On the performance of nonlinear dynamical systems under parameter perturbation

We present a method for analysing the deviation in transient behaviour between two parameterised families of nonlinear ODEs, as initial conditions and parameters are varied within compact sets over which stability is guaranteed. This deviation is taken to be the integral over time of a user-specified, positive definite function of the difference between the trajectories, for instance the L 2 norm. We use sum-of-squares programming to obtain two polynomials, which take as inputs the (possibly differing) initial conditions and parameters of the two families of ODEs, and output upper and lower bounds to this transient deviation. Equality can be achieved using symbolic methods in a special case involving Linear Time Invariant Parameter Dependent systems. We demonstrate the utility of the proposed methods in the problems of model discrimination, and location of worst case parameter perturbation for a single parameterised family of ODE models.

[1]  Graziano Chesi,et al.  On the role of homogeneous forms in robustness analysis of control systems , 2003 .

[2]  Fernando D. Bianchi,et al.  Gain scheduling control of variable-speed wind energy conversion systems using quasi-LPV models , 2005 .

[3]  Xiping Zhang,et al.  Parameter-Dependent Lyapunov Functions and Stability Analysis of Linear Parameter-Dependent Dynamical Systems , 2003 .

[4]  Ufuk Topcu,et al.  Robust Region-of-Attraction Estimation , 2010, IEEE Transactions on Automatic Control.

[5]  S. Prajna,et al.  On Model Reduction of Polynomial Dynamical Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[6]  Guido Buzzi-Ferraris,et al.  A new sequential experimental design procedure for discriminating among rival models , 1983 .

[7]  George H. Hines,et al.  Equilibrium-independent passivity: A new definition and numerical certification , 2011, Autom..

[8]  Antonis Papachristodoulou,et al.  Robust nonlinear stability and performance analysis of an F/A‐18 aircraft model using sum of squares programming , 2013 .

[9]  S. Bittanti,et al.  Affine Parameter-Dependent Lyapunov Functions and Real Parametric Uncertainty , 1996 .

[10]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[11]  Dimitri Peaucelle,et al.  Robust performance analysis with LMI-based methods for real parametric uncertainty via parameter-dependent Lyapunov functions , 2001, IEEE Trans. Autom. Control..

[12]  Antonis Papachristodoulou,et al.  Guaranteed error bounds for structured complexity reduction of biochemical networks. , 2012, Journal of theoretical biology.

[13]  A. Packard,et al.  Induced L/sub 2/-norm control for LPV system with bounded parameter variation rates , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[14]  A. Packard.  Gain scheduling via linear fractional transformations , 1994 .

[15]  S. Behtash,et al.  Design of controllers for linear parameter-varying systems by the gain scheduling technique , 1992 .

[16]  Antonis Papachristodoulou,et al.  On validation and invalidation of biological models , 2009, BMC Bioinformatics.

[17]  Pablo A. Parrilo,et al.  SOSTOOLS Version 3.00 Sum of Squares Optimization Toolbox for MATLAB , 2013, ArXiv.

[18]  James Anderson,et al.  Region of attraction analysis via invariant sets , 2014, 2014 American Control Conference.

[19]  Antonis Papachristodoulou,et al.  A Converse Sum of Squares Lyapunov Result With a Degree Bound , 2012, IEEE Transactions on Automatic Control.

[20]  Philipp Rumschinski,et al.  Methodology article Set-base dynamical parameter estimation and model invalidation for biochemical reaction networks , 2010 .

[21]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[22]  Steven P. Asprey,et al.  On the design of optimally informative dynamic experiments for model discrimination in multiresponse nonlinear situations , 2003 .

[23]  S. Prajna,et al.  SOS-based solution approach to polynomial LPV system analysis and synthesis problems , 2005 .

[24]  Pierre-Alexandre Bliman,et al.  A Convex Approach to Robust Stability for Linear Systems with Uncertain Scalar Parameters , 2003, SIAM J. Control. Optim..

[25]  Fen Wu,et al.  Induced L2‐norm control for LPV systems with bounded parameter variation rates , 1996 .

[26]  Pramod P. Khargonekar,et al.  On the control of linear systems whose coefficients are functions of parameters , 1984 .

[27]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[28]  Pablo A. Parrilo,et al.  Selecting a monomial basis for sums of squares programming over a quotient ring , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[29]  L. Perko.  Differential Equations and Dynamical Systems , 1991 .

[30]  John Doyle,et al.  Model validation: a connection between robust control and identification , 1992 .

[31]  Ufuk Topcu,et al.  Local Stability Analysis for Uncertain Nonlinear Systems , 2009, IEEE Transactions on Automatic Control.

[32]  Andrew Packard,et al.  Local robust performance analysis for nonlinear dynamical systems , 2009, 2009 American Control Conference.

[33]  Fernando D. Bianchi,et al.  Wind Turbine Control Systems: Principles, Modelling and Gain Scheduling Design , 2006 .

[34]  A. Papachristodoulou,et al.  A tutorial on sum of squares techniques for systems analysis , 2005, Proceedings of the 2005, American Control Conference, 2005..

[35]  F. Doyle,et al.  A benchmark for methods in reverse engineering and model discrimination: problem formulation and solutions. , 2004, Genome research.

[36]  G. Chesi.  Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems , 2009 .

[37]  Philipp Rumschinski,et al.  Set-base dynamical parameter estimation and model invalidation for biochemical reaction networks , 2010, BMC Systems Biology.

[38]  A. Packard,et al.  Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback , 1994 .

[39]  Andrew Packard,et al.  Stability Region Analysis Using Polynomial and Composite Polynomial Lyapunov Functions and Sum-of-Squares Programming , 2008, IEEE Transactions on Automatic Control.

[40]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[41]  Pierre-Alexandre Bliman,et al.  An existence result for polynomial solutions of parameter-dependent LMIs , 2004, Syst. Control. Lett..

[42]  Roy S. Smith,et al.  A generalization of the structured singular value and its application to model validation , 1998, IEEE Trans. Autom. Control..

[43]  Philipp Rumschinski,et al.  Combining qualitative information and semi‐quantitative data for guaranteed invalidation of biochemical network models , 2012 .

[44]  G. Chesi.  Domain of Attraction: Analysis and Control via SOS Programming , 2011 .

[45]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[46]  William G. Hunter,et al.  Designs for Discriminating Between Two Rival Models , 1965 .