Real-time nonlinear feedback control of pattern formation in (bio)chemical reaction-diffusion processes: a model study.

Theoretical and experimental studies related to manipulation of pattern formation in self-organizing reaction-diffusion processes by appropriate control stimuli become increasingly important both in chemical engineering and cellular biochemistry. In a model study, we demonstrate here exemplarily the application of an efficient nonlinear model predictive control (NMPC) algorithm to real-time optimal feedback control of pattern formation in a bacterial chemotaxis system modeled by nonlinear partial differential equations. The corresponding drift-diffusion model type is representative for many (bio)chemical systems involving nonlinear reaction dynamics and nonlinear diffusion. We show how the computed optimal feedback control strategy exploits the system inherent physical property of wave propagation to achieve desired control aims. We discuss various applications of our approach to optimal control of spatiotemporal dynamics.

[1]  L. G. Stern,et al.  Fractional step methods applied to a chemotaxis model , 2000, Journal of mathematical biology.

[2]  R. Bellman.  Dynamic programming. , 1957, Science.

[3]  J. L. Hudson,et al.  Chemical complexity: Spontaneous and engineered structures , 2003 .

[4]  H. Petty,et al.  Imaging sustained dissipative patterns in the metabolism of individual living cells. , 2000, Physical review letters.

[5]  S. Sinha,et al.  Targeting spatiotemporal patterns in extended systems with multiple coexisting attractors. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  H. Berg,et al.  Dynamics of formation of symmetrical patterns by chemotactic bacteria , 1995, Nature.

[7]  T Sakurai,et al.  Experimental and theoretical studies of feedback stabilization of propagating wave segments. , 2002, Faraday discussions.

[8]  Moritz Diehl,et al.  Real-Time Optimization for Large Scale Nonlinear Processes , 2001 .

[9]  Hans G. Othmer,et al.  The Diffusion Limit of Transport Equations II: Chemotaxis Equations , 2002, SIAM J. Appl. Math..

[10]  Oppo,et al.  Stabilization, Selection, and Tracking of Unstable Patterns by Fourier Space Techniques. , 1996, Physical review letters.

[11]  D Lebiedz,et al.  Manipulation of surface reaction dynamics by global pressure and local temperature control: a model study. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Benno Hess,et al.  Spatial Dissipative Structures in Yeast Extracts , 1980 .

[13]  Johannes P. Schlöder,et al.  An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects , 2003, Comput. Chem. Eng..

[14]  Milos Dolnik,et al.  Oscillatory cluster patterns in a homogeneous chemical system with global feedback , 2000, Nature.

[15]  Dirk Lebiedz,et al.  Dynamic control and information processing in chemical reaction systems by tuning self-organization behavior. , 2004, Chaos.

[16]  Michael A. Henson,et al.  Nonlinear model predictive control: current status and future directions , 1998 .

[17]  Daniel Walgraef,et al.  Spatio-temporal pattern formation , 1996 .

[18]  F. Finocchi,et al.  Chemical Reactions and Dust Destruction in Protoplanetary Accretion Disks , 1996 .

[19]  Dirk Lebiedz,et al.  Manipulation of self-aggregation patterns and waves in a reaction-diffusion system by optimal boundary control strategies. , 2003, Physical review letters.

[20]  Johannes P. Schlöder,et al.  An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization: Part II: Software aspects and applications , 2003, Comput. Chem. Eng..

[21]  H Maurer,et al.  External optimal control of self-organisation dynamics in a chemotaxis reaction diffusion system. , 2004, Systems biology.

[22]  S. Sinha,et al.  Adaptive control of spatially extended systems: targeting spatiotemporal patterns and chaos , 1998 .

[23]  Philip Ball,et al.  The Self-Made Tapestry: Pattern Formation in Nature , 1999 .

[24]  W. Baxter,et al.  Spiral waves of excitation underlie reentrant activity in isolated cardiac muscle. , 1993, Circulation research.

[25]  Alexander S. Mikhailov,et al.  Controlling Chemical Turbulence by Global Delayed Feedback: Pattern Formation in Catalytic CO Oxidation on Pt(110) , 2001, Science.

[26]  MORITZ DIEHL,et al.  A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control , 2005, SIAM J. Control. Optim..

[27]  Lorenz T. Biegler,et al.  Efficient Solution of Dynamic Optimization and NMPC Problems , 2000 .

[28]  Grégoire Nicolis,et al.  Self-Organization in nonequilibrium systems , 1977 .

[29]  J. Keener,et al.  Singular perturbation theory of traveling waves in excitable media (a review) , 1988 .

[30]  H. Petty,et al.  Dynamic Chemical Instabilities in Living Cells May Provide a Novel Route in Drug Development , 2004, Chembiochem : a European journal of chemical biology.

[31]  Hans Bock,et al.  An Efficient Algorithm for Nonlinear Model Predictive Control of Large-Scale Systems Part I: Description of the Method (Ein effizienter Algorithmus für die nichtlineare prädiktive Regelung großer Systeme Teil I: Methodenbeschreibung) , 2002 .

[32]  Celso Grebogi,et al.  Using small perturbations to control chaos , 1993, Nature.

[33]  宁北芳,et al.  疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A , 2005 .

[34]  Wolfgang Dahmen,et al.  Introduction to Model Based Optimization of Chemical Processes on Moving Horizons , 2001 .

[35]  H. Berg,et al.  Chemotaxis of bacteria in glass capillary arrays. Escherichia coli, motility, microchannel plate, and light scattering. , 1990, Biophysical journal.

[36]  Grigory V. Osipov,et al.  CONTROLLED MOVEMENT AND SUPPRESSION OF SPIRAL WAVES IN EXCITABLE MEDIA , 1998 .

[37]  J. Murray,et al.  Model and analysis of chemotactic bacterial patterns in a liquid medium , 1999, Journal of mathematical biology.

[38]  Cornelia Denz,et al.  Manipulation, Stabilization, and Control of Pattern Formation Using Fourier Space Filtering , 1998 .

[39]  Dirk Lebiedz,et al.  Specific external forcing of spatiotemporal dynamics in reaction-diffusion systems. , 2005, Chaos.

[40]  Malur K. Sundareshan,et al.  Design of Decentralized Observation Schemes for Large-Scale Interconnected Systems: Some New Results , 1989, 1989 American Control Conference.

[41]  R Martin,et al.  Controlling pattern formation and spatio-temporal disorder in nonlinear optics. , 1997, Optics express.

[42]  Moritz Diehl,et al.  Efficient NMPC of unstable periodic systems using approximate infinite horizon closed loop costing , 2004, Annu. Rev. Control..

[43]  M. Diehl,et al.  Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations , 2000 .

[44]  H. Berg,et al.  Complex patterns formed by motile cells of Escherichia coli , 1991, Nature.

[45]  M. Diehl,et al.  Nominal stability of real-time iteration scheme for nonlinear model predictive control , 2005 .

[46]  R Neubecker,et al.  Manipulation and removal of defects in spontaneous optical patterns. , 2003, Physical review letters.

[47]  Johannes P. Schlöder,et al.  Real-Time Optimization for Large Scale Processes: Nonlinear Model Predictive Control of a High Purity Distillation Column , 2001 .

[48]  A G Papathanasiou,et al.  Spatiotemporal Addressing of Surface Activity , 2001, Science.

[49]  Howard R. Petty,et al.  Dissipative metabolic patterns respond during neutrophil transmembrane signaling , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[50]  H. Petty,et al.  Apparent role of traveling metabolic waves in oxidant release by living neutrophils , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[51]  A. V. Mamaev,et al.  SELECTION OF UNSTABLE PATTERNS AND CONTROL OF OPTICAL TURBULENCE BY FOURIER PLANE FILTERING , 1998 .

[52]  Kenneth Showalter,et al.  Design and Control of Wave Propagation Patterns in Excitable Media , 2002, Science.