Adaptive learning control for nonlinear systems with extended matching unstructured uncertainties

The output tracking control problem via state feedback is addressed for a class of single input-single output nonlinear systems which are affected by extended matching unstructured uncertainties. Under the assumption that the output reference signal is sufficiently smooth and periodic with known period, a robust adaptive learning control is designed, which learns the unstructured unknown periodic disturbance signals due to system uncertainties by identifying the Fourier coefficients of any truncated approximation while guaranteeing L₂ and L_¿ transient performances. For any initial condition of the system in an arbitrary given compact set, by properly setting the control parameters: i) the output tracking error exponentially converges to a residual set which may be arbitrarily reduced by increasing the number of terms in the truncated Fourier series expansions; ii) when the unknown periodic disturbances can be represented by finite Fourier series expansions, the output tracking error exponentially converges to zero.