In-domain control of a heat equation: an approach combining zero-dynamics inverse and differential flatness

This paper addresses the set-point control problem of a heat equation with in-domain actuation. The proposed scheme is based on the framework of zero dynamics inverse combined with flat system control. Moreover, the set-point control is cast into a motion planing problem of a multiple-input, multiple-out system, which is solved by a Green's function-based reference trajectory decomposition. The validity of the proposed method is assessed through convergence and solvability analysis of the control algorithm. The performance of the developed control scheme and the viability of the proposed approach are confirmed by numerical simulation of a representative system.


Introduction
Control of parabolic partial differential equations (PDEs) is a long-standing problem in PDE control theory and practice. There exists a very rich literature devoted to this topic, and it is continuing to draw a great attention for both theoretical studies and practical applications. In the existing literature, the majority of work is dedicated to 5 boundary control, which may be represented as a standard Cauchy problem to which functional analytic setting based on semigroup and other related tools can be applied proposed in, e.g., [21,22,11] The system model used in this work is taken from [19]. In order to perform control 40 design based on the principle of superposition, we present the original system in a form of parallel connection. As the control with multiple actuators located in the domain leads to a multiple-input, multiple-output (MIMO) problem, we introduce a Green's function-based reference trajectory decomposition scheme that enables a simple and computational tractable implementation of the proposed control algorithm. 45 The remainder of the paper is organized as follows. Section 2 describes the model of the considered system and its equivalent settings. Section 3 presents the detailed control design. Section 4 deals with motion planning and addresses the convergence and the solvability of the proposed control scheme. A simulation study is carried out in Section 5, and, finally, some concluding remarks are presented in Section 6. 50

Problem Setting
In the present work, we consider a scaler parabolic equation describing one-dimensional heat transfer with boundary and in-domain control, which is studied in [19]. Denote by z(x, t) the heat distribution over the one-dimensional space, x, and the time, t. The derivatives of z(x, t) with respect to its variables are denoted by z x and z t , respectively.
The considered heat equation with boundary and in-domain control in a normalized coordinate is of the form: where for a function f (·) and a point x ∈ [0, 1] we define with x − and x + denoting, respectively, the usual meaning of left and right hand limits to x. The initial condition is specified in (1b) with φ(x) ∈ L 2 (0, 1). It is assumed that in System (1), we can control the heat flow at the points x j for j = 1, . . . , m, i.e., Note that in (1) Problem 1. The considered regulation problem for set-point control is to find a dy-60 namic control u(t) such that the regulation error satisfies e(t) → 0 as t → ∞.
For the purpose of control design, we introduce an equivalent formulation of the in-domain control problem described in (1) by replacing the jump conditions in (1e) by pointwise controls as source terms. The resulting system will be of the following form where δ(x − x j ) is the Dirac delta function supported at the point x j , denoting the position of control support, and α j : t → R, j = 1, . . . , m, are the in-domain control signals.
Remark 1. It is noticed that the approximate controllability of the heat equation with pointwise control may be lost if the support of the control is located on a nodal set of eigenfunctions (see, e.g., [23,24]). Nevertheless, this situation will not happen to the considered system. Indeed, the eigenfunctions of System (2a) with the boundary conditions (2c) are given by [25] ψ n (x) = cos(µ n x) + k 0 µ n sin(µ n x), n = 1, 2, . . .

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To establish in-domain control at every actuation point, we will proceed in the way of parallel connection, i.e., for every x j ∈ (0, 1), consider the following two systems Similarly, System (7) and (8) are equivalent provided z ∈ H 1 (0, 1) and x ] xj for all j = 1, 2, ..., m, where z j denotes the solution to System (8). One may directly check is a solution to System (1). Therefore, throughout this paper, we assume x ] xj for all j = 1, 2, ..., m. Due to the equivalences of System (1) and (2), and System (7) and (8), we may consider (2) and System (7) in the following parts.

Control Design Based on Zero-Dynamics Inverse and Differential Flatness
In the framework of zero-dynamics inverse, the in-domain control is derived from the so-called forced zero-dynamics. To work with the parallel connected system (7), we first split the reference signal as: where γ j (x, x j ) will be determined in Theorem 5 (see Section 4). Denoting by ε j (t) = the regulation error corresponding to System (7), the zero-dynamics can be obtained by replacing the input constraints in (7e) by the requirement that the regulation errors vanish identically, i.e., ε j (t) = 0. Thus, we obtain for a fixed j: Note that (10) and (11) form a dynamic control scheme. Note also that (11) implies that the in-domain control signals can be derived from either the system trajectory or the solution of the zero-dynamics. The convergence of regulation errors with ZDI-based control is given in following theorem.

Theorem 2. ([19]) The regulation error corresponding to System
The proof of Theorem 2 is detailed in [19]. Note that a key fact used in the proof of this theorem is that the system given in (7) with null interior control is exponentially Obviously, to find the control signals, we need to solve the corresponding zerodynamics (10). To this end, we leverage the technique of flat systems [27,11,9]. In particular, we apply a standard procedure of Laplace transform-based method to find the solution to (10). Henceforth, we denote by f (x, s) the Laplace transform of a function f (x, t) with respect to the time variable. Then, for fixed x j ∈ (0, 1), the transformed equations of (10) in the Laplace domain read as We divide (12) into two sub-systems, i.e., for fixed x j ∈ (0, 1), considering and Let ξ j − (x, s) and ξ j + (x, s) be the general solutions to (13) and (14), respectively, and denote their inverse Laplace transforms by ξ j − (x, t) and ξ j + (x, t). The solution to (10) can be written as Then at each point x i ∈ (0, 1), by (11) and the argument of "parallel connection" (see In the following steps, we present the computation of the solution to System (10), 95 ξ j . Issues related to the generation reference trajectory z D (x, t) for System (1) will be addressed in Section 4.
Note that ξ j − (x, s) and ξ j + (x, s), the general solutions to (13) and (14), are given by We obtain by applying (13c) and (13d) which can be written as We obtain   C 1 Therefore, the solution to (13) can be expressed as We may proceed in the same way to deal with (14). Indeed, letting we get from (14) Applying modulus theory [28,29] to (16) and (18), we may choose y j (x j , s) as the basic output such that Then, using the property of hyperbolic functions, we obtain from (17) and (19) that Note that is a solution to (12). Using the fact we obtain the time-domain solution to (10), which is given by Furthermore, by a direct computation we get It follows from (15) that Finally, provided z d j (x j , t) = ξ j (x j , t), for j = 1, . . . , m, the reference trajectory z D (x, t) can be determined from (9) and (25).

Motion Planning
For control purpose, we have to choose appreciate reference trajectories, or equivalently the basic outputs. Denote now byz D (x) the desired steady-state profile. Without loss of generality, we consider a set of basic outputs of the form: where ϕ j (t) is a smooth function evolving from 0 to 1. Motion planning amounts then to deriving y(x j ) fromz D (x) and to determining appropriate functions ϕ j (t), for j = 1, . . . , m.
To this aim and due to the equivalence of the systems (1) and (2), we consider the steady-state heat equation corresponding to System (2): Based on the principle of superposition for linear systems, the solution to the steady-state heat equation (29) can be expressed as: where G(x, ζ) is the Green's function corresponding to (29), which is of the form Indeed, it is easy to check that G xx (x, ζ) = δ(x − ζ) and G(x, ζ) satisfies the boundary conditions, G x (0, ζ) − k 0 G(0, ζ) = 0 and G x (1, ζ) + k 1 G(1, ζ) = 0, the joint condition, G(ζ + , ζ) = G(ζ − , ζ), and the jump condition, Taking m distinguished points along the solution to (29),z(x 1 ), . . . ,z(x m ), we get Note that in (32), the matrix formed by the Green's function defined an input-output map in steady-state, which is also called the influence matrix.
In steady-state, we can obtain from (27) that Finally, y(x j ) can be computed by (33) and (40) for a givenz D (x).
It is worth noting that (33) provides a simple and straightforward way to compute the static control from the prescribed steady-state profile. Indeed, a direct computation 125 can show that applying (33) will result in the same static control obtained in [19] where a serially connected model is used.

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PROOF. We prove the convergence of the power series (25) and (27) Then the convergence of the series in (25) and (27) follows easily using the same argument.
We recall that the bounds of Gevrey functions of order σ are given by [30] ∃K, Denote in (42) Then, (42) converges if lim sup n→∞ n |b n | < 1. Now b n can be estimated by (43) Therefore lim sup n→∞ n |b n | ≤ lim sup where in the second inequality we applied Stirling's formaula n √ n! (n/e). We can conclude by Cauchy-Hadamard Theorem that (42) converges for σ < 2, and for σ = 2 if 2K > 1. The series (42) diverges if σ > 2.

Remark 2.
For any x ∈ (0, 1), replace x i by x in the proof of Theorem 5, we can get 140 |z(x, t) − z D (x, t)| → 0 as t → ∞, which shows that the solution z(x, t) of System (1) converges to the reference trajectory z D (x, t) at every point x ∈ (0, 1).

Simulation Study
In the simulation, we implement System (2)  The basic outputs ϕ j (t) used in the simulation are Gevrey functions of the same order.
In order to meet the convergence condition given in Lemma 4, the parameter of the Gevrey function is set to ε = 1.1. The feedback boundary control gains are chosen as 150 k 0 = k 1 = 10. The initial condition in simulation is set to z(x, 0) = cos(πx).
The desired steady-state heat distribution is a piecewise linear curve, depicted in Fig. 1-(a), which is a solution to (29). The corresponding static controls, α 1 , . . . , α 12 , are shown in Fig. 1-(b). Note that the dynamic control signals, α i (t), are smooth functions connecting 0 to α i for i = 1, . . . , 12. The evolution of heat distribution 155 with static and dynamic control, as well as the corresponding regulation errors with respect to the static profile defined as e(x, t) = z(x, t) − z D (x), are depicted in Fig. 2.
The simulation results show that the system performs well with the developed control scheme. It can also be seen that the dynamic control provides a faster response time compared to the static one.

Conclusion
This paper presented a solution to the problem of set-point control of heat distribution with in-domain actuation described by an inhomogeneous parabolic PDE. To apply the principle of superposition, the system is presented in a parallel connection form. The dynamic control problem introduced by the ZDI design is solved by using 165 the technique of flat systems motion planning. As the control with multiple in-domain actuators results in a MIMO problem, a Green's function-based reference trajectory decomposition is introduced, which considerably simplifies the control design and implementation. Convergence and solvability analysis confirms the validity of the control algorithm and the simulation results demonstrate the viability of the proposed ap-170 proach. Finally, as both ZDI design and flatness-based control can be carried out in a systematic manner, we can expect that the approach developed in this work may be applicable to a broader class of distributed parameter systems.