This chapter has three objectives: the formulation of parametric models for the approximation problem; the design of online learning schemes; and the derivation of parameter estimation algorithms with certain stability and robustness properties. The perspective of this chapter is motivated in Section 4.1, where we use examples to develop some intuition into the adaptive approximation problem for unknown nonlinear functions that appear in the state equation model of a dynamical system. This section includes a formal definition of the adaptive approximation problem and a discussion of various key issues in parametric estimation. In the subsequent sections of this chapter, we describe in detail the procedure for designing online learning algorithms, which consists of three steps: (i) derivation of parametric models; (ii) design of online learning scheme; and (iii) derivation of parameter estimation algorithms. The overall learning approach is developed in a continuous-time framework, where it is assumed that the original dynamical system as well as the adaptive law evolve in continuous-time. The focus on this chapter is parameter estimation methods for adaptive function approximation, not adaptive approximation based control. The methods that are developed here will provide a foundation for the adaptive approximation based control approaches that are developed in Chapters 6 and 7.

Section 4.2 considers the derivation of parametric models. The objective in deriving a suitable parametric model is to rewrite the nonlinear differential equation model in a structured way such that the uncertainty appears in a desired fashion. Specifically, any unknown functions in the state variable model are replaced by approximators (potentially, of any form described in Chapter 3), such that the uncertainty is now converted into two components that will be treated differently:

• parameter uncertainty - unknown "optimal" weights of the approximator;
  
  
• functional approximation error - due to the approximator not being able to represent exactly the unknown function.

Based on the derived parametric model, in Section 4.3 we consider the design of online learning schemes. This step constructs an architecture for adaptive approximation. The architecture is tightly related to the parametric model derived in Section 4.2. Two types of online learning schemes will be investigated: the error filtering online learning scheme, and the regressor filtering online learning scheme. The final step of the design procedure, described in Section 4.4, deals with deriving adaptive laws for updating the parameter estimates (weights) that reside in the function approximator.

The stability and convergence properties of the learning architecture (under certain conditions) are formally analyzed in Section 4.5. In Section 4.6, we examine the case where the functional approximation error is nonzero, or there are external time-varying disturbances and/or measurement noise terms that cannot be approximated by the adaptive approximation scheme. In this situation, we consider the modification of the learning algorithms, leading to so called robust learning algorithms, and consider the stability and convergence properties of robust learning schemes. Finally, Section 4.7 provides some concluding remarks.