15.5.2.2 First-order controllable systems

The approach developed for the nonholonomic integrator generalizes to systems of the form

$${\dot x}_i = u_i \qquad \mbox{for $i$\ from $1$\ to $m$}$$

$${\dot x}_{ij} = x_i u_j - x_j u_i \qquad \mbox{for all $i,j$\ so that $i < j$\ and $1 \leq j \leq m$}$$ (15.155)

and

$${\dot x}_i = u_i \qquad \mbox{for $i$\ from $1$\ to $m$}$$

$${\dot x}_{ij} = x_i u_j \qquad \mbox{for all $i,j$\ such that $i < j$ and $1 \leq j \leq m$} .$$ (15.156)

Brockett showed in [142] that for such first-order controllable systems, the optimal action trajectory is obtained by applying a sum of sinusoids with integrally related frequencies for each of the $m$ action variables. If $m$ is even, then the trajectory for each variable is a sum of $m/2$ sinusoids at frequencies $2 \pi$, $2 \cdot 2\pi$, $\ldots$, $(m/2)\cdot 2\pi$. If $m$ is odd, there are instead $(m-1)/2$ sinusoids; the sequence of frequencies remains the same. Suppose $m$ is even (the odd case is similar). Each action is selected as

$$u_i = \sum_{k=1}^{m/2} a_{ik} \sin 2 \pi k t + b_{ik} \cos 2 \pi k t .$$ (15.157)

The other state variables evolve as

$$x_{ij} = x_{ij}(0) + \frac{1}{2} \sum_{k=1}^{m/2} \frac{1}{k} (a_{jk} b_{ik} - a_{ik} b_{jk}) ,$$ (15.158)

which provides a constraint similar to (15.153). The periodic behavior of these action trajectories causes the $x_i$ variables to return to their original values while steering the $x_{ij}$ to their desired values. In a sense this is a vector-based generalization in which the scalar case was the nonholonomic integrator.

Once again, a two-phase steering approach is obtained:

1. Apply any action trajectory that brings every $x_i$ to its goal value. The evolution of the $x_{ij}$ states is ignored in this stage.
2. Apply sinusoids of integrally related frequencies to the action variables. Choose each $u_i(0)$ so that $x_{ij}$ reaches its goal value. In this stage, the $x_i$ variables are ignored because they will return to their values obtained in the first stage.

This method has been generalized even further to second-order controllable systems:

$${\dot x}_i = u_i \qquad \mbox{for $i$\ from $1$\ to $m$}$$

$${\dot x}_{ij} = x_i u_j \qquad \mbox{for all $i,j$\ such that $i < j$ and $1 \leq j \leq m$}$$ (15.159)

$${\dot x}_{ijk} = x_{ij} u_k \qquad \mbox{for all $(i,j,k) \in J$,}$$

in which $J$ is the set of all unique triples formed from distinct $i,j,k \in \{1,\ldots,m\}$ and removing unnecessary permutations due to the Jacobi identity for Lie brackets. For this problem, a three-phase steering method can be developed by using ideas similar to the first-order controllable case. The first phase determines $x_i$, the second handles $x_{ij}$, and the third resolves $x_{ijk}$. See [727,846] for more details.