• 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 ,