The Control Lie Algebra (CLA)

For a given state transition equation of the form (8.1), consider the set of all vector fields that can be generated by taking Lie brackets, $[\alpha^i(x)$, $\alpha^j(x)$, of vector fields $\alpha^i(x)$ and $\alpha^j(x)$ for $i \not = j$. Next, consider taking Lie brackets of the new vector fields with each other, and with the original vector fields. This process can be repeated indefinitely by iteratively applying the Lie bracket operations to new vector fields. The resulting set of vector fields can be considered as a kind of algebraic closure with respect to the Lie bracket operation. Let the control Lie algebra, $CLA(\triangle)$, denote the set of all vector fields that are obtained by this process.

In general, $CLA(\triangle)$ can be considered as a vector space, in which the basis elements are the vector fields $\alpha^1(x)$, $\dots$,$\alpha^m(x)$, and all new, linearly-independent vector fields that were generated from the Lie bracket operations.

The process of finding the basis of $CLA(\triangle)$ is generally a tedious process. There are several systematic approaches for generating the basis, one of which is called the Phillip-Hall basis. The vector fields that should be generated for the first two steps of the Phillip-Hall approach are given below. Each Lie bracket has the opportunity to generate a vector field that is linearly-independent; however, it is not guaranteed to generate one. In fact, all Lie bracket operations may fail to generate a vector field that is independent of the original vector fields. Consider for example, the case in which the original vector fields, $\alpha^i$, are all constant. All Lie brackets will be zero.