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 L2 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.