PDE Backstepping Boundary Observer Design for Microfluidic Systems

In this brief, we explore the use of conformal mapping theory to reduce the complexity in PDE backstepping boundary observer design. The technique is applied to a genetic analysis microchip that features a collocated sensor-actuator architecture in which the temperature of the reaction chamber is the spatially distributed control variable. The size and structure of the microchip do not allow for sensor placement within the reaction chamber, making temperature estimation mandatory. The PDE backstepping boundary observer design is chosen to provide real-time data from the temperature inside the microchip. The standard PDE backstepping boundary observer design results in a partial differential equation for the kernel function with double the spatial dimension of the original problem, which makes the design intractable for the problems with dimensions higher than one. We show that the spatial domain of the original problem can be reduced with the use of the conformal mapping. The resulting observer is tested and experimentally validated, shown excellent performance with respect to the spacial L2 norm of the estimation error.

[1]  S. Rosenberg.  The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds , 1997 .

[2]  Hye Jung Cho,et al.  Precise temperature control and rapid thermal cycling in a micromachined DNA polymerase chain reaction chip , 2002 .

[3]  Ramon Miranda,et al.  Observer design for a class of parabolic PDE via sliding modes and backstepping , 2010, 2010 11th International Workshop on Variable Structure Systems (VSS).

[4]  Jana Lauzon,et al.  An inexpensive and portable microchip-based platform for integrated RT-PCR and capillary electrophoresis. , 2008, The Analyst.

[5]  Robin Fortt,et al.  Thermal optimisation of the Reimer-Tiemann reaction using thermochromic liquid crystals on a microfluidic reactor. , 2005, Lab on a chip.

[6]  K. Goodson,et al.  Transient liquid crystal thermometry of microfabricated PCR vessel arrays , 1998 .

[7]  J. W. Brown,et al.  Complex Variables and Applications , 1985 .

[8]  J P Landers,et al.  Polymerase chain reaction in polymeric microchips: DNA amplification in less than 240 seconds. , 2001, Analytical biochemistry.

[9]  M. Athans,et al.  Toward a practical theory for distributed parameter systems , 1970, IEEE Transactions on Automatic Control.

[10]  W. Su,et al.  Output-feedback boundary control of an uncertain heat equation with noncollocated observation: A sliding-mode approach , 2010, 2010 5th IEEE Conference on Industrial Electronics and Applications.

[11]  M. Amouroux,et al.  Sensors and observers in distributed parameter systems , 1988 .

[12]  J. W. Brown,et al.  Fourier series and boundary value problems , 1941 .

[13]  Yonghao Zhang,et al.  Microfluidic DNA amplification--a review. , 2009, Analytica chimica acta.

[14]  Christopher J. Backhouse,et al.  Nonlinear Controller Designs for Thermal Management in PCR Amplification , 2012, IEEE Transactions on Control Systems Technology.

[15]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[16]  M. Krstić,et al.  Backstepping observers for a class of parabolic PDEs , 2005, Syst. Control. Lett..

[17]  T Kitamori,et al.  Non-contact photothermal control of enzyme reactions on a microchip by using a compact diode laser. , 2000, Journal of chromatography. A.

[18]  Y. A. LlU,et al.  Observer theory for distributed-parameter systems , 1976 .

[19]  Mikhail Belkin,et al.  Towards a theoretical foundation for Laplacian-based manifold methods , 2005, J. Comput. Syst. Sci..

[20]  M. A. Northrup,et al.  DNA Amplification with a Microfabricated Reaction Chamber , 1993 .

[21]  Shinji Hara,et al.  Backstepping observer using weighted spatial average for 1-dimensional parabolic distributed parameter systems ⋆ , 2011 .

[22]  G V Kaigala,et al.  A scalable and modular lab-on-a-chip genetic analysis instrument. , 2010, The Analyst.

[23]  Tobin A. Driscoll,et al.  Schwarz-Christoffel Toolbox User''s Guide , 1994 .

[24]  Christopher J. Backhouse,et al.  PDE Modeling of a Microfluidic Thermal Process for Genetic Analysis Application , 2013, J. Appl. Math..

[25]  Lloyd N. Trefethen,et al.  Schwarz-Christoffel Mapping , 2002 .

[26]  Miroslav Krstic,et al.  Closed-form boundary State feedbacks for a class of 1-D partial integro-differential equations , 2004, IEEE Transactions on Automatic Control.

[27]  Engin Yaz,et al.  Receding window observer and dynamic feedback control of discrete infinite dimensional systems , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[28]  Govind V Kaigala,et al.  Automated screening using microfluidic chip‐based PCR and product detection to assess risk of BK virus‐associated nephropathy in renal transplant recipients , 2006, Electrophoresis.

[29]  M. Krstić,et al.  Boundary Control of PDEs , 2008 .

[30]  R. Gressang,et al.  Observers for systems characterized by semigroups , 1975 .

[31]  James P Landers,et al.  Advances in polymerase chain reaction on microfluidic chips. , 2005, Analytical chemistry.

[32]  M. Krstić.  Boundary Control of PDEs: A Course on Backstepping Designs , 2008 .

[33]  Darwin R. Reyes,et al.  Microwave dielectric heating of fluids in an integrated microfluidic device , 2007 .

[34]  Masataro Nishimura,et al.  Observer for distributed-parameter diffusion systems , 1972 .