Output regulation for coupled linear parabolic PIDEs

Automatica 100 (2019) 360–370

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Output regulation for coupled linear parabolic PIDEs✩ Joachim Deutscher *, Simon Kerschbaum Lehrstuhl für Regelungstechnik, Universität Erlangen-Nürnberg, Cauerstraße 7, D–91058 Erlangen, Germany

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Article history: Received 30 May 2018 Received in revised form 8 August 2018 Accepted 7 November 2018

Keywords: Distributed-parameter systems Parabolic systems Output regulation Backstepping Boundary control Observer

a b s t r a c t This paper presents a backstepping-based solution of the output regulation problem for coupled parabolic partial integro-differential equations (PIDEs) with spatially-varying coefficients and distinct diffusion coefficients. The considered setup assumes in-domain as well as boundary disturbances, while the output to be controlled can be defined distributed in-domain, pointwise in-domain or at the boundaries and need not be measured. By assuming a finite-dimensional signal model, which may also be non-diagonalizable, a systematic solution of the output regulation problem is presented by making use of observer-based feedforward control. Existence conditions for the corresponding regulator are formulated in terms of the plant transfer behaviour. For the resulting closed-loop system, exponential stability with a prescribed decay rate is verified. The regulator design for two unstable coupled parabolic PIDEs demonstrates the results of the paper. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction In the last years the output regulation problem for distributedparameter systems (DPSs) with boundary inputs and pointwise outputs has attracted the attention of many researchers. On the one hand, this system class typically arises in applications if the underlying dynamics require the modelling w.r.t. space and time. On the other hand, the output regulation problem, i.e., the design of stabilizing compensators with the ability to track reference inputs in the presence of disturbances, is of fundamental importance in control theory. In this respect, a common design paradigm is to assume that the reference and disturbance inputs belong to a class of functions which can be represented as the solutions of ODEs. Then, the corresponding initial values parametrize this function family leading to a model for the exogenous signals. With this assumption, the mathematical foundations of the output regulation problem in question can be found in Natarajan, Gilliam, and Weiss (2014) and Paunonen and Pohjolainen (2014). Besides the different methods to exploit these results (see, e.g., Aulisa & Gilliam, 2016 for an overview), the combination of the output regulation theory with the backstepping approach (see, e.g., Krstic & Smyshlyaev, 2008) led to systematic procedures for the regulator design. Currently, results are available for large classes of coupled hyperbolic systems (see Anfinsen & Aamo, 2017; ✩ 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] (S. Kerschbaum).

*

https://doi.org/10.1016/j.automatica.2018.11.033 0005-1098/© 2018 Elsevier Ltd. All rights reserved.

Deutscher, 2017; Deutscher & Gabriel, 2018). As far as parabolic systems are concerned, the backstepping-based regulator design dealing with a single PDE was first proposed in Deutscher (2015) for observer-based feedforward controllers and the solution of the robust output regulation problem on the basis of the internal model principle was considered in Deutscher (2016). These results were extended in Zhou and Weiss (2017) to regulate the output of a Schrödinger equation. Recently, the backstepping design of regulators for hyperbolic–parabolic PDE–PDE cascades, that result from parabolic systems with input delay, was developed in Gu and Wang (2018a). The case of two cascaded parabolic PDEs was dealt with in Kang and Guo (2016), in which the coupling appears at the boundaries. In Gu and Wang (2018b) a disturbance rejection problem for a parabolic ODE–PDE–PDE cascade with an indomain coupling was tackled utilizing backstepping methods. To the best knowledge of the authors no results can be found in the literature concerning the output regulation problem for bidirectionally in-domain coupled systems of parabolic type, though they arise frequently in applications. In particular, typical examples are encountered in chemical and biochemical engineering problems (see Atkinson, 1974; Jakobsen, 2014). In the last few years a significant progress has been made in the solution of backstepping problems for coupled parabolic systems. Firstly, the constant coefficients case was considered in Baccoli, Pisano, and Orlov (2015) and Orlov, Pisano, Pilloni, and Usai (2017). The extension of these results to spatially-varying systems was presented in Deutscher and Kerschbaum (2018) and Vazquez and Krstic (2017) leading to new powerful tools for the stabilization of coupled parabolic PDEs. This suggests to formulate the backstepping-based output regulation for this important system class, too, which is the topic of this contribution.

J. Deutscher and S. Kerschbaum / Automatica 100 (2019) 360–370

In this paper the output regulation problem is considered for coupled parabolic partial integro-differential equations (PIDEs) with spatially-varying parameters and distinct diffusion coefficients. In contrast to coupled parabolic PDEs, the consideration of PIDEs also includes, for example, the modelling of crystallization processes (see Christofides, 2002) or they result from a singular perturbation of separate subsystems with different time-scales (see Meurer, 2016). It is assumed that disturbances act in-domain, at both boundaries and on the output to be controlled. The latter can be defined distributed in-domain, pointwise in-domain or at the boundaries. In order to determine an output feedback regulator for this general setup, the results for scalar parabolic PDEs in Deutscher (2015), i.e., SISO systems, are extended to MIMO systems. While in the latter reference only diagonalizable finite-dimensional signal models were considered, which restrict the class of exogenous signals, the finite-dimensional signal models considered in this paper are of general form, i.e., need not be diagonalizable. Consequently, also polynomial type signals can be considered along with trigonometric signals and linear combinations thereof. In order to solve the output regulation problem, an observer-based feedforward controller is designed. This also allows its solution for DPSs with in-domain outputs, that often cannot be measured. By utilizing the results in Deutscher and Kerschbaum (2018), the regulator equations are formulated in the backstepping coordinates. Hence, they attain a simple form, for which solvability conditions can easily be derived on the basis of the plant transfer behaviour. In order to implement the resulting state feedback regulator, a lumpedparameter reference observer and a distributed-parameter disturbance observer are designed. For this, the results in Deutscher and Kerschbaum (2018) are extended to the backstepping design of state observers for PIDEs with spatially-varying coefficients and Robin boundary conditions. To our best knowledge, a solution of this problem in the literature is only available for two coupled parabolic PDEs with constant diffusion coefficients and spatiallyvarying reaction in Camacho-Solorio, Vazquez, and Krstic (2017). The resulting backstepping observer is extended to determine a disturbance observer estimating the plant and disturbance model states. These results are formulated using the considered general class of signal models. The backstepping-based solution allows to formulate the existence conditions for these observers on the basis of the plant disturbance behaviour. For the compensator consisting of the state feedback regulator and the observers, the separation principle is verified. With this, it is possible to assign a desired stability margin for the closed-loop system, while ensuring output regulation. As a result, a systematic method for the design of output feedback regulators for a large class of coupled parabolic systems is obtained. Finally, the proposed regulator design is demonstrated for two unstable coupled parabolic PIDEs subject to in-domain and boundary disturbances, while the output to be controlled is defined pointwise and distributed in-domain. The next section states the considered output regulation problem. Then, the state feedback regulator problem is solved in Section 3. In order to obtain an output feedback regulator, the design of reference and disturbance observers is presented in Section 4. The paper is concluded by demonstrating the proposed regulator design for two coupled unstable parabolic PIDEs. 2. Problem formulation Consider the coupled parabolic PIDEs

∂t x(z , t) = Λ(z)∂z2 x(z , t) + A[x(t)](z) + G1 (z)d(t) θ0 [x(t)](0) = G2 d(t), t>0 θ1 [x(t)](1) = u(t) + G3 d(t), t>0 η(t) = x(0, t), t≥0 y(t) = C [x(t)] + G4 d(t), t≥0

(1a) (1b) (1c) (1d) (1e)

361

with the formal operator A[x(t)](z) = A(z)x(z , t) + A0 (z)x(0, t) +

z

∫

F (z , ζ )x(ζ , t)dζ .

(2)

0

The PIDEs in (1a) are defined on (z , t) ∈ (0, 1) × R+ with the state x(z , t) ∈ Rn , n > 1, and the input u(t) ∈ Rn . Furthermore, Λ(z) ∈ Rn×n is the diagonal matrix

Λ(z) = diag(λ1 (z), . . ., λn (z))

(3)

with λi ∈ C [0, 1], i = 1, 2, . . . , n, satisfying λ1 (z) > · · · > λn (z) > 0. It is assumed that A ∈ (C 1 [0, 1])n×n , A0 ∈ (C 1 [0, 1])n×n and F ∈ (C 1 ([0, 1]2 ))n×n . While the disturbance d(t) ∈ Rq is 2

unknown, the corresponding disturbance input locations characterized by G1 ∈ (C [0, 1])n×q and Gi ∈ Rn×q , i = 2, 3, 4, are known. Furthermore, decoupled Robin boundary conditions (BCs) are considered at both boundaries, which means that the matrices Bi , i = 0, 1, in the formal matrix differential operators

θi [h] = dz h + Bi h,

i = 0, 1

(4)

are given by the diagonal matrices Bi = diag(b1i , . . ., bni ),

i = 0, 1.

(5)

The initial condition (IC) of the system is x(z , 0) = x0 (z) ∈ R . For the observer design the anti-collocated measurement η is available, while the output to be controlled y need not be measured. It is modelled by the formal output operator n

C [ h] =

l ∑

1

∫

C (ζ )h(ζ )dζ

Ci h(zi ) +

(6)

0

i=1

with h(z) ∈ Cn , Ci ∈ Rn×n , zi ∈ [0, 1], i = 1, 2, . . . , l, and C ∈ (L2 (0, 1))n×n . This includes outputs defined at the boundaries as well as mixed pointwise and distributed in-domain outputs. Remark 1. General parabolic systems described by the PIDE ∂t x(z , t) = ∂z (Λ(z)∂z x(z , t)) + Φ (z)∂z x(z , t) + A[x(t)](z) + G1 (z)d(t) with the diffusion matrix Λ(z) of the form (3) and the convection matrix

Φ (z) = diag(Φ1 (z), . . ., Φn (z))

(7)

with Φ ∈ (C [0, 1]) can be traced back to (1a) by making use of the boundedly invertible Hopf–Cole-type state transformation 1

n×n

xˇ (z , t) = exp(

1

z

∫

2

Λ−1 (ζ )(Λ′ (ζ ) + Φ (ζ ))dζ )x(z , t).

(8)

0

Remark 2. A collocated measurement, i.e., a measurement at z = 1, requires A[x(t)](z) = A(z)x(z , t) in (2). This is due to the fact that the related Volterra-type backstepping transformation for the observer design is spatially anti-causal ∫ z in this case and thus spatially causal terms A0 (z)x(0, t) and 0 F (z , ζ )x(ζ , t)dζ are not allowed. Therefore, an anti-collocated setup is investigated in this paper. It is assumed that the reference input r(t) ∈ Rn and the disturbance d can be represented by the solution of the finitedimensional signal model

v˙ (t) = S v (t),

t > 0, v (0) = v0 ∈ Rnv

(9a)

d(t) = Pd v (t) = P¯ d vd (t),

t≥0

(9b)

r(t) = Pr v (t) = P¯ r vr (t),

t≥0

(9c)

with Pd ∈ R and Pr ∈ R . Therein, the spectrum σ (S) of the matrix S only contains eigenvalues on the imaginary axis. This allows the modelling of a large class of persistently acting exogenous signals. In particular, they can be constant or trigonometric q×nv

n×nv

362

J. Deutscher and S. Kerschbaum / Automatica 100 (2019) 360–370

functions of time as well as linear combinations of both signal forms. Since the signal model may also be non-diagonalizable, polynomials and trigonometric functions with amplitudes that are polynomials in t are included. By introducing S = bdiag(Sd , Sr ) and v = col(vd , vr ) one obtains the disturbance model v˙ d (t) = Sd vd (t), vd (0) = vd,0 ∈ Rnd and the reference model v˙ r (t) = Sr vr (t), vr (0) = vr ,0 ∈ Rnr , nd + nr = nv . Consequently, P¯ d ∈ Rq×nd and P¯ r ∈ Rn×nr has to hold in (9). Furthermore, the matrix pairs (P¯ d , Sd ) and (P¯ r , Sr ) are required to be observable. For the design of the regulator the knowledge of the signal model (9) and of the reference input r is assumed. In this paper the output regulation problem is solved by utilizing output feedback control. This amounts to designing a stabilizing compensator such that lim ey (t) = lim (y(t) − r(t)) = 0

t →∞

(10)

t →∞

holds independently of the initial values w.r.t. the plant (1), the signal model (9) and the controller. 3. State feedback regulator

to satisfy the kernel equations

Λ(z)Kzz (z , ζ ) − (K (z , ζ )Λ(ζ ))ζ ζ = B[K ](z , ζ )

u(t) = −Rv v (t) − R1 x(1, t) −

1

∫

R2 (ζ )x(ζ , t)dζ 0

= R[v (t), x(t)]

(11) n×nv

n×n

with the feedback gains Rv ∈ R , R1 ∈ R and R2 (z) ∈ Rn×n . Representing the exogenous signals by (9), applying (11) to (1) and introducing the abbreviations ˜ Gi = Gi Pd , i = 1, 2, 3, 4, leads to the closed-loop system

v˙ (t) = S v (t)

(12a)

∂t x(z , t) = Λ(z)∂z2 x(z , t) + A[x(t)](z) + ˜ G1 (z)v (t) θ0 [x(t)](0) = ˜ G2 v (t)

(12b)

θ1 [x(t)](1) = R[v (t), x(t)] + ˜ G3 v (t) ˜ ey (t) = C [x(t)] + (G4 − Pr )v (t),

(12d)

(12c) (12e)

z

∫

K (z , ζ )x(ζ , t)dζ = Tc [x(t)](z)

(13)

0

Λ(z)K ′ (z , z) + Λ(z)Kz (z , z) + Kζ (z , z)Λ(z) + K (z , z)Λ′ (z) = −(A(z) + µc I) K (z , z)Λ(z) − Λ(z)K (z , z) = 0 K (0, 0) = 0.

R1 = −K (1, 1)

(14a)

R2 (ζ ) = −θ1 [K (·, ζ )](1) = −∂z K (1, ζ ) − B1 K (1, ζ )

(14b)

the closed-loop system (12) is mapped into the target system (15a)

˜[˜x(t)](z) + H1 (z)v (t) ∂t x˜ (z , t) = Λ(z)∂z2 x˜ (z , t) + A ˜ θ0 [˜x(t)](0) = G2 v (t)

(15b)

θ1 [˜x(t)](1) = (−Rv + ˜ G3 )v (t) −1 ey (t) = CTc [˜x(t)] + (˜ G4 − Pr )v (t)

(15d)

(15c) (15e)

with H1 (z) = K (z , 0)Λ(0)˜ G2 − Tc [˜ G1 ](z) and the formal operator

˜[h](z) = −µc h(z) − ˜ A A0 (z)h(0).

(17c) (17d) (17e)

Therein, (17a) is defined on 0 < ζ < z < 1 and the formal operator B[K ](z , ζ ) = K (z , ζ )(A(ζ ) + µc I)

−F (z , ζ ) +

z

∫ ζ

K (z , ζ¯ )F (ζ¯ , ζ )dζ¯

(18)

was utilized. The strictly lower triangular matrix ˜ A0 (z) in (16) is given by

... ... .. .. . . .. .. . . ˜ . . . An n−1 (z)

0

⎤

0

.. ⎥ .⎥ ⎥ .. ⎥ . .⎦

(19)

0

For the resulting kernel equations (17) it is shown in Deutscher and Kerschbaum (2018) that a piecewise C 2 -solution exists. The elements ˜ Aij (z), i > j, in (19) are determined by the kernel. The corresponding inverse backstepping transformation reads x(z , t) = x˜ (z , t) +

z

∫

KI (z , ζ )x˜ (ζ , t)dζ = Tc−1 [˜x(t)](z).

(20)

0

The kernel KI (z , ζ ) ∈ Rn×n appearing in (20) is the solution of the kernel equations

Λ(z)KI ,zz (z , ζ ) − (KI (z , ζ )Λ(ζ ))ζ ζ = E [KI ](z , ζ )

(21a)

KI ,ζ (z , 0)Λ(0) + KI (z , 0)(Λ′ (0) + Λ(0)B0 ) z

∫

KI (z , ζ )˜ A0 (ζ )dζ

= A0 (z) + ˜ A0 (z) +

(21b)

Λ(z)KI′ (z , z) + Λ(z)KI ,z (z , z) + KI ,ζ (z , z)Λ(z) + KI (z , z)Λ′ (z) = −(A(z) + µc I) KI (z , z)Λ(z) − Λ(z)KI (z , z) = 0 KI (0, 0) = 0

(21c) (21d) (21e)

with (21a) defined on 0 < ζ < z < 1 and

is utilized so that with the feedback gains

v˙ (t) = S v (t)

(17b)

0

in which ey is the tracking error defined in (10). In order to obtain a simple form of the regulator equations to be solved for determining the gain Rv in (11), the backstepping transformation x˜ (z , t) = x(z , t) −

K (z , ζ )A0 (ζ )dζ 0

⎢ ⎢˜ A21 (z) ˜ A0 (z) = ⎢ ⎢ .. ⎣ . ˜ An1 (z)

Consider the state feedback regulator

z

∫ =˜ A0 (z) + A0 (z) −

⎡ 3.1. Regulator equations in backstepping coordinates

(17a)

Kζ (z , 0)Λ(0) + K (z , 0)(Λ′ (0) + Λ(0)B0 )

(16)

With similar calculations as in Deutscher and Kerschbaum (2018) it can be verified that the integral kernel K (z , ζ ) ∈ Rn×n in (13) has

E [KI ](z , ζ ) = −(A(z) + µc I)KI (z , ζ ) − F (z , ζ ) z

∫ − ζ

F (z , ζ¯ )KI (ζ¯ , ζ )dζ¯ .

The existence proof for a piecewise C 2 -solution KI (z , ζ ) and a method for solving (21) can be found in Deutscher and Kerschbaum (2018). In order to map (15) into

v˙ (t) = S v (t)

˜[˜ε(t)](z) ∂t ε˜ (z , t) = Λ(z)∂z2 ε˜ (z , t) + A θ0 [˜ε (t)](0) = 0 θ1 [˜ε (t)](1) = 0 ey (t) = CTc−1 [˜ε (t)],

(22a) (22b) (22c) (22d) (22e)

the boundedly invertible change of coordinates

ε˜ (z , t) = x˜ (z , t) − Π (z)v (t)

(23)

J. Deutscher and S. Kerschbaum / Automatica 100 (2019) 360–370

with Π (z) ∈ Rn×nv is introduced. Therein, ε˜ (z , t) is the error between the transformed state x˜ (z , t) and the steady state value Π (z)v (t), which is needed to achieve (10). With this, output regulation can be traced back to the stability of the PDE subsystem (22b)– (22e) describing the tracking error dynamics. Differentiating (23) w.r.t. time, inserting (15) in the result and choosing Rv = ˜ G3 − θ1 [Π ](1)

(24)

yields (22) if Π (z) is the solution of the regulator equations

(25b)

[Π ] = Pr − ˜ G4

(25c)

−1

CTc

(25a)

with (25a) defined on z ∈ (0, 1). Their solvability depends on the plant transfer behaviour from u to the output to be controlled y. In particular, the regulator equations have a solution if the eigenmodes of the signal model (9) can be transferred to y in order to compensate the disturbances and to establish the reference value in the steady state. This is the result of the next lemma, for which it is convenient to define the matrices

[ ]

I T1 = n 0

∈R

2n×n

[ ] and T2 =

0 In

∈ R2n×n .

(26)

Lemma 3 (Regulator Equations). Let N(s) be the numerator of the transfer matrix Fu (s) = N(s)D−1 (s) of (1) from u to y. The regulator equations (25) have a unique solution Π (z) = [Πij (z)] ∈ Rn×nv with Πij piecewise C 2 -functions, i = 1, 2, . . . , n, j = 1, 2, . . . , nv , iff det N(µ) ̸ = 0, ∀µ ∈ σ (S). Furthermore, the numerator is given by N(s) = CTc−1 [M(s)] with

T1⊤ Ψ (z , ζ , s)T2 Λ−1 (ζ )˜ A0 (ζ )dζ

(27)

0

and Ψ (z , ζ , s) : [0, 1]2 × C → C2n×2n being the fundamental matrix solving the initial value problem (IVP) ∂z Ψ (z , ζ , s) = Υ (z , s)Ψ (z , ζ , s), Ψ (ζ , ζ , s) = I, where

[

Υ (z , s) =

]

0 (s + µc )Λ−1 (z)

I . 0

(28)

Proof. Assume that the matrix S ∈ Rnv ×nv has the Jordan blocks Ji , i = 1, 2, . . . , r, to which the Jordan chains (1)

= µi ϕi(1)

(k) i

(k) i i

S ϕi Sϕ

=µϕ

(29a) (k−1) i

+ϕ

,

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

(29b) (k)

are associated and l1 + · · · + lr = nv . In (29) the vectors ϕi are the generalized eigenvectors of S w.r.t. the eigenvalue µi , i = 1, 2, . . . , r, in which r is the number of eigenvalues of S with linearly independent eigenvectors (see, e.g., Lancaster & Tismenetsky, 1985, Ch. 6 for details on the related Jordan canonical form). The generalized (k) eigenvectors ϕi , i = 1, 2, . . . , r, k = 1, 2, . . . , li , are determined such that they form a basis for Rnv , which is always possible. (k) Postmultiplying (25) by the generalized eigenvectors ϕi , defining (k) (k) (k) (k) (k) (k) (k) (k) πi = Π ϕi , h1,i = −H1 ϕi , g˜2,i = ˜ G2 ϕi and ri = (Pr − ˜ G4 )ϕi lead to the one-sided coupled BVPs (k)

(k)

(k)

˜[πi ](z) + µi πi (z) d2z πi (z) = Λ−1 (z)(−A + πi(k−1) (z) + h(k) 1,i (z)) (k) i

(k) g2,i (k) ri

θ0 [π ](0) = ˜ −1

CTc

(k) i

[π ] =

(31)

with M(z , s) defined in (27) and (k)

m(k) (z , µi ) = T1⊤ Ψ (z , 0, µi )T2 g˜2,i

(32)

z

∫

(k)

+

(k−1)

T1⊤ Ψ (z , ζ , µi )T2 Λ−1 (ζ )(h1,i (ζ ) + πi

(ζ ))dζ .

Inserting this in (30c) gives (k)

(k)

CTc−1 [M(µi )]πi (0) = ri

− CTc−1 [m(k) (µi )].

(33)

Thus, iff the condition of the lemma holds, then (33) is uniquely (k) solvable for πi (0). Hence, by making use of the fact that the gen(k) eralized eigenvectors ϕi are linearly independent, the solution (l )

(1)

Π (z) = [π1 (z) . . . π1 1 (z) . . . πr(lr ) (z)] (l )

·[ϕ1(1) . . . ϕ1 1 . . . ϕr(lr ) ]−1

(34)

of (25) can be obtained. This is the piecewise classical solution (k) as h1,i (z) are piecewise C 1 -functions (see (32)). In order to derive the transfer matrix Fu (s) = N(s)D−1 (s), apply the backstepping transformation x˜ (z , t) = Tc [x(t)](z) to (1) (see (13)). For d(t) ≡ 0 this leads to

˜[˜x(t)](z) ∂t x˜ (z , t) = Λ(z)∂z2 x˜ (z , t) + A θ0 [˜x(t)](0) = 0

(35b)

θ1 [˜x(t)](1) = u(t) − K (1, 1)Tc−1 [˜x(t)](1) ∫ 1 − θ1 [K (·, ζ )](1)Tc−1 [˜x(t)](ζ )dζ

(35c)

0 CTc−1

(35a)

[˜x(t)]

(35d)

with (35a) defined on (z , t) ∈ (0, 1) × R . Since the transfer matrix Fu (s) only has to describe the transmission of the eigenmodes u(t) = u0 est , s ∈ C, u0 ∈ Cn , t ≥ 0, to the output, the related transfer matrix Fu (s) is readily determined for this system in the sense of Zwart (2004). As a result the numerator N(s) of the lemma is obtained. □ +

z

+

πi(k) (z) = M(z , µi )πi(k) (0) + m(k) (z , µi )

y(t) =

M(z , s) = T1⊤ Ψ (z , 0, s)(T1 − T2 B0 )

∫

(0)

with (30a) defined on z ∈ (0, 1) and πi (z) ≡ 0. The solution of (30a) and (30b) is

0

˜[Π ](z) − Π (z)S = −H1 (z) Λ(z)d2z Π (z) + A θ0 [Π ](0) = ˜ G2

363

(30a)

Remark 4. In view of the assumption for Λ(z) in Section 2, the matrix Λ−1 ∈ (C 2 [0, 1])n×n exists so that the fundamental matrix Ψ (z , ζ , s) is the unique solution of the related IVP in Lemma 3 (see, e.g., Kailath, 1980, Ch. 9.1). In general, only the numerical computation of Ψ (z , ζ , s) is possible. However, if the coefficients λi in Λ(z) (see (1)) are constant, i.e., λi (z) = λi = const ., i = 1, 2, . . . , n, then the fundamental matrix is given explicitly by the matrix exponential Ψ (z , ζ , s) = eΥ (s)(z −ζ ) . 3.2. Stability of the tracking error dynamics The next theorem shows that for suitable µc the system (22b)– (22d) describing the tracking error dynamics is exponentially stable, which implies output regulation. Theorem 5 (State Feedback Regulator). Assume that αc = µmax − µc < 0, in which µmax is the largest eigenvalue of (22b)–(22d) for ˜[˜x(t)](z) ≡ 0. Let K (z , ζ ) and Π (z) be the solutions of the kernel A equations (17) and of the regulator equations (25), respectively. Then, the state feedback regulator (11) with the feedback gains (14) and (24) achieves output regulation (10). The dynamics of the tracking error e(t) = {e(z , t) = x(z , t) − Tc−1 [Π ](z)v (t), z ∈ [0, 1]} are ex-

(30b)

ponentially stable in the L2 -norm ∥h∥ = ( i.e.,

(30c)

∥e(t)∥ ≤ Mc e(αc +c)t ∥e(0)∥,

t≥0

∫1 0

1

∥Λ− 2 (ζ )h(ζ )∥2Cn dζ )1/2 ,

(36)

364

J. Deutscher and S. Kerschbaum / Automatica 100 (2019) 360–370

for all e(0) ∈ (H 2 (0, 1))n satisfying the BCs of the tracking error dynamics, an Mc ≥ 1 and any c > 0 such that αc + c < 0. For the proof see Appendix A.

which is readily obtained from (1), (9) and (39). The design of the disturbance observer (39) is based on transforming (40) into a stable PDE–ODE cascade. To this end, the backstepping transformation ex (z , t) = e˜ x (z , t) −

z

∫

3.3. State feedback regulator design

PI (z , ζ )e˜ x (ζ , t)dζ = To−1 [˜ex (t)](z)

(41)

0

is introduced with The design of (11) requires to solve the kernel equations (17) and (21). With this, the feedback gains (14) can be determined. Finally, solving the regulator equations (25) to compute the feedback gain (24) completes the state feedback regulator design. In practical applications the state of the plant (1) and of the disturbance model contained in (9) are not available for measurement. In addition, if only the current value of the reference input r(t) is available, then also the state of the reference model is unknown for nr > 1. Hence, a reference and disturbance observer have to be designed, which is shown in the next section.

L1 = PI (0, 0)

(42)

to map (40) into the intermediate target system e˙ d (t) = Sd ed (t) − Ld e˜ x (0, t)

(43a)

∂t e˜ x (z , t) = Λ(z)∂z2 e˜ x (z , t) − µo e˜ x (z , t) + H¯ 1 (z)P¯ d ed (t) − ˜ L(z)e˜ x (0, t) θ0 [˜ex (t)](0) = G¯ 2 ed (t) ∫ 1 ¯ 3 ed (t). θ1 [˜ex (t)](1) = − A¯ 0 (ζ )e˜ x (ζ , t)dζ + G

(43b) (43c) (43d)

0

4. Reference and disturbance observer

In (43b) ˜ L(z) is an auxiliary observer gain defined by

4.1. Reference observer The state vr of the reference model can be estimated from the available reference input r(t) by the reference observer

v˙ˆ r (t) = Sr vˆ r (t) + Lr (r(t) − P¯ r vˆ r (t)),

t>0

(37)

with the IC vˆ r (0) = vˆ r ,0 ∈ R for nr > 1. In view of (P¯ r , Sr ) being observable, there always exists an observer gain Lr ∈ Rnr ×n nr

ensuring an exponentially stable observation error dynamics e˙ r (t) = (Sr − Lr P¯ r )er (t)

(38)

4.2. Disturbance observer In what follows the disturbance observer (39a)

∂t xˆ (z , t) = Λ(z)∂ ˆ , t) + A[ˆx(t)](z) + G1 (z)P¯ d vˆ d (t) + L(z)(η(t) − xˆ (0, t)) θ0 [ˆx(t)](0) = G¯ 2 vˆ d (t) + L1 (η(t) − xˆ (0, t))

(39b)

θ1 [ˆx(t)](1) = u(t) + G¯ 3 vˆ d (t)

(39d)

2 z x(z

(39c)

is designed to estimate the state vd of the disturbance model and the state x of the plant. Therein, (39b) evolves on the domain (z , t) ∈ (0, 1) × R+ , (39a), (39c) and (39d) are defined for t > 0 ¯ i = Gi P¯ d , i = 2, 3, is utilized. This observer has the as well as G ICs vˆ d (0) = vˆ d,0 ∈ Rnd , xˆ (z , 0) = xˆ 0 (z) ∈ Rn as well as the observer gains Ld ∈ Rnd ×n , L(z) ∈ Rn×n and L1 ∈ Rn×n . It results from extending the disturbance observer in Deutscher (2015) for a single parabolic PDE to the coupled PIDEs (1). In what follows a new backstepping approach for its design is proposed by making use of the results in Deutscher and Kerschbaum (2018). By introducing the error states ed (t) = vd (t) − vˆ d (t) and ex (z , t) = x(z , t) − xˆ (z , t), the observer error dynamics are described by e˙ d (t) = Sd ed (t) − Ld ex (0, t)

∂t ex (z , t) = Λ(z)∂z2 ex (z , t) + A[ex (t)](z) + G1 (z)P¯ d ed (t) − L(z)ex (0, t) θ0 [ex (t)](0) = G¯ 2 ed (t) − L1 ex (0, t) θ1 [ex (t)](1) = G¯ 3 ed (t),

Therein, To [·] is the backstepping transformation e˜ x (z , t) = ex (z , t) +

z

∫

P(z , ζ )ex (ζ , t)dζ = To [ex (t)](z)

(40a)

(40b) (40c) (40d)

(45)

0

into the new coordinates e˜ x (z , t), i.e., the inverse of (41). Utilizing the relation To [PI (·, 0)](z) = P(z , 0) implied by the reciprocity of the kernels PI (z , ζ ) and P(z , ζ ) (see Krstic & Smyshlyaev, 2008, Ch. ¯ 1 (z) in (43b) reads 4.5), the matrix H

¯ 1 (z) = To [G1 ](z) − P(z , 0)Λ(0)G2 . H

with er (t) = vr (t) − vˆ r (t).

v˙ˆ d (t) = Sd vˆ d (t) + Ld (η(t) − xˆ (0, t))

˜ L(z) = To [L](z) − To [A0 ](z) − To [PI (·, 0)](z)(Λ(0)B0 + Λ′ (0)) −To [PI ,ζ (·, 0)](z)Λ(0). (44)

(46)

In comparison to (40), the PDE subsystem (43b)–(43d) attains a simple structure in the new coordinates. Thus, the decoupling into the PDE–ODE cascade on the basis of this subsystem leads to BVPs which are easier to solve and analyse. In order to obtain (43), differentiate (41) w.r.t. time as well as utilize (40) and (43). Then, similar calculations as in Deutscher and Kerschbaum (2018) show that the integral kernel PI (z , ζ ) ∈ Rn×n in (41) has to be the solution of the kernel equations

Λ(z)PI ,zz (z , ζ ) − (PI (z , ζ )Λ(ζ ))ζ ζ = F [PI ](z , ζ ) PI ,z (1, ζ ) + B1 PI (1, ζ ) = −A¯ 0 (ζ ) Λ(z)PI′ (z , z) + Λ(z)PI ,z (z , z) + PI ,ζ (z , z)Λ(z) + PI (z , z)Λ′ (z) = µo I + A(z) PI (z , z)Λ(z) − Λ(z)PI (z , z) = 0 PI (1, 1) = 0

(47a) (47b) (47c) (47d) (47e)

with F [PI ](z , ζ ) = −(µo I + A(z))PI (z , ζ )

+ F (z , ζ ) −

∫ ζ

z

F (z , ζ¯ )PI (ζ¯ , ζ )dζ¯

(48)

and (47a) is defined on the triangular spatial domain 0 < ζ < z < 1. The matrix A¯ 0 (ζ ) ∈ Rn×n in (47b) has the form

⎡

0

⎢. ⎢ ..

A¯ 0 (ζ ) = ⎢ ⎢. .

⎣. 0

A¯ 12 (ζ )

..

..

.

. ...

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

A¯ 1n (ζ )

.. .

A¯ n−1 n

⎤

⎥ ⎥ ⎥, ⎥ (ζ ) ⎦

(49)

0

in which the elements A¯ ij (ζ ), i < j, are determined by the solution of (47). In Appendix B it is verified that there exists a piecewise

J. Deutscher and S. Kerschbaum / Automatica 100 (2019) 360–370

C 2 integral kernel PI (z , ζ ) ∈ Rn×n , which follows from the results in Deutscher and Kerschbaum (2018). Similarly, the piecewise C 2 integral kernel P(z , ζ ) ∈ Rn×n determining (45) follows from solving the kernel equations

Λ(z)Pzz (z , ζ ) − (P(z , ζ )Λ(ζ ))ζ ζ = G [P ](z , ζ ) ∫ 1 ¯ Pz (1, ζ ) + B1 P(1, ζ ) = −A0 (ζ ) − A¯ 0 (ζ¯ )P(ζ¯ , ζ )dζ¯

(50a) (50b)

Lemma 7 (Observability of (Γ (0), Sd )). Let Nd (s) be the numerator of 1 the transfer matrix Fd (s) = D− d (s)Nd (s) of (1) from d to η . The pair (Γ (0), Sd ) is observable iff (1)

Nd (νi )P¯ d ψi

̸= 0,

i = 1, 2, . . . , rd ,

(1)

(50d)

where

(50e)

R(z , s) = T1⊤ Φ (z , 0, s)T2 G2 z

Φ (z , ζ , s)T2 Λ−1 (ζ )H¯ 1 (ζ )dζ

)

(58)

0

G [P ](z , ζ ) = P(z , ζ )(µo I + A(ζ )) z

ζ

(57)

(

∫

P(z , ζ¯ )F (ζ¯ , ζ )dζ¯

A¯ 0 (ζ )R(ζ , s)dζ + G3 , 0

(50c)

−

∫

1

∫ Nd (s) = −θ1 [R(·, s)](1) −

with

+ F (z , ζ ) +

(56)

wherein ψi is the eigenvector of Sd w.r.t. the eigenvalue νi . Furthermore, the numerator is given by

ζ

Λ(z)P ′ (z , z) + Λ(z)Pz (z , z) + Pζ (z , z)Λ(z) + P(z , z)Λ′ (z) = µo I + A(z) P(z , z)Λ(z) − Λ(z)P(z , z) = 0 P(1, 1) = 0

365

(51)

and Φ (z , ζ , s) resulting from the fundamental matrix Ψ (z , ζ , s) of ¯ 1 (z) defined in (46). Lemma 3 by replacing µc with µo and H

and (50a) is defined on the triangular spatial domain 0 < ζ < z < 1. For the proof of well-posedness see Appendix B. In the final step, the new coordinates

Proof. In order to verify that Nd (s) in (57) is the numerator of the transfer matrix Fd (s), apply the backstepping transformation x(z , t) = To−1 [˜x(t)](z) to (1) (see (41)) and set u(t) ≡ 0. This leads to the result

ε˜ x (z , t) = e˜ x (z , t) − Γ (z)ed (t)

∂t x˜ (z , t) = Λ(z)∂z2 x˜ (z , t) + H¯ 0 (z)x˜ (0, t) + H¯ 1 (z)d(t) θ0 [˜x(t)](0) = PI (0, 0)x˜ (0, t) + G2 d(t) ∫ 1 θ1 [˜x(t)](1) = − A¯ 0 (ζ )x˜ (ζ , t)dζ + G3 d(t)

(52)

with Γ (z) ∈ R are introduced to decouple the ε˜ x -system from the ed -system in (43). This maps the intermediate target system (43) into the PDE–ODE cascade n×nd

∂t ε˜ x (z , t) = Λ(z)∂z2 ε˜ x (z , t) − µo ε˜ x (z , t) θ0 [˜εx (t)](0) = 0 ∫ 1 θ1 [˜εx (t)](1) = − A¯ 0 (ζ )ε˜ x (ζ , t)dζ

(53a) (53b) (53c) (53d)

with (53a) defined on (z , t) ∈ (0, 1) × R+ as well as (53b), (53c) and (53d) for t > 0. After differentiating (52) w.r.t. time and inserting (43), the target system (53) results if Γ (z) is the solution of

Λ(z)d2z Γ (z) − µo Γ (z) − Γ (z)Sd = −H¯ 1 (z)P¯ d , z ∈ (0, 1) θ0 [Γ ](0) = G¯ 2 ∫ 1 ¯3 θ1 [Γ ](1) = − A¯ 0 (ζ )Γ (ζ )dζ + G

(59b) (59c)

0

0

e˙ d (t) = (Sd − Ld Γ (0))ed (t) − Ld ε˜ x (0, t)

(59a)

(54a) (54b)

η(t) = x˜ (0, t)

(59d)

¯ 0 (z) = To [A0 (·) + PI (·, 0)(Λ (0) − Λ(0)(PI (0, 0) − B0 )) + with H PI ,ζ (·, 0)Λ(0)]. From this, the transfer matrix Fd (s) with the numerator (57) can be calculated in terms of the fundamental matrix Φ (z , ζ , s). The pair (Γ (0), Sd ) is observable iff ′

(1)

Γ (0)ψi

= γi(1) (0) ̸= 0,

i = 1, 2, . . . , rd

(60)

(see Kailath, 1980, Th. 6.2.5 and the proof of Lemma 6). In view of (C.5) and (57) the condition (60) can be rewritten as

γi(1) (0) = Ω −1 (νi )Nd (νi )P¯ d ψi(1) ̸= 0.

(61)

Hence, (60) is satisfied iff (56) holds. □ (54c)

0

4.3. Disturbance observer design

and the auxiliary observer gain satisfies

˜ L(z) = Γ (z)Ld .

(55)

The next lemma clarifies the solvability of (54). Lemma 6 (PDE–ODE Cascade). Denote the spectrum of the PDE subsystem (53a)–(53c) by σo . Then, the BVP (54) has a unique solution Γ (z) = [Γij (z)] ∈ Rn×nd with Γij piecewise C 2 -functions, i = 1, 2, . . . , n, j = 1, 2, . . . , nd , iff σo ∩ σ (Sd ) = ∅. For the proof see Appendix C. A prerequisite for a stable PDE– ODE cascade (53) is the stability of the ODE subsystem (53d). This amounts to determining a stabilizing observer gain Ld , which exists if (Γ (0), Sd ) is observable. This property is directly related to the plant disturbance transfer matrix. In particular, estimation of the disturbance states vd is only possible if the transmission of the disturbance model eigenmodes to the measurement η is not blocked by the corresponding transfer behaviour. This is the existence condition for the disturbance observer (39), which is the result of the next lemma.

From the presented results the following observer design procedure can be derived. Firstly, the observer gain Ld is determined to ensure a Hurwitz matrix Sd − Ld Γ (0) giving ˜ L(z) via (55), for which the kernel equations (47), (50) and the BVP (54) have to be solved. Subsequently, the observer gain L(z) is obtained from L(z) = To−1 [˜ L](z) + A0 (z) + PI (z , 0)(Λ(0)B0 + Λ′ (0))

+ PI ,ζ (z , 0)Λ(0)

(62)

(see (44)). Together with the observer gain L1 given by (42) this completes the disturbance observer design. 4.4. Stability of the disturbance observer error dynamics In order to simplify the stability analysis of the observer error dynamics (40), the transformation

εd (t) = ed (t) −

1

∫

Q (ζ )ε˜ x (ζ , t)dζ 0

(63)

366

J. Deutscher and S. Kerschbaum / Automatica 100 (2019) 360–370

with Q (z) ∈ Rnd ×n is introduced to map (53) into the target system

Zwart, 1995, Lem. A.3.60). As a result, the PDE subsystem (64a)– (64c) can be represented by the abstract IVP

∂t ε˜ x (z , t) = Λ(z)∂z2 ε˜ x (z , t) − µo ε˜ x (z , t) θ0 [˜εx (t)](0) = 0 ∫ 1 θ1 [˜εx (t)](1) = − A¯ 0 (ζ )ε˜ x (ζ , t)dζ

ε˙˜ x (t) = Ao ε˜ x (t),

(64a) (64b) (64c)

0

ε˙ d (t) = (Sd − Ld Γ (0))εd (t).

(64d)

Hence, the stability result for (53) can be traced back to the independent stability analysis for the decoupled ODE subsystem (64d) and the PDE subsystem (64a)–(64c). This significantly simplifies the subsequent derivations. After differentiating (63) w.r.t. time and inserting (53) it is straightforward to verify that Q (z) has to be the solution of

− (µo I + Sd − Ld Γ (0))Q (z) = Q (1)A¯ 0 (z) dz Q (0)Λ(0) + Q (0)(Λ′ (0) + Λ(0)B0 ) = −Ld dz Q (1)Λ(1) + Q (1)(Λ′ (1) + Λ(1)B1 ) = 0 d2z (Q (z)Λ(z))

(65a) (65b) (65c)

with (65a) defined on z ∈ (0, 1). With a similar reasoning as in Appendix C it can be shown that (65) has a piecewise C 2 -solution if σo ∩ σ (Sd − Ld Γ (0)) = ∅ (see Lemma 6). If the conditions of Lemma 7 are satisfied, then the eigenvalues of Sd −Ld Γ (0) are freely assignable. Consequently, solvability of (65) can always be ensured by a suitable eigenvalue assignment for Sd − Ld Γ (0). With this, the stability result of the next theorem can easily be verified. Theorem 8 (Disturbance Observer). Let the observer gains L(z), L1 be given by (62) and (42). Assume that σo ∩ σ (Sd − Ld Γ (0)) = ∅, αx = µmax − µo < 0 (see Theorem 5) and that Sd − Ld Γ (0) is a Hurwitz matrix implying αo = max(αd , αx ) < 0 with αd = maxλ∈σ (Sd −Ld Γ (0)) Re λ. Then, the dynamics of the observer error eo (t) = col(ed (t), ex (t)) with ex (t) = {ex (z , t), z ∈ [0, 1]} are 1 exponentially stable in the norm ∥ · ∥Xo = (∥ · ∥2Cnd + ∥ · ∥2 ) 2 with ∥ · ∥ defined in Theorem 5, i.e.,

∥eo (t)∥Xo ≤ Mo e(αo +c)t ∥eo (0)∥Xo ,

t ≥ 0,

(66)

for all eo (0) ∈ Cnd ⊕ (H 2 (0, 1))n satisfying the BCs of the observer error dynamics, an Mo ≥ 1 and any c > 0 such that αo + c < 0. Proof. In order to reformulate the PDE subsystem (64a)–(64c) as an abstract IVP, consider the operator Ao h = Λd2z h − µo h

(67)

(71)

having the state ε˜ x (t) = {˜εx (z , t), z ∈ [0, 1]} in the state space X = (L2 (0, 1))n with the usual weighted inner product. The properties of Ao imply that this IVP is well-posed and exponentially stable with growth bound αx . Then, the result of the theorem follows from the bounded invertibility of the related transformations by utilizing standard arguments. □ 5. Output feedback regulator In order to obtain the output feedback regulator, the estimates

vˆ r , vˆ d and xˆ provided by the observers (37) and (39) are utilized in the state feedback regulator (11). With vˆ = col(vˆ d , vˆ r ) this yields the output equation u(t) = R[ˆv (t), xˆ (t)]

(72)

of the output feedback regulator, which consists of (37), (39) and (72). The next theorem shows that this compensator ensures output regulation for the closed-loop system on the basis of the separation principle. For this, the stability w.r.t. the closed-loop state xcl = col(er , ed , ex , xˆ − Π vˆ ) is investigated, in which er = vr − vˆ r , ed = vd − vˆ d and ex = x − xˆ are the errors of the reference observer (37) and the disturbance observer (39) with the corresponding growth bounds αr and αo . Furthermore, the growth bound αc achieved by the state feedback regulator is utilized. Theorem 9 (Output Feedback Regulator). Consider the output feedback regulator (37), (39) and (72) with the feedback gains (14), (24) and the observer gains (42) and (62). Assume that αcl = max(αc , αr , αo ) < 0 with αr = maxλ∈σ (Sr −Lr P¯r ) Re λ. Then, output regulation (10) is achieved and the dynamics of the closed-loop state xcl = col(er , ed , ex , xˆ − Π vˆ ) are exponentially stable in the norm 1

∥ · ∥Xcl = (∥ · ∥2Cnr + ∥ · ∥2Cnd + ∥ · ∥2 + ∥ · ∥2 ) 2 with ∥ · ∥ defined in Theorem 5, i.e.,

˜ (αcl +c)t ∥xcl (0)∥Xcl , t ≥ 0, ∥xcl (t)∥Xcl ≤ Me

(73)

for all xcl (0) ∈ C ⊕ C ⊕ (H (0, 1)) ⊕ (H (0, 1)) satisfying the ˜ ≥ 1 and any c > 0 such that BCs of the closed-loop dynamics, an M αcl + c < 0. nr

nd

2

n

2

n

For the proof see Appendix D.

with D(Ao ) = {h ∈ (H 2 (0, 1))n | θ0 [h](0) = 0,

θ1 [h](1) = −

t > 0, ε˜ x (0) ∈ D(Ao ),

6. Example

1

∫

A¯ 0 (ζ )h(ζ )dζ }.

(68)

0

A straightforward calculation shows that the adjoint of Ao is

¯⊤ A∗o h = Λd2z h − µo h − ΛA 0 h(1)

(69)

To illustrate the results of this paper, an unstable system consisting of two coupled parabolic PIDEs is considered. The parameters of the plant (1) are chosen as λ1 (z) = 32 + z 2 cos(2π z), λ2 (z) = 12 − 41 sin(2π z),

[

and

A(z) =

D(Ao ) = {h ∈ (H (0, 1)) | θ0 [h](0) = θ1 [h](1) = 0}. 2

∗

n

(70)

The operator Ao has the same structure as the operator A + ∆ in the abstract IVP (A.4) of Appendix A, because A¯ ⊤ 0 (z) is a strictly lower triangular matrix (see (49)). This implies that A∗o is the infinitesimal generator of an exponentially stable and analytic C0 -semigroup T ∗ (t) on X with the growth bound αx . Consequently, Ao is the infinitesimal generator of the C0 -semigroup T (t) (see Curtain & Zwart, 1995, Th. 2.2.6). In view of ∥T ∗ (t)∥ = ∥T (t)∥, t ≥ 0, the latter semigroup also has the growth bound αx (see Curtain & ∗

F (z , ζ ) =

1 1

[2

+z

ez +ζ 1 − eζ −z

1+z 1

] , A0 (z) =

ez −ζ e−(z +ζ )

[

z z

]

1−z , 1−z

(74)

] (75)

with the boundary matrices B0 = diag(−2, −1), B1 = 0 (see (4)). This leads to an unstable system with its largest eigenvalue at 2.87, which is calculated numerically on the basis of a finite-dimensional approximation model. A disturbance d(t) ∈ R acts in-domain as well as at the boundaries which is described by the disturbance input vectors G1 (z) = [1 1]⊤ , G2 = G3 = [1 1]⊤ and G4 = 0. The

J. Deutscher and S. Kerschbaum / Automatica 100 (2019) 360–370

Fig. 1. Closed-loop reference tracking behaviour for r1 (t) = 2 + r2 (t) = −5 + t − 6⌊ 6t ⌋.

1 t 2

367

− 4.5⌊ 9t ⌋,

output to be controlled (6) is defined in-domain pointwise as well as distributed, leading to

[

1 C [x(t)] = 0

]

0 x(0.25, t) + 0

1

∫

[

0

]

0 0

0 x(ζ , t)dζ . 1

(76)

The available anti-collocated measurement is given by (1d). Output regulation problem. The output feedback regulator should achieve the tracking of the ramp reference signals r1 (t) = 2r10 + 12 r10 t, r2 (t) = −5r20 + r20 t, t > 0, r10 , r20 ∈ R, which can be periodically reset to achieve sawtooth waves. Furthermore, the present sinusoidal disturbances d(t) = d0 sin(4t + ϕ ), t > 0, d0 ∈ R, ϕ ∈ R, shall be rejected. This leads to a non-diagonalizable signal model (9) described by

[

]

0 Sd = −4

⎡

0

⎢0 Sr = ⎣ 0 0

4 , 0 1 0 0 0

0 0 0 0

P¯ d = 1

[

⎤

0 0⎥ , 2⎦ 0

]

0

⎡

(77) 1

⎢ P¯ r = ⎣ 2 0

⎤ 2 0

0 1 2

0 ⎥

−5

⎦.

(78)

State feedback regulator design. The design parameter of the state feedback controller is chosen as µc = 5 > µmax = −0.34. The kernel equations (17) are solved in Matlab by implementing the method of successive approximations. With the resulting inverse backstepping transformation (20) the numerator N(s) of the transfer matrix from u to y can be checked to have a non-zero determinant for all eigenvalues of the signal model, i.e., det N(0) = 0.89, det N(±j4) = −0.47 ± j2.2. Thus, the regulator equations are solvable so that the resulting signal model state feedback gain Rv can be computed. The fundamental matrix needed for the solution of the regulator equations is computed by discretizing the ζ -coordinate with 201 points and solving the resulting ODEs in the interval [ζ , 1] using Matlab’s solver ode15s. Observer design. The design of an output feedback controller requires the implementation of the reference and disturbance observers (37) and (39). For the observable reference model, the observer results from an eigenvalue placement specified by σ (Sr − Lr P¯ r ) = {−4, −5, −6, −7}. The state observer is parametrized by µo = 10 and the corresponding kernel equations are solved using successive approximations. The condition of Lemma 6 is fulfilled because σ (Sd ) = {±j4} ∩ σo = ∅ so that (54) is solvable. With their solution, the observer gain Ld is computed by minimizing a quadratic cost functional. The result places the eigenvalues of Sd − Ld Γ (0) at {−7, −7}. The infinite-dimensional part of the observer (39b)–(39d) is approximated using the finite element method (FEM) with 13 grid points for each state. Simulation results. In the simulation, the plant is approximated by an FEM model with 201 grid points for each state. Fig. 1 shows the

Fig. 2. Closed-loop disturbance behaviour for d(t) = sin(4t) (upper plot); disturˆ = P¯ d vˆ d (t) (lower plot). bance d(t) and its estimation d(t)

resulting tracking behaviour for two different sawtooth reference inputs r1 (t) = 2 + 21 t − 4.5⌊ 9t ⌋ and r2 (t) = −5 + t − 6⌊ 6t ⌋ as well as d(t) ≡ 0, where ⌊·⌋ = floor(·) denotes the floor function. It can be seen that both outputs track their reference signal asymptotically. However, the resetting of the reference model leads to initial value disturbances explaining the resulting deviations at the edges of the reference input. Similarly, Fig. 2 shows the disturbance behaviour (i.e., r(t) ≡ 0) when applying the sinusoidal disturbance d(t) = sin(4t), as well as the corresponding reconstruction of the disturbance by the observer. After the convergence of the observer, the outputs both converge to zero verifying asymptotic disturbance rejection. 7. Concluding remarks With the results in Deutscher and Kerschbaum (2018) and more involved technical developments, the proposed regulator design can also be extended to the more general setup, where both Dirichlet and Robin/Neumann BCs are present. Furthermore, the consideration of a collocated setting, i.e., a measurement at z = 1, poses no new problems and can be solved by directly extending the presented approach. Then, however, only coupled parabolic PDEs can be considered (see Remark 2). Furthermore, the digital implementation of the output feedback regulator and its analysis are important for applications. This is an interesting topic for future work, where the results in Karafyllis and Krstic (2018) are of interest. Appendix A. Proof of Theorem 5 The target system (22b)–(22d) coincides with the one considered in Deutscher and Kerschbaum (2018) so that the stability result of the theorem directly follows from this reference. Since this result is also utilized in the proof of Theorem 8, the main steps of the proof are reviewed in the following, in order to render the paper self contained. The main idea of the well-posedness and stability proof is to represent the system operator of (22b)–(22d) as a sum of an operator A with known properties and a perturbation ∆. Then, the

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properties of A + ∆ follow from analysing the perturbation ∆. In order to determine A, consider (22b)–(22d) without the local term ˜ A0 (z)ε˜ (0, t) giving rise to n decoupled parabolic PDEs. Then, define the operators Ai h = λ

2 i dz h

− µc h,

i = 1, 2, . . . , n,

(A.1)

(17). A straightforward calculation yields

¯ ζ ) + µc I) Λ(z)K¯ zz (z , ζ ) − K¯ ζ ζ (z , ζ )Λ(ζ ) = K¯ (z , ζ )(A( ∫ z + Λ(z)F¯ (z , ζ ) + K¯ (z , ζ¯ )F¯ (ζ¯ , ζ )dζ¯

(B.1a)

K¯ ζ (z , 0)Λ(0) + K¯ (z , 0)B0

(B.1b)

ζ

on

z

∫ = dz h(1) + bi1 h(1) = 0}

0

(A.2)

in the Hilbert space H = L2 (0, 1) with the usual weighted inner product. With this, the operator Ah = [A1 h1 . . . An hn ]⊤

(A.3)

and D(A) = D(A1 ) ⊕ · · · ⊕ D(An ) is the system operator of (22b)– (22d) without the local term. The latter is taken into account by ∆h = −˜ A0 h(0) with D(∆) = D(A). Consequently, the whole system (22b)–(22d) can be represented by the abstract IVP

ε˙˜ (t) = (A + ∆) ε˜ (t),

t > 0, ε˜ (0) = ε˜ 0 ∈ D(A),

t ≥ 0,

Λ(z)K¯ ′ (z , z) + Λ(z)K¯ z (z , z) + K¯ ζ (z , z)Λ(z) + K¯ (z , z)Λ′ (z) = −(A(z) + µc I)

(B.1c)

K¯ (z , z)Λ(z) − Λ(z)K¯ (z , z) = 0

(B.1d)

K¯ (0, 0) = B0 Λ(0)

(B.1e)

¯ ζ ) = Λ−1 (ζ )A(ζ )Λ(ζ ) and F¯ (z , ζ ) = Λ−1 (z)F (z , ζ )Λ(ζ ). with A( Similarly, setting P¯ I (z , ζ ) = PI (z , ζ )Λ(ζ ) in (47) gives Λ(z)P¯ I ,zz (z , ζ ) − P¯ I ,ζ ζ (z , ζ )Λ(ζ ) = −(A(z) + µo I)P¯ I (z , ζ ) ∫ z + F (z , ζ )Λ(ζ ) − F (z , ζ¯ )P¯ I (ζ¯ , ζ )dζ¯

(B.2a)

P¯ I ,z (1, ζ ) + B1 P¯ I (1, ζ ) = −A¯ 0 (ζ )Λ(ζ )

(B.2b)

ζ

(A.4)

where ε˜ (t) = {˜ε (z , t), z ∈ [0, 1]} is the state in the state space X = (L2 (0, 1))n with the weighted L2 -inner product. In view of (A.3) and (A.1) the operator −A consists of the n completely decoupled Sturm–Liouville operators −Ai , i = 1, 2, . . . , n (see Delattre, Dochain, & Winkin, 2003). Hence, A is the generator of an analytic 1 C0 -semigroup with compact resolvent, because the operators A− i exist for µc > µmax (i.e., 0 ∈ ρ (Ai ) holds) and are compact (see Kato, 1995, Th. III/6.29 and Naylor & Sell, 1982, Th. 7.5.4). Furthermore, since the perturbation ∆ in (A.4) is relatively bounded w.r.t. A and the evaluation operator E h = h(0), D(E ) = D(∆), has finite-dimensional range, it is relatively compact (see Kato, 1995, Rem. IV/1.13). As a consequence, A + ∆ is the generator of an analytic C0 -semigroup and has compact resolvent (see Engel & Nagel, 2000, Th. III/2.10 & Lem. III/2.16 and Kato, 1995, Th. IV/3.17). This implies that the spectrum determined growth assumption holds and that A + ∆ has a discrete spectrum (see Triggiani, 1975 and Kato, 1995, Th. III/6.29). Consequently, (A.4) is exponentially stable for αc = µmax − µc < 0, i.e.,

˜c e(αc +c)t ∥˜ε(0)∥, ∥˜ε (t)∥ ≤ M

K¯ ζ (z , ζ )A0 (ζ )dζ

=˜ A0 (z) + A0 (z) −

D(Ai ) = {h ∈ H 2 (0, 1) | dz h(0) + bi0 h(0)

Λ(z)P¯ I′ (z , z) + Λ(z)P¯ I ,z (z , z) + P¯ I ,ζ (z , z)Λ(z)

− P¯ I (z , z)Λ′ (z) = (A(z) + µo I)Λ(z) P¯ I (z , z)Λ(z) − Λ(z)P¯ I (z , z) = 0

(B.2d)

P¯ I (1, 1) = 0.

(B.2e)

(B.2c)

Introducing the change of variables zˇ = 1 − ζ and ζˇ = 1 − z for (B.2) as well as transposing the result yields new kernel equations, which have the same form as (B.1) with A0 (z) ≡ 0. Hence, the well-posedness of (17) implies the same for (47). With the same reasoning a similar result can also be derived for the kernel equations (50). Appendix C. Proof of Lemma 6 Assume that the matrix Sd ∈ Rnd ×nd has the Jordan blocks Ji , i = 1, 2, . . . , rd , to which the Jordan chains (1)

= νi ψi(1)

(k)

= νi ψi(k) + ψi(k−1) ,

(A.5)

S d ψi

˜c ≥ 1 and any c > 0 such that for all ε˜ (0) = ε˜ 0 ∈ D(A), an M αc + c < 0. This is implied by the triangular structure of A + ∆ yielding the growth bound αc = max{Re λ | λ ∈ σ (A + ∆)} for the C0 -semigroup generated by A + ∆ and a real point spectrum. Therein, the growth bound αc is the greatest lower bound for all decay rates αc + c in (A.5). With (A.5) established and the fact that A + ∆ is the generator of an analytic semigroup, output regulation limt →∞ ey (t) = limt →∞ CTc−1 [˜ε (t)] = 0 (see (22e)) follows for the considered unbounded output operators defined by (6) from the same reasoning as in Deutscher (2015). In order to obtain the stability result (36) in the original coordinates, consider the tracking error e(z , t) = Tc−1 [˜ε (t)](z) = x(z , t) − Tc−1 [Π ](z)v (t) in view of (20) and (23). With this, the stability result (36) can be verified by taking the bounded invertibility of the changes of coordinates (13) and (23) into account (see also Deutscher, 2015).

S d ψi

(C.1a) k = 2, 3, . . . , li ,

(C.1b) (k)

are associated and l1 + · · · + lrd = nd . In (C.1) the vectors ψi , i = 1, 2, . . . , rd , k = 1, 2, . . . , li , are the generalized eigenvectors of Sd w.r.t. the eigenvalue νi , i = 1, 2, . . . , rd , in which rd is the number of eigenvalues of Sd with linearly independent eigenvectors. Postmultiplying (54) by these vectors and introducing the (k) (k) (k) ¯ 1 P¯ d ψ (k) , h(k) = G¯ 2 ψ (k) and abbreviations γi = Γ ψi , h1,i = −H i i 2 ,i (k)

(k)

¯ 3ψ h3,i = G i

leads to the one-sided coupled BVPs

(k)

(k)

d2z γi (z) = (µo + νi )Λ−1 (z)γi (z) (k−1) + Λ−1 (z)(h(k) (z)) 1,i (z) + γi (k) i

θ0 [γ ](0) =

(k) h2,i

θ1 [γi(k) ](1) = −

(C.2a) (C.2b)

1

∫

(k)

0

(k)

A¯ 0 (ζ )γi (ζ )dζ + h3,i

(C.2c)

(0)

Appendix B. Well-posedness of the kernel equations (47) In order to verify the well-posedness of the kernel equations (47), introduce the transformed kernel K¯ (z , ζ ) = K (z , ζ )Λ(z) for

with (C.2a) defined on z ∈ (0, 1) and γi (z) ≡ 0. Let Φ (z , ζ , s) be the fundamental matrix of Lemma 3 with µc replaced by µo . Then, the solution of (C.2a) and (C.2b) is

γi(k) (z) = M(z , νi )γi(k) (0) + m(k) (z , νi )

(C.3)

J. Deutscher and S. Kerschbaum / Automatica 100 (2019) 360–370

with M(z , s) = T1⊤ Φ (z , 0, s)(T1 − T2 B0 ) and (k) i )T2 h2,i

m (z , νi ) = T1 Φ (z , 0, ν (k)

⊤

(C.4) (k)

+ 0

(k−1)

T1⊤ Φ (z , ζ , νi )T2 Λ−1 (ζ )(h1,i (ζ ) + γi

observers. Hence, the separation principle is satisfied for the closedloop system. Closed-loop output regulation. The relative boundedness of the output operator w.r.t. the closed-loop state can be verified, which implies output regulation (see Deutscher, 2015).

z

∫

369

(ζ ))dζ .

Inserting (C.3) in (C.2c) gives (k)

Ω (νi )γi (0) = ω(k) (νi )

(C.5)

with Ω (νi ) = θ1 [M(·, νi )](1) +

∫1

∫1 0

A¯ 0 (ζ )M(ζ , νi )dζ and ω(k) (νi ) =

θ1 [m (·, νi )](1) − 0 A¯ 0 (ζ )m(k) (ζ , νi )dζ + h(k) 3,i . It is not difficult to verify that the solutions of det Ω (s) = 0, s ∈ C, are the eigenvalues (k)

of the PDE subsystem (53a)–(53c) . Thus, iff the condition of the (k) lemma holds, then (C.5) is uniquely solvable for γi (0). As the gen(k) eralized eigenvectors ψi are linearly independent, the solution (1)

(lr )

(l )

Γ (z) = [γ1 (z) . . . γ1 1 (z) . . . γrd d (z)] (l )

(lr )

·[ψ1(1) . . . ψ1 1 . . . ψrd d ]−1

(C.6)

of (54) follows. This is the piecewise classical solution as the ¯ 1 (z) in (C.2a) are piecewise C 1 -functions (see (C.4)). elements of H Appendix D. Proof of Theorem 9 The proof follows the same lines as the proof of Theorem 6 in Deutscher (2015). Therefore, only its main steps are sketched. Abstract closed-loop IVP. The closed-loop system can be represented in the coordinates er , εd , ε˜ x and xˆ (cf. (38), (64) and (39)). Apply the backstepping transformation w (z , t) = Tc [ˆx(t)](z) (see (13)) to the xˆ -system and introduce the new coordinates ε¯ (z , t) = w(z , t) − Π (z)vˆ (t), in which Π (z) satisfies the regulator equations (25). This results in a cascade, in which the ε¯ -system is driven by the er -, εd - and ε˜ x -system. The BC of the ε¯ -system at z = 0 depends on the states of the εd - and ε˜ x -system. In order to remove this inhomogeneous BC, introduce the transformation εˇ (z , t) = ε¯ (z , t) − F (z)r(z , t), that yields the εˇ -system with homogeneous BCs. Therein, r(z , t) is defined by the inhomogeneity and F (z) is a suitable matrix function F (z) ∈ Rn×n . Hence, one can represent the closed-loop system by the abstract IVP x˙˜ cl (t) = (Acl + ∆cl )x˜ cl (t),

t > 0, x˜ cl (0) ∈ D(Acl ),

(D.1)

having the state x˜ cl = col(er , εd , ε˜ x , εˇ ) in the Hilbert space Xcl = Cnr ⊕ Cnd ⊕ (L2 (0, 1))n ⊕ (L2 (0, 1))n with the standard weighted inner product. Well-posedness and closed-loop stability. First, it is easy to verify that the C0 -semigroup T (t) generated by the operator Ao in the proof of Theorem 8 is analytic. This follows from the property of A∗o , a direct calculation by utilizing the definition of analytic semigroups in Tucsnak and Weiss (2009, Def. 5.4.5) and the properties of adjoints (see Curtain & Zwart, 1995, Lem. A.3.60 & A.3.65). Since the εˇ -system has the same system operator as (22b)–(22d), the proof of Theorem 5 shows that its system operator generates an analytic C0 -semigroup. These results imply that the operator Acl is also the generator of an analytic C0 -semigroup. The latter property is preserved for Acl + ∆cl , because ∆cl is relatively bounded with Acl bound zero (see Deutscher, 2015 for the latter result). Furthermore, it can be verified that Acl + ∆cl has a compact resolvent implying a discrete spectrum (see Deutscher, 2015). Consequently, the IVP (D.1) is well-posed and the spectrum determined growth assumption holds. This and the fact that Acl + ∆cl has a triangular structure yields the stability result (73), when taking the boundedness of the corresponding transformations into account (see also Deutscher, 2015). Therein, the growth bound αcl is directly determined by the growth bounds achieved by the state feedback controller and the

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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 to 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). At present he serves as Associate Editor for Automatica.

Simon Kerschbaum received the B.Sc. and M.Sc. degrees in electrical engineering from the Friedrich-Alexander University Erlangen-Nuremberg (FAU), Germany, in 2012 and 2014. Since then, he is a Ph.D. student at the Chair of Automatic Control (FAU) in the research group of Prof. Deutscher. His research interests include backstepping methods for coupled parabolic systems and output regulation for distributed-parameter systems.