State Feedback Regulation Problem to the Reaction-Diffusion Equation †

: In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded input, output, and disturbance operators, a ﬁnite-dimensional exosystem (exogenous system), and the state of the exosystem as the state to the feedback law. As is well-known, the SFRP can be solved only if the so-called Francis (regulator) equations have solution. In our work, we try with the solution of the Francis equations from the 1-D R-D equation following given criteria to the eigenvalues from the exosystem and transfer function of the system, but the state operator is here deﬁned in terms of the Sturm–Liouville differential operator (SLDO). Within this framework, the SFRP is then solved for the 1-D R-D equation. The numerical simulation results validate the performance of the regulator.


Introduction
Some physical quantities from applications involving the diffusion and structural vibrations depend on both position and time. Systems whose dynamic evolves in an infinite-dimensional Hilbert space are modeled by partial differential equations (PDEs). Such systems are called infinite-dimensional systems. Because these systems reflect the spatial distribution of a physical quantity, these systems are called distributed parameter systems (DPSs). The aim when designing the controller for infinite-dimensional systems is that the control system is stable and robust in the presence of parametric uncertainties and external perturbations.
The design of classical controllers is commonly based on the transfer function of the system. Infinite-dimensional systems have transfer function. Unlike transfer functions from finite-dimensional systems, transfer functions of infinite-dimensional systems are not rational functions. If the transfer function from an infinite-dimensional system is provided, then the controller can be directly designed. A drawback from this last approach is that the controller will be infinite-dimensional and it may be approximated by a finite-dimensional system. For some practical applications, it could be that a transfer function is not available. Subsequently, a finite-dimensional model of the system must be drawn from which the controller design may be based. This latter approach is the most used methodology to design controllers of systems modeled by PDEs [1]. the heat equation; this latter as an exponentially stable target system. To the best of our knowledge, there is no work regarding the solution of the SFRP for harmonic tracking with harmonic disturbance rejection to the 1-D R-D equation.
In this work, although it could be argued that the 1-D R-D equation is be among the simplest PDEs systems, our application is concerned with the SFRP to the design of a harmonic tracking regulator, but, in contrast with our previous work recently reported in [36], with harmonic disturbance rejection for this kind of system. It is well known that the SFRP is solved only if the Francis (regulator) equations have a solution. In our work, the main contribution is that the state operator to the Francis equations, corresponding to the 1-D R-D equation, here is defined in terms of the Sturm-Liouville differential operator (SLDO) to then solve them.
The manuscript is organized, as follows. In Section 1, the SFRP, modeling of DPS, control of PDEs, and applications of the R-D equation are summarized; the problem statement is formulated in Section 2; the regulator design is detailed at Section 3; the simulation results are included in Section 4; and, the conclusion is drawn at the end.

Sturm-Liouville Boundary Value Problem
Let us consider a differential equation of the form with symmetric boundary conditions, also known as Sturm-Liouville boundary conditions, at the endpoints. The Sturm-Liouville Equation (1) can be rewritten as with ∆ the Laplacian operator [26,37], assuming that p(x) is a C 1 function, q(x) and r(x) are continuous functions, p(x) > 0, and r(x) > 0 [38]. In this case, the boundary value problem is said to be regular. It should be noticed that these latter conditions are satisfied in many problems in mathematical physics.
is known as the SLDO [39]. Let us consider the linear homogeneous differential operator Accordingly, the Equation (1) can be written as Each one of the boundary conditions (2) involves only one of the boundary points, and that is why are said to be separated. In order to impose any restriction on y, at least one of the parameters α 1 and α 2 must be different from zero to the first boundary condition. Alike, at least one of the parameters β 1 and β 2 must differ from zero [40]. The boundary value problem consisting of (1) and (2) is called the Sturm-Liouville boundary value problem (SLBVP) [41].
Modeling that involves ordinary differential equations (ODEs) usually leads to initial value problems or boundary value problems, these latter written in the form of the SLBVPs, which are eigenvalue problems that involve a parameter λ related to frequencies, energies, or other physical quantities [42]. In fact, the next theorem summarizes the way about how to convert a linear second order differential equation A(x)y + B(x)y + C(x)y into one of the form (1/r(x))((p(x)y ) + q(x)y). If p(x), q(x), and r(x) are given, then A(x) = p(x)/r(x), B(x) = p(x) /r(x), and C(x) = q(x)/r(x). Theorem 1. [43] Suppose that A(x) > 0, B(x) and C(x) are analytic real-valued functions in the (finite or infinite) interval a < x < b, then exist functions p(x) > 0, q(x), and r(x) > 0 likewise analytic and real valued in a < x < b and Proof. See [43].
The expression on the right from this last equation is referred as the Sturmian form of the differential equation, also called self-adjoint form. Solutions of SLBVPs, called eigenfunctions, have many general properties in common, as the orthogonality property, useful in eigenfunctions expansions in terms of Fourier series, Legendre polinomials, Bessel functions, and other eigenfunctions [41,44].

Abstract Control System Model
Let us consider an abstract distributed parameters control system in the Hilbert space Z in the form where the state operator A refers to an unbounded densely defined operator, z(t) stands for the state of the system, z 0 is for the initial condition, z t is the derivative of z(t) with respect to time, u(t) ∈ U is the control input, and y(t) ∈ Y is the measured output. U , Y are Hilbert spaces of either finite or infinite dimension. B in is an input operator, B d ∈ L(U , Z ) is a disturbance operator, d(t) refers to the disturbance, and C is an output operator. The operator A is given in terms of a linear elliptic partial differential operator L in the Hilbert space Z = L 2 (Ω) of infinite dimension with bounded domain Ω ∈ R n with piecewise smooth boundary. To the second order case, where a(x) ≥ 0 for all x ∈ Ω, a ij ∈ C ∞ (Ω), a ij (x) = a ji (x), and Ω is the closure of Ω. If there exist constants 0 < c 1 < c 2 < ∞, by uniform ellipticity it means that for all ξ ∈ R n and x ∈ Ω, where | · | represents the Euclidean norm in R n .
In our work, from (5) and (11), the state operator is defined here as given in the form of the SLDO, namely, The output operator C ∈ L(Z, Y ) is a set of bounded operators C i given by for some Ω j of the domain Ω, with Lebesgue measure In the same setting, and indicator function Accordingly,

Exogenous System (Exosystem)
Let us consider a finite-dimensional exosystem which generates both reference output y r (t) and disturbance d(t), W is the state space of the exosystem, S ∈ L(W ), Q ∈ L(W, Y ), and P ∈ L(W, Z ). Let us consider the system given by z(x, 0) = z 0 (x).
If there exist positive constants M and α, such that with z 0 ∈ Z, then the system (20) and (21) is exponentially stable. From the above, it is said that the state operator A generates an exponentially stable C 0 semigroup in Z [45], i.e., A is stable. Accordingly, here it is assumed that the uncontrolled system (20) and (21), i.e., to the case when u = 0 and d = 0, is exponentially stable. Additionally, it is assumed that the exogenous system (16)-(19) is neutrally stable. To the linear case, in the Lyapunov sense, this is equivalent to the origin being stable implying that σ(S) ⊂ iR and S has no nontrivial Jordan blocks. ρ(T) refers to the resolvent set of an operator T and σ(T) refers to the spectrum of T. Let us consider the error signal The main task for the regulator is to force the output of the system to track a reference signal in presence of a disturbance d(t), i.e., e(t) → 0 as t → ∞. Accordingly, the problem is stated as follows.

Problem 1. State Feedback Regulator Problem (SFRP).
The SFRP consists in to find a control law in function of the state of the exosystem with Γ ∈ L(W, U ), such that for the system which corresponds to the interconnection of (8)-(10) with (16)- (19), the error norm satisfies e(t) → 0 as t → ∞, for any z 0 ∈ Z and w 0 ∈ W.
Being e(t) a finite dimension vector, then all l p norms in (30) are equivalent. Because exponential stability for the system (20) and (21) has been assumed, then a state feedback control law is not involved. In this work, our main focus is with tracking and harmonic disturbance rejection, so, the stabilization problem is out of scope of our proposal. Thus, the solvability of the SFRP is stated in the next theorem.
then the linear SFRP has solution. The feedback control law that solves the SFRP is given by Proof. The proof can be carried out along the same lines as in [16]. From this last theorem, from the assumptions that e At is an exponentially stable semigroup and that the exosystem is neutrally stable, if (31) holds then e(t) → 0 as t → ∞ for all z 0 ∈ Z, w 0 ∈ W, if and only if [CΠ − Q] = 0.

Regulator Design
Let us consider the 1-D R-D equation that is given by where Dz xx (x, t) is referred as the diffusion term, with diffusion coefficient (constant) D > 0, and λz(x, t) is the reaction (source) term, with λ an arbitrary constant. Refer to [37] to have an idea about typical values that D may take when trying with heat conduction or molecular diffusion.
The system (34)- (38) is formulated in the form (8)-(10) in the Hilbert state space Z = L 2 (0, 1). z x refers to the partial derivative with respect to space and z xx refers to the second partial derivative with respect to space also. The maximal elliptic operator is given by L = d 2 /dx 2 in (11) with domain D(L) = H 2 (0, 1), the Sobolev space of functions ζ ∈ Z with dζ/dx both continuous on (0, 1) and d 2 ζ/dx 2 ∈ Z. The state operator (12) is a self-adjoint operator in Z, i.e., The spectrum of A denoted by where λ k = −µ 2 k with µ k = k − 1 2 π, is purely discrete with a corresponding set of orthonormal eigenvectors The operator A is assumed as an infinitesimal generator of an exponentially stable C 0 semigroup in Z in terms of the eigenfunction expansion It should be noticed that the system under consideration is a single-input/single-output system with scalar input and output B in and C, respectively, and B d is for a bounded disturbance. Additionally, the input, output, and disturbance operators are bounded operators acting in the interior of the domain.
The input to the system is spatially uniform over a small interval about a fixed point and The input operators B in and B d are given as where u j (t) and d j (t) are scalar control inputs and disturbances, respectively. B To guarantee that B j in ∈ Z, here it is assumed that |Ω j | > 0. The output is the average transport reaction over a small interval about a point x out = x 1 ∈ (0, 1), i.e., Because Cζ = ζ, c , C is a bounded linear observation functional on Z. In our work, d(t) = A d sin (βt) ∈ R entering across the whole interval is considered, so B d = 1. In our proposal, the Francis Equations (31) and (32) take the form with Because of the block diagonal structure of S, the Francis equations can be decoupled into two separate parts, the first one to work with the harmonic tracking task and the last one to work with the rejection of a harmonic disturbance.

Harmonic Tracking
Accordignly, along the same lines from [16], for the first part on which it is desirable to track a harmonic signal y r (t) = A r sin (αt) consider an exosystem given as Taking P = [0 0] and Q = [1 0] then y r (t) = Qw = A r sin (αt).

Harmonic Disturbance Rejection
Now, for the last part where is desirable to reject a sinusoidal disturbance such as d(t) = A d sin (βt), let us chose an exosystem that is governed by the harmonic oscillator Taking Q = [0 0] and P = [1 0], then y r (t) = Qw = 0.
Again, to this case W = R 2 so, we look for The Francis equations that are applied to the vector w = [w 3 w 4 ] T ∈ W result in the system Expanding the above matrix multiplications, the regulator equation from the left becomes Because (55) has to hold for w, now consider the case for w 3 = 1 and w 4 = 0 to then consider that for w 3 = 0 and w 4 = 1 resulting By multiplying (57) by i = √ −1 and adding the result to (56), we get Noting that iβ ∈ ρ(A), applying (iβI − A) −1 to both sides of (58), we get By multiplying both sides of (59) by C, from (54) implies thus, we get where we have used the notation Finally, solving for Γ we have Consequently, by combining (52) and (60) we get Although it could be argued that the 1-D R-D equation was so easy to find an explicit formula for the transfer function G(s) = C(sI − A) −1 B in , then there is no reason for which the state feedback regulator problem to this type of PDE had not been reported in the literature in spite of its importance to a myriad of applications.

Simulation Results
In our numerical simulation, we have set A d = 2, A r = 2, α = 4, β = 4, x 0 = 0.75, x 1 = 0.25 and ν 0 = ν 1 = 0.25. From Figures 1-3, it can be seen that the controlled output y(t) tracks the reference signal y r (t) from an initial condition ϕ(x) = 4 cos (πx). Additionally, it can be seen that the error signal e(t) tends to zero as time tends to infinity. The performance of the regulator is shown for different λ values. The convergence of y(t) to y r (t) is faster for small λ values. From Figures 4-6, the solution surface is shown for every λ value. In mass balance the source term λz can be interpreted as the rate of production (destruction) of species per unit volume. From (34), the sign from the diffusion term agrees with the observation that mass flows from high to low mass concentration. In general, pressure gradients, temperature, and external forces affect the mass flux but their effects can be neglected and take the diffusion coefficient D > 0 to be constant. In fact, the units may be chosen in a convenient way in order to make D = 1. In our work, the diffusion coefficient is set to the unit value.  Figure 2. Performance of the regulator for λ = 0, i.e., to the case for which the R-D equation is reduced to the Fick's second law of diffusion (also called simply as diffusion equation), for whose case Γ 1 = 2.4674, Γ 2 = 1, Γ 3 = −0.5, and Γ 4 = 0.  Figure 3. Performance of the regulator for λ = −2.5, for whose case Γ 1 = 3.0924, Γ 2 = 1, Γ 3 = −0.5, and Γ 4 = 0.

Conclusions
In [16], from an abstract control system model for the 1-D heat equation, the SFRP was solved in terms of the transfer function of the system, but with feedback control law in function of the state of the exosystem. The SFRP to the 1-D heat equation was solved for harmonic tracking and rejection of a constant disturbance. The SLDO (5) is present for most of the R-D equation types [34]. In our work, we propose the state operator for an abstract control system model to the 1-D R-D equation, in contrast with that in [16], namely, (13), as given in the form of the SLDO (12). Accordingly, the 1-D R-D equation is characterized, along the same lines as in [16], in terms of the Francis (regulator) equations, but with state operator (12) and then these are solved. The SFRP to the 1-D R-D equation is solved for harmonic tracking with harmonic disturbance rejection. The simulation results validate the performance of the regulator, i.e., the error tends to zero as time tends to infinity. From all of the above, we conclude that our proposal performs well, i.e., the SFRP for the 1-D R-D equation has solution. As future work, our proposal may be extended to those approaches in [16], and so on, on which the abstract control system model to the R-D equation could falls.