# 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 ,