The paper treats a question of adaptive proportional-integral (PI) and adaptive proportional-integral-derivative (PID) controller design for nonlinear systems as well as the design of a universal adaptive controller which is an extension of the adaptive PI (PID) control scheme. The presented design methodology guarantees desired output transient performance indices by inducing of two-time-scale motions in the closed-loop system where the controller dynamics is a singular perturbation with respect to the system dynamics. Stability conditions imposed on the fast and slow modes and sufficiently large mode separation rate between fast and slow modes can ensure that the full-order closed-loop nonlinear system achieves the desired properties in such a way that the output transient performances are desired and insensitive to external disturbances and plant's parameter variations. The novelty in the paper is that the high-frequency-gain online identification and adaptive gain tuning are incorporated in the control system in order to maintain the two-time-scale structure in the closed-loop system trajectories and stability of fast-motion transients for a large range of plant's parameter variations. The singular perturbation method is used through-out the paper in order to get explicit expressions for evaluation of the controller parameters. Numerical example and simulation results are presented.