Transient bounds for adaptive control systems

Adaptive control systems are essentially nonlinear and mechanisms to analyze their stability and transient response typically derive from more general nonlinear theories such as small gain arguments or passivity. In this note, we consider how these theories may be applied to adaptive control to quantify transient response. This involves the explicit description of the constants appearing in the passivity theorem and/or the small gain theorem which characterize both system gains and initial condition effects. The result is a fundamental connection between transient response bounds and uniform plant controllability, which connects initial state conditions with the input-output analysis. Applying these general theorems to adaptive control, we are able to interpret the uniform controllability condition as a persistency of excitation requirement and thereby to provide local bounds on transient response. The implication of these sufficient results is that without both bounds upon initial conditions and guarantees of excitation, potentially extreme transient excursions of system variables are possible even though global convergence and asymptotic performance are guaranteed