Adaptive/Robust Backstepping Techniques
- Uncertainties
Structured(or parametric) uncertainties
Corresponds to inaccuracies on terms actually included in the model
Unstructured uncertainties(or unmodeled dynamics)
Corresponds to inaccuracies on(i.e., underestimation of) the system order
Adaptive Backstepping
Consider a nonlinear system
where the parameter is unknown, we assume that is constant.
Objective: as
For this system the adaptive controller design is in two steps.
Step 1
We introduce and and consider as a control to stabilize the -system with respect to the Lyapunov function
The -system and the corresponding are:
let , and if
Objective:
Substitute and
we obtain:
where should be taken off.
Step 2
With , the original system has been transformed into
Choose Lyapunov Function
Objective:
where
Robust Backstepping
Consider a nonlinear system
we assume that the unknown parameter belongs to a known interval
Objective: as
For this system the robust controller design is in two steps.
Step 1
To apply backstepping, we design a -controller for
Choose Lyapunov function
Several different designs are possible:
where is a smooth switching function.
Step 2
We now employ backstepping to design -controllers for the second order system.
Choose Lyapunov function:
Choose
and we have
Objective:
We should choose to satisfy :
- When ,
- When ,