13.2.3 Nonlinear Systems

Although many powerful control laws can be developed for linear systems, the vast majority of systems that occur in the physical world fail to be linear. Any differential models that do not fit (13.37) or (13.40) are called nonlinear systems. All of the models given in Section 13.1.2 are nonlinear systems for the special case in which $X = {\cal C}$.

One important family of nonlinear systems actually appears to be linear in some sense. Let $X$ be a smooth $n$-dimensional manifold, and let $U = {\mathbb{R}}^m$ for some $m \leq n$. Using a coordinate neighborhood, a nonlinear system of the form

$${\dot x}= f(x) + \sum_{i=1}^m g_i(x) u_i$$ (13.41)

for smooth functions $f$ and $g_i$ is called a control-affine system or affine-in-control system.^13.7 These have been studied extensively in nonlinear control theory [478,846]. They are linear in the actions but nonlinear with respect to the state. See Section 15.4.1 for further reading on control-affine systems.

For a control-affine system it is not necessarily possible to obtain zero velocity because $f$ causes drift. The important special case of a driftless control-affine system occurs if $f \equiv 0$. This is written as

$${\dot x}= \sum_{i=1}^m g_i(x) u_i .$$ (13.42)

By setting $u_i = 0$ for each $i$ from $1$ to $m$, zero velocity, ${\dot x}= 0$, is obtained.

Example 13..6 (Nonholonomic Integrator)   One of the simplest examples of a driftless control-affine system is the nonholonomic integrator introduced in control literature by Brockett in [142]. It is sometimes referred to as Brockett's system, or the Heisenberg system because it arises in quantum mechanics [112]. Let $X = {\mathbb{R}}^3$, and let the set of actions $U = {\mathbb{R}}^2$. The state transition equation for the nonholonomic integrator is

$$\begin{split}{\dot x}_1 & = u_1 \\ {\dot x}_2 & = u_2 \\ {\dot x}_3 & = x_1 u_2 - x_2 u_1 . \end{split}$$ (13.43)

$\blacksquare$

Many nonlinear systems can be expressed implicitly using Pfaffian constraints, which appeared in Section 13.1.1, and can be generalized from C-spaces to phase spaces. In terms of $X$, a Pfaffian constraint is expressed as

$$g_1(x) {\dot x}_1 + g_2(x) {\dot x}_2 + \cdots + g_n(x) {\dot x}_n = 0 .$$ (13.44)

Even though the equation is linear in ${\dot x}$, a nonlinear dependency on $x$ is allowed.

Both holonomic and nonholonomic models may exist for phase spaces, just as in the case of C-spaces in Section 13.1.3. The Frobenius Theorem, which is covered in Section 15.4.2, can be used to determine whether control-affine systems are completely integrable.