A Lyapunov Approach to Robust Regulation of Distributed Port-Hamiltonian Systems

This paper studies robust output tracking and disturbance rejection for boundary controlled infinite-dimensional port--Hamiltonian systems including second order models such as the Euler--Bernoulli beam. The control design is achieved using the internal model principle and the stability analysis using a Lyapunov approach. Contrary to existing works on the same topic no assumption is made on the external well-posedness of the considered class of PDEs. The results are applied to robust tracking of a piezo actuated tube used in atomic force imaging.


Introduction
We consider robust output regulation for a class of linear partial differential equations (PDEs) with boundary control and observation, namely, port-Hamiltonian systems (PHS) [13,11] ∂x ∂t (z, t) = P 2 ∂ 2 ∂z 2 H(z)x(z, t) + P 1 ∂ ∂z H(z)x(z, t) (1a) + (P 0 − G 0 ) (H(z)x(z, t)) + B d (z)w dist,1 (t), (1b) on a one-dimensional spatial domain [a, b] (see Section 2 for detailed assumptions). In robust regulation, the purpose of the control u(t) ∈ R p is to achieve the asymptotic convergence of the output y(t) ∈ R p of (1) to a given reference signal y ref (t), i.e., y(t) − y ref (t) → 0 as t → ∞, despite external disturbance signals w dist (t) := (w dist,1 (t), w dist,2 (t), w dist,3 (t)). The signals y ref (t) and w dist (t) are assumed to have the forms for known frequencies 0 = ω 0 < ω 1 < · · · < ω q and unknown amplitudes {a 1 k } q k=0 , {a 2 k } q k=1 ⊂ R p , and {b 1 k } q k=0 , {b 2 k } q k=1 ⊂ R n d,1 +p+n d, 3 . Several recent articles have considered output regulation for individual linear PDEs, such as 1D heat equations [4], beam equations [12] and wave equations [6]. In this paper we solve the control problem for a class of boundary controlled 1D PDEs (1), which covers many particular hyperbolic PDE systems such as boundary controlled wave equations, Schrödinger equations, Timoshenko and Euler-Bernoulli beam models with spatially varying physical parameters, and is used in modeling and control of flexible structures, heat exchangers, and chemical reactors. We focus here on impedance passive PHS (1), and solve the output regulation problem using a finite-dimensional dynamic error feedback controlleṙ where J c is skew-symmetric, B c ∈ R p×nc , and D c ∈ R p×p satisfies D c ≥ 0. Finally, δ c > 0 is a gain parameter. In studying the class (1) of PDEs our aim is to design the controller (3) under assumptions that can be verified directly based on the properties of the original PDE (1) and the matrices (P 0 , P 1 , P 2 , G 0 , W 1 , W 2 ,W ), without the need to reformulate (1) as an abstract system.
Our results for the class (1) are based on the theoretical results on robust output regulation of abstract boundary control and observation systems [3,21] presented in this paper. They extend the theory related to internal model based controllers for passive well-posed linear systems and PHS in [18,7,8,10,9], and they compose the main technical contributions of the paper. In particular, we introduce a new Lyapunov-type argument for the stability analysis of the closed-loop system consisting of the boundary control system and the controller (extending our earlier results in [16] for PHS with distributed control and observation). In addition, the controller design is done without assuming well-posedness of the original control system (which was assumed in [18]) and the analysis is completed directly in the abstract boundary control system framework (whereas in [10,9] the boundary control inputs were first reformulated as distributed inputs using a state extension). The class (1) includes models which are not wellposed (in the sense of [20,Sec. 2]). The stability analysis of the closed-loop system is also related to references [17,14] studying the stability of coupled impedance passive systems in a different context i.e. when the infinite dimensional system is undamped and the controller strictly input passive.
The paper is organised as follows. In Section 2 we define the considered class of boundary controlled PHS and state our main result for the PDEs (1) (these are proved later in Section 5). In Sections 3-5 we present our main results for abstract boundary control systems. The results are applied in solving a concrete output regulation problem in Section 6. The paper ends with some conclusions and perspectives. Notation. If X and Y are Banach spaces and A : X → Y is a linear operator, we denote by D(A), N (A) and R(A) the domain, kernel and range of A, respectively. The space of bounded linear operators from X to Y is denoted by L(X, Y ). If A : X → X, then σ(A) and ρ(A) denote the spectrum and the resolvent set of A, respectively. For λ ∈ ρ(A) the resolvent operator is R(λ, A) = (λ − A) −1 . The inner product on a Hilbert space is denoted by ·, · . For T ∈ L(X) on a Hilbert space X we define

The Main Results for PHS
In this section we summarise our main results for the class (1) of boundary controlled PDEs. The parameters P 2 , P 1 , P 0 , G 0 ∈ R n×n are assumed to satisfy is a bounded and Lipschitz continuous matrix-valued function such that H(z) = H(z) T and H(z) ≥ κI, with κ > 0, for all z ∈ [a, b]. The distributed disturbance input profile is assumed to satisfy B d (·) ∈ L 2 (a, b; R n×n d,1 ) and can be unknown.
We consider first and second order PHS by assuming that either P 2 is invertible (the system (1) is of order N = 2) or P 2 = 0 and P 1 is invertible (the system is of order N = 1). The boundary inputs and ouputs are determined using the following boundary port variables.
Definition 2.1. The boundary port variables f ∂ (t) and e ∂ (t) associated to the system (1) are defined as The input u(t) ∈ R p , output y(t) ∈ R p (the numbers of inputs and outputs are the same) and the disturbance inputs w dist (t) = (w dist,1 (t), w dist,2 (t), w dist,3 (t)) T ∈ R n d,1 +p+n d,3 of the system are defined as in (1). We assume the matrices W 1 , W 2 , andW determining the inputs and outputs satisfy the following (concrete and checkable) conditions. As shown later in Lemma 5.3, part (b) of Assumption 2.2 guarantees that (1) is impedance passive. Assumption 2.2. Denote Σ := 0 I I 0 ∈ R 2nN ×2nN . We assume W 1 ∈ R p×2nN and W 2 ∈ R n d,3 ×2nN with n d,3 = nN − p andW ∈ R p×2nN satisfy the following (a) W := W 1 W 2 ∈ R nN ×2nN has full rank and W ΣW T ≥ 0 . Our second assumption concerns stabilizability properties of (1). The system (1) is exponentially stable if there exist M, α > 0 such that with u(t) ≡ 0 and w dist (t) ≡ 0 we have for all x(·, 0) ∈ L 2 (a, b; R n ) such that Hx(·, 0) ∈ H N (a, b; R n ) and for which (1c) hold for t = 0. Assumption 2.3. For any K ∈ R p×p , K > 0, system (1) becomes exponentially stable with output feedback u(t) = −Ky(t).
The output feedback u(t) = −Ky(t) alters the boundary conditions of the PDE (1) by changing W 1 in (1c) to W 1 + KW . By [10, Lem. 7] Assumption 2.3 holds in particular if W 1 ∈ R nN ×2nN (i.e., (1) has p = nN inputs) and if Assumption 2.2 holds. For further results on stability of (1), see [1]. Definition 2.4 contains the construction of the controller (3). The controller has an internal model of the frequencies in (2) in the sense that {±iω k } q k=1 ∪ {0} are eigenvalues of J c with geometric multiplicities equal to p (see also Section 4).
The controller is robust in the sense that the tracking (5) is achieved (with a modified α > 0) also if the parameters (P 2 , P 1 , P 0 , G 0 , W 1 , W 2 ,W , H, B d ) of (1) are perturbed in such a way that Assumption 2.2 continues to hold and the closed-loop system remains exponentially stable.
The proof of Theorem 2.5 is presented in Section 5. If ω 0 = 0 is a transmission zero, then J 0 c and B 0 c can be removed from the controller parameters in (4) and Theorem 2.5 holds for y ref (t) and w dist (t) with a 0 = 0 and b 0 = 0.

Background on Boundary Control Systems
Our main abstract results are formulated for the general class of boundary control and observation systems [19,3] on a Hilbert space X. We present these abstract results only in the case D c = 0. This simplification does not result in loss of generality, because if D c = 0, then (6b) becomes (which has the same structure as (6b)) whereũ(t) is the control produced by the controller (3) with D c = 0. We make the following standard assumptions on the parameters of (6).
Assumption 3.1. We assume X and Z ⊂ X are (complex) Hilbert spaces 3 ) and C ∈ L(Z, C p ) have the properties: (a) The operator A : By [15,Thm. 3.4] part (c) of Assumption 3.1 is equivalent to the system (6) being impedance passive in the sense that 1 2 We also denote A : For λ ∈ ρ(A) we denote the transfer function (from the input u(t) to the output y(t)) of the system (1) by P (λ). By [3, Thm. 2.9], for any u ∈ U and λ ∈ ρ(A) we have P (λ)u = Cx where x ∈ Z is such that (λ − A)x = 0 and Bx = u. If we denote Re T = 1 2 (T + T * ), then the passivity of the system implies that Re P (iω) ≥ 0 for all iω ∈ ρ(A) ∩ iR, see [20].
We assume the controller (3) on X c = C nc satisfies J * c = −J c ∈ C nc×nc , B c ∈ C nc×p , D c ∈ C p×p with D c ≥ 0 and δ c > 0 (as mentioned above, in Sections 3-5 we let D c = 0). We now show that the closed-loop system consisting of (6) and the controller (3) on X c = C nc leads to a well-defined closed-loop state x e (t) := (x(t), x c (t)) T and regulation error e(t) for all reference and disturbance signals in (2). The closed-loop system (with D c = 0) has the forṁ  3 ) and for all initial states x(0) ∈ Z and x c (0) ∈ X c satisfying the compatibility conditions and Proof. The closed-loop system is a boundary control and observation system on the spaces Z × X c and X e = X × X c . The operator B e is surjective due to Assumption 3.1(b). Our aim is to show that A e generates a contraction semigroup on X e . Since C e ∈ L(Z × X c , C p ) and B d and B c are bounded, the properties of the closed-loop system's state then follow (due to linearity) from [ Therefore A e is dissipative, and it remains to show that λ − A e is surjective for some λ > 0. Let λ > 0, y 1 ∈ X, and y 2 ∈ X c be arbitrary. We will construct x e := (x, x c ) T ∈ N (B e ) such that (y 1 , y 2 ) T = (λ − A e )x e . Recall that P (λ) is the transfer function of (A, B, C) and denote P c (λ) = δ 2 c B * c R(λ, J c )B c . Since λ > 0 is real, we have P c (λ) ≥ 0 and P (λ) ≥ 0, and it can be shown that Q 1 (λ) := I + P (λ)P c (λ) and Q 2 (λ) := I +P c (λ)P (λ) are boundedly invertible. Denote R λ = R(λ, A) and R c λ = R(λ, J c ) for brevity. Due to the theory in [3], [21, Ch. 10] the "abstract elliptic problem"

Robust tracking and disturbance rejection
In this section we formulate the robust output regulation problem and present a general condition for a controller (3) to solve this problem.
The Robust Output Regulation Problem. Let 0 < ω 1 < · · · < ω q . Choose a controller (3) in such a way that the following hold.
(a) The semigroup T e (t) generated by A e = A e | N (Be) is exponentially stable. (b) There exists α > 0 such that for all y ref (t) and w dist (t) of the form (2) and for all initial states x(0) ∈ Z and x c (0) ∈ X c satisfying the boundary conditions of (6) the regulation error satisfies (6) are perturbed in such a way that Assumption 3.1 is satisfied and the perturbed closed-loop operator generates an exponentially stable semigroup, then (b) continues to hold for someα > 0.
The robust output regulation problem only has a solution if the control system does not have transmission zeros at {±iω k } q k=0 (a transmission zero at λ ∈ ρ(A) is equivalent to P (λ) ∈ C p×p being singular). For impedance passive systems it is natural to make the following stronger assumption.
The following theorem shows that a controller incorporating an internal model (in the sense of conditions (8) below) will solve the robust output regulation problem provided that the closed-loop system is exponentially stable. The result generalises [10,Thm. 4] by removing the assumption of regularity (and well-posedness) of the closed-loop system, and the proof is completed without reformulating (6) as a system with extended state and distributed inputs.
Theorem 4.2. Let 0 = ω 0 < ω 1 < · · · < ω q . A controller (3) with J * c = −J c , D c = 0 and δ c > 0 solves the robust output regulation problem if A e = A e | N (Be) generates an exponentially stable semigroup and Then there exists α > 0 such that for any y ref (t) and w dist (t) of the form (2) and for all x(0) ∈ Z and x c (0) ∈ X c satisfying the compatibility conditions Proof. Assume the closed-loop system is exponentially stable and (8) are satisfied. Then there exist M e , ω e > 0 such that T e (t) ≤ M e e −ωet . Let {µ k } q k=−q be such that µ k = ω k for k > 0, µ 0 = 0, and µ k = −ω |k| for k < 0. We can then write for some constant elements {y k r } k , {w 1k } k , {w 2k } k , and {w 3k } k . Since iµ k ∈ ρ(A e ) for all k, we have from [21, Sec. 10.1] that we can choose Σ k ∈ Z such that Consider initial conditions x(0) ∈ Z and x c (0) ∈ X c satisfying the compatibility conditions due to (9a). For all t ≥ 0 we also have from (9b) that Thus If we write Σ k = Π k Γ k ∈ Z × X c , then (9a) and the conditions (8) imply Finally, since C e A −1 e ∈ L(X, C p ) for boundary control systems, we have and thus e αt e(t) → 0 as t → ∞ for any 0 < α < ω e .
Since the proof can be repeated analogously for any perturbations of (A 0 , B, B d , B d , C) for which Assumption 3.1 is satisfied and the closed-loop semigroup is exponentially stable, the controller satisfies part (c) of the robust output regulation problem.

A Passive Robust Controller
In this section we prove that if the system (6) is exponentially stable and the parameters of the controller (3) on X c = C p(2q+1) are chosen as (real) matrices D c = 0, then the controller solves the robust output regulation problem for a range of gain parameters δ c > 0. The following theorem is the main abstract result of the paper, and it is also used in proving Theorem 2.5 at the end of this section.
The main part of the proof of Theorem 5.1 consists of showing the exponential stability of the closed-loop system for δ c ∈ (0, δ * c ), and for this we use a new Lyapunov argument. Similar methods have been used in study of stability of coupled PHS especially in [17,14]. Our situation is different from the previous references due to the fact that the infinitedimensional system (6) is exponentially stable and the unstable controller (3) is finite-dimensional. The proof of Theorem 5.1 begins with the definition of a component H ∈ L(X c , X) of the Lyapunov candidate function in Lemma 5.2. For the proofs we define a block-diagonal similarity transform T = blockdiag(T 0 , T 1 , . . . , T q ) ∈ C nc×nc where n c = p(2q + 1) such that for k ∈ {1, . . . , q} Lemma 5.2. Let Assumption 4.1 hold and assume A generates an exponentially stable semigroup on X. Let X c = C p(2q+1) and let J c and B c be as in (10). Then there exists H ∈ L(X c , X) satisfying R(H) ⊂ Z such that and we have CH ∈ L(X c , C p ). Moreover, there exist constants δ * 0 , M c > 0 such that for any δ c ∈ (0, δ * 0 ) we can choose P c0 > 0 such that P c0 ≤ M c and Thus H ∈ L(X c , X) and R(H) ⊂ Z. We further have from [3, Thm. 2.9] that CH ±k y = Cz ±k = −P (±iω k )y for all y ∈ C p and k ∈ {0, . . . , q}. Because of this, we have CHT = − P (iω 0 ), P (iω 1 ), P (−iω 1 ), . . . , P (iω q ), P (−iω q ) , which in particular implies CH ∈ L(X c , C p ).
Let H ∈ L(X c , X) and δ * 0 , M c > 0 be as in Lemma 5.2, and let δ c ∈ (0, δ * 0 ). We choose the Lyapunov function candidate V e for the closed-loop system by where x = x(t) and x c = x c (t) are the states of the plant and the controller, respectively, and P and P c will be chosen later. Since the coordinate transform (x, x c ) → (x + δ c Hx c , x c ) is boundedly invertible, V e is a valid Lyapunov function candidate whenever P > 0 and P c > 0.
Proof of Theorem 2.5. To apply Theorem 5.1 we rewrite the feedthrough D c > 0 as in (7), in which case the boundary control system has the input operator B + D c C and the controller (3) has no feedthrough. This corresponds to preliminary output feedback u(t) = −D c y(t) +ũ(t). Denote by By Lemma 5.3, the original system is impedance passive, and since D c > 0, the output feedback preserves impedance passivity. The operator A Dc is dissipative, and straightforward perturbation arguments (similar to those in the proof of Proposition 3.2) show that R(1 − A Dc ) = X. Thus A Dc generates a contraction semigroup by the Lumer-Phillips Theorem and this semigroup is exponentially stable by Assumption 2.3 (with K = D c ). As shown in [9, Prop. II.4]), C = C| D(A Dc ) is admissible with respect to the semigroup generated by A Dc .

Application to Atomic Force Microscopy
As application example we consider the output tracking trajectory problem for a piezo actuated tube used in positioning systems for Atomic Force Microscopy (see Figure 1 (left)).  . Ces techniques diées pour la commande monovariable, c'est à dire pour les actionneurs m ou pour les actionneurs multi-axes mais en étudiant chaque axe de l'action [16,169,126].

Objectifs et plan du rapport
Ce travail de thèse s'articule autour de quatre principaux objectifs : 1. Mettre en place des modèles multivariables pour les actionneurs p This actuator provides the high positioning resolution and the large bandwidth necessary for the trajectory control during scanning processes. The active part situated at the tip of the flexible tube is composed of three concentric layers: piezo material in between two cylindric electrodes (Figure 1 (right)). The deformation of the active material subject to an external voltage results in an torque applied at the extremity of the tube.
We consider the motion of the tube in one direction. In this case the structure of the system behaves as a clamped-free beam, represented by the Timoshenko beam model and actuated through boundary control stemming from the piezoelectric action at the tip of the beam. By choosing as state variables the energy variables, namely the shear displacement x 1 (t) = ∂w ∂z (·, t) − φ(·, t), the transverse momentum distribution x 2 (t) = ρ ∂w ∂t (·, t), the angular displacement x 3 (t) = ∂φ ∂z (·, t) and the angular momentum distribution x 4 (t) = I ρ ∂φ ∂t (·, t) for t ≥ 0, where w(z, t) is the transverse displacement and φ(z, t) the rotation angle of the beam, the port-Hamiltonian model of the uncontrolled Timoshenko beam has the form (1a)-(1b) with H(·) ≡ diag K, 1 ρ , EI, 1 Iρ ∈ R 4 , and G 0 = diag(0, b w , 0, b φ ) [13]. Here ρ, I ρ , E, I and K are the mass per unit length, the angular moment of inertia of a cross section, Young's modulus of elasticity, the moment of inertia of a cross section, and the shear modulus respectively, b w , b φ the frictious coefficients. From Definition 2.1 considering that N = 1 and Q = P 1 we get The beam is clamped at point a, i.e., 1 ρ x 2 (a, t) = 1 Iρ x 4 (a, t) = 0 for t ≥ 0 and free/actuated at point b, i.e., Kx 1 (b, t) = 0 and EIx 3 (b, t) = u(t) for t ≥ 0. The angular velocity ∂φ ∂t (b, t) at the tip of the beam is measured. The input and output of the system are then of the form (1) with has full rank and W ΣW T = 0. Furthermore (W T 1W +W T W 1 − Σ)g, g = 0 for all g ∈ N (W 2 ), the system is then impedance passive satisfying Assumption 2.2. The system is also exponentially stable and Assumption 2.3 holds. From Proposition 3.2 the closed loop system has a solution and the regulation error is well defined.
We now build a controller to achieve the robust output tracking for the Piezoelectric tube model. We use the numerical values given in Table 1 to achieve a realistic approximation of the dynamics of the piezo actuated tube.
with two pairs of frequencies ±ω k where ω i > 0, k ∈ {1, 2}. As an input disturbance signal we consider 50 Hz AC noise coming from the electrical network, hence w dist,2 (t) = c sin(2π50t + θ) with unknown c ∈ R and θ ∈ [0, 2π]. Since the piezo-actuated tube is a single-input single-output system, we can use a controller of the form (with e(t) = y ref (t) − y(t)) x c (t) =  For simulation the Timoshenko beam model was discretized using a structure preserving method based on the Mixed Finite Element Method [5,2]. We denote by N f the number of basis elements, and consequently the full finite dimensional system has order 4N f . All the numerical values of the parameters related to the simulation can be found in table 1. Figure 2 depicts the output tracking performance for the zero initial states of the system and the controller, and exhibits steady convergence of the tracking error to zero. Due to robustness the output tracking is achieved even if the physical parameters of the piezo actuated tube model contain uncertainties or experience changes, as long as the closed-loop system stability is preserved.

Conclusions
In this paper we have proposed a constructive method for the design of impedance passive controllers for robust output regulation of port-Hamiltonian systems with boundary control and observation. Our results use Lyapunov techniques and extend previous results on this topic by removing the assumption of wellposedness, which is often highly challenging to verify for concrete PDE models. Future research topics include the design of robust controllers for nonlinear PHS.