Comparison of Non-Deterministic Nonlinear Systems

We characterize a notion of system comparison, termed as $(T_e,\gamma,\delta)$-similarity, for non-deterministic nonlinear systems. Building on a similar notion recently proposed for stable linear systems, the proposed notion characterizes the dissimilarity between the outputs, measured using the $L_2$ norm, of two nonlinear dynamical systems in terms of their inputs and disturbances. By establishing a relationship between $(T_e,\gamma,\delta)$-similarity and differential dissipativity, we establish equivalence between $(T_e,\gamma,\delta)$-similarity of nonlinear systems and the $(T_e,\gamma,\delta)$-similarity of their differential dynamics. We characterize the $(T_e,\gamma,\delta)$-similarity for nonlinear systems as a Linear Matrix Inequality feasibility problem and also provide necessary and sufficient conditions for solving this feasibility problem. We demonstrate the utility of the proposed notion through its use in two applications: (i) robust hierarchical control applied to a planar aircraft and (ii) the improvement (or design) of abstract models applied to the Moore-Greitzer model and an electronic circuit.