Nonlinear model reduction for control of distributed systems: A computer‐assisted study

Shvartsman, Stanislav Y.; Kevrekidis, Ioannis G.

doi:10.1002/aic.690440711

Cited by 156 publications

(67 citation statements)

References 92 publications

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“…Such "further reduced" models are often based on modal representations of the dynamics, the modes coming from the leading part of the linearized problem ( [8]), from their Krylov subspace approximations ( [44]), from empirically determined eigenfunctions ( [43], and references therein), or from an appropriate (dissipative) part of the linear problem operator, such as the eigenfunctions of the Stokes operator in the case of the Navier-Stokes equations ( [17], [73], [74]). Beyond the qualitative similarity with singular perturbation methods that construct invariant manifolds and exploit them in controller design for finite dimensional systems ( [57]), there is an extensive interest in the use of inertial manifolds and approximate inertial manifolds for closed loop dynamics analysis and controller design (see, for example, [12,15,69,70]). Motivated by the theory of Inertial Manifolds and Approximate Inertial Manifolds, we will also explore a second (Nonlinear Galerkin, NLG) model reduction step.…”

Section: Introductionmentioning

confidence: 99%

Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

Cao¹,

Kevrekidis²,

Titi³

2001

Indiana Univ. Math. J.

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ABSTRACT. This paper introduces an explicit numerical criterion for the stabilization of steady state solutions of the NavierStokes equations (NSE) with linear feedback control. Given a linear feedback controller that stabilizes a steady state solution to the closed-loop standard Galerkin (or nonlinear Galerkin) NSE discretization, it is shown that, if the number of modes involved in the computation is large enough, this controller stabilizes a nearby steady state of the closed-loop NSE. We provide an explicit estimate, in terms of the physical parameters, for the number of modes required in order for this criterion to hold. Moreover, we provide an estimate for the distance between the stabilized numerical steady state and the actually stabilized steady state of the closed-loop Navier-Stokes equations. More accurate approximation procedures, based on the concept of postprocessing the Galerkin results, are also presented. All the criterion conditions are imposed on the computed numerical solution, and no a priori knowledge is required about the steady state solution of the full PDE. This criterion holds for a large class of unbounded linear feedback operators and can be slightly modified to include certain nonlinear parabolic systems such as reaction-diffusion systems.

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Section: Introductionmentioning

confidence: 99%

Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

Cao¹,

Kevrekidis²,

Titi³

2001

Indiana Univ. Math. J.

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“…which constitutes the basis for the reduced order model (ROM) development (Shvartsman and Kevrekidis, 1998;Alonso et al, 2004b). Usually the number of modes necessary to obtain a good approximation is much lower than the number of equations required in classical methods such as finite elements or finite differences.…”

Section: The Kirchhoff Transform (3) Applied To This System Results Imentioning

confidence: 99%

“…This approach takes advantage of the spatial differential operator structure and uses the Galerkin method to approximate the system by a low-dimensional set of ODEs and to design the controller (Shvartsman and Kevrekidis, 1998). Christofides and coworkers -see, for example (Christofides and Daoutidis, 1996;Shi et al, 2006)-employed this approach to derive stabilizing controllers based on feed-back linearization and applied it to chemical systems such as tubular reactors or particulate processes, among others.…”

Section: Introductionmentioning

confidence: 99%

Robust feed-back control of distributed chemical reaction systems

Vilas¹,

García²,

Banga³

et al. 2007

Chemical Engineering Science

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There are many distributed processes in the chemical industry as it is the case of tubular reactors in which the parameters or the structure of the reaction terms are only a rough approximation of reality. In order to efficiently control this kind of systems, it is important to take into account this lack of detailed information (robustness). In this work, we make use of the classical theory on the robust nonlinear control for finite dimensional systems and extend it to distributed process systems by taking advantage of the special nature of dissipative systems. In this way, theoretical issues related to the nonlinearity of the diffusion terms and inhomogeneous boundary conditions are handled by means of the Kirchhoff and state transformations, respectively. In addition, and for practical reasons, the problem of controller saturation is considered. The different aspects of the methodology will be illustrated through a number of computational experiments concerning non-isothermal tubular reactors with convection and/or diffusion terms.

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“…As discussed by many authors before -see for instance (Shvartsman and Kevrekidis, 1998) and references therein-the FHN system has a rich variety of dynamic behaviors as its solution is highly dependent on the values of the diffusion and kinetic parameters.…”

Section: Description Of the Reaction-diffusion System And Control Promentioning

confidence: 99%

“…Alternatively, and in the context of chemical systems, Shvartsman and Kevrekidis (1998) and Shvarstman et al (2000) made use of a particular reduced order representation of the original one dimensional distributed FHN system to carry out bifurcation analysis. The reduced order dynamic model was also employed to design a simple feedback controller that drove the system to a pre-defined limit cycle.…”

Section: Introductionmentioning

confidence: 99%

Stabilization of inhomogeneous patterns in a diffusion–reaction system under structural and parametric uncertainties

Vilas¹,

García²,

Banga³

et al. 2006

Journal of Theoretical Biology

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Many phenomena such as neuron firing in the brain, the travelling waves which produce the heartbeat, arrythmia and fibrillation in the heart, catalytic reactions or cellular organization activities, among others, can be described by a unifying paradigm based on a class of nonlinear reaction-diffusion mechanisms. The FitzHughNagumo (FHN) model is a simplified version of such class which is known to capture most of the qualitative dynamic features found in the spatiotemporal signals. In this paper, we take advantage of the dissipative nature of diffusion-reaction systems and results in finite dimensional nonlinear control theory to develop a class of nonlinear feed-back controllers which is able to ensure stabilization of moving fronts for the FHN system, despite structural or parametric uncertainty. In the context of heart or neuron activity, this class of control laws is expected to prevent cardiac or neurological disorders connected with spatiotemporal wave disruptions. In the same way, biochemical or cellular organization related with certain functional aspects of life could also be influenced or controlled by the same feed-back logic. The stability and robustness properties of the controller will be proved theoretically and illustrated on simulation experiments.

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Nonlinear model reduction for control of distributed systems: A computer‐assisted study

Cited by 156 publications

References 92 publications

Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

Numerical criterion for the stabilization of steady states of the Navier-Stokes equations

Robust feed-back control of distributed chemical reaction systems

Stabilization of inhomogeneous patterns in a diffusion–reaction system under structural and parametric uncertainties

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