  We discuss a design approach for nonlinear discrete-time adaptive observer.
This involves transforming a nonlinear system into a quasi-LPV (Linear
Parameter Varying) polytopic model in Takagi-Sugeno (T-S) form using nonlinear
embedding and sector nonlinearity (SNL) transformation. We then develop a
discrete-time counterpart for a joint state and parameter estimation, based on
design strategies developed for continuous time models in the existing
literature. The design uses a Lyapunov approach and provides an error bounded
by $\mathbb{L}_2$ gain. Based on this strategy, we propose a design for
adaptive observers for nonlinear systems whose T-S form can have unmeasured
premise variables.
