Boundary observers for coupled diffusion–reaction systems with prescribed convergence rate

doi:10.1016/j.sysconle.2019.104586

Systems & Control Letters

Volume 135, January 2020, 104586

https://doi.org/10.1016/j.sysconle.2019.104586 Get rights and content

Abstract

Following recent results on the boundary stabilization of coupled first-order hyperbolic equations by means of integral transformations, here a new result is presented for the problem of state estimation of coupled linear reaction–diffusion PDEs with Neumann boundary conditions from boundary measurements. For this purpose, an observer is constructed with a prescribed convergence rate. The stability of the estimation error system is derived by mapping the estimation error system to a stable target system using a pair of integral transformations. Our method is applicable as well to the dual problem of boundary stabilization of coupled linear reaction–diffusion PDEs. A numerical scheme, based on power series approximations of the kernels is formulated, taking into account the fact that the kernels are only piecewise differentiable.

Introduction

The problems of stabilization and estimation for coupled linear parabolic equations have been addressed recently, by means of the backstepping method for PDEs [1], in a series of publications. First, the stabilization and estimation problems for coupled reaction–diffusion equations, with constant parameters and equal diffusion coefficients, were solved in [2], [3] and [4]. The extension to allow distinct diffusion coefficients was proposed later for coupled reaction–diffusion equations with constant coefficients in [5], [6]. Then, boundary stabilization for coupled reaction–diffusion equations, with a spatially varying reaction, was solved in [7], in a relative general way. The generality allowed for subsequent results on the boundary estimation of coupled reaction–diffusion equations, with a spatially varying reaction, in [8], and on the boundary stabilization problem for coupled reaction–advection–diffusion equations with spatial variation in all parameters in [9]. Likewise, the problem of boundary stabilization and output regulation for one-dimensional coupled parabolic PIDEs with spatially varying coefficients and with Dirichlet, Neumann, and Robin boundary conditions was addressed in [10] and in [11], respectively. More recently, stabilization for a pair of coupled diffusion–reaction equations with unknown parameters was studied in [12]. The estimation and stabilization problems are closely related. In the estimation problem, one commonly designs an observer which guarantees some stability property for the origin of the estimation error system. The stability of the estimation error system then implies the convergence of the state estimate to the unknown state.

Briefly speaking, in the backstepping method, one seeks for an invertible transformation to map a, possibly unstable, PDE to a carefully selected stable target system. The transformation is typically an integral transformation and the main difficulty arises when trying to solve the PDEs verified by the kernels in the integral transformation. In [7], [9] Volterra integral transformation (of second kind) was employed for a system of $n$ coupled (advection)-reaction–diffusion equations. The kernels in [7], [9] satisfy $n^{2}$ coupled second-order hyperbolic equations in a triangular domain and were solved by deriving an equivalent system of $2 n^{2}$ coupled first-order hyperbolic equations, noticing a resemblance with the kernel equations appearing in the boundary stabilization problem of coupled systems of first-order hyperbolic equations [13], [14]. A similar approach was followed in [8], but making use of a more recent solution of the boundary stabilization problems of coupled systems of first-order hyperbolic equations [15].

The contribution of this paper is twofold, we provide a pair of integral transformations to decoupled the equation in the estimation error system and device a numerical method to compute the kernel equations.

First, motivated again by advances on the problem of boundary stabilization for coupled first-order hyperbolic equations [16] where a decoupling technique is applied, we propose a new solution to the state estimation problem for coupled reaction–diffusion equations from boundary measurements. We show that a pair of integral transformations allows us to map the estimation error system into a simple stable target system, with uncoupled equations. Previous methods [5], [6], [7], [8], [9], [10], [11] lead to target systems with coupled equations, convoluting the assignment of an exact convergence rate or the formulation of robustness with respect to measurement disturbances [17]. Compared with [2], [3], [4], the result in this paper is not restricted to systems with equal diffusivity. The case with equal diffusion coefficients is less involved; in particular, a solution to the kernel equations can be found following the same method used in the problems with a single PDE. The result in this paper is not restricted to the problem of state estimation from boundary measurements. Actually, due to the similarity of the kernel equations in the problems of boundary stabilization and boundary estimation, this result is also applicable to problem of boundary stabilization of coupled linear reaction–diffusion PDEs. We derive and solve the equations for the kernels of each transformation; the first one over a triangular domain and the second one over a square of unit area. The solutions are constructed by the method of characteristics; where nontrivial partitions of the domains are required. We show that for both transformations, the kernel equations are second-order coupled hyperbolic, with a coupling between some of the kernels at the boundaries.

Second, we provide a simple numerical method to solve the kernel equations. The numerical scheme is based on polynomial approximations of the kernels; taking into account the fact that the kernels are piecewise differentiable. The problem of approximating solution of kernel equations by polynomials was studied previously in [18], where the authors formulate the approximation problem as an optimization problem, but it has not been applied to kernels with discontinuous derivatives.

The structure of the paper is as follows. In Section 2 the estimation problem is introduced. The main result in presented in Section 3. The solution to the kernel equations is derived in Section 4. A numerical scheme to compute the kernels is presented in Section 5; together with an example of the numerical computation 6. Finally, we conclude the paper with some remarks in Section 7.

Section snippets

Notation

•
For a function $f : [0, 1] \mapsto R^{n}$ , with $f (x) = {[f_{1} (x), \dots, f_{n} (x)]}^{T}$ , such that $f_{i} \in L^{2} (0, 1)$ , for $i \in {1, \dots, n}$ , we use the following norm notation ${‖ f ‖}_{L^{2}}^{2} = \int_{0}^{1} {| f (x) |}_{2}^{2} d x, {| f (x) |}_{2}^{2} = \sum_{i = 1}^{n} {| f_{i} (x) |}^{2},$
•
A function $f : [0, 1] \mapsto R^{n}$ belong to the space $L^{2} (0, 1; R^{n})$ if ${‖ f ‖}_{L^{2}} < \infty,$

Coupled parabolic reaction diffusion systems

Consider a linear reaction diffusion equation $u_{t} (x, t) = Σ u_{x x} (x, t) + Λ (x) u (x, t),$ with coefficients $Σ = [\begin{matrix} ϵ_{1} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & ϵ_{n} \end{matrix}], Λ (x) = [\begin{matrix} λ_{11} (x) & \dots & λ_{1 n} (x) \\ ⋮ & ⋱ & ⋮ \\ λ_{n 1} (x) & \dots & λ_{n n} (x) \end{matrix}] .$ for $x \in (0, 1)$ , $t \in (0, T]$ , with $λ_{i j} \in C^{1} (0, 1)$ for all $i, j \in {1, 2, \dots, n}$ and distinct diffusivities $ϵ_{i} > 0$ , for all $i \in {1, 2, \dots, n}$ . The state $u (x, t) \in R^{n}$

Stability of the estimation error system

Theorem 1

The origin of the estimation error system (10)–(11), with initial condition ${\tilde{u}}_{0} \in L^{2} (0, 1)$ and observer gains computed from (13), is exponentially stable, that is, for any prescribed $σ > 0$ , there exists a positive constant $κ$ , such that ${‖ \tilde{u} (\cdot, t) ‖}_{L^{2}} \leq κ exp [- σ t],$ for all $t > 0$ .

In the proof of Theorem 1, the main question is if the kernel PDEs (14)–(18) and (20)–(22) do indeed have a solution, as implicitly assumed in the theorem’s statement, The next result answers this question.

Theorem 2

Both systems of kernel equations

Kernel equations for first transformation

The coefficients in the diagonal of $K (x, s)$ satisfy the equation $ϵ_{i} K_{x x}^{i i} (x, s) - ϵ_{i} K_{s s}^{i i} (x, s) = - c_{i} K^{i i} (x, s) - \sum_{l = 1}^{l = n} λ_{i l} (x) K^{l i} (x, s),$ for $i \in {1, 2, \dots, n}$ , with boundary conditions $\frac{d}{d x} [K^{i i} (x, x)] = - \frac{c_{i} + λ_{i i} (x)}{2 ϵ_{i}},$ $K_{x}^{i i} (0, s) = 0, K^{i i} (0, 0) = 0 .$ The coefficients in the upper triangular part of $K (x, s)$ satisfy the equation $ϵ_{i} K_{x x}^{i j} (x, s) - ϵ_{j} K_{s s}^{i j} (x, s) = - c_{j} K^{i j} (x, s) - \sum_{l = 1}^{l = n} λ_{i l} (x) K^{l j} (x, s),$ for $i \in {1, 2, \dots, n - 1}$ and $i < j$ , with boundary conditions $K_{x}^{i j} (x, x) = \frac{λ_{i j} (x)}{ϵ_{j} - ϵ_{i}}, K_{s}^{i j} (x, x) = \frac{λ_{i j} (x)}{ϵ_{i} - ϵ_{j}},$ $K^{i j} (x, x) = 0, K_{x}^{i j} (0, s) = 0, K^{i j} (0, 0) = 0 .$

The coefficients

A numerical method to compute kernels

The numerical approximation of the kernels is based on a piecewise polynomial approximation that takes into account the piecewise differential nature of the kernels. For the approximation of coefficients in $K (x, s)$ , the domain is divided according to the intersection of the sets $A_{1}^{i j}$ , $A_{2}^{i j}$ , $B_{1}^{i j}$ and $B_{2}^{i j}$ , defined in Section 4 (Fig. 2, Fig. 3) corresponding to all the coefficients within the same column; due to the column-wise coupling in Eqs. (48), (51), (54). For the approximation of

Kernel functions

For a pair of coupled reaction–diffusion equation, a total of five kernel functions have to be computed. Fig. 8 shows a plot of the polynomial approximation of the non-zero element in the kernel matrix $\overset{ˇ}{K}$ and Fig. 9 shows a plot of the polynomial approximation of the element $K_{12}$ The order of polynomial approximation is $m = 10$ , and the parameters in the problem are the following $Σ = [\begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}], Λ (x) = [\begin{matrix} 1 & x \\ x & 1 \end{matrix}], C = [\begin{matrix} 5 & 0 \\ 0 & 11 \end{matrix}] .$

Observer

To evaluate the performance of the observer, we consider an unstable pair of coupled

Conclusion

This paper details the design of observers for coupled systems of diffusion–reaction equations. The converge of the estimate follows from the stability of the estimation error system; derived by mapping the estimation error system to a stable target system using a pair of integral transformations. The target system is a set of $n$ decoupled equations. The simple target system is advantageous to precisely assign designer-chosen convergence rates. Future work includes the adaptive estimation

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Leobardo Camacho-Solorio acknowledges financial support from UC MEXUS, United States of America and CONACYT, Mexico .

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Boundary observers for coupled diffusion–reaction systems with prescribed convergence rate

Abstract

Introduction

Section snippets

Notation

Coupled parabolic reaction diffusion systems

Stability of the estimation error system

Kernel equations for first transformation

A numerical method to compute kernels

Kernel functions

Observer

Conclusion

Declaration of Competing Interest

Acknowledgments

Boundary Control of PDEs

Anticollocated backstepping observer design for a class of coupled reaction–diffusion PDEs

J. Control Sci. Eng.

Boundary control of coupled reaction–diffusion processes with constant parameters

Automatica

Output feedback stabilization of coupled reaction–diffusion processes with constant parameters

SIAM J. Control Optim.

Boundary control of coupled reaction-advection-diffusion systems with spatially-varying coefficients

IEEE Trans. Automat. Control

Backstepping control of coupled linear parabolic PIDEs with spatially-varying coefficients

IEEE Trans. Automat. Control