (t)](z) + NI (z)w0 (t)

t>0

(1e)

t≥0

(1f)

(5a)

0

(5b)

with T2 [ϑ (t)](z) = ϑ (z , t) + −1

z



PI (z , ζ )ϑ (ζ , t)dζ 0

(6)

474

J. Deutscher et al. / Automatica 95 (2018) 472–480

In view of Assumption (A4), the feedback

map (1) into the ODE–PDE–ODE system

w ˙ 1 (t) = F1 w1 (t) + B1 x2 (1, t) + Bu(t) ∂t ϑ (z , t) = Λ(z)∂z ϑ (z , t) + H0 (z)ϑ1 (0, t) ϑ2 (0, t) = Q0 ϑ1 (0, t) ϑ1 (1, t) = C1 w1 (t) + C [w0 (t), x(t), x2 (1, t)] w ˙ 0 (t) = (F0 − B0 K0 )w0 (t) + B0 ϑ1 (0, t), in which ϑi = Ei ϑ , i = 1, 2, K0 ∈ R ⊤ ⊤ n×p [H ⊤ with 1 (z) H 2 (z)] ∈ R ⊤



0

⎢ ⎢h21 (z) H1 (z) = ⎢ ⎢ . ⎣ .. hp1 (z)

... .. . .. . ...

... .. . .. .

0

hp p−1 (z)

0

p×n0

(7a) (7c) (7d) (7e) as well as H0 (z) =

(8)

V (z , ζ )x(ζ , t)dζ − N(z)w0 (t) 0

= T2 T1 [x(t)](z) − T2 [NI ](z)w0 (t)

E1⊤ (V (1, 0)Λ(0)E2 C0 − N(1)F0 )w0 (t)

+ (Q1 E2⊤ A(1) − E1⊤ V (1, 1)Λ(1))(E1 Q1 + E2 )x2 (1, t) + Q1 E2⊤ Λ(1)E2 ∂z x2 (1, t) ∫ 1 ((V (1, z)Λ(z))′ − V (1, z)A(z))x(z , t)dz . + E1⊤

(15)

0

Hence, the feedback (14) also depends on the local terms x1 (0, t), x2 (1, t) and ∂z x2 (1, t). In view of (12), nη = n1 − p remaining state components η have to be introduced so that (7a) can be represented in new coordinates. For this, consider

η(t) = Π1 w0 (t) + Tη w1 (t) +



1

R1 (z)ϑ (z , t)dz

(16)

0

∫ ζ

z

P(z , ζ¯ )K (ζ¯ , ζ )dζ¯

(10)

∫z

following from the definition T2 [˜x(t)](z) = x˜ (z , t) − 0 P(z , ζ )x˜ · (ζ , t)dζ of the inverse of (6). With that, the formal operator C [w0 (t), x(t), x2 (1, t)] = −E1⊤ N(1)w0 (t) 1

E1⊤ V (1, z)x(z , t)dz + Q1 x2 (1, t)



C [w ˙ 0 (t), ∂t x(t), x˙ 2 (1, t)] =

(9)

based on (5), with the kernel



with the new input u¯ (t) ∈ Rp achieves ξ˙ (t) = u¯ (t). By taking (1) into account and utilizing an integration by parts one obtains

+ E1⊤ (V (1, 0)Λ(0)(E1 + E2 Q0 ) − N(1)B0 )x1 (0, t)

.. ⎥ .⎥ p×p ⎥ .. ⎥ ∈ R ⎦ .

z

V (z , ζ ) = K (z , ζ ) + P(z , ζ ) −

(14)

+ (Q1 E2⊤ A(1) − E1⊤ V (1, 1)Λ(1))E1 C1 w1 (t)



and H2 (z) ∈ R . The latter matrices are determined by NI (z) and PI (z , ζ ). In order to find C in (7d), it is convenient to consider the composite transformation



− C [w ˙ 0 (t), ∂t x(t), x˙ 2 (1, t)] + u¯ (t))

(7b)

m×p

ϑ (z , t) = x(z , t) −

u(t) = (C1 B)−1 (−C1 F1 w1 (t) − C1 B1 x2 (1, t)

(11)

0

in (7d) is derived. It is verified in Deutscher et al. (2018) that the equations to be solved for determining (5) are well-posed and that they can be traced back to solving the kernel equations found in Hu et al. (2015) and Volterra integral equations of the second kind. In particular, the kernel K (z , ζ ) follows from solving the kernel equations for the state feedback design in Hu et al. (2015). Furthermore, the kernel PI (z , ζ ) and the matrix NI (z) are obtained by determining the solution of the inverse decoupling equations. With this, the kernel P(z , ζ ) results from PI (z , ζ ) by utilizing the corresponding reciprocity relation. The result of the transformation (5) is that the BC (7c) of the PDE subsystem (7b)–(7d) is decoupled from the w0 -system (cf. (1b)).

with Π1 ∈ Rnη ×n0 , Tη ∈ Rnη ×n1 and R1 (z) ∈ Rnη ×n . In Isidori (1995, Prop. 5.1.2) it is shown that Tη can always be found such that

[ ] det

C1 Tη

= det T1 ̸= 0

(17)

and Tη B = 0 hold. This implies that (12) and (16) qualify as a change of coordinates for (7a). In what follows, Π1 and R1 (z) in (16) are determined such that the resulting η-system takes the form

η˙ (t) = Fη η(t) + ˜ P ξ (t)

(18)

with the matrices Fη and ˜ P to be determined. For this, the ξ -coordinates (12) and the w1 -system are mapped into ∫z ϑ -coordinates using the inverse x(z , t) = ϑ (z , t) + 0 VI (z , ζ ) · ϑ (ζ , t)dζ + T1−1 [NI ](z)w0 (t) of the composite transformation (9), with the kernel VI (z , ζ ) resulting from V (z , ζ ) and the corresponding reciprocity relation. Therein, T1−1 is the inverse backstepping transformation (see (5a) and Hu et al., 2015). By that, (12) reads

ξ (t) = C1 w1 (t) + C˜[w0 (t), ϑ (t), ϑ2 (1, t)]

(19)

with the formal operator

˜[w0 (t), ϑ (t), ϑ2 (1, t)] = (Q1 E2⊤ − E1⊤ )M1 w0 (t) C

3.2. Decoupling of the w1 -system

+ (Q1 E2⊤ − E1⊤ )

In what follows, new coordinates are introduced for the w1 system in (7a) so that a closed-loop system in the form of an ODE–PDE–ODE cascade can be achieved. In particular, the BC (7d) suggests to introduce the new state components

ξ (t) = C1 w1 (t) + C [w0 (t), x(t), x2 (1, t)]

(12)

for the w1 -system in (7a) with ξ (t) ∈ R in order to decouple this system from the ϑ - and the w0 -system. Time differentiation of (12) yields p

ξ˙ (t) = C1 F1 w1 (t) + C1 B1 x2 (1, t) + C1 Bu(t) + C [w ˙ 0 (t), ∂t x(t), x˙ 2 (1, t)].

1



M2 (z)ϑ (z , t)dz + Q1 ϑ2 (1, t),

(20)

0

in which M1 = T1−1 [NI ](1) and M2 (z) = VI (1, z) were utilized. Moreover, apply the transformation T1−1 to map x2 (1, t) in x˜ coordinates, utilize (5b) and change the order of integration in the result. With this, (7a) becomes

w ˙ 1 (t) = F1 w1 (t) + B1 E2⊤ M1 w0 (t) + B1 ϑ2 (1, t) ∫ 1 + B1 E2⊤ M2 (z)ϑ (z , t)dz + Bu(t).

(21)

0

In view of Tη F1 T1−1 = [P Fη ] (see (17)) and (13)

˜ P = P + R1 (1)Λ(1)E1 ,

(22)

J. Deutscher et al. / Automatica 95 (2018) 472–480

time differentiation of (16), utilizing (1), (7e) and (21) as well as an integration by parts show that (18) is obtained if Π1 and R1 (z) are the solution of Fη Π1 − Π1 (F0 − B0 K0 ) = CM1

(23a)

(R1 (z)Λ(z)) + Fη R1 (z) = CM2 (z),

z ∈ (0, 1)



R1 (0)Λ(0)(E1 + E2 Q0 ) = Π1 B0 +

(23b)

1



R1 (z)H0 (z)dz

(23c)

0

R1 (1)Λ(1)E2 = PQ1 − Tη B1 ⊤



(23d)

u¯ (t) = −Kξ ξ (t) − Kη η(t)

(24)

in (14) this leads to the ODE–PDE–ODE cascade

⇒ z =1

Σ∞ (ϑ )

⇒ z =0

Σn0 (w0 )

having the representation

ξ˙ (t) = −Kξ ξ (t) − Kη η(t) η˙ (t) = Fη η(t) + ˜ P ξ (t) ∂t ϑ (z , t) = Λ(z)∂z ϑ (z , t) + H0 (z)ϑ1 (0, t) ϑ2 (0, t) = Q0 ϑ1 (0, t) ϑ1 (1, t) = ξ (t) w ˙ 0 (t) = (F0 − B0 K0 )w0 (t) + B0 ϑ1 (0, t).

(25a) (25b) (25c) (25d) (25e) (25f)

The next lemma presents the conditions for the solvability of (23), i.e., the existence of the mapping into the ODE–PDE–ODE cascade (25). Lemma 1. If the spectra σ (F0 − B0 K0 ) of F0 − B0 K0 and σ (Fη ) of Fη satisfy σ (F0 − B0 K0 ) ∩ σ (Fη ) = ∅, then (23) has a unique solution with the elements of R1 (z) piecewise C 1 -functions. For the proof see Appendix. Since the pair (F0 , B0 ) is controllable by Assumption (A1), the condition of Lemma 1 can always be fulfilled. In addition, the (ξ , η)-system has to be stabilizable, which can be checked with the condition presented in the next lemma. Lemma 2. If vi ⊤ R1 (1)Λ(1)E1 ̸ = −vi ⊤ P, i = 1, 2, . . . , q, holds for all eigenvalues µi , i = 1, 2, . . . , q, of Fη in the closed right halfplane and corresponding left eigenvectors vi , then the (ξ , η)-system is stabilizable. This result directly follows from applying the PHB eigenvector tests (see, e.g., Kailath, 1980, Th. 6.2-5). Remark 3. Since the eigenvalues of Fη coincide with the invariant zeros of (C1 , F1 , B) (see, e.g., Isidori, 1995, Ch. 4.3), the condition of Lemma 2 is only an issue for a non-minimum phase system (C1 , F1 , B) and thus is always fulfilled in the minimum phase case. The next theorem asserts the stability of (25), which implies a stable closed-loop system in the original coordinates. For this, piecewise differentiable ICs x0 (z) are assumed so that the spatial derivative appearing in (14) and (15) is well-defined. Theorem 4 (State Feedback Controller). Assume that (25a)–(25b) is exponentially stable and that F0 − B0 K0 is a Hurwitz matrix. Then, the closed-loop system, i.e., (1) with state feedback (14) and (24), is asymptotically stable pointwise in space for all piecewise differentiable ICs x(z , 0) = x0 (z) as well as the ICs w0 (0) = w0,0 ∈ Rn0 and w1 (0) = w1,0 ∈ Rn1 . More specifically, the convergence of ∑ ∥x(t)∥∞ = supz ∈[0,1] ∥x(z , t)∥Rn for t ≥ tc with tc = pi=+11 |φi (1)|

and φi (z) =

∫z 0

The proof of this theorem can be found in Deutscher et al. (2018), where the stability of the same type of ODE–PDE–ODE cascade is investigated. This can easily be checked on the basis of the cascade (25), the results in Deutscher et al. (2018) and the corresponding inverse transformations. In the latter reference it is also verified that the rate of exponential convergence can be specified by an eigenvalue assignment for (25a)–(25b) and F0 − B0 K0 . Finally, the feedback operator K can be represented in the original coordinates by utilizing (12), (15) and (16) in the overall control law (14) and (24), as well as taking (9) in (16) into account.



with C = Tη B1 E2 − P(Q1 E2 − E1 ). Together with the feedback

Σn1 (ξ , η)

475

(dζ /λi (ζ )) is exponential.

4. Collocated observer design Consider the observer

∂t xˆ (z , t) = Λ(z)∂z xˆ (z , t) + A(z)xˆ (z , t) + L(z)(y(t) − xˆ 2 (1, t)) xˆ 2 (0, t) = Q0 xˆ 1 (0, t) + C0 w ˆ 0 (t) xˆ 1 (1, t) = Q1 y(t) + C1 w ˆ 1 (t) ˙ˆ 0 (t) = F0 w w ˆ 0 (t) + B0 xˆ 1 (0, t) + L0 (y(t) − xˆ 2 (1, t))

(26d)

˙ˆ 1 (t) = F1 w w ˆ 1 (t) + B1 y(t) + Bu(t) + L1 (y(t) − xˆ 2 (1, t))

(26e)

(26a) (26b) (26c)

for (1) with the PDEs for xˆ 1 = E1 x defined on (z , t) ∈ [0, 1) × R+ , the PDEs for xˆ 2 = E2⊤ xˆ on (z , t) ∈ (0, 1] × R+ and (26b)–(26e) for t > 0. Therein, L(z) ∈ Rn×m , L0 ∈ Rn0 ×m and L1 ∈ Rn1 ×m are the observer gains, while xˆ (z , 0) = xˆ 0 (z) ∈ Rn , z ∈ [0, 1], w ˆ 0 (0) = w ˆ 0,0 ∈ Rn0 and w ˆ 1 (0) = w ˆ 1,0 ∈ Rn1 are the observer ICs. By introducing the error states εx = x − xˆ , ε0 = w0 − w ˆ 0 and ε1 = w1 − w ˆ 1 , the observer error dynamics reads ⊤ˆ

∂t εx (z , t) = Λ(z)∂z εx (z , t) + A(z)εx (z , t) − L(z)εx2 (1, t) εx2 (0, t) = Q0 εx1 (0, t) + C0 ε0 (t) εx1 (1, t) = C1 ε1 (t) ε˙ 0 (t) = F0 ε0 (t) + B0 εx1 (0, t) − L0 εx2 (1, t) ε˙ 1 (t) = F1 ε1 (t) − L1 εx2 (1, t)

(27a) (27b) (27c) (27d) (27e)

in view of (1) and (26), in which εxi = Ei⊤ εx , i = 1, 2, is utilized. 4.1. Backstepping transformation of the observer error PDE subsystem Consider the invertible backstepping transformation

εx (z , t) = ε˜ x (z , t) −

1



JI (z , ζ )ε˜ x (ζ , t)dζ = T3−1 [˜εx (t)](z)

(28)

z

with the integral kernel JI (z , ζ ) ∈ Rn×n and the auxiliary observer gain

˜ L(z) = T3 [L](z) − T3 [JI (·, 1)](z)Λ(1)E2

(29)

in order to map the error system (27) into

∂t ε˜ x (z , t) = Λ(z)∂z ε˜ x (z , t) + G(z)ε1 (t) − ˜ L(z)ε˜ x2 (1, t) ∫ 1 ε˜ x2 (0, t) = Q0 ε˜ x1 (0, t) − S(z)ε˜ x (z , t)dz + C0 ε0 (t)

(30a) (30b)

0

ε˜ x1 (1, t) = C1 ε1 (t) ε˙ 0 (t) = F0 ε0 (t) + B0 ε˜ x1 (0, t) ∫ 1 − B0 E1⊤ JI (0, ζ )ε˜ x (ζ , t)dζ − L0 ε˜ x2 (1, t)

(30c)

(30d)

0

ε˙ 1 (t) = F1 ε1 (t) − L1 ε˜ x2 (1, t).

(30e)

Therein, the states ε˜ xi = Ei⊤ ε˜ x , i = 1, 2, are utilized and G(z) = T3 [JI (·, 1)](z)Λ(1)E1 C1 holds. Furthermore, the transformation T3 in (29) is a Volterra integral transformation of the type (28). Its kernel J(z , ζ ) can be obtained from the reciprocity relation between JI (z , ζ ) and J(z , ζ ) (see Deutscher et al., 2018). In (30b) the

476

J. Deutscher et al. / Automatica 95 (2018) 472–480

matrix S(ζ ) = [S1 (ζ ) S2 (ζ )] consists of S1 (ζ ) ∈ Rm×p and the strictly lower triangular matrix S2 (ζ ) = [sij (ζ )], which has the same form as (8). This is required in order to obtain well-posed kernel equations (see Hu et al., 2016, 2015). By utilizing similar calculations as in Hu et al. (2016) it can be shown that the kernel JI (z , ζ ) has to satisfy the kernel equations

Proof. The explicit solution of (35a)–(35c) can be obtained from the results in Deutscher et al. (2018), where the same type of IVP is considered in the state feedback design. With this, the Sylvester equation (35d) can be solved. The corresponding solution exists and is unique, because σ (F0 − ˜ L0 C0 ) ∩ σ (F1 ) = ∅ (see, e.g., Lancaster & Tismenetsky, 1985, Ch. 12.5, Th. 2). □

Λ(z)∂z JI (z , ζ ) + ∂ζ (JI (z , ζ )Λ(ζ )) = −A(z)JI (z , ζ )

In view of Assumption (A2) the condition of Lemma 5 can always be fulfilled by a suitable choice of ˜ L0 .

(E2 − Q0 E1 )JI (0, ζ ) = −S(ζ )

(31a)



(31b)

JI (z , z)Λ(z) − Λ(z)JI (z , z) = A(z)

(31c)



with (31a) defined on 0 < z < ζ < 1. By transposing the kernel equations (31) and interchanging the independent variables z and ζ , it can be verified that (31) can be traced back to the kernel equations for the state feedback design (see, e.g., Deutscher et al., 2018). With this, the existence of a unique piecewise C 1 -solution JI (z , ζ ) of (31) can be inferred from Hu et al. (2015). After solving (31), the matrix S1 (ζ ) and the elements sij (ζ ), i > j, of S2 (ζ ) are determined by the resulting kernel JI (z , ζ ). 4.2. Decoupling of the observer error dynamics w.r.t. the w0 -system In order to decouple the observer error dynamics (30d) from the other subsystems, the change of coordinates

ε˜ 0 (t) = ε0 (t) − Π2 ε1 (t) −

1



R2 (z)ε˜ x (z , t)dz

(32)

0

with Π2 ∈ Rn0 ×n1 and R2 (z) ∈ Rn0 ×n is considered. In particular, (30) is mapped into (33a)

∂t ε˜ x (z , t) = Λ(z)∂z ε˜ x (z , t) + G(z)ε1 (t) − ˜ L(z)ε˜ x2 (1, t) ∫ 1 ˜ S(z)ε˜ x (z , t)dz ε˜ x2 (0, t) = Q0 ε˜ x1 (0, t) −

(33b)

0

+ C0 ε˜ 0 (t) + C0 Π2 ε1 (t) ε˜ x1 (1, t) = C1 ε1 (t) ε˙ 1 (t) = F1 ε1 (t) − L1 ε˜ x2 (1, t),

(33c) (33d) (33e)

where

˜ S(z) = S(z) − C0 R2 (z).

(34)

Time differentiation of (32), utilization of (30) and an integration by parts yields the decoupling equations (R2 (z)Λ(z))′ + (F0 − ˜ L0 C0 )R2 (z) = −˜ L0 S(z) + B0 E1⊤ JI (0, z)

(35a)

R2 (0)Λ(0)(E1 + E2 Q0 ) = −B0

(35b)

R2 (0)Λ(0)E2 = −˜ L0

(35c)

(F0 − ˜ L0 C0 )Π2 − Π2 F1 = R2 (1)Λ(1)E1 C1 ∫ 1 + R2 (z)G(z)dz

ε˜ x (z , t) = ex (z , t) −

1



Ω (z , ζ )ex (ζ , t)dζ = T4−1 [ex (t)](z)

(37a)

z

e˜ x (z , t) = ex (z , t) − Γ (z)ε1 (t)

(37b)

with Ω (z , ζ ) ∈ Rn×n and Γ (z) ∈ Rn×n1 are introduced in order to map (33) into a stable ODE–PDE–ODE cascade. In particular, for determining the transformation (37a), consider the intermediate target system

ε˙˜ 0 (t) = (F0 − ˜ L0 C0 )ε˜ 0 (t)

(38a)

¯ ε1 (t) − L(z)e ¯ ∂t ex (z , t) = Λ(z)∂z ex (z , t) + G(z) x2 (1, t) ∫ 1 ¯ S(z)e ex2 (0, t) = Q0 ex1 (0, t) − x (z , t)dz

(38b)

(38c) (38d) (38e)

¯ with G(z) = T4 [Ω (·, 1)](z)Λ(1)E1 C1 + T4 [G](z) and

˜ L(z) = Ω (z , 1)Λ(1)E2 + T4−1 [L¯ ](z)

(39)

¯ as well as S(z) = [S¯1 (z) S¯2 (z)], S¯1 (z) ∈ Rm×p and the matrix S¯2 (z) = [¯sij (z)] being of the form (8). This ensures that the PDE subsystem in the final target system is finite-time stable (see, e.g., Hu et al., 2016 for further details). The transformation T4 can be obtained in the same way as T3 (see Section 4.1). Differentiate (37a) w.r.t. time and insert (38) in the result. Then, an integration by parts and applying the Leibniz differentiation rule yields the kernel equations Λ(z)∂z Ω (z , ζ ) + ∂ζ (Ω (z , ζ )Λ(ζ )) = 0 ∫ ζ ¯ ζ) ˜ (E2⊤ − Q0 E1⊤ )Ω (0, ζ ) = ˜ S(ζ ) − S(ζ¯ )Ω (ζ¯ , ζ )dζ¯ − S( Ω (z , z)Λ(z) − Λ(z)Ω (z , z) = 0

(35d)

with (35a) defined on z ∈ (0, 1) and 1

R2 (z)˜ L(z)dz − R2 (1)Λ(1)E2 + Π2 L1

+ C0 ε˜ 0 (t) + C0 Π2 ε1 (t) ex1 (1, t) = C1 ε1 (t) ε˙ 1 (t) = F1 ε1 (t) − L1 ex2 (1, t)

(40a) (40b)

0

0



Since (34) implies that the matrix ˜ S(z) in (33c) does not have the form of S(ζ ) and since the PDE subsystem (33b)–(33d) still depends on ε1 , the transformations

0

ε˙˜ 0 (t) = (F0 − ˜ L0 C0 )ε˜ 0 (t)

L0 =

4.3. Stabilization and decoupling of the transformed PDE observer error subsystem

(36)

0

so that the resulting change of coordinates (32) maps (30) into (33). The solvability of the decoupling equations (35) and thus the existence of the corresponding change of coordinates (32) is clarified in the next lemma. Lemma 5. If the spectra of F0 − ˜ L0 C0 and F1 satisfy σ (F0 − ˜ L0 C 0 ) ∩ σ (F1 ) = ∅, then the decoupling equations (35) have a unique solution with the elements of R2 (z) piecewise C 1 -functions.

(40c)

for Ω (z , ζ ) in (37a) with (40a) defined on 0 < z < ζ < 1. In what follows, it is shown that the solution of (40) can be traced back to solving simple Volterra integral equations of the second kind. For this, assume

[

0 Ω (z , ζ ) = 0

]

0 Ω1 (z , ζ )

(41)

with the upper triangular matrix

⎡ ω11 (z , ζ ) ⎢ ⎢ 0 Ω1 (z , ζ ) = ⎢ ⎢ .. ⎣ . 0

... .. . .. . ...

... .. . .. . 0

⎤ ω1m (z , ζ ) ⎥ .. ⎥ . ⎥ ∈ Rm×m . ⎥ .. ⎦ . ωmm (z , ζ )

(42)

J. Deutscher et al. / Automatica 95 (2018) 472–480

Next, postmultiply (40b) by E1 E1⊤ + E2 E2⊤ = In and equate the result w.r.t. E1⊤ and E2⊤ . With this and taking the triangular form of S¯2 (ζ ) ¯ in (38c) is determined by into account, the matrix S(z) S¯1 (ζ ) = ˜ S(ζ )E1 ζ



˜ S(ζ¯ )E2 Ω1 (ζ¯ , ζ )dζ¯ ]i>j ,

0

ζ

In order to map (38) into the ODE–PDE–ODE cascade

Σn0 (ε˜ 0 )

(43b)

⇒ z =0

˜ S(ζ¯ )E2 Ω1 (ζ¯ , ζ )dζ¯ ]i≤j = [˜ S(ζ )E2 ]i≤j

(44)

[Ω1 (z , z)]i
(45)

Substituting (41) in (40a) and collecting the results (44) and (45) shows that Ω1 (z , ζ ) has to satisfy the BVP

Λ2 (z)∂z Ω1 (z , ζ ) + ∂ζ (Ω1 (z , ζ )Λ2 (ζ )) = 0 [Ω1 (z , z)]i
= [˜ S(ζ )E2 ]ij ,

(46a)

i≤j

(47)

with monotonically increasing and thus invertible functions ∫z φi (z) = 0 (dζ /λi+p (ζ )). Then, by utilizing the method of characteristics, the elements ωij (z , ζ ) of the kernel Ω1 (z , ζ ) solving (46a)– (46b) are 1

λ (ζ ) ⎧i+p ⎨0,

fii (σii (z , ζ ))

(48a)

φi (z) > φj (ζ )

1

λj+p (ζ )

fij (σij (z , ζ )),

φi (z) ≤ φj (ζ )

(48b)

for i ≤ j and 1 ≤ i, j ≤ m (see (42)) and some piecewise continuous functions fij (z). By inserting (48) in (46c) the integral equations j ∫ ∑

ζ

s˜ik (ζ¯ )fkj (σkj (ζ¯ , ζ ))dζ¯ = λj+p (ζ )s˜ij (ζ )

(49)

0

for i ≤ j result, in which s˜ik (z) = [˜ S(z)E2 ]ik was utilized. From this, the substitution ξ = σjk (ζ , ζ¯ ) finally yields the Volterra integral equations of the second kind f˜ij (ζ ) +

j ∫ ∑ k=1

ζ

(51c) (51d)

ε˙ 1 (t) = (F1 − L1 E2 Γ (1))ε1 (t) − L1 e˜ x2 (1, t) ⊤

(51e)

consider the transformation (37b). Time differentiation of the latter and inserting (38) in the result yields (51) if Γ (z) is the solution of the decoupling equations

¯ , Λ(z)Γ ′ (z) − Γ (z)F1 = −G(z) (E2⊤ − Q0 E1⊤ )Γ (0) = C0 Π2 −

z ∈ (0, 1)

(52a)

1



¯ Γ (z)dz S(z)

(52b)

0

E1⊤ Γ (1) = C1

(52c)

and (46c)

σij (z , ζ ) = φi−1 (φi (z) − φj (ζ ))

k=1

e˜ x1 (1, t) = 0

(46b)

with Λ2 (z) = E2⊤ Λ(z)E2 and (46a) defined on 0 < z < ζ < 1. For its solution, introduce

fij (σij (0, ζ )) +

(51a) (51b)

0

k=1



Σn1 (ε1 )

0

for Ω1 (z , ζ ) is readily obtained. Furthermore, inserting (41) in (40c) and taking λi (z) ̸ = λj (z), i ̸ = j, i, j = 1, 2, . . . , n, into account directly yields

ωij (z , ζ ) =

⇒ z =1

ε˙˜ 0 (t) = (F0 − ˜ L0 C0 )ε˜ 0 (t) ∂t e˜ x (z , t) = Λ(z)∂z e˜ x (z , t) ∫ 1 ¯ e˜ x (z , t)dz + C0 ε˜ 0 (t) S(z) e˜ x2 (0, t) = Q0 e˜ x1 (0, t) −

0

ωii (z , ζ ) =

Σ∞ (e˜ x )

with the representation

in which [M ]i>j and [M ]i≤j are the elements mij of a matrix M = [mij ] satisfying i > j and i ≤ j, respectively. From the same result, the BC

[Ω1 (0, ζ ) +

Lemma 6. The kernel equations (40) have a unique piecewise C 1 -solution.

(43a)

[S¯2 (ζ )]i>j = [˜ S(ζ )E2 −



477

κjk (ζ , ξ )f˜kj (ξ )dξ = λj+p (ζ )s˜ij (ζ )

(50)

0

for f˜ij (ζ ) = fij (σij (0, ζ )), i ≤ j, and κjk (ζ , ξ ) = −(s˜ik (φk−1 (φj (ζ ) − φj (ξ )))λk+p (φk−1 (φj (ζ ) − φj (ξ ))))/λj+p (ξ ) after a simple calculation. Since s˜ij (z) is a piecewise C 1 -function (see (34) and the related results in Section 4.1), the solution of (50) uniquely exists and is piecewise C 1 (see, e.g., Linz, 1985, Th. 3.2). As a result, one obtains the following lemma for the solvability of the kernel equations (40).

¯ = Γ (z)L1 L(z)

(53)

holds. The next lemma asserts the solvability of (52). Lemma 7. The decoupling equations (52) have a unique solution with the elements of Γ (z) piecewise C 1 -functions. After transposing (52) it is easily inferred that the result has the same structure as (23b)–(23d). Hence, the proof of Lemma 5 follows from the same reasoning as in Appendix. The stability of the ODE–PDE–ODE cascade (51) requires the matrix F1 − L1 E2⊤ Γ (1) to be Hurwitz. This can be ensured by a suitable choice of L1 if (E2⊤ Γ (1), F1 ) is observable. The condition for this can be found in the next lemma. Lemma 8. The pair (E2 ⊤ Γ (1), F1 ) is observable, if ((E2 ⊤ − Q0 E1 ⊤ )ri (0) +

1



¯ S(z)r i (z)dz)vi ̸ = C0 Π2 vi

(54)

0

holds for all eigenvectors vi , i = 1, 2, . . . , n1 , of F1 w.r.t. the eigenvalues νi , in which ri (z) = Φ (z , 1, −νi )E1 C1 + ∫corresponding 1 ¯ ζ )dζ and Φ (z , ζ , s) = diag(es(ψ1 (z)−ψ1 (ζ )) , Φ (z , ζ , −ν )Λ−1 (ζ )G( i z ∫1 s(ψn (z)−ψn (ζ )) ...,e ) as well as ψi (z) = z (dζ /λi (ζ )). Proof. By the PHB eigenvector tests (see, e.g., Kailath, 1980, Th. 6.2-5), the observability of (E2⊤ Γ (1), F1 ) is equivalent to E2⊤ Γ (1)vi ̸ = 0,

i = 1, 2, . . . , n1 .

(55)

With the same reasoning as in Appendix one can show that the solution of (52) satisfies

Γ (1)vi = Mi−1 (−(E2⊤ − Q0 E1⊤ )ri (0) ∫ 1 ¯ − S(z)r i (z)dz + C0 Π2 )vi

(56)

0

for i = 1, 2, . . . , n1 with some nonsingular matrices Mi . As a consequence of the latter property, the result (54) is implied by (55) and (56). □

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J. Deutscher et al. / Automatica 95 (2018) 472–480

The next theorem presents the conditions for the stability of the ODE–PDE–ODE cascade (51), which implies that (26) is an observer for (1). Theorem 9 (Observer). Assume that F0 − ˜ L0 C0 and F1 − L1 E2 ⊤ Γ (1) are Hurwitz matrices. Then, the observer error dynamics (27) with the gains L(z) resulting from (53), (39) and (29) as well as the gain L0 following from (36) is asymptotically stable pointwise in space for all ICs xˆ (z , 0) = xˆ 0 (z) with xˆ 0 piecewise differentiable, w ˆ 0 (0) = w ˆ 0,0 ∈ Rn0 and w ˆ 1 (0) = w ˆ 1,0 ∈ Rn1 . More specifically, the convergence of ∥εx (t)∥∞ = supz ∈[0,∑ 1] ∥εx (z , t)∥Rn and εi (t), i ∫= 0, 1, is exponential z n for t ≥ to with to = i=p |φi (1)| and φi (z) = 0 (dζ /λi (ζ )). Proof. Consider the transformation e¯ x (z , t) = e˜ x (z , t) − Π (z)ε˜ 0 (t), differentiate it w.r.t. time and insert (51) in the result. This yields the new PDE subsystem

∂t e¯ x (z , t) = Λ(z)∂z e¯ x (z , t) ∫ e¯ x2 (0, t) = Q0 e¯ x1 (0, t) −

(57a) 1

¯ e¯ x (z , t)dz S(z)

(57b)

0

e¯ x1 (1, t) = 0

(57c)

if Π (z) is the solution of the decoupling equations

Λ(z)Π ′ (z) − Π (z)(F0 − ˜ L0 C0 ) = 0 ∫ 1 ¯ Π (z)dz (E2⊤ − Q0 E1⊤ )Π (0) = C0 − S(z)

(58a) (58b)

0

E1⊤ Π (1) = 0

(58c)

with (58a) defined for z ∈ (0, 1). This is the same type of BVP as in (52) so that its solvability follows from Lemma 5. With the method of characteristics it is easy to show that e¯ x is bounded for t ∈ [0, to ) and e¯ x (z , t) ≡ 0 for t ≥ to . Hence, the resulting error dynamics are governed by the finite-dimensional system

[ ] [ F0 − ˜ L0 C0 ε˙˜ 0 (t) = ε˙ 1 (t) −L1 E2⊤ Π (1)

][

0 F1 − L1 E2⊤ Γ (1)

] ε˜ 0 (t) , ε1 (t)

(59)

Proof. Apply the feedback u(t) = K[w ˆ 0 (t), w ˆ 1 (t), xˆ (t)] = K[w0 (t), w1 (t), x(t)] − K[ε0 (t), ε1 (t), εx (t)] to (1). Then, utilize the change of coordinates (5) as well as (12) and (16). With the results in Deutscher et al. (2018) for the ODE–PDE–ODE cascade (25) and the proof of Theorem 9, the closed-loop dynamics can be described by the finite-dimensional system



⎤ ⎡ −Kξ ξ˙ (t) ˜ ⎣ η˙ (t) ⎦ = ⎣ P w ˙ 0 (t) B0 E1⊤ Σ (0) [

A(z) = ⎢

Theorem 10 (Observer-based Compensator). Let the conditions for the state feedback and the observer design be satisfied (see Theorems 4 and 9). Then, the corresponding observer-based compensator (26) and (60) asymptotically stabilizes the resulting closed-loop system pointwise in space for all ICs defined in Theorems 4 and 9. More specifically, the convergence of ∥x(t)∥∞ = supz ∈[0,1] ∥x(z , t)∥Rn and ∥εx (t)∥∞ = supz ∈[0,1] ∥εx (z , t)∥Rn as well as of wi (t) and εi (t), i = 0, 1, for t ≥ tc + to is exponential.

(61)

A numerical example of a 4 × 4 unstable heterodirectional hyperbolic system serves to illustrate the efficiency of the observer-based compensator (26) and (60). In accordance with the nomenclature in (1), the matrices Λ = diag(8, 7, −1, −2) and

⎡ ⎢ ⎢ ⎣

1

e8−

cos z 8

0

0

0

0 0

0 0

e− 7 0

0 e−z

0

0

0

1

ez − 8 +

cos z 8

2z

−z

⎤ ⎥ ⎥ ⎥ ⎦

(62a)

are chosen in (1a). Hence, the state x1 (z , t) = [x1 (z , t), x2 (z , t)]⊤ propagates in negative z-direction and x2 (z , t) = [x3 (z , t), x4 (z , t)]⊤ in positive z-direction as well as u(t), y(t) ∈ R2 . The ODE subsystem in (1b) and (1d) at the boundary z = 0 is defined in terms of the matrices

(60)

can be implemented by the observer (26) resulting in the observerbased compensator. The stability of the corresponding closed-loop system is stated in the following theorem.

]

Observe that the homogeneous part of (61) has a triangular structure and that this system is cascaded with the triangular system (59). This implies that the eigenvalue assignment for the related matrices also describe the closed-loop dynamics for t ≥ tc + to . In this sense, the separation principle holds for the design of the observer-based compensator (26) and (60), which allows a systematic controller design.

The proof of Theorem 4 implies that the exponential convergence of ∥εx (t)∥∞ can be specified by an eigenvalue assignment for F0 − ˜ L0 C0 and F1 − L1 E2⊤ Γ (1) (see (59)).

u(t) = K[w ˆ 0 (t), w ˆ 1 (t), xˆ (t)]

Fη 0

]

and (59) for t ≥ tc + to . In addition, the states ϑ and e¯ x of the corresponding PDE subsystems are bounded for t ∈ [0, tc + to ) (see Deutscher et al., 2018 and the proof of Theorem 9). The matrix Σ (0) ∈ Rp×p in (61) follows from the decoupling approach considered for the ODE–PDE–ODE cascade (25) in Deutscher et al. (2018). Furthermore, the matrices W0 ∈ Rp×n0 and W1 ∈ Rp×n1 result from representing the feedback operator in (4) w.r.t. the coordinates ε˜ 0 , ε1 and e˜ x as well as by taking e˜ x (z , t) = Π (z)ε˜ 0 (t) for t ≥ to into account (see Section 4 and the proof of Theorem 9). If the conditions of the theorem are satisfied, then (61) is exponentially stable and is driven by the exponentially stable system (59). Hence, all related states converge exponentially to zero. This also implies the convergence of x and εx in the ∞-norm, which follows from a similar reasoning as in the proof of Theorem 9. Finally, the assumed smoothness of the ICs imply that the feedback operator K can be evaluated unambiguously. □

6. Example

By making use of the estimates w ˆ 0 and w ˆ 1 and xˆ in (4), the feedback

⎤[

C1 B(W0 ε˜ 0 (t) + W1 ε1 (t)) 0 0



which is exponentially stable, because F0 −˜ L0 C0 and F1 − L1 E2⊤ Γ (1) are Hurwitz matrices. By utilizing (37) and (28) one obtains εx (z , t) = T3−1 T4−1 [Π ](z)ε˜ 0 (t) + T3−1 T4−1 [Γ ](z)ε1 (t) for t ≥ to . Since the solution of (59) decays exponentially, this shows the exponential convergence of εx to the origin in the ∞-norm. Hence, in view of (28) and (32), the observer errors ε0 and ε1 (see (59)) also decay exponentially to zero. □

5. Observer-based compensator

0 ξ (t) ⎦ η(t) 0 w0 (t) F0 − B0 K0

− Kη

[ F0 =

] −2

1 2

0

[

[

1 1

, B0 =

0

]

−1

, C0 =

]

0 Q0 = 2

0 , 0

[

1 1

]

0 , 0

(62b)

the ODE subsystem at z = 1 by means of



1

0 Q1 = ⎣ 1



[ 10 C1 =

e

5 − 72

0

− 97

2e e

2

2e− 7

e

1 − 7+cos 8

1−cos 1 8

0

⎤ ⎥ ⎦, (62c)

]

0 2

−e− 7

J. Deutscher et al. / Automatica 95 (2018) 472–480

479

Fig. 2. Euclidean norm of the plant’s lumped states w0 (t) and w1 (t) in the closedloop system.

7. Concluding remark

Fig. 1. Profiles of the plant’s distributed state components xi (z , t), i = 1, . . . , 4, in the closed-loop system.

in (1c) and



1

⎢ F1 = ⎣ 2 0 0



2 2 0

⎡ 1 ⎥ ⎣ 0 , B = 1 0⎦ 1

1

2



0 −e−1 ⎦ , B = 2e−1

1 1 0

[

] −1 1 0

(62d)

in (1e). It can easily be verified, that this ODE–PDE–ODE system satisfies the Assumptions (A1)–(A4). Note that the third-order ODE at z = 1 is not fully actuated as u(t) ∈ R2 . Furthermore, all eigenvalues of F1 have positive real parts and the ODE subsystem (C1 , F1 , B) is non-minimum phase with an invariant zero at 1. Choosing the eigenvalues −1 and −2 for F0 − B0 K0 in (7e), based on the associated matrix K0 , the kernels K (z , ζ ) and P(z , ζ ) of the transformations (5a) as well as NI (z) are calculated numerically in the same manner as in Deutscher et al. (2018), i.e., by successive approximations. With the analytic solution of (23b)–(23d), Lemma 2 is verified to be fulfilled as ˜ P ̸ = 0⊤ in (22). The feedback gains Kξ and Kη are chosen such that all three eigenvalues of the (ξ , η)subsystem in (25a)–(25b) are placed at −10. The kernel JI (z , ζ ) in the transformation (28), required in the implementation of the observer (26), is found in the same manner as K (z , ζ ). The observer gain˜ L0 is chosen such that the eigenvalues of F0 −˜ L0 C0 are −10 and −12. The kernel Ω (z , ζ ) in the transformation (37a) is found by solving three Volterra integral equations (cf. (50)) via the method of successive approximations. With the aid of the analytic solution of (52), the condition in Lemma 8 is verified to be fulfilled. By that, the pair (E2⊤ Γ (1), F1 ) is observable and the observer gain L1 is chosen such that the eigenvalues of F1 − L1 E2⊤ Γ (1) are −8, −10 and −12. The compensator was implemented in MATLAB. There, the simulation using finite differences yields the results depicted in Figs. 1 1 and 2, in which the initialization x1 (z , 0) = (25(z − 10 )(z − 12 ))2 1 1 1 i for z ∈ [ 10 , 2 ] and x (z , 0) = 0 otherwise, x (z , 0) = 0, i = 2, 3, 4, w0 (0) = 0, w1 (0) = 0 of the plant and xˆ (z , 0) = 0, w ˆ 0 (0) = 0, w ˆ 1 (0) = 0 of the observer is considered. The large magnitudes of the states at the beginning underline the unstable nature of the open-loop system. However, the observer-based compensator manages to stabilize the plant. The observer error εx (z , t) decays 23 exponentially for t ≥ to with to = 17 + 1 + 21 = 14 (cf. Theorem 9), the plant state x(z , t) for t ≥ tc + to with tc =

1 8

+

1 7

+1 =

71 56

(cf. Theorem 4). Both, the plant’s distributed state in Fig. 1 and the Euclidean norm of the plant’s lumped states w0 and w1 in Fig. 2, are close to zero for t > 2. As the dynamics of the closed-loop system are only due to the ODEs for t ≥ tc + to ≈ 2.91, the simulation revealed, that the output feedback controller renders all states basically zero for t > 8.

In principle, ODEs at z = 1 with a vector relative degree greater than one or which are even not decouplable can also be considered. This problem requires much more technical developments and thus is an interesting topic for future work. The observer design for an anticollocated setup seems to be a somewhat simpler problem and can be dealt with by utilizing the results in Deutscher et al. (2018). An important topic for further research is to investigate the robustness of the proposed compensator. For this, the results in Auriol, Aarsnes, Martin, and Di Meglio (2018) are of interest. Appendix. Proof of Lemma 1 The Sylvester equation (23a) has a unique solution because

σ (F0 − B0 K0 ) ∩ σ (Fη ) = ∅ (see, e.g., Lancaster & Tismenetsky, 1985, Ch. 12.5, Th. 2). Assume that the matrix Fη ∈ Rnη ×nη has r Jordan blocks Ji , i = 1, 2, . . . , r, to which the Jordan chains ⊤ ϕ⊤ i(1) Fη = µi ϕ i(1)

(A.1a)

⊤ ⊤ ϕ⊤ i(k) Fη = µi ϕ i(k) + ϕ i(k−1) ,

k = 2, 3, . . . , li ,

(A.1b)

are associated and l1 + · · · + lr = nη . In (A.1) the vectors ϕi(k) are the generalized left eigenvectors of Fη w.r.t. the eigenvalue µi , i = 1, 2, . . . , r, in which r is the number of eigenvalues of Fη with linearly independent eigenvectors (see, e.g., Lancaster & Tismenetsky, 1985, Ch. 6 for details on the related Jordan canonical form). The generalized eigenvectors ϕi(k) , i = 1, 2, . . . , r, k = 1, 2, . . . , li , are determined such that they form a basis of Rnη , which is always possible. Premultiplying (23b)–(23d) by the generalized eigenvectors ⊤ ⊤ ⊤ ⊤ ⊤ ϕ⊤ i(k) and defining ri(k) (z) = ϕi(k) R1 (z), hi(k) (z) = ϕi(k) (Tη B1 E2 M2 (z) − ⊤ ⊤ ⊤ ⊤ −1 ˜ P C (z))Λ (z), πi(k) = ϕi(k) Π1 as well as bi(k) = ϕi(k) (PQ1 − Tη B1 ) lead to the one-sided coupled BVPs ⊤ −1 dz r ⊤ (z) i(k) (z) = r i(k) (z)(−µi I − dz Λ(z))Λ

(A.2a)

−1 − r⊤ (z) + h⊤ z ∈ (0, 1) i(k−1) (z)Λ i(k) (z), ∫ 1 ⊤ r⊤ r⊤ i(k) (0)Λ(0)(E1 + E2 Q0 ) = πi(k) B0 + i(k) (z)H0 (z)dz

(A.2b)

0

⊤ r⊤ i(k) (1)Λ(1)E2 = bi(k)

(A.2c)

for i = 1, 2, . . . , r, k = 1, 2, . . . , li with r i(0) (z) ≡ 0 . The fundamental matrix Ψ (z , ζ , s) : [0, 1]2 × C → Cn×n , satisfying ∂z Ψ (z , ζ , s) = (−sI − dz Λ(z))Λ−1 (z)Ψ (z , ζ , s), Ψ (ζ , ζ , s) = I, is ⊤



ζ ) −s(φ1 (z)−φ1 (ζ )) ζ ) −s(φn (z)−φn (ζ )) e e Ψ (z , ζ , s) = diag( λλ1 ((z) , . . . , λλnn((z) ), 1

(A.3)

∫z

where φi (z) = 0 (dζ /λi (ζ )), i = 1, 2, . . . , n. Then, in view of ⊤ ⊤ ⊤ −1 ri(k) (1) = (ri(k) (1)Λ(1)E1 E1⊤ + b⊤ (1) (see (A.2c)) the IVP i(k) E2 )Λ resulting from (A.2a) and (A.2c) has the solution ⊤ ⊤ ri(k) (z) = ri(k) (1)Λ(1)E1 P(z , µi ) + m⊤ i(k) (z)

(A.4)

480

J. Deutscher et al. / Automatica 95 (2018) 472–480

⊤ ⊤ with P(z , µi ) = E1⊤ Λ−1 (1)Ψ (z , 1, µi ) and m⊤ i(k) (z) = bi(k) E2 ∫z ⊤ ⊤ −1 −1 Λ (1)Ψ (z , 1, µi ) − 1 (r i(k−1) (ζ )Λ (ζ ) − hi(k) (ζ ))Ψ (z , ζ , µi )dζ . Inserting this in (A.2b) yields ⊤ (1)Λ(1)E1 Ξi = β ⊤ ri(k) i(k) ,

in which Ξi = P(0, µi )Λ(0)(E1 + E2 Q0 ) −

(A.5)

∫1 0

P(z , µi )H0 (z)dz and

∫1 β i(k) = πi(k) B0 − mi(k) (0)Λ(0)(E1 + E2 Q0 ) + 0 m⊤ i(k) (z)H0 (z)dz. Since Ψ (z , 1, µi ) in P(z , µi ) is diagonal and the first p rows of H0 (z) form a strictly lower triangular matrix (see (8)), the Ξi is ∏pmatrix µi φj (1) lower triangular. Hence, its determinant det Ξi = ̸= j=1 e 0, i = 1, 2, . . . , r, can be directly calculated. Consequently, one ⊤ can solve (A.5) uniquely for ri(k) (1)Λ(1)E1 . With this, the linear independence of the generalized eigenvectors ϕi(k) , i = 1, 2, . . . , r, k = 1, 2, . . . , li , implies the solvability of (23b)–(23d). Since h⊤ i(k) (θ ) ⊤



Luo, Z.-H., Guo, B.-Z., & Morgul, O. (1999). Stability and Stabilization of Infinite Dimensional Systems with Applications. London: Springer-Verlag. Sagert, C., Di Meglio, F., Krstic, M., & Rouchon, P. (2013). Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling. In Proc. IFAC Symp. System Structure and Control, Grenoble, France, 779–784. Tang, S., & Xie, Ch. (2011). State and output feedback boundary control for a coupled PDE-ODE system. Systems & Control Letters, 60, 540–545.



in (A.2a) is only piecewise C 1 , the BVP (23b)–(23d) has a unique piecewise C 1 -solution. References Auriol, J., Aarsnes, U., Martin, P., & Di Meglio, F. (2018). Delay-robust control design for two heterodirectional linear coupled hyperbolic PDEs. IEEE Transactions on Automatic Control (in press). Bastin, G., & Coron, J.-M. (2016). Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Birkhäuser. Bou Saba, D., Bribiesca-Argomedo, F., Di Loreto, M., & Eberard, D. (2017). Backstepping stabilization of 2 × 2 linear hyperbolic PDEs coupled with potentially unstable actuator and load dynamics. Proc. CDC, Melbourne, Australia, 2498–2503. Deutscher, J., Gehring, N., & Kern, R. (2018). Output feedback control of general linear heterodirectional hyperbolic PDE-ODE systems with spatially-varying coefficients. International Journal of Control (in press). Di Meglio, F., Argomedo, F. Bribiesca, Hu, L., & Krstic, M. (2018). Stabilization of coupled linear heterodirectional hyperbolic PDE-ODE systems. Automatica, 87, 281–289. 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, 3301–3314. Hu, L., Vazquez, R., Di Meglio, F., & Krstic, M. (2015). Boundary exponential stabilization of 1-D inhomogeneous quasilinear hyperbolic systems Submitted for Publication; Available at arXiv:1512.03539. Isidori, A. (1995). Nonlinear Control Systems. London: Springer-Verlag. Kailath, T. (1980). Linear Systems. Englewood Cliffs, NJ: Prentice Hall. Krstic, M. (2009). Delay Compensation for Nonlinear, Adaptive and PDE Systems. Boston: Birkhäuser. Krstic, M., & Smyshlyaev, A. (2008). Boundary Control of PDEs —A Course on Backstepping Designs. Philadelphia: SIAM. Lancaster, P., & Tismenetsky, M. (1985). The Theory of Matrices. San Diego: Academic Press. Linz, P. (1985). Analytical and Numerical Methods for Volterra Equations. Philadelphia: SIAM.

Joachim Deutscher received the Dipl.-Ing. (FH) degree in electrical engineering from the University of Applied Sciences Würzburg-Schweinfurt-Aschaffenburg, Germany, in 1996, the Dipl.-Ing. Univ. degree in electrical engineering, the Dr.-Ing. and the Dr.-Ing. habil. degrees both in automatic control from the Friedrich-Alexander University Erlangen-Nuremberg (FAU), Germany, in 1999, 2003 and 2010, respectively. From 2003–2010 he was a Senior Researcher at the Chair of Automatic Control (FAU), in 2011 he was appointed Associate Professor and since 2017 he is a Professor at the same university. Currently, he is head of the Infinite-Dimensional Systems Group at the Chair of Automatic Control (FAU). His research interests include control of distributed-parameter systems and control theory for nonlinear lumped-parameter systems with applications in mechatronic systems and robotics. He has co-authored a book on state feedback control for linear lumped-parameter systems: Design of Observer-Based Compensators (Springer, 2009) and is author of the book: State Feedback Control of Distributed-Parameter Systems (in German) (Springer, 2012).

Nicole Gehring received the Dipl.-Ing. degree in electrical engineering from Dresden University of Technology, Germany, in 2007. Following a two-year stint of industrial work in the control of power plants and motor vehicles, in 2015, she finished her doctoral thesis at Saarland University, Germany, and received the Dr.-Ing. degree. As a Postdoc, she stayed with the Technical University of Munich, Germany, for two year. Currently, she is with Johannes Kepler University Linz, Austria. Her research mainly focuses on linear distributedparameter systems and linear time-delay systems, especially in the context of control and observer design, as well as parameter identification.

Richard Kern received the M.Sc. degree in mechanical engineering from the Technical University of Munich (TUM), Germany, in 2014. Since June 2014, he is a Ph.D. student at the Chair of Automatic Control at TUM. His research interests include modelling and control of pneumatic systems with distributed parameters.