Elsevier

Journal of Process Control

Volume 23, Issue 9, October 2013, Pages 1362-1379
Journal of Process Control

Boundary model predictive control of thin film thickness modelled by the Kuramoto–Sivashinsky equation with input and state constraints

https://doi.org/10.1016/j.jprocont.2013.03.009Get rights and content

Highlights

  • Boundary model predictive control of thin film thickness.

  • Kuramoto–Sivashinsky (K–S) equation.

  • Model modal predictive controller (MPC).

  • Input and PDE states constraints satisfaction.

Abstract

In this work, a model modal predictive control (MMPC) strategy is proposed to stabilize the falling liquid film thickness in the vertical tubes modelled by the Kuramoto–Sivashinsky (K–S) equation in the presence of naturally present state and input constraints. The novel features of proposed synthesis is the development of an infinite dimensional PDE state representation which incorporates the exact transformation of boundary into the distributed control setting and benefits arising from the property of decoupled boundary applied input actuation and K–S PDE modal states. Furthermore, the dissipative structure of the K–S spectral operator provides the foundation for the model modal based predictive controller (MMPC) synthesis which utilizes the finite dimensional state representation to formulate the quadratic objective function while the infinite dimensional K–S PDE state constraints are appropriately defined and cast in the form of a constrained quadratic programme. Finally, we demonstrate that if feasible, the MMPC achieves stabilization of the thin film thickness and satisfies naturally present state and input constraints. Numerical simulation of the boundary applied actuation evaluates the proposed method's performance.

Introduction

The dynamics of the falling film process in a vertical tube has been modelled by the Kuramoto–Sivashinsky equation (KSE) which attracted a lot of attention in the scientific community due to its complexity and ability to describe a large variety of physical phenomena like stability of the falling film fronts [1], [2], [3], unstable drift waves in plasma [4], unstable flame fronts [5], [6], [7], phase turbulence in the Belousov–Zhabotinsky reaction diffusion system [8], [9], and interfacial instabilities between two viscous fluids [10]. This fourth order dissipative partial differential equation (PDE), has been extensively studied in the context of a dynamical system and in particular the pioneering works [11], [12] provided the first description of complex dynamical features, such as global attractors and inertial manifolds for the K–S equation.

On the other hand, although there is no shortage of the research about dynamical features of the K–S equation, very few publications consider the control problem realizations associated with the K–S equation. The primary reason for the scarcity of the control and application relevant results are on one side the issues of the actuator design and/or measurement sensor placement in such complex systems, while on the other the issues of the appropriate controller synthesis that can cope with the real-time dynamical features of the K–S equation. One of the pioneering works on the control realization associated with the K–S equation is presented in [13], [14], in which spatially distributed actuators stabilize the spatially uniform system state through the full state and/or output gain based feedback structure. Along the same theme of contributions, the [15] explored reduced order techniques to synthesize linear and nonlinear quadratic regulators for the K–S equation spatially distributed control.

In the aforementioned spatially distributed feedback control realizations the periodic boundary conditions have been imposed in the KSE model, which is suitable from the stand point of mathematical tractability, but it removes the possibility of naturally present irregularities and wave interactions since periodic boundary conditions correspond to the monochromatic wave description in which the waves retain their wavelength as they grow in their amplitudes [1], [2]. However, considering that in most cases, the actuators can be only placed at the boundary of the system's domain, it is more practical to develop the boundary control scheme instead of the distributed one and to account for the wave interaction in the K–S equation model description. Therefore, motivated by the notion of the boundary applied actuation, which is quite frequently present in the engineering practice, it is of interest to address the issue of boundary applied actuation in the context of the K–S equation model describing the two-phase annular flow, see Fig. 1 [1], [2]. In particular, in the seminal work [1], [2] the height of the falling liquid film in the vertical tube is regulated by the upwind gas flow at the tube's inlet, and both the steady-state and experimental correlations are provided in order to relate the upwind gas flow at the tube's inlet and the steady-state thin film heights. We are motivated to explore a set of more physically meaningful boundary conditions, such as Derichlet and/or Neumann, since the upwind gas flow and the angle of the upwind gas flow front can be regarded as two boundary applied inputs that may effectively influence the dynamics of the K–S equation, see Fig. 2 and be possible explored in physical actuator realizations.

The control theory of boundary controlled distributed parameter systems, described by linear parabolic and/or hyperbolic PDEs, has been well established [16], [17], [18], [19], and the general framework of necessary conditions for the boundary applied actuation and stabilization has been provided. In particular, the issue of stabilization for a class of dissipative distributed parameter system is realized by the state space decomposition based on system modes, as the stabilization of a finite number of unstable modes yields the stabilization of the original infinite-dimensional system [20]. In the context of the K–S equation model [21], Liu and Krstic explored the problem of global boundary control of the K–S equation in the case when actuation appears under Dirichlet and Neumann boundary conditions. However, an important notion of the realizability and optimality of the boundary applied actuation within the K–S equation setting has not been fully explored, either due to the model complexity associated with the highly dissipative numerically destabilizing fourth-order spatial operator and distributed nature of the KSE on one side, or due to the complexity of formulating optimal boundary control problem in the infinite-dimensional setting that also accounts for the constraints on the allowable actuator effort and K–S equation state. In recent works, the optimal control strategy formulated as a model modal predictive control (MMPC) synthesis is developed for distributed applied actuation in the context of the K–S equation [22], and has been extended to the special case of boundary actuation considered in [23] where the stabilization and suboptimal control realization of the K–S controlled equation in the presence of constraints is realized.

Building on the aforementioned contributions, we explore the model modal predictive control (MMPC) synthesis for the K–S equation in this paper, due to its superior ability to handle states and inputs constrained control problem, which is common in practice since physically relevant actuators suffer from physical limitations on allowable inputs, and the key variables in the system are usually subjected to some constraints for safety and/or performance consideration. In particular, the discrete system based MMPC solves the finite horizon optimal control problem by solving the quadratic optimization problem, as the control vector moves achieving both unstable states stabilization and constraints satisfaction over the finite horizon length by the application of a specific feature of the MMPC which incorporates the state constraints relaxation method [24], [25], [26].

In this paper, we explore the issue of the physically meaningful boundary conditions that incorporate the upward air flow rate, and the features of possible actuator degrees of freedom that may be explored in the feedback control structure. In particular, we account for the upward air flow rate as an independent boundary applied input that is directly related to the steady-state thin film thickness on the basis of experimental and mathematical correlations provided in [1], [2], and we account for the possible actuator realization that relates the angle at which the air flow front enters the annular entrance which yields the input associated with the Newman boundary conditions. Our assumption is based on experimental correlations among the tangential stress that air front exerts on the relative film thickness height at the annual entrance and along the tube, and on the notion that the average film thickness in the moving coordinates has to be equal to the base steady state value as a unique conservation property. More important, the boundary conditions imposed in this work allow for the model representation of the wave interactions and irregularities inevitably present in physical system and we demonstrate the MMPC control framework that accounts successfully for actuator physical constraints.

This work, compared with the previous contributions, provides a complete and detailed analysis of the spectrum of the linearized operator of the K–S equation, and the K–S spatial operator features are explored by calculations of associated eigenvalues and eigenfunctions which are obtained as a function of physical parameters. In the considered model realization, for stabilization of the height of the falling film process, two available inputs are applied at the system's boundary. First, the upstream air flow rate μ(t) and a variation of the air front angle ω(t) at which the blown air is entering the annulus, see Fig. 2. An appropriately designed exact transformation is initially used to reformulate the original boundary control problem as an abstract boundary control problem and to decouple the influence of boundary applied inputs on the modal states of the K–S PDE. The transformation provides the coupling among the infinite-dimensional K–S equation modal states only through the input injection. The abstract boundary control problem is reformulated as an infinite-dimensional discrete system, used in the synthesis of a low-dimensional model modal predictive controller (MMPC). The model modal predictive controller benefits from the dominant diagonal feature of the K–S operator spectrum and from the separation among the finite number of unstable modes and the infinite-dimensional stable modal complement. The feasibility of constrained optimization constructed by the MMPC controller synthesis leads to stabilization, input and K–S PDE state constraint satisfaction, and we provide an insight into the features of the state constraint implementation by accounting for the evolution of the fast modal exponentially stable high frequency waves content in the travelling wave evolution associated with the thin film thickness. The proposed control problem formulation has been evaluated through simulations in the case of a full-state feedback control synthesis. The paper is organized as follows: the preliminaries section provides the information about the model setting; an infinite-dimensional state space is developed based on the boundary transformation and its formulation is presented in Section 3; in Section 4, the model modal predictive controller synthesis is demonstrated and discussion regarding constructions of the approximate cost function and applied input and state constraints satisfaction is presented. Finally, the simulation study for the K–S equation with the applied controller in Section 5 shows the performance of the proposed approach which is followed by the conclusion in Section 6.

Section snippets

Kuramoto–Sivashinsky equation

The K–S equation is a fourth order partial differential equation, given by: xt+vxζζζζ+xζζ+xζx=0y(t)=0lδ(ζζc)x(ζ,t)dζ with the boundary and initial conditions given as follows: x(0,t)=0,x(l,t)=μ(t)1+u2(t),xζ(0,t)=0,xζ(l,t)=u(t),x(ζ,0)=0 and subject to the following constraints: yminy(t)ymaxuminu(t)umaxμminμ(t)μmax x(ζ, t), denotes the thin film thickness and it is considered as the state variable in the separable Hilbert space H; ζ  [0, l] is the spatial coordinate and t  [0, ∞) is the

Infinite-dimensional state space model

In this section, we obtain the linear partial differential equation model for the falling thin film dynamics, xt=vxζζζζxζζB100B2x(ζ,t)=I00tx(ζ,t)|l=d1d201u(t)μ(t) which is obtained by linearization of the K–S equation Eq. (1) around the spatially uniform base steady state x(ζ, t) = 0. Moreover, the boundary conditions are also linearized such that the linearization of the x(l,t)=μ(t)/1+u(t)2 yields the following expression x(l, t) = d1u(t) + d2μ(t), where coefficients d1 and d2 are calculated

The model modal predictive controller synthesis MMPC

The model modal predictive controller for the dissipative distributed parameter system given by the K–S PDE benefits from the spectral separation of the operator eigenspectrum into few dynamically dominant modes and an infinite dimensional stable complement. Therefore, the PDE modes are separated into two subspaces: the slow modal space, whose diagonal eigenvalue matrix is a subset of Λ, denoted by Λs, and it is spanned by Hs=span{ϕ1,ϕ2,,ϕns} where ns is the number of slow modal states,

Simulation study

The K–S equation (1), (2) with the following parameters v=0.4 and l = 9 is considered in the simulation study as the relevant physical representation of the liquid film surface in vertical tubes with nonlinear wave interaction. In particular, it can be observed from Fig. 4 that the eigenfunctions of the K–S represent the spatial wave characteristics which do not posses periodicity (see Fig. 4, ϕ1(ζ) which is not periodic within domain) and therefore accounts for irregularities and wave

Summary

In this work, a boundary model predictive control scheme is proposed to regulate the falling thin film process modelled by the K–S equation. The original PDE system is transferred by the exact state transformation to the infinite state space model and the reduction technique is employed to obtain a finite dimensional model where the slow modal is captured for the purpose of controller synthesis. The model modal predictive control formulation is developed, explicitly incorporating output, input

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