The approach to implementation and debugging presented in this appendix has been defined based on interactions with numerous students and colleagues. The objective is to correctly implement a working adaptive approximation based controller.

1. Derive a state space model for the plant that is of interest. Relative to the model, clearly record which portions are known and which are not. Denote the unknown functions by _iwhere i counts over the number of unknown functions.
   
   
2. Choose a control design approach. For this approach, assume for a moment that all portions of the model are known. Derive a control law applicable to this known system that is provably stable. Note the stability properties that are expected.
   
   
3. Implement a simulation of the state space system. Also, implement the controller equations. In the controller, let the symbol _irepresent the approximation to _i. Make sure that the controller implements _ias a clearly distinguishable entity as it will be replaced later. For this step in the debugging process, assume some reasonable function for each _iand let _i = _i. With this perfect modeling, the stability properties provable in the previous step should hold exactly.
   
   
4. Run the simulation from various initial conditions and with various commanded trajectories. Make sure that all proven stability properties hold. For example, if you have proven that the derivative of a function V is negative definite, then make sure that it is in the simulation. If any proven stability properties do not hold, even intermittently, then debugging is required. If any bugs are not removed at this step, then they may lead to misinterpretations or instability later.
   
   
5. Parameterize each unknown function: _i= (θ*)^T ø(x, σ*) + e_i(x).
   
   
6. Derive parameter adaptation laws for θ and σ such that the adaptive closed-loop system has the desired set of stability properties required for the application conditions.
   
   
7. Modify the simulation from Step 3 so that _i = θ^T ø(x, σ) where θ and σ are estimated by the methods determined in Step 6. It is particularly important that relative to the working simulation from Step 3, the only changes should be those required to change the _ifunctions to the form required for adaptive approximation.
   
   
8. Run the simulation from various initial conditions and with various commanded trajectories. Make sure that all proven stability properties hold. Assuming that the simulation was properly debugged in Step 3, this step should only involve tuning and debugging of the approximator and parameter estimation routines.
   
   
9. Translate the adaptive approximation based controller resulting from the above process to the platform required for actual implementation.

It is important to not skip Steps 3 and 4. Skipping those steps can result in bugs in the basic control law implementation being misinterpreted as problems or bugs in the adaptive approximation process. The above stepwise derivation and debugging approach decomposes the problem into pieces that can be separably solved, analyzed, and debugged.