Control design for suppressing transversal patterns in reaction–(convection)–diffusion systems

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Abstract

The stabilization of planar stationary fronts solutions in a two-dimensional rectangular or cylinder domain, in which a diffusion–convection–reaction process occurs, is studied by reducing the original two-variable PDEs model to an approximate one-dimensional model that describes the behavior of the front line. We consider the control strategy based on sensors placed at the designed front line position and actuators that are spatially-uniform or space dependent. We present a systematic control design that determines the number of required sensors and actuators, their position and their form. The control used linear analysis of a lumped truncated model and concepts of finite and infinite zeros of linear multidimensional systems.

Introduction

Nonlinear parabolic partial differential equations (PDEs), which typically describe reaction–diffusion (R–D) systems, may admit spatially-dependent solutions like stationary fronts as well as spatiotemporal patterns. The latter can often be described as composed of slow-moving fronts separated by domains of moderate changes. Propagating fronts and patterned states may emerge in several technologies including catalytic reactors [1], distillation processes [2], flame propagation and crystal growth [3] (see also [4] for references) as well as in physiological systems like the heart [5]. Our interest lies in catalytic reactors, in which stationary or moving fronts and spatiotemporal patterns have been observed and simulated in various systems like flow through a catalyst [6], fixed-bed reactors [7], [8], reactors with flow reversal [9] and loop reactors [10]. The instabilities emerge due to thermal effects in exothermic reactions, due to self-inhibition by a reactant and due to slow reversible modifications of the surface. The construction of a controller that can stabilize a certain inhomogeneous solution in an one-dimensional (1-D) R–D or a reaction–convection–diffusion (R–C–D) system is currently a subject of intensive investigation [11], [12], [13], [14]. Yet, most catalytic reactors, as well as physiological systems like the heart, exhibit a behaviour that can be properly described by two- or even three-dimensional patterns.

In the present work we are interested in stabilizing stationary (planar) fronts in a two-dimensional (2-D) domain in which a chemical R–D or R–C–D process occurs; i.e., we want to suppress patterns that are transversal to the main direction. The kinetic model can be described by coupled fast-activator and a slow-inhibitor. This terminology will be elaborated on below. Previous studies of 1-D systems demonstrated that the simplest approach is by applying a point-sensor control in which a single space-independent actuator responds to a sensor that is located at the front position [12]. However, a single point sensor control cannot stabilize planar fronts in a wide 2-D system. To overcome this problem we have recently studied control by several sensor/actuators situated at the front line [15]. We have also proposed there a new systematic strategy for input–output selection, which assures that the control system obtained is a minimal-phase system1 (i.e., its finite zeros [16], [17] lie in the left-half part of the complex plane). This property allows in turn to use the root-locus technique [18] for control design. The concepts of finite [16], [17] and infinite zeros [18] of a linear multivariable system are explored to realize the above strategy.

This formal approach, however, lacks insight into the wave behavior, an insight that can be used to suggest efficient modes of control. Here we employ an approximate model reduction to a one-dimensional form that follows the front position while approximating the front velocity. Such an approximate reduced model allows us to qualitatively analyze various control strategies. We show that this formalism accounts for a large class of problems and we apply it to R–D and R–C–D problems. Finally we compare the results of these formal and approximate approaches.

The root-locus method, first introduced by Evans in 1948, has been a powerful tool for the analysis and design of feedback system with single input/single output systems as well as for multi-input/multi-output systems (e.g., see [18]). Unfortunately no control design applications that use the multivariable root-locus technique (with system zeros analysis) have been reported to systems that are described by partial differential equations. Such systems are common in reaction–diffusion and in fluid-flow processes [14], [19], [20], [21], [22].

To explain the issues involved in suppression of transversal patterns let us review mechanisms that induce such patterns. Stationary fronts in one-variable one-dimensional R–D systems are typically structurally unstable and exist only as a boundary between conditions that lead to expansion of one (say, hot) or the other (cold) state. Changing a parameter will monotonically change front velocity and can cause a sign-change. Two-variable 1-D such systems with a fast diffusing-activator and a slow non-diffusing inhibitor (like the FitzHugh-Nagumo model, see below) also cannot produce stationary fronts and fronts will eventually disappear out of the system. The change in front-velocity, upon changing a parameter, may be more complex and involve a hysteresis loop [23]. As stated above fronts can be pinned to their position in 1-D systems by measuring the state variable at the front and using it to attenuate one of the parameters that affect front velocity. Such point-sensor control of planar fronts in 2-D systems will lead to transversal patterns in a sufficiently wide system. That calls for control that uses spatial actuators, that are discussed below. Stationary fronts and patterns may emerge in 1- or 2-D activator–inhibitor systems when the inhibitor diffusivity is sufficiently large (the Turing mechanism), or due to global-coupling [24], to end effects or to inhomogeneities (see [25] for a recent review); these effects are not discussed here.

The behavior of fronts in R–C–D systems is somewhat similar to R–D system after accounting for the effect of convection. Decades of investigation into front propagation in catalytic reactors, using a thermo-kinetic model that accounts for reactant concentration and reactor temperatures as its variables, have shown that for the physically-common case that reactant diffusivity is small or negligible the problem can be reduced to an one-variable presentation and stationary fronts exist only as a boundary between expanding hot and cold zones [7], [26], [27]. Front velocity is extremely slow, due to the high heat capacity, and near the stationary front conditions it is linear with fluid velocity. Thus, the simplest approach to control stationary fronts in 1-D reactor is to use the flow rate or feed conditions as an actuator.

Transversal patterns may emerge in R–C–D systems with oscillatory kinetics, in which the thermo-kinetics model is coupled with a slow non-diffusing inhibitor. Symmetry breaking in the azimuthal direction, of a cylindrically-shaped thin catalytic reactor, was simulated when the perimeter is sufficiently large [27] in realistic reactors models of high Le, ratio of solid- to fluid-phase heat capacities, and high Pe, ratio of convection to conduction numbers. An approximate description of front position was developed there [27] and is employed here. Transversal patterns may also emerge in simple thermo-kinetic (i.e., non-oscillatory) model but apparently only with a sufficiently large reactant diffusivity (PeC/PeT < 1) due to Turing-like interaction between the temperature and the concentration [28], [29]. However, in most situations PeT < PeC.

Section snippets

Problem statement

Consider the R–C–D problem, in the (z, s) rectangular or cylindrical shell domain of length L and width S, which is described by a nonlinear parabolic PDEs of the formyt+(V+λV)yz-yzz-yss=P(y,θ)+λ,θt=εQ(y,θ)where y = y(z, s, t) is typically an activator (e.g., temperature), undergoing diffusion and convection, and θ = θ(z, s, t) is the slow inhibitor that is a local (surface) property; λ and λV are the control variables; ε is the ratio of time scales (the diffusivity here is set to unity since it was

Approximate model

Consider the following approximations (see [15] for a detailed derivation): When the activator (y) is fast while the inhibitor (θ) is slow (i.e., ε ≪ 1) we can study, to a first approximation, the velocity of the activator front for frozen θ profile. The opposite inclination of the activator and inhibitor profiles (Fig. 1a) is the source of instability, leading to front motion in the one-dimensional system. If Z(s, t) is the front position, and c = c∞(θo(Z(s)), λ) is the approximate velocity of a

Root-locus control design

To design control (15) we use the multivariable root-locus technique [18] that is based on analysis of finite zeros and infinite zeros [16], [17] of the open-loop system. For realization of the root-locus technique we need at first to estimate the minimal number of sensors (η) that assures stabilization of the closed-loop system (14), (15). This η coincides with number of inputs and outputs of the open-loop system (Eq. (14)). Then we need to find matrix H of full rank

Polynomial model (Cubic source-function, Eq. (3))

Let us apply this procedure to design additive control λ(λV = 0) that will stabilize the (analytical) steady-state solution of R–D system (Eqs. (1), (2), (3), V = 0) with ε → 0 in the rectangular domain. The steady state yo and c∞ at the front position are calculated as yo=1-γtanh((z-Zf)(1-γ)/2), c∞f≅3(θo+λ)/2[15] with ∂c∞∂θf≅3/2 and ∂y∂zf=(1-γ)/2 (see details in [32]). We will compare our results with previous ones [15] that were based on the zero structure of exact 2-D model (i.e., for any ε).

The

Conclusions

We have developed a new methodology to stabilize planar stationary front solutions in a two-dimensional rectangular and cylinder domain, in which a diffusion–convection–reaction systems occurs. To that end we use an approximate 1-D model that describes the behavior of a front line. This reduction considerably simplified the search of a control structure in comparison with the exact procedure applied to the original 2-D model. We use a finite-dimensional point-sensor feedback control that is

Acknowledgements

Work supported by the US-Israel Binational Science Foundation. MS is a member of the Minerva Center for Nonlinear Physics of Complex Systems. ON acknowledges partial support by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel.

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