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Finite-dimensional observer-based PI regulation control of a reaction-diffusion equation

This paper investigates the output feedback setpoint regulation control of a reaction-diffusion equation by means of boundary control. The considered reaction-diffusion plant may be open-loop unstable. The proposed control strategy consists of the coupling of a finite-dimensional observer and a PI controller in order to achieve the boundary setpoint regulation control of various system outputs such as the Dirichlet and Neumann traces. In this context, it is shown that the order of the finite-dimensional observer can always be selected large enough, with explicit criterion, to achieve both the stabilization of the plant and the setpoint regulation of the system output.


I. INTRODUCTION
The problem of controlling the output of a system so as to achieve asymptotic tracking of prescribed trajectories is one of the most fundamental problems in control theory. In the general context of finite-dimensional linear time-invariant (LTI) control systems, the problem of setpoint regulation control is very classical and has been widely investigated. One possible way to solve this problem is based on the augmentation of the state-space representation of the plant with an integral component of the tracking error and the use of the separation principle by exploiting separately a Luenberger observer (which allows the estimation of the state based on the measure only) and a stabilizing full-state feedback (see, e.g., [10]). Even if this approach has reached a very high level of maturity for finitedimensional systems, its possible extension to infinite-dimensional systems, as those considered in this paper, is still an open problem.
Infinite-dimensional systems emerge in many practical applications due to the occurrence of delays, reaction-diffusion dynamics, or even flexible behavior (see, e.g., [12], [16], [17] for introductory textbooks on dedicated control theory for infinite-dimensional systems). While many efficient control design methods have been reported for the stabilization of distributed parameter systems, very few have been extended to the problem of output regulation. The main reason is that all techniques that have been developed for finite-dimensional LTI systems cannot be easily generalized to infinite-dimensional plants. For instance, the frequency domain approach has been generalized to the infinite-dimensional setting, but it requires to deal with an infinite number of poles, yielding an infinite-dimensional pole allocation problem. The state-space approach is followed in this work.
We propose, for the first time, an output feedback control design procedure to achieve the setpoint regulation control of reactiondiffusions system by means of a finite-dimensional observer coupled with a PI controller. The considered reaction-diffusion plant, which might be unstable, is modeled by a Sturm-Liouville operator as those classical introduced in the context of parabolic partial differential Hugo Lhachemi is with Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des signaux et systèmes, 91190, Gif-sur-Yvette, France (e-mail: hugo.lhachemi@centralesupelec.fr). His work has been partially supported by ANR PIA funding: ANR-20-IDEES-0002.
Christophe Prieur is with Université Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, F-38000, Grenoble, France (e-mail: christophe.prieur@gipsalab.fr). His work has been partially supported by MIAI@Grenoble Alpes (ANR-19-P3IA-0003) equation (PDE). The case of PI regulation of this system by means of a state feedback was reported in [14] (see also [3], [4], [8], [14], [15], [20], [24]- [26], [28] for various approaches about PI control design for different types of PDEs). Here we go beyond by designing an output feedback PI control strategy. Even if the proposed procedure also applies to bounded control inputs and bounded observations, we focus the presentation on boundary controls and boundary measurements. This is because these configurations are the most interesting for practical applications and also the most challenging since they involve unbounded control and observation operators (see, e.g., [7] for further explanations). We study several cases for the input-to-output map, covering Dirichlet control inputs (easily extendable to Neumann control inputs as discussed in conclusion) along with Dirichlet and/or Neumann to-be-regulated outputs and measured outputs. We also show that our procedure can be used to regulate a system output that is distinct of the measured one. Therefore, our approach gives a complete framework to study every associated input-to-output maps.
The proposed control design strategy consists of an adequate integral component coupled with a finite-dimensional observer. The design of finite-dimensional observer-based controllers for distributed parameter plants is challenging due to the fact that the separation principle, that is classically used for finite-dimensional systems, does not apply for infinite-dimensional systems [2], [6], [9], [22]. Taking advantage of spectral reduction approaches [5], [21] and using the control architecture initially reported in [22], a LMI-based procedure for solving this stabilization problem for reaction-diffusion PDEs was reported in [11] in the case were the either control or observation operator is bounded. This approach was extended in [13] to the case were both control and observation operators are unbounded, including both Dirichlet and Neumann settings. The present work, taking advantage of [13], goes beyond the simple problem of closedloop stabilization by embracing the issue of output setpoint regulation control. Since the designed observer only estimates a finite number of modes of the infinite-dimensional system, there is an inherent mismatch between the actually measured system output and its estimation as soon as the output is to be regulated to a non-zero value. Hence, one of the main challenges is to account for this mismatch in the dynamics of the observer and then in the subsequent stability analysis. An other challenge is to couple this finite-dimensional observer with a suitable integral component, inspired by the one described in [14] for a state-feedback, in an output feedback setting. Our approach is based on Lyapunov direct methods and the main results take the form of explicit sufficient conditions ensuring the both stability and setpoint regulation control of the closed-loop plant. We assess that these conditions are always feasible provided the order of the observer is selected large enough. Therefore, we show in a constructive manner that the setpoint regulation control of reactiondiffusion PDEs can always be achieved by the coupling of a PI and a finite-dimensional observer.
The paper is organized by considering successively different inputoutput maps for the reaction-diffusion equation depending on the selected boundary measured output, the to-be-regulated output, and the control input. After recalling classical notations and properties for the Sturm-Liouville operators in Section II, the case of a Dirichlet observation and a Dirichlet control input is considered in Section III. Then the case of a Neumann measurement and a Dirichlet control input is considered in Section IV. While the to-be-regulated output and the measured output are the same in the two latter sections, a crossed configuration is considered in Section V. The regulation problem is solved for a Dirichlet measured output, a Neumann tobe-regulated output, and a Dirichlet control input. This final result completes the picture and gives a full study of the different cases for the input-to-output map of the considered class of distributed parameter systems. Some numerical simulations are given in Section VI for this final result. Section VII collects some concluding remarks.

II. NOTATION AND PROPERTIES
Spaces R n are endowed with the Euclidean norm denoted by · . The associated induced norms of matrices are also denoted by · . Given two vectors X and Y , col(X, Y ) denotes the vector [X , Y ] . L 2 (0, 1) stands for the space of square integrable functions on (0, 1) and is endowed with the inner product f, g = 1 0 f (x)g(x) dx with associated norm denoted by · L 2 . For an integer m ≥ 1, the Sobolev space of order m is denoted by H m (0, 1) and is endowed with its usual norm denoted by · H m . For a symmetric matrix P ∈ R n×n , P 0 (resp. P 0) means that P is positive semi-definite (resp. positive definite).

III. DIRICHLET MEASUREMENT AND REGULATION CONTROL
We consider the reaction-diffusion system with Dirichlet boundary observation described for t > 0 and x ∈ (0, 1) by in the case p ∈ C 2 ([0, 1]). Here qc ∈ R is a constant, u(t) ∈ R is the command input, y(t) ∈ R is the measurement, z0 ∈ L 2 (0, 1) is the initial condition, and z(t, ·) ∈ L 2 (0, 1) is the state. The control design objective is to design a finite-dimensional observer-based controller to achieve both stabilization and setpoint regulation control of y(t) to some prescribed reference signal r(t). By setpoint tracking, we mean that our objective is to ensure that y(t) → re when t → +∞ as soon as r(t) → re when t → +∞ for any arbitrarily given re ∈ R.

A. Spectral reduction
As classically done in the context of boundary control systems (see [7,Sec. 3.3] for details), we start by transforming the nonhomogeneous problem (3) into an equivalent homogeneous one by introducing the change of variable: that gives the equivalent representation: We introduce the coefficients of projection wn(t) = w(t, ·), φn , an = a, φn , and bn = b, φn . Considering classical solutions associated with any z0 ∈ H 2 (0, 1) and any u(0) ∈ R such that z 0 (0) = 0 and z0(1) = u(0) (their existence for the upcoming closed-loop dynamics is an immediate consequence of [19, Chap. 6, Thm. 1.7]), we have w(t, ·) ∈ D(A) for all t ≥ 0 and we infer thaṫ

B. Control design
Let δ > 0 be the desired exponential decay rate for the setpoint regulation. We fix N0 ≥ 1 so that −λn + qc < −δ < 0 for all n ≥ N0 + 1. The integer N0, which is definitely fixed for the rest of the control design procedure, can be interpreted as the number of modes that will be "actively" modified by the feedback. We now introduce an arbitrary integer N ≥ N0 + 1 which will be further constrained later. Inspired by [22], we design as in [13] an observer to estimate the N first modes of the plant while the state-feedback is performed on the N0 first modes of the plant. In this framework, the estimation of the modes ranging from N0 + 1 to N will solely be used to improve the estimate of the system output (see (11)). Introducing Our objective is to introduce an integral component to achieve the setpoint regulation control of the system output y(t). To do so, we first consider the following classical integral component: This zi-dynamics, which involves all the modes wn for n ≥ 1, cannot be embedded into the reduced model (8) that only involves the modes wn for 1 ≤ n ≤ N0. To circumvent this issue, we follow the idea developed in [14] by introducing The main benefit is that the ξp-dynamics only involves the modes wn for 1 ≤ n ≤ N0 while achieving the same equilibrium condition than the zi-dynamics. However, in this work and in sharp contrast with the state-feedback setting of [14], the modes wn are not measured. Hence we need to replace them in the dynamics of the integral component by their estimated versionŵn which will be described below. Hence, the employed integral component is described by: We now define for 1 ≤ n ≤ N the observer dynamics: and where ln ∈ R are the observer gains. We set ln = 0 for N0 +1 ≤ n ≤ N . Compared to the stabilization problem studied in [13], we introduce the additional term −α1u(t) in the observer dynamics (11). This term is added to compensate the inherent steady-state mismatch between the actually measured system outputỹ(t) and its estimation N i=1 φi(0)ŵi(t), obtained from the observer that estimates the only N first modes of the plant, as soon as the output is to be regulated to a non-zero value. Note that this latter estimate of the output improves as the dimension of the observer N increases.
In the sequel we select, once for all and independely of the dimension N of the observer, the gains K ∈ R 1×(N 0 +2) and L ∈ R N 0 so that A1 + B1K and A0 − LC0 are Hurwitz with eigenvalues that have a real part strictly less than −δ < 0.
with ∆X ∈ R 2N +2 and ∆w ∈ D(A). The first term accounts for the dynamics of the truncated model (25) while the series, which in view of (2) is related to the H 1 norm of the PDE trajectories, is used to handle the modes wn for n ≥ N + 1 of the PDE plant. Proceeding exactly as in [13] but taking into account the extra contribution of the reference signal appearing in (25), we obtain for t ≥ 0 thaṫ with Γn = 2γ − 1 − 1 α λn + qc + δ + βM 1,φ for n ≥ N + 1, α > 1 and β > 0 arbitrary, and where, with a slight abuse of notation,V (t) denotes the time derivative of V (X(t), w(t)) along the system trajectories (31). Since α > 1 we have Γn ≤ Θ2 ≤ 0 for all n ≥ N + 1. From (32a), there exists > 0 such that Θ1 − I. Hence the assumptions imply thatV (t) + 2δV (t) ≤ − ∆X(t) 2 − 2∆X(t) P Lr∆r(t) ≤ P Lr 2 ∆r(t) 2 where Young's inequality has been used to derive the latter estimate. After integration, we obtain for any η ∈ [0, 1) the existence of a constant M1 > 0 such that The claimed estimate (33) easily follows from the definition (34) of the Lyapunov function, the use of (2), Poincaré's inequality, and the change of variable (31a).
Remark 1: Constraints (32) are nonlinear w.r.t. the decision variables. However, for a given value of N ≥ N0 + 1 and arbitrarily fixing α > 1, constraints (32) become LMI conditions w.r.t. the decision variables P 0 and β, γ > 0. As shown in the proof of Theorem 1, these latter LMI conditions are always feasible provided the order of the observer N is selected large enough. Similar remarks apply to Theorems 3 and 5.

IV. NEUMANN MEASUREMENT AND REGULATION CONTROL
We now consider the reaction-diffusion system with Neumann boundary observation described for t > 0 and x ∈ (0, 1) by in the case p ∈ C 2 ([0, 1]).

B. Equilibirum condition and dynamics of deviations
Proceeding similarly to Section III, we can characterize the equilibrium condition of the closed-loop system composed of the reactiondiffusion system (38), the auxiliary command input dynamics (6a), the integral action (42), the observer dynamics (44), and the statefeedback (18). In particular, setting r(t) = re ∈ R, it can be shown that there exists a unique solution to: 0 = (−λn + qc)wn,e + anue + bnve = 0, n ≥ 1, (46a) Moreover we can define we n≥1 wn,eφn ∈ D(A). Introducing the change of variable ze = we + xue, ze is a static solution of (38a-38b) associated with the constant control input u(t) = ue. Denoting by ye z e (0), we also infer that ye = re, achieving the desired reference tracking. This allows the introduction of the dynamics of deviation of the different quantities w.r.t. the equilibrium condition characterized by re ∈ R. We have: (47f)
We are now in position to assess the setpoint regulation control of the left Dirichlet trace.

CONTROL
We now consider the reaction-diffusion system described by (3a-3c), still in the case p ∈ C 2 ([0, 1]), but this time with the boundary measurement ym(t) and the distinct and unmeasured output to-beregulated yr(t) described by:

C. Stability analysis and regulation assessment
The proof of the following theorem directly follows from the proofs reported in the previous sections.

VI. NUMERICAL ILLUSTRATION
We illustrate the result of Section V for Dirichlet measurement and Neumann regulation using a modal approximation that captures the 50 dominant modes of the reaction-diffusion PDE. We set p = 1, q = 0, and qc = 3 for which the open-loop plant is unstable. Selecting δ = 0.5, we obtain N0 = 1, the feedback gain K = −10.4134 −11.3747 2.3100 , and the observer gain L = 1.4373. The conditions of Theorem 5 are found feasible for N = 3. The time-domain evolution of the closed-loop system trajectories are depicted in Fig. 1, confirming the theoretical predictions.

VII. CONCLUSION
We proposed the design of a finite-dimensional observer-based PI controller to achieve the both output stabilization and regulation control of reaction-diffusion PDEs. Even if presented for a Dirichlet boundary control input, the presented results easily extend to Neumann/Robin boundary control (by modifying the change of variable formula to obtain an homogeneous PDE, giving different a, b ∈ L 2 (0, 1)). In-domain measurements can also be handled with the same approach provided the satisfaction of adequate observability conditions. While we have adopted in this paper an early lumping approach, future research directions for finite-dimensional PI regulation control of reaction-diffusion PDEs may be concerned with the study of late lumping approaches [1] in the framework of backstepping control design for PDE-ODE cascades [23], [27].