On the gain margin of nonlinear and optimal regulators

The robustness of nonlinear regulators for nonlinear systems with respect to variations in gain is investigated. It is shown that there exist regulators that produce asymptotically stable closed-loop systems, but do not tolerate any variation in gain without instability. However, if the linearized closed-loop system is also asymptotically stable, then there is always some gain margin. For a wide class of optimal regulators, it is shown that the gain margin is infinite with respect to increases in gain and that decreases down to 0.5 can be tolerated. The robustness properties of linear quadratic control laws are thus generalized.