Stability problems associated with the transverse feedback linearization normal form

Transverse feedback linearization (TFL) has previously been proposed as a method to solve set stabilization problems. TFL allows a designer to separately design a control law to stabilize the target set and a controller to achieve the desired motion on the set. However, care must be taken to ensure that the designed controllers work as expected when combined together. Two new results on the exponential stability of cascade connected systems are provided. Two examples are used to illustrate how these results can be applied to the stability analysis of set stabilization problems using TFL.