An enhanced feedback adaptive observer for nonlinear systems with lack of persistency of excitation

The design of exponentially convergent adaptive observers is addressed for linear observable systems which are perturbed by linearly parameterized nonlinearities depending on measured signals (inputs and outputs). When there is a lack of persistency of excitation a new robust adaptive observer is presented, which performs an additional feedback depending on the kernel of the Gramian of the regressor vector, which is computed online, and generates state variables estimates whose estimation errors are exponentially convergent to zero, provided that a design parameter is chosen to be sufficiently small. The boundedness of the parameter and observer estimation errors is always guaranteed. Parameter estimates do not converge to their true values unless the regressor vector is persistently exciting (i.e., the Gramian of the regressor vector is nonsingular). In this case, a well-known exponentially convergent adaptive observer is reobtained, since the additional feedback is zero.