A contraction approach to input tracking via high gain feedback

This paper adopts a contraction approach to study exogenous input tracking in dynamical systems under high gain proportional output feedback. We give conditions under which contraction of a nonlinear system's tracking error implies input to output stability from the input signal's time derivatives to the tracking error. This result is then used to demonstrate that the negative feedback connection of plants composed of two strictly positive real subsystems in cascade can follow external inputs with tracking errors that can be made arbitrarily small by applying a sufficiently large feedback gain. We utilize this result to design a biomolecular feedback regulation scheme for a synthetic genetic sensor model, making it robust to variations in the availability of a cellular resource required for protein production.

[1]  F. Hoppensteadt.  Singular perturbations on the infinite interval , 1966 .

[2]  Domitilla Del Vecchio,et al.  Retroactivity Attenuation in Bio-Molecular Systems Based on Timescale Separation , 2011, IEEE Transactions on Automatic Control.

[3]  Thomas Lorenz,et al.  Mutational Analysis: A Joint Framework for Cauchy Problems in and Beyond Vector Spaces , 2010 .

[4]  Domitilla Del Vecchio,et al.  Modularity, context-dependence, and insulation in engineered biological circuits , 2015 .

[5]  Jean-Jacques E. Slotine,et al.  On partial contraction analysis for coupled nonlinear oscillators , 2004, Biological Cybernetics.

[6]  Eduardo D. Sontag,et al.  Lyapunov Characterizations of Input to Output Stability , 2000, SIAM J. Control. Optim..

[7]  H. Khalil,et al.  Adaptive stabilization of a class of nonlinear systems using high-gain feedback , 1987 .

[8]  Ali Saberi,et al.  Adaptive stabilization of a class of nonlinear systems using high-gain feedback , 1986, 1986 25th IEEE Conference on Decision and Control.

[9]  D. Vecchio,et al.  Biomolecular Feedback Systems , 2014 .

[10]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[11]  Domitilla Del Vecchio,et al.  A Contraction Theory Approach to Singularly Perturbed Systems , 2013, IEEE Transactions on Automatic Control.

[12]  Eduardo Sontag.  Contractive Systems with Inputs , 2010 .

[13]  Petar V. Kokotovic,et al.  Singular perturbations and time-scale methods in control theory: Survey 1976-1983 , 1982, Autom..

[14]  W. Marsden.  I and J , 2012 .

[15]  Domitilla Del Vecchio,et al.  Limitations and trade-offs in gene expression due to competition for shared cellular resources , 2014, CDC.

[16]  Ali Saberi.  Output-feedback control with almost-disturbance-decoupling property—a singular perturbation approach , 1987 .

[17]  Peddapullaiah Sannuti,et al.  Direct singular perturbation analysis of high-gain and cheap control problems , 1983, Autom..

[18]  Domitilla Del Vecchio,et al.  Effective interaction graphs arising from resource limitations in gene networks , 2015, 2015 American Control Conference (ACC).

[19]  Nathan van de Wouw,et al.  Convergent dynamics, a tribute to Boris Pavlovich Demidovich , 2004, Syst. Control. Lett..

[20]  C. D. Goldstein.  Asymptotic Unbounded Root Loci-. Formulas and Computation , 2001 .

[21]  Hassan K. Khalil,et al.  Nonlinear Output-Feedback Tracking Using High-gain Observer and Variable Structure Control, , 1997, Autom..

[22]  Ali Saberi,et al.  Output feedback control with almost disturbance decoupling property: A singular perturbation approach , 1984, The 23rd IEEE Conference on Decision and Control.

[23]  Eduardo D. Sontag,et al.  Passivity-based Stability of Interconnection Structures , 2008, Recent Advances in Learning and Control.

[24]  Vadim I. Utkin,et al.  A singular perturbation analysis of high-gain feedback systems , 1977 .

[25]  Eduardo D. Sontag,et al.  Diagonal stability of a class of cyclic systems and its connection with the secant criterion , 2006, Autom..

[26]  P. Kokotovic.  Applications of Singular Perturbation Techniques to Control Problems , 1984 .

[27]  U. Shaked,et al.  Asymptotic behaviour of root-loci of linear multivariable systems , 1976 .

[28]  Eduardo Sontag.  Input to State Stability: Basic Concepts and Results , 2008 .

[29]  R. Sousa,et al.  T7 RNA polymerase. , 2001, Progress in nucleic acid research and molecular biology.

[30]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[31]  Hassan K. Khalil,et al.  Singular perturbation methods in control : analysis and design , 1986 .

[32]  Eduardo Sontag,et al.  Input‐to‐state stability with respect to inputs and their derivatives , 2003 .

[33]  Arjan van der Schaft,et al.  On differential passivity of physical systems , 2013, 52nd IEEE Conference on Decision and Control.

[34]  Domitilla Del Vecchio,et al.  Mitigation of resource competition in synthetic genetic circuits through feedback regulation , 2014, 53rd IEEE Conference on Decision and Control.

[35]  R. Sousa,et al.  T 7 RNA polymerase. , 2001, Uirusu.

[36]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[37]  Frank Allgöwer,et al.  Adaptive lambda-tracking for nonlinear higher relative degree systems , 2005, Autom..

[38]  J. Edmunds,et al.  Multivariate root loci: a unified approach to finite and infinite zeros , 1979 .

[39]  C. Desoer,et al.  The measure of a matrix as a tool to analyze computer algorithms for circuit analysis , 1972 .

[40]  P. Olver.  Nonlinear Systems , 2013 .

[41]  Zahra Aminzare,et al.  Contraction methods for nonlinear systems: A brief introduction and some open problems , 2014, CDC.

[42]  R. Marino.  High-gain feedback in non-linear control systems† , 1985 .

[44]  Domitilla Del Vecchio,et al.  Modularity, context-dependence, and insulation in engineered biological circuits. , 2015, Trends in biotechnology.

[45]  Eduardo Sontag,et al.  Notions of input to output stability , 1999, Systems & Control Letters.

[46]  P. Kokotovic,et al.  On stability properties of nonlinear systems with slowly varying inputs , 1991 .