Gap Metrics, Representations, and Nonlinear Robust Stability

Abstract

Various alternative definitions for the nonlinear H2 -, L2 -, and $\nu$-gap metrics are studied. The concept of $\beta$-conjugacy and multiplicative homogeneity are introduced to relate the metrics to each other and to compare the stability margins of nonlinear feedback loops expressed in terms of the norms of complementary parallel projections. Left and right representations for the graph of a nonlinear system are studied. A new definition of "normalized" is introduced for left representations. Formulas for the gap metrics as the norm of the product of left and right representations are derived. The problem of controller synthesis for input-affine nonlinear systems to achieve norm bounds on the parallel projection operators is studied for input-affine nonlinear systems. The duality between the optimization of the two parallel projections is highlighted. State-space realizations for the normalized left and right representations are derived using nonlinear $H_\infty$ synthesis methods.