Nonautonomous Controllers and Output Regulation of Unknown Harmonic Signals for Regular Linear Systems

We introduce general results on well-posedness and output regulation of regular linear systems with nonautonomous controllers. We present a generalization of the internal model principle for time-dependent controllers with asymptotically converging parameters. This general result is utilized in controller design for output tracking and disturbance rejection of harmonic signals with unknown frequencies. Our controller can be flexibly combined with different frequency estimation methods. The results are illustrated in rejection of unknown harmonic disturbances for a 1-D boundary controlled heat equation.

In the classical output regulation problem the reference and disturbance signals are assumed to have the forms where the frequencies 0 = ω 0 < ω 1 < . . .< ω q are assumed to be known and the amplitudes (a k ) q k=0 , (b k ) q k=1 , (c k ) q k=0 , Lassi Paunonen is with the Mathematics, Faculty of Information Technology and Communication Sciences, Tampere University, 33101 Tampere, Finland (e-mail: lassi.paunonen@tuni.fi).
Digital Object Identifier 10.1109/TAC.2023.3324523and (d k ) q k=1 may be unknown.In this article, we study a more challenging version of the control problem where also the frequencies (ω k ) q k=1 are unknown.For finite-dimensional linear and nonlinear systems the case of unknown frequencies has been studied in [8], [20], [31], [42], [43] using adaptive internal models.Controllers have also been designed for selected PDE models, in particular, 2 × 2 hyperbolic systems [2], a Kirchoff plate [28], and a 1-D boundary controlled heat equation [15].Our focus is on output regulation for linear distributed parameter systems, and for such systems our problem has only been studied in [38] and [39] under restrictive structural assumptions.We study a considerably larger class of systems, namely regular linear systems [32], [40] x(0) = x 0 (2a) on a Hilbert space X (see Section II for detailed assumptions).
Regular linear systems can be used in controller design for a wide range of PDE models with boundary control and observation, e.g., 1-D convection-diffusion equations, wave equations, beam equations, as well as multidimensional heat equations [5].
As our ultimate contribution we introduce a controller design method for output regulation of signals (1) with unknown frequencies, amplitudes, and phases.In particular, we introduce a dynamic error feedback controller with a time-varying internal model based on estimates (ω k (t)) q k=1 of the frequencies in (1).The controller design leads to a nonautonomous dynamic error feedback controller on a Hilbert space Z.Here, G 2 (•) ∈ L ∞ (0, ∞; L(C p , Z)) and and G 1 (t) may contain an unbounded time-varying part (see Assumption 2.2).The analysis of well-posedness of the closed-loop system consisting of ( 2) and ( 3) is highly nontrivial for unbounded operators B and C.
As our first main result we prove that the closed-loop system has a well-defined mild state and output determined by bounded input/output maps.We achieve this result by expressing the timevarying closed-loop system as a nonautonomous output feedback of an autonomous regular linear system and by employing the nonautonomous feedback theory developed by Schnaubelt in [29].Besides output regulation, the well-posedness result in Section III is also applicable in the study of other control problems with nonautonomous controllers.
In Section IV we introduce general theory for output regulation in the situation where the controller parameters (G 1 (t), G 2 (t), K(t))-especially the internal model-converge to a limit (G ∞ 1 , G ∞ 2 , K ∞ ) as t → ∞.As our main result we show that if the autonomous "limit controller" (G ∞ 1 , G ∞ 2 , K ∞ ) contains an internal model [14], [26] of the true frequencies (ω k ) k and the closed-loop system is exponentially stable, then the controller achieves output regulation.
In Section V, we introduce our controller for output regulation of signals (1) with unknown frequencies.We begin by introducing a general controller structure with a time-varying internal model and an observer part for closed-loop stabilization.The controller also includes an auxiliary output y aux (t) for estimation of the frequencies (ω k ) k in (1).One of the key features of our controller is that y aux (t) is by design independent of the timevarying parts of the controller, and therefore the convergence of the frequency estimates in the internal model can be completed separately of the analysis of the closed-loop dynamics.In particular, our controller is not restricted to a single estimation method, but can instead be combined with any method, which can identify (ω k ) k based on the output y aux (t).As the final part of the controller design we present an online tuning algorithm for the stabilization of the nonautonomous closed-loop system and for guaranteeing the output tracking.In Section V-C, we analyze the robustness properties of the controller with respect to perturbations in the system (A, B, C, D).The detailed robustness properties depend on the chosen frequency estimation method due to the effect of the perturbations on y aux (t).Our main result shows that for sufficiently long update intervals our controller achieves approximate output regulation despite small perturbations provided that the frequency estimates approximate the true frequencies (ω k ) k with sufficient accuracy.We illustrate the controller design in Section VI in adaptive output regulation for a boundary controlled heat equation with uncertainty.
The main difference compared to [2], [15], [28], which have studied control of individual PDE models is that our results are applicable for abstract regular linear systems and various PDEs within this class.Moreover, in [2] (ω k ) q k=1 are estimated from the tracking error signal.Our use of the auxiliary output y aux (t) is inspired by the "residual generator" in [8,Sec. 4] (a similar signal is used in [15]).A preliminary version of Theorem 4.2 was presented in [1].
Notation: If X and Y are Hilbert spaces, then the space of bounded linear operators A : X → Y is denoted by L(X, Y ).The domain, kernel, and range of A : D(A) ⊂ X → Y are denoted by D(A), N (A), and R(A), respectively.The resolvent operator of A : D(A) ⊂ X → X is defined as R(λ, A) = (λI − A) −1 for those λ ∈ C for which the inverse is bounded.The inner product on X is denoted by •, • X .By L p (0, τ; X) and L ∞ (0, τ; X) we denote, respectively, the spaces of p-integrable and essentially bounded measurable functions f : (0, τ) → X.
generates a strongly continuous semigroup T (t) on X, we define X 1 = D(A) equipped with the graph norm of A and define X −1 as the completion of X with respect to the norm x −1 := (λ 0 − A) −1 x X for a fixed λ 0 ∈ ρ(A).Then, A extends to an operator X → X −1 (also denoted by A) and this extension generates a semigroup [also denoted by T (t)] on X −1 [34,Sec. 2.10].

A. Background on Regular Linear Systems
Let X, U , and Y be Hilbert spaces and consider on X, where the operator A : D(A) ⊂ X → X is assumed to generate a strongly continuous semigroup T (t) on X and The operator B is an admissible input operator for the semigroup [35,Sec. 3].
Assumption 2.1: For some Hilbert spaces X, U , and Y the operators (A, B, C) have the following properties.
1) 4) is a regular linear system by [35,Thm. 5.6] and its transfer function is given by P (λ) = C Λ R(λ, A)B + D. In this situation we write "(A, B, C, D) is a regular linear system."

B. Assumptions on the System and the Controller
We consider a regular linear system of the form (2) on a Hilbert space X, where x(t) ∈ X, u(t) ∈ C m , y(t) ∈ C p , and w dist (t) ∈ C n d are the system's state, input, output, and external disturbance, respectively.In particular, the number of outputs of the system is p ∈ N, and and D d ∈ C p×n d are allowed to be unknown.We assume that ) is a regular linear system.The transfer function of (2) (from u to y) is denoted by We make the following assumptions on the parameters of the dynamic error feedback controller (3).
Assumption 2.2: For almost every t ≥ 0 we have for some Hilbert space U c is an admissible input operator for this semigroup, and In Section V, the controller will contain additional dynamics for estimation of (ω k ) k in (1), but in our control scheme the convergence of the frequency estimates is analyzed separately.
We can formally express the closed-loop system consisting of ( 2) and (3) with state x e (t) = [x(t), z(t)] T ∈ X e := X × Z

and input w
where The existence of well-defined mild state x e (t) and output e(t) of ( 5) are proved in Section III.

III. WELL-POSEDNESS OF THE CLOSED-LOOP SYSTEM
In this section we will prove that the closed-loop system is well-posed in the sense that for the initial state x e0 = [x 0 , z 0 ] T ∈ X e and for w e ∈ L 2 loc (0, ∞; C n d +p ) the ( 5) have a well-defined mild state and output x e (t) = U e (t, 0)x e0 + Φ We prove the closed-loop well-posedness using the nonautonomous feedback theory in [29,Sec. 4].More precisely, we will express the system (5) as a part of a system obtained from an autonomous regular linear system under a combination of (see Fig. 1): 1) autonomous output feedback with feedback operator Δ 0 ; 2) parallel interconnection with a feedthrough operator D add e ; 3) nonautonomous feedback with feedback operators Δ(t).
is a regular linear system with input space U ee and output space Y ee .The operator The results in [40] and [32,Sec. 7.5] imply that first applying output feedback with operator Δ 0 and subsequently adding a parallel connection with the (constant) transfer function D add eo produces a regular linear system where We denote by T e (t) the strongly continuous semigroup generated by A ∞ e and by F ∞ ee the extended input-output map of and define P out = I, 0, 0 ∈ L(Y ee , Y ).A direct computation shows that for a.e.t ≥ 0 the operator I − D ∞ ee Δ(t) ∈ L(Y ee ) is boundedly invertible and where A e (t) is as in (5).This identity confirms that A e (t) are (at this stage formally) associated to the system obtained from with the nonautonomous feedback Δ(t).
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Proof: The input-output map We have ) and the inverse is uniformly bounded with respect to s ≥ 0 and 0 < t 0 ≤ 1.Since K ∞ and I in C ∞ eo are bounded operators, it is easy to verify that the restrictions of the input-output maps F c3 , F c4 , F c5 , and and Δ K are essentially bounded the L(L 2 (s, s + t 0 ; Y ee ))-operator norm of the second term of I − F ∞ ee Δ(•) converges to zero as t 0 → 0 uniformly with respect to s ≥ 0. Because of this, for a sufficiently small t 0 > 0 the operators I − F ∞ ee Δ(•) ∈ L(L 2 (s, s + t 0 ; Y ee )) for s ≥ 0 have uniformly bounded inverses L 2 (s, s + t 0 ; Y ee ).By [29,Lem. 4.2] the same is then true for all t 0 > 0.
We define The following theorem shows that the closed-loop system (5) has a well-defined strongly continuous evolution family U e (t, s) and an input map Φ t,s e defined in (7a).Moreover, since a direct computation shows that B ee (t)P in = B e (t) and P out C ee (t) = C e (t) for a.e.t ≥ 0, the mappings defined in (7) and in the following theorem satisfy Ψ s e = P out Ψ s ee and F s e = P out F s ee .Because of this, Theorem 3.2 implies that for x 0 ∈ X, z 0 ∈ Z, w dist (t), and y ref (t), the closed-loop system (5) has a well-defined mild state x e (t) and output e(•) ∈ L 2 loc (0, ∞; C p ) determined by (6).The integral equation associates (A e (t)) t≥0 to the evolution family U e (t, s).
Theorem 3.2: Let Assumption 2.2 hold and let Δ(•) be as in (10).There exists a strongly continuous evolution family U e (t, s) such that for all x ∈ X e and s ≥ 0 x for every t 0 > 0 and some γ(t 0 ) > 0 (depending only on t 0 > 0), and U e (t, s)x = T e (t − s)x , and C ee (•) defined in (11) are "admissible input and output operators for the evolution family (T e (t − s)) t≥s≥0 " in the sense of [29,Def. 3.3 s, t; U ee ) and x ∈ X e , then (T e , Φ t,s 0 , Ψ s 0 , F s 0 ) t≥s≥0 is a well-posed nonautonomous system by [29,Lem. 3.9].
We will now show that ) is an admissible feedback for (T e , Φ t,s 0 , Ψ s 0 , F s 0 ) t≥s≥0 .By construction, the system (T e , Φ t,s 0 , Ψ s 0 , F s 0 ) t≥s≥0 has the properties in the first part of [29,Thm. 3.11] u ∈ L2 loc (s, ∞; Y ee ) for all s ≥ 0 and u ∈ L 2 loc (s, ∞; U ee ).Because of this, the proof of [29,Thm. 4.4(a)] shows that there exists a strongly continuous evolution family U e (t, s) which satisfies the integral equation in the claim, and loc (s, ∞; Y ee ) for all x ∈ X e and s ≥ 0. The proof of [29,Thm. 4.4(a)] also shows that C ee (t) = (I − D ∞ ee Δ(t)) −1 C ∞ ee , a.e.t ≥ 0, are admissible observation operators for U e (t, s).If Ψ s ee is defined as in the claim, then by [29, Lem.2.5] the pair (U e , Ψ s ee ) is a "nonautonomous observation system" in the sense of [29,Def. 2.2].Since B e (t) ∈ L(C n d +p , X e ) for a.e.t ≥ 0, Φ t,s e can be defined as in (7a), and (U e , Φ t,s e ) is a "nonautonomous control system" in the sense of [29,Def. 3 , the properties of (U e , Ψ s e ) and [29,Prop. 2.11] also imply that there exist κ, t 1 > 0 such that for all s ≥ 0 and w e ∈ L 2 loc (0, ∞; C n d +p ) we have Φ t,s e w e ∈ D(C ee (t)) for a.e.t ≥ s and e w e L 2 (s,s+t 1 ) ≤ κ w e L 2 (s,s+t 1 ) .Thus, [29,Lem. 3.9] implies that (U e , Φ t,s e , Ψ s ee , F s ee,0 ) t≥s≥0 with F s ee,0 := C e (•)Φ •,s e is a well-posed nonautonomous system, and since F s ee w e = F s ee,0 w e + D ∞ ee P in w e , the same is finally true also for (U e , Φ t,s e , Ψ s ee , F s ee ) t≥s≥0 .In particular, Φ t,s e , Ψ s ee , F s ee have the boundedness properties in the claim.Finally, we will show that ) −1 and denoting u = P in w e for brevity we get e , Ψ s ee , F s ee ) t≥s≥0 is a well-posed nonautonomous system by Theorem 3.2.Thus, for every w e ∈ L 2 loc (0, ∞; C n d +p ) the closed-loop state x e (t) in ( 6) satisfies x e (•) ∈ C([0, ∞); X e ) by [29, Def.2.1 and Prop.3.5 (2)].Since 1 The identity (4.13) does not require "absolute regularity" and it extends to L 2 loc (0, ∞; X e ) since (U e , Ψ s ee ) is a nonautonomous observation system.
Therefore, the properties of regular linear systems imply that if w dist (t) is as in (1), then (2) has a well-defined mild state x(t) satisfying x(t) ∈ D(C Λ ) for a.e.t ≥ 0, and the output y(t) is determined by (2b) for a.e.t ≥ 0.

IV. REGULATION WITH CONVERGING CONTROLLERS
In this section we introduce general results on output regulation with a nonautonomous controller (G 1 (t), G 2 (t), K(t)) satisfying Assumption 2.2.Our first main result in Theorem 4.2 is applicable in the situation where the controller parameters have well-defined asymptotic limits in the sense that 2 Our second main result in Theorem 4.4 considers a more general situation where the abovementioned norms become small as t → ∞ but do not necessarily converge to zero.The main condition in our results is that the part ≥ p for all k ∈ {0, . . ., q}, where p ∈ N is the number of outputs of (2).
Our first result states that if the controller parameters converge, if G ∞ 1 has an internal model of the frequencies of y ref (t) and w dist (t) and if the closed-loop system is exponentially stable, then the controller achieves output regulation.Exponential stability of U e (t, s) means that there exists M, α > 0 such that U e (t, s) ≤ Me −α(t−s) for t ≥ s ≥ 0. Theorem 4.2: Assume y ref (t) and w dist (t) in (1) and the initial states x 0 ∈ X and z 0 ∈ Z are such that there exist G 1 (•), G 2 (•) and K(•) satisfying Assumption 2.2, and for some 1 has an internal model of (ω k ) q k=0 in (1), then If ess sup t≥0 e αt δ G (t) < ∞ for some α > 0, then there exists In Theorem 4.2 the controller parameters and their limits are allowed to depend on the initial states of the system and the controller and of w dist (t) and y ref (t).This possibility is motivated by our controller design for output regulation with unknown frequencies in Section V.If (G 1 (t), G 2 (t), K(t)) are independent of the initial states and w dist (t) and y ref (t), the claims of Theorem 4.2 (and Theorem 4.4) hold for all x 0 ∈ X, z 0 ∈ Z, w dist (t), and y ref (t).
The proof of Theorem 4.2 utilizes the feedback structure introduced in Section III.To this end we use the notation in Section III and in particular denote The "if"-part of the following lemma also follows from [29, Thm.5.6] (see also [10,Sec. 4]).
Moreover, Theorem 3.2 implies that for every t 0 > 0 there exists γ(t 0 ) > 0 such that sup s≥0 C ∞ ee U e (s + •, s)x L 2 (s,s+t 0 ) ≤ γ(t 0 ) x for all x ∈ X e and s ≥ 0. Because of this, the integral equation in Theorem 3.2 together with the admissibility of B ∞ ee for the semigroup T e (t) imply that for any fixed t 0 > 0 there exists A direct computation using (9) shows that this system has the form of the closed-loop system in [24, Sec.II] corresponding to the reg- (13c) As shown in the proof of Theorem 3.2, F s ee is an input-output map of a nonautonomous well-posed system with an exponentially stable evolution family U e (t, s).Thus, Lemma A.1(a) implies sup t≥0 F 0 ee w e L 2 (t,t+1) < ∞ and ee is exponentially stable.Thus, Lemma A.1(c) applied to this autonomous system and u = Δ(•)F 0 ee w e together with (13) show that F 0 e w e − F ∞ e w e L 2 (t,t+1) → 0 as t → ∞.This completes the proof of (12).
Finally, let α > 0 be such that ess sup Exponential stability of U e (t, s) and Assumption 2.2 imply sup t≥0 Φ e (t, 0)w e < ∞ for all w e ∈ L ∞ (0, ∞; C n d +p ) and thus x e (t) in Theorem 4.2 satisfies sup t≥0 x e (t) < ∞.
The following result generalizes Theorem 4.2 to the situation where the parameters of the controller do not necessarily converge as t → ∞, or where the limit of G 1 (t) has an internal model of frequencies which are only close to (ω k ) q k=0 .In these cases the asymptotic tracking error will be small provided that the asymptotic error in the frequencies is sufficiently small.Theorem 4.4: Assume that x 0 ∈ X, z 0 ∈ Z, y ref (t), and w dist (t) in (1) are such that there exist G 1 (•), G 2 (•), and K(•) satisfying Assumption 2.2 and U e (t, s) is exponentially stable.Moreover, assume G ∞  1 has an internal model of Proof: As shown in the proof of Theorem 4.2, we have e(•) = e 0 (•) + (F 0 e w e − F ∞ e w e ), where w e (t) = [w dist (t) T , y ref (t) T ] T and t → e α 0 t e 0 (t) ∈ L 2 (0, ∞) for some α 0 > 0.Moreover, the identity (13) and Lemma A.1(b) imply that there exists where Since F ∞ ee is the extended input-output map of the regular linear system Since the definition of Δ(t) in (10) By Theorem 3.2, (U e , Φ t,s e , Ψ s ee , F s ee ) t≥s≥0 is an exponentially stable nonautonomous well-posed system.We have from [29,Def. 3.6] that for a.e.t ≥ t 0 (F 0 ee w e )(t) = (Ψ t 0 ee Φ t 0 ,0 e w e )(t) + (F 0 ee w t 0 )(t) where and the properties with convergence in L 2 loc (0, ∞; Y ee ).The choice of M 0 > 0 and w e L 2 (t,t+1) ≤ w e ∞ for t ≥ 0 imply that for all n ∈ N sup t≥0 Thus, (15) implies sup t≥0 F 0 ee w t 0 L 2 (t,t+1) ≤ 2M 0 w e ∞ .Since Ψ t 0 ee Φ t 0 ,0 e w e L 2 (t,t+1) → 0 as t → ∞ and since V. CONTROLLER DESIGN FOR OUTPUT REGULATION WITH UNKNOWN FREQUENCIES In this section we introduce a controller for output regulation of y ref (t) and w dist (t) with unknown frequencies.Our controller contains a time-varying internal model of estimates (ω k (t)) q k=1 of (ω k ) q k=1 in ( 1) and an observer-based part for achieving closed-loop stability.The frequency estimates are formed based on an auxiliary output y aux (t) of the controller which contains the information on (ω k ) k but is by design independent of the time-varying parts of the controller.Therefore, our controller can be combined with any estimation method, which can asymptotically estimate the frequencies (ω k ) k from y aux (t).We solve the problem under Assumptions 5.1 and 5.3.
Assumption 5.1: There exist K ∈ L(X, C m ) and L ∈ L(C p , X) such that the semigroups generated by A + B K : Assumption 5.3: Assume y ref (t) and w dist (t) are of the form (1) with 0 = ω 0 < ω 1 < . . .< ω q and (A, B, C, D) does not have transmission zeros at {0} ∪ {±iω k } q k=1 .We begin by introducing a general controller structure in Section V-A and present general conditions for output regulation in the situation where (ω k (t)) q k=1 converge to (ω k ) k in (1).We analyze the structure of y aux (t) in Lemma 5.6 and in Remark 5.7 we list selected methods which can be used to estimate the frequencies based on y aux (t).In Section V-B we present our controller tuning algorithm for constructing the controller parameters in order to achieve closed-loop stability and output regulation under Assumptions 5.1 and 5.3.The tuning is based completely on design of autonomous feedback and output injections.Finally, in Section V-C we analyze the robustness properties of our controller.

A. Controller With a Time-Varying Internal Model
Our error feedback controller has the form where e(t) = y(t) − y ref (t) is the regulation error.The controller structure in Definition 5.4 generalizes the autonomous Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
robust controller in [17,Sec. 7] and the adaptive internal modelbased controller scheme in [8,Sec. 4], where a separate "residual generator" system was used to construct y aux (t).Definition 5.4: The controller (G , where 0 p , I p ∈ R p×p are the zero and identity matrices and The function G 1 (•) is the time-varying internal model, which contains the estimates (ω k (t)) q k=1 of the frequencies in w dist (t) and y ref (t).By construction, for every t ≥ 0 the pair (G For any The feedback theory for regular linear systems in [40] im- as t → ∞ for all k and for some K ∞ ∈ L(Z, C m ).If the semigroup generated by a direct computation shows that By assumption the semigroups generated by A + LC : D(A) ⊂ X → X and A s + B s K ∞ are exponentially stable.Moreover, B s is an admissible input operator for the semigroup generated by A s + B s K ∞ by the results in [40,Sec. 7].Thus, the semigroup generated by e is exponentially stable and similarity implies the same for T e (t).The claims now follow from Theorem 4.2 with Lemma 4.3 and Remark 4.6.
We conclude this section by analysing y aux (t).Lemma 5.6 in particular shows that y aux (t) is independent of the time-varying parameters (ω k (t)) q k=1 and K(t).The form of y aux (t) involves B dL = [B d + LD d , −L] and the transfer function of the regular linear system (A + LC, B dL , C, [D d , −I]).Lemma 5.6: Let x 0 ∈ X and z 0 = (z 10 , z 20 ) ∈ Z and let y ref (t) and w dist (t) be as in (1).Consider the controller (G 1 (t), G 2 , K(t)) in Definition 5.4, let T L (t) be the semigroup generated by A + LC and denote ω −k := −ω k for k ∈ Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.{1, . . ., q}.Then, for a.e.t ≥ 0 y aux (t) = y 0 (t) + q k=−q e iω k t P tot,L (iω k )c k e (22) with for k ∈ {1, . . ., q}, and We have t → e αt y 0 (t) ∈ L 2 (0, ∞; C p ) for some α > 0.Moreover, if C ∈ L(X, C p ), if x 0 − 20 ∈ D(A), or if A generates an analytic semigroup, then y 0 (•) is continuous and e αt y 0 (t) → 0 as t → ∞ for some α > 0. Proof: Since Assumption 2.2 holds the closed-loop state x e (t) = [x(t), z 1 (t), z 2 (t)] T and the regulation error e(t) in ( 6) are welldefined by Theorem 3.2.With B s ∈ L(U, Z 0 × X −1 ) in (19) and Q e in (20) we have has a block triangular form for all t ≥ s ≥ 0. Applying the similarity transformation Q e in (20) to (6a) and (7a), therefore, shows that with w e (t) = [w dist (t) for a.e.t ≥ 0. Thus, y aux (t) is the output of the regular linear system (A L , B dL , C, [D d , −I]) with initial state x(0) = x 0 − z 20 ∈ X and input w e (t).When x(0) = 0 and w e (t) = e iω k t w 0 for some w 0 ∈ C n d +p , [32, Cor.4.6.13]implies y aux (t) = e iω k t P tot (iω k )w 0 − C Λ T L (t)R(iω k , A L )B dL w 0 .Finally, linearity implies that for x(0) = x 0 − z 20 and w e (t) = [w dist (t) T , y ref (t) T ] T the output y aux (t) has the form in (22) with the given {c k e } q k=−q and y 0 (t).Since C is admissible with respect to the exponentially stable semigroup T L (t), we have t → e αt y 0 (t) ∈ L 2 (0, ∞; C p ) for some α > 0.
In the last claim, if C ∈ L(X, C p ), then pointwise convergence of y 0 (t) follows directly from stability of T L (t).In the other cases Remark 5.7 (Methods for frequency estimation): Multifrequency estimators based on dynamical adaptive observers have been developed by several authors in, e.g., [3], [7], [9], [19], [23], [36], [37], [41]. 3Our controller requires frequency estimation in the presence of the nonsmooth decaying part y 0 (t) of y aux (t) (i.e., the estimator is required to be input-to-state stable).One such adaptive estimator was introduced in [7] (see [7,Rem. 3]).The estimator in [7] is compatible with our control scheme and it only requires knowing the number q of frequencies and nonzero amplitudes.Several estimators are also capable of online estimation of q and have desirable transient performance and robustness properties [3], [8], [9].
Estimation of (ω k ) q k=1 from y aux (t) requires that all frequencies appear in the nondecaying part of y aux (t).This is generically true since the amplitudes corresponding to ±ω k 0 are zero only if a k 0 , b k 0 , c k 0 , and d k 0 in (1) are related in a very specific way through the identity P tot (±iω k 0 )c ±k 0 e = 0.

B. Controller Tuning Algorithm
In this section we introduce an algorithm for constructing (ω k (•)) q k=1 and K(•) in the controller.Even though several estimators provide continuous-time estimates of (ω k ) q k=1 , we choose the estimates (ω k (•)) q k=1 in the internal model to be piecewise constant functions, which are updated at predefined time instances 0 = t 0 < t 1 < t 2 < • • • (via sample-and-hold).This way we can guarantee stable closed-loop behavior during the update intervals [t j , t j+1 ] despite possible rapid changes in the frequency estimates.The algorithm utilizes the estimate admissibility condition (EAC) defined in the following.
Definition 5.8: Let ε f > 0 and M f > 0. We say that (ω k ) q k=1 satisfy the EAC(M f , ε f ) for the system (A, B, C, D) if the following hold.1) 3) |iω k ± iλ| ≥ ε f for every transmission zero iλ ∈ iR of (A, B, C, D) and for all k.The algorithm uses Theorem 5.9 in the following to stabilize the pairs (A s (t j ), B s ), where: with G 1 (t) = diag(0 p , ω1 (t)Ω p , . . ., ωq (t)Ω p ).The result guarantees that the stabilizing feedback gains K j are a priori bounded and the stabilized semigroups satisfy uniform decay estimates with M s , α s > 0 independent of t j .Moreover, in this method the stabilizing gain of the infinite-dimensional pair (A, B) does not need to be recomputed when the frequencies in G 1 (t) are updated.The assumptions of the theorem will be guaranteed by our tuning algorithm.A similar method has been previously used in [17] and [24] (also [22,Thm. 3.7]) for internal 3 While many estimators are introduced only for scalar-valued signals, they can also be used if p > 1 by replacing y aux (t) with r T y aux (t) where r ∈ R p is a fixed random vector.The randomness of r guarantees the presence of all frequency components in r T y aux (t) with probability 1.
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models with fixed frequencies.The result uses notation where R K (λ) = R(λ, A + BK 21 ), C K = C Λ + DK 21 , and . . .2q+1) where Π 1j ∈ L(C p(2q+1) ) is the unique nonnegative solution of for some M K > 0 independent of t j .Moreover, the semigroup T j s (t) generated by A s (t j ) + B s K j is exponentially stable so that for any 0 < α s < min{r, −ω 0 (T K (t))} there exists M s , M B > 0 (independent of t j ) such that , and H j .
The proof of Theorem 5.9 is presented in the Appendix.In the tuning algorithm we denote by 0 < μ 1 (t) < • • • < μ q (t) the estimated frequencies computed based the signal y aux (t) by the separate frequency estimator.We make the following assumptions on the parameters of the algorithm.
3) The frequency overlap parameter ε f > 0 is suitably small and the upper bound M f > 0 for the frequencies is suitably large.4) The operator K 21 ∈ L(X, C m ) is such that the semigroup generated by A + BK 21 is exponentially stable.5) The initial frequency estimates (μ k (0)) q k=1 satisfy EAC(M f , ε f ) for (A, B, C, D).The controller tuning algorithm in the following constructs r > 0, and K 21 as in Assumption 5.10.Set j = 0.
Step 2 (Frequency update).Set ωk (t as in Theorem 5.9 and set K(t) ≡ K j for t ∈ [t j , t j+1 ).Increment j to j + 1 and go to Step 1.
Since the initial frequency estimates (μ k (0)) q k=1 are assumed to satisfy EAC(M f , ε f ), the tuning algorithm will proceed to Step 2 when j = 0, and therefore G 1 (•) and K(•) are welldefined on [0, ∞).Our main result in the following shows that if the estimates (μ k (t)) k converge to the true frequencies (ω k ) k in (1), then the controller constructed with the abovementioned algorithm achieves output regulation of y ref (t) and w dist (t).
The controller satisfies for all k and t ≥ 0 by construction and K j ≤ M K for all j ∈ N 0 by Theorem 5.9.Thus, for any frequency estimates (μ k (•)) k .Thus, by Lemma 5.6, y aux (t) has the form (22) and is independent of G 1 (•) and K(•).Assume now that x 0 ∈ X, z 0 ∈ Z, w dist (t), and k=1 satisfy EAC(M f , ε f ) for all j ≥ N .Thus, the algorithm will go to Step 2 for all j ≥ N , and the frequency estimates of the internal model satisfy ωk (t) → ω k as t → ∞ for all k.By Theorem 5.9 we have , and H j .The claims will follow from Theorem 5.5 provided that the semigroup generated by satisfy EAC(M f , ε f ) by assumption, the stability of this Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
semigroup follows directly from Theorem 5.9 when we replace , and H j .
The following table summarizes the parameters of the controller (17).
The following lemma shows that for sufficiently large sampling intervals the evolution family U e (t, s) is always exponentially stable (independently of the behavior of the frequency estimates (μ k (t)) k ).Note that the required size of τ 1 > 0 depends on the other tuning parameters in Assumption 5.10.
Lemma 5.12: Let Assumptions 5.1 and 5.10 hold.There exists τ 1 > 0 such that if t j − t j−1 ≥ τ 1 for all j ∈ N in the controller tuning algorithm, then there exist M e , α e > 0 such that U e (t, s) ≤ M e e −α e (t−s) for all t ≥ s ≥ 0 and for any y ref (t) and w dist (t) in (1) and for any x 0 ∈ X and z 0 ∈ Z.
Proof: Since G 1 (•) and K(•) are piecewise constant we have A e (t) ≡ A e (t j ) for all t ∈ [t j , t j+1 ) and j ∈ N 0 .Thus, if we denote by T j e (t) the semigroup generated by A e (t j ), then for all t ≥ s ≥ 0 we have U e (t, s) = T j e (t − s) if t, s ∈ [t j , t j+1 ) for some j ∈ N 0 , and otherwise s) where j, ∈ N 0 are such that s ∈ [t , t +1 ) and t ∈ [t j , t j+1 ).Since 0 < τ 1 ≤ t j+1 − t j ≤ τ 2 for all j ≥ 0 by assumption, the evolution family U e (t, s) is exponentially stable provided that there exists M e0 ≥ 0 such that T j e (t) ≤ M e0 for all t ≥ 0 and j ∈ N 0 and sup j≥0 T j e (t j+1 − t j ) < 1. Theorem 5.9 and the Hille-Yosida theorem imply the existence of M s , α s , M B , M K > 0 such that for K j in the controller tuning algorithm we have R(λ, A s (t and K j ≤ M K for all λ ∈ C + and j ∈ N 0 Using the similarity transform Q e in (20) we have [similarly as in (21)] The similarity, the triangular structure of Q e A e (t j )Q −1 e and the norm estimates previously imply that there exists M R > 0 such that sup λ∈C + R(λ, A e (t j )) ≤ M R for all j ≥ 0. By the Gearhart-Prüss-Greiner theorem [13, Thm.V.1.11]there exist M e0 , α e0 > 0 such that T j e (t) generated by A e (t j ) satisfy T j e (t) ≤ M e0 e −α e0 t for all t ≥ 0 and j ∈ N 0 (this uniform bound can be deduced, e.g., by applying [13, Thm.V. 1.11] to the semigroup diag(T 0 e (t), T 1 e (t), . ..) on the Hilbert space 2 (X e )).This further implies that if we choose τ 1 > 0 such that M e0 e −α e0 τ 1 < 1, then also T j e (t j+1 − t j ) ≤ M e0 e −α e0 τ 1 < 1 and U e (t, s) is exponentially stable.Since M e0 and α e0 do not depend on x 0 , z 0 , w dist (t), and y ref (t), we can choose M e , α e > 0 as in the claim.

C. Robustness Analysis
We conclude this section by analysing the robustness properties of the controller constructed in the controller tuning algorithm.The robustness properties depend on the chosen frequency estimation method-especially on its capability of handling small persistent errors in y aux (t)-but we can nevertheless present a general result for robustness analysis.Throughout the section we assume that Assumptions 5.1, 5.3, and 5.10 are satisfied.We consider a perturbed regular linear system ( Ã, B, C, D) with parameters , and δ D ∈ C p×m .We do not need to consider perturbations in B d and D d since these parameters were allowed to be unknown.We begin by describing the effects of the perturbations on y aux (t).Lemma 5.13: Consider the perturbed system ( Ã, B, C, D) in ( 26) and the controller in Definition 5.4.Assume K(•) ∈ L ∞ (0, ∞; L(Z, C m )) and ωk (•) ∈ L ∞ (0, ∞) for all k are piecewise constant.Then, the auxiliary output ỹaux (t) corresponding to the perturbed system satisfies ỹaux (t) = y aux (t) + y pert (t) for a.e.t ≥ 0, where y aux (t) is as in Lemma 5.6 and (27) where δ AC = δ A + Lδ C , δ BD = δ B + Lδ D , and x e (t) is the state of the (perturbed) closed-loop system (5).
Proof: Since ( Ã, B, C, D) is a regular linear system, Theorem 3.2 implies that the closed-loop system consisting of the perturbed system and the controller in Definition 5.4 has a well-defined mild state x e (t).Denote by A e (t) and Ãe (t) the closed-loop system operators corresponding to the nominal system (A, B, C, D) and the perturbed system ( Ã, B, C, D), respectively.Then, D( Ãe (t)) = D(A e (t)) and Ãe (t) = A e (t) + δ e (t) for a.e.t ≥ 0 where If we apply the similarity transform Q e in (20) to the perturbed closed-loop system (5) with initial condition x(0) = x 0 − z 20 and with A L = A + LC.
We will now prove that x(t) is indeed a mild solution of (28).
Denote by U e (t, s) and Ũe (t, s) the evolution families in Theorem 3.2 corresponding to the nominal and perturbed systems, respectively.Let 0 = t 0 < t 1 < t 2 < • • • be such that A e (t) ≡ A e (t j ) and δ e (t) ≡ δ e (t j ) for t ∈ [t j , t j+1 ).If we denote by T j e (t) and T j e (t) the semigroups generated by A e (t j ) and Ãe (t j ), respectively, then U e (t, s) and Ũe (t, s) are of the form given in the proof of Lemma 5.12.Moreover, since Ãe (t j ) = A e (t j ) + δ e (t j ) for all j ∈ N 0 , the perturbation formula in [13, Cor.III.1.7]and a direct computation show that Ũe (t, s)x = U e (t, s)x + t s U e (t, r)δ e (r) Ũe (r, s)xdr (29) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
for all x ∈ X e and t ≥ s ≥ 0. Applying the similarity transformation Q e to ( 6) and (7a) and using the relationship (29) between Ũe (t, s) and U e (t, s) it is straightforward to confirm that x(t) is the mild solution of (28).Since x(t) ∈ D(C Λ ) for a.e.t ≥ 0, analogous arguments as in the proof of Lemma 5.6 show y aux (t) = C Λ x(t) + [D d , −I]w e (t).Comparing ( 23) and (28) shows that y aux (t) = y aux (t) + y pert (t) for a.e.t ≥ 0.
Our main result below shows that for sufficiently long sampling intervals in the controller tuning algorithm the effect of small perturbations on y aux (t) will be small.Moreover, if the frequencies can be estimated with a sufficiently small asymptotic error, then the controller achieves output tracking in an approximate sense, i.e., with a small asymptotic error.
There exist ε stab , M pert > 0 such that if then for all x 0 ∈ X, z 0 ∈ Z, w dist (t), and y ref (t) we have ỹaux (t) = y aux (t) + y pert (t), where y aux (t) is as in Lemma 5.6 and (1) satisfy EAC( Mf , εf ) with some εf > ε f and 0 < Mf < M f .For any ε err > 0 there exists δ err > 0 such that if the perturbations satisfy (30) and x 0 ∈ X, z 0 ∈ Z, w dist (t), and y ref (t) are such that (μ k (t)) q k=1 satisfy max for some τ 0 > 0, then lim sup Proof: Fix x 0 ∈ X, z 0 ∈ Z, w dist (t), and y ref (t) and let M K > 0 be as in Theorem 5.9.Then, G 1 (•) and K(•) constructed in the controller tuning algorithm satisfy and thus by Theorem 3.2 the closedloop system corresponding to the perturbed system ( Ã, B, C, D) has a well-defined evolution family Ũe (t, s) and state x e (t).We begin by introducing some notation.We denote by A e (t) and Ãe (t) the closed-loop operators for (A, B, C, D) and ( Ã, B, C, D), respectively.Moreover, we denote by T j e (t) and T j e (t) the semigroups generated by A e (t j ) and Ãe (t j ), respectively.For the proof of the last claim we additionally assume that (ω k ) k satisfy EAC( Mf , εf ).We then define G ∞ 1 , G ∞ 11 and Δ G 1 (t) as in (18) with as in Theorem 5.9.Finally, we denote by A ∞ e and Ã∞ e the operators in (9a) for (A, B, C, D) and ( Ã, B, C, D), respectively, and denote the semigroups generated by these two operators with T e (t) and Te (t), respectively.Note that G ∞ 1 , G ∞ 11 , G 2 , and K ∞ are independent of x 0 ∈ X, z 0 ∈ Z, y ref (t), and w dist (t).
If M e0 , α e0 > 0 are as in the proof of Lemma 5.12, we have T j e (t) ≤ M e0 e −α e0 t for all t ≥ 0 and j ∈ N 0 and the choice of τ 1 > 0 implies M e0 e −α e0 τ 1 < 1.By construction A ∞ e has the same structure as A e (t j ), j ∈ N 0 , with (ω k (t j )) k and K j replaced with (ω k ) k and K ∞ , respectively, and K ∞ in Theorem 5.9 is chosen similarly as K j .Under the additional assumption that (ω k ) k satisfy EAC( Mf , εf ), the frequencies (ω k ) k satisfy the assumptions of Theorem 5.9 with the same parameters as (ω k (t j )) k , j ∈ N 0 [both satisfy EAC(M f , ε f ) for (A, B, C, D)].Therefore, we can deduce as in the proof of Lemma 5.12 that also T e (t) ≤ M e0 e −α e0 t for all t ≥ 0.
Since Ãe (t) ≡ Ãe (t j ) for t ∈ [t j , t j+1 ) and j ∈ N 0 , the structure of Ũe (t, s) is analogous to that in the proof of Lemma 5.12.Because the abovementioned estimates for T j e (t) and Te (t j+1 − t j ) are uniform with respect to (δ A , δ B , δ C , δ D ) satisfying (30), we have (similarly as in the proof of Lemma 5.12) that there exist M e , α e > 0 such that Ũe (t, s) ≤ M e e −α e (t−s) for all t ≥ s ≥ 0 and for any (δ A , δ B , δ C , δ D ) for which (30) holds.
We will now prove the claims concerning y pert (t).If M e , α e > 0 are as previous, (6a) and (7a) and the structure of (constant) B e (t) ≡ B e ∈ L(C n d +p , X e ) imply that the state x e (t) ≤ M 2 ( x e0 + w e (•) ∞ ) for a constant M 2 ≥ 0 independent of x 0 , z 0 , w dist (t), and y ref (t).Lemma 5.13 shows that y pert (t) = (F L u)(t) for a.e.t ≥ 0, where F L is the input-output map of the exponentially stable regular linear system (A + LC, I, C, 0) and where u( for all τ ≥ 0 and for a constant M 3 ≥ 0 depending only on L , the first estimate for y pert (•) for some M pert ≥ 0 follows from Lemma A.1(a).If C ∈ L(X, C p ) and if M L , α L > 0 are such that T L (t) ≤ M L e −α L t for all t ≥ 0, then the second claim follows from (27) and a direct estimate: To prove the last claim we will apply Theorem 4.4 to the perturbed system ( Ã, B, C, D).Assume (ω k ) k satisfy EAC( Mf , εf ) and let ε stab > 0 be as previous.It is easy to see that the perturbations (26) lead to bounded perturbations of 9) with norm bounds depending on c δ , M K , and L .Moreover, by the choice of ε stab the perturbation Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

VI. ADAPTIVE REGULATION FOR A HEAT EQUATION
In this example we study a 1-D boundary controlled reactiondiffusion equation on ξ ∈ (0, 1) where v(ξ, t) describes temperature at the point ξ ∈ (0, 1) and at time t > 0. We construct a controller for output regulation of y ref (t) = 0.2 sin(0.5t+ 0.5) + 0.4 sin(6t + 0.5) and w dist (t) = [cos(1.5t+ 0.5), sin(0.5t+ 0.2), cos(1.5t− 0.4)] T , both assumed to be unknown.In the simulation we consider b d (ξ) = cos(3ξ) (this is not used in controller design).The stabilizing parameters are chosen as K 21 x = −2 1 0 x(ξ)dξ for x ∈ L 2 (0, 1), L ≡ −4 ∈ L 2 (0, 1).The system does not have transmission zeros on iR.We use ε f = 0.2 and M f = 30 in the EAC, and Fig. 2. Controlled heat equation with the estimator in [7].r = 0.2, R = 1 ∈ R, and Q = I ∈ R 3×3 .The frequencies will also not be updated if (μ k (t j )) 3 k=1 are complex or negative.Since G 1 (t) and K(t) are piecewise constant, the results in [25, Sec.III and V] imply that for t ∈ [t j , t j+1 ) the controller state , where K j 1 , H j , and the estimates (ω k (t j )) q k=1 in G 1 (t j ) are obtained from the tuning algorithm.Since the true frequencies satisfy EAC(0.3,10), by Theorem 5.12 this controller stabilizes the unperturbed closedloop system (without the unmodeled reaction term) for all sufficiently long update intervals.
In this example we assume that the number of nonzero frequencies q = 3 is known and use the adaptive estimator from [7] with parameters "γ 1 = 0.005," "γ 2 = 10," and "{k i } i " being the coefficients of the Hurwitz polynomial (λ + 2) 2•3−1 .The input-to-state stability of the estimator and Theorem 5.14 shows that for sufficiently long update intervals the controller achieves closed-loop stability and approximate output tracking for any reaction term with a small r(•) L 2 .In the case r(ξ) ≡ 0, closed-loop stability and perfect output tracking follow from Theorem 5.11.Fig. 2 shows the behavior of the frequency estimates and the regulation error for the reaction profile r(ξ) = 1.5 sin(0.5πξ), the update sequence t j = 6j, j ∈ N 0 , and initial states x 0 (•, 0) ≡ 0 and z 0 = 0 ∈ Z.The simulations are implemented using finite difference with 100 points on [0,1].Initial frequency estimates are chosen as μ k (0) = k ∈ R for k ∈ {1, 2, 3}.Our controller can be compared to the ODE-PDE controller in [15], which similarly includes an adaptive estimator, but has continuously time-varying parameters and different general structure.
Lemma A.2: Let (A, B, C, D) be a regular linear system.Assume that there exist M, α > 0 such that the semigroup T (t) generated by A satisfies T (t) ≤ Me −αt for t ≥ 0. If we denote the extended input, output and input-output maps of a perturbed system ( Ã, B, C, D) by Φ, Ψ, and F , respectively, then for any ε ∈ (0, α/M) and κ > 0 we have sup Since δ A ≤ ε and δ B + δ C + δ D ≤ κ, the claim follows directly from (32).
The following corollary of the continuity of the solutions of Riccati equations is essential for the proof of Theorem 5.9.To the best of our knowledge, this result is new.
Lemma A.3: Let r > 0, Q ∈ C n×n and R ∈ C m×m satisfy Q > 0 and R > 0. Let Ω ⊂ R q be a compact set and let δ → A δ : Ω → C n×n and δ → B δ : Ω → C n×m be continuous functions such that the pair (A δ , B δ ) is controllable for all δ ∈ Ω.If we define K δ = −R −1 B * δ Π δ , δ ∈ Ω, where Π δ ∈ C n×n are the unique nonnegative solutions of δ Π δ = −Q then there exist M, M K > 0 such that K δ ≤ M K and e (A δ +B δ K δ )t ≤ Me −rt for all t ≥ 0 and δ ∈ Ω.
Proof: The claim is trivially true if Ω is empty.Let δ ∈ Ω.The assumptions imply that Π δ exists and is unique, and for all x ∈ C n we have 2 Re (rI Moreover, under our assumptions Π δ is positive definite.Therefore, rI + A δ + B δ K δ is dissipative with respect to the inner product •, • δ := •, Π δ • C n on C n .Thus, if we define x δ := Π 1/2 δ x C n for x ∈ C n , then e (rI+A δ +B δ K δ )t x δ ≤ x δ for all t ≥ 0. The definition of • δ now implies that e (A δ +B δ K δ )t ≤ Π 1/2 δ Π −1/2 δ e −rt for all t ≥ 0.
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received 22 December 2022; revised 27 April 2023; accepted 18 September 2023.Date of publication 13 October 2023; date of current version 28 June 2024.This work was supported by the Academy of Finland under Grant 298182, Grant 310489, and Grant 349002 held by L. Paunonen.Recommended by Associate Editor Y. Le Gorrec.(Corresponding author: Lassi Paunonen.)
Applying the identity (4.13) in the proof of [29, Thm.4.4(b)] 1 to these two systems and f = B e (•)w e ∈ L 2 loc (0, ∞; X e ) and using B e (•) = B ee (•)P in we get F s ee w e − D ∞ ee P in w e = F s ee,0 w e = C ee (•) • s U e (•, r)B e (r)w e (r)dr

Remark 4 . 5 :
The proof ofTheorem 4.4  shows that M err , δ 0 > 0 depend on the norm F ∞ ee of the extended input-output map of the autonomous system (A ∞ e , B ∞ ee , C ∞ ee , D ∞ ee ) and on M 0 , M 1 > 0 in Lemma A.1(a)-(b) corresponding to this system.Moreover, the proof of Lemma A.1, implies that M 0 , M 1 > 0 are determined by constants M e , α e > 0 such that T e (t) ≤ M e e −α e t for t ≥ 0 and by upper bounds for the norms of the input, output, and input-output map of(A ∞ e , B ∞ ee , C ∞ ee , D ∞ ee )on the time interval [0,1].Remark 4.6: By [24, Thm.7] the internal model of G ∞ 1 in Theorems 4.2 and 4.4 can be replaced with the conditions
[29,2.4],where T e (t) is the semigroup generated by A ∞ e .For t ≥ s ≥ 0 and u ∈ L 2 (s, t; U ee ) we define the mapping K s as in[29, Def.3.3] by s) is exponentially stable if and only if the semigroup T e (t) generated by A ∞ e is exponentially stable.Proof: In the notation of Section III, a direct computation shows M D := ess sup s≥0 ) is the regulation error corresponding to zero initial states of the system and the controller and the signals w dist (t) and y ref (t).Since U e (t, s) is exponentially stable, there exists α 2 > 0 such that t → e α 2 t (Ψ 0 (10)t) ∈ L 2 (0, ∞) for some α 1 > 0. e x e0 )(t) ∈ L 2 (0, ∞).It remains to analyze the term F 0e w e − F ∞ e w e .Note that sup t≥0 w e L 2 (t,t+1;C n d +p ) < ∞.For Δ(•) in(10)we have ess sup t≤s≤t+1 L ∞ (t,t+1) F 0 ee w e L 2 (t,t+1) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.lim sup t→∞ e(•) L 2 (t,t+1) ≤ lim sup t→∞ F 0 e w e − F ∞ e w e L 2 (t,t+1) ≤ M 1 lim sup t→∞ Δ(•) and U e (t, s) is exponentially stable.If ess sup t≥0 e αt |ω k (t) − ω k | < ∞ and ess sup t≥0 e αt K(t) − K ∞ < ∞ for some α > 0 and for all k, then there exists α e > 0 such that t → e α e t (y(t) − y ref (t)) ∈ L 2 (0, ∞; Y ).Proof: Let w dist (t), y ref (t), x 0 ∈ X, and z 0 ∈ Z be such that the assumptions hold.If we define G [24,hown in the proof of[24,  Thm.15], the pair (G ∞ 1 , G 2 ) satisfies the "G-conditions" (16).In view of Remark 4.6, the claims follow from Theorem 4.2 and Lemma 4.3 once we show that the semigroup T e (t) generated by A ∞ e is exponentially stable.The operator A ∞ e is exactly the operator A e (t) with G 1 (t) and K(t) replaced with G ∞ 1 and K ∞ , respectively, i.e., Definition 5.4.The condition EAC(M f , ε f ) in Definition 5.8 is used in Step 1 to detect if the frequency estimates nearly overlap or are close to the transmission zeros of (A, B, C, D).In both cases the closed-loop stabilization becomes difficult, and therefore the the algorithm does not update the frequencies of the internal model if EAC(M f , ε f ) is violated (Step 2 vs. Step 3).This update strategy guarantees that the assumptions of Theorem 5.9 are satisfied in Step 2.
and K(•) in the observer-based controller in B e w e (t) + Q e δ e (t)x e (t)whereQ e δ e (•)x e (•) ∈ L 2 loc (0, ∞; X e ) by Remark 3.3.The triangular structure of Q e A e (t)Q −1e therefore implies that x(t) : ).Since (ω k ) k satisfy EAC( Mf , εf ) with Mf < M f and εf > ε f , by choosing a sufficiently small δ err > 0 we can guarantee that if (31) holds, then(μ k (t j )) k satisfy EAC(M f , ε f ) for all j ∈ N such that t j ≥ τ 0 .Such choice guarantees that (ω k (t j )) k and K(t j ) are updated whenever t j ≥ τ 0 , and thus we also have max k |ω k (t) − ω k | ≤ δ err for all t ≥ τ 0 + τ 2 .Theorem 5.9 shows thatlim j→∞ K j = K ∞ if lim j→∞ max k |ω k (t) − ω k | = 0 and thus δ G (t) for t ∈ [t j , t j+1) can be made arbitrarily small by requiring that max k |ω k (t j ) − ω k | is small.For any ε err > 0 we can now combine the abovementioned properties to choose δ (31)> 0 [independent of x 0 , z 0 , w dist (t), and y ref (t)] such that if(31)holds for some τ 0 > 0, then lim sup t→∞ [11,boundary input u(t) acts at ξ = 0, the output y(t) is the temperature measurement at ξ = 1, and d 2 (t) and d 3 (t) are boundary disturbances.The reaction term with profile r(•) is unmodeled and we consider it as perturbation in the system.The disturbance input profile b d (ξ) is unknown.The system defines a regular linear system on X = L 2 (0, 1) with state x(t) = v(•, t).The full disturbance input is defined asw dist (t) = [d 1 (t), d 2 (t), d 3 (t)] T ∈ R 3 .Since the boundary disturbances d 2 (t) and d 3 (t) are smooth functions, we can apply a change of variables as in[11, Sec.10.1, Ex. 10.1.7]toexpress the heat equation as a regular linear system with bounded B d , D d ∈ R 1×3 and a modified initial state.