15.4.3.5 Handling Control-Affine Systems with Drift

Determining whether a system with drift (15.52), is STLC is substantially more difficult. Imagine a mechanical system, such as a hovercraft, that is moving at a high speed. Due to momentum, it is impossible from most states to move in certain directions during an arbitrarily small interval of time. One can, however, ask whether a system is STLC from a state $x \in X$ for which $h_0(x) = 0$. For a mechanical system, this usually means that it starts at rest. If a system with drift is STLC, this intuitively means that it can move in any direction by hovering around states that are close to zero velocity for the mechanical system.

The Lie algebra techniques can be extended to determine controllability for systems with drift; however, the tools needed are far more complicated. See Chapter 7 of [156] for more complete coverage. Even if $\dim({\cal L}({\triangle})) = n$, it does not necessarily imply that the system is STLC. It does at least imply that the system is accessible, which motivates the definition given in Section 15.1.3. Thus, the set of achievable velocities still has dimension $n$; however, motions in all directions may not be possible due to drift. To obtain STLC, a sufficient condition is that the set of possible values for ${\dot x}$ contains an open set that contains the origin.

The following example clearly illustrates the main difficultly with establishing whether a system with drift is STLC.

Example 15..20 (Accessible, Not STLC)   The following simple system clearly illustrates the difficulty caused by drift and was considered in [741]. Let $X = {\mathbb{R}}^2$, $U = {\mathbb{R}}$, and the state transition equation be

$$\begin{split}{\dot x}_1 & = u \\ {\dot x}_2 & = x_1^2 . \\ \end{split}$$ (15.111)

This system is clearly not controllable in any sense because $x_2$ cannot be decreased. The vector fields are $h_0(x) = [0 \;\; x_1^2]^T$ and $h_1(x) = [1 \;\; 0]^T$. The first independent Lie bracket is

$$[h_1,[h_0,h_1]] = [0 \;\; -2] .$$ (15.112)

The two-dimensional Lie algebra is

$${\cal L}({\triangle}) = \operatorname{span}\{ h_1, [h_1,[h_0,h_1]] \} ,$$ (15.113)

which implies that the system is accessible. It is not STLC, however, because the bracket $[h_1,[h_0,h_1]]$ was constructed using $h_0$ and was combined in an unfortunate way. This bracket is indicating that changing $x_2$ is possible; however, we already know that it is not possible to decrease $x_2$. Thus, some of the vector fields obtained from Lie brackets that involve $h_0$ may have directional constraints. $\blacksquare$

In Example 15.20, $[h_1,[h_0,h_1]]$ was an example of a bad bracket [925] because it obstructed controllability. A method of classifying brackets as good or bad has been developed, and there exist theorems that imply whether a system with drift is STLC by satisfying certain conditions on the good and bad brackets. Intuitively, there must be enough good brackets to neutralize the obstructions imposed by the bad brackets [156,925].