L₂-Suboptimal Control for Nonlinear Systems via Convex Optimization

Analogous to ${\mathcal {H}}_{\infty }$ -methods for linear time invariant systems, ${\mathcal {L}}_{2}$ -gain is an important input-output characterization of robustness for nonlinear systems. The Hamilton-Jacobi Inequality can be used to establish ${\mathcal {L}}_{2}$ -gain if an appropriate storage function can be identified. Continuous piecewise affine storage functions for small-signal ${\mathcal {L}}_{2}$ -stability analysis of nonlinear systems have previously been applied to open-loop analysis. Here, they are used to develop a suboptimal controller synthesis method for nonlinear systems that minimizes the local, closed-loop small-signal ${\mathcal {L}}_{2}$ -gain. The method selects a piecewise affine state feedback controller using convex optimization.