Nonlinear systems and linearization

Thus far, we’ve mostly considered linear system models. Many of the analytic tools we’ve developed—ODE solution techniques, superposition, eigendecomposition, stability analysis, impedance modeling, transfer functions, frequency response functions—do not apply to nonlinear systems. In fact, analytic solutions are unknown for most nonlinear system ODEs. And even basic questions are relatively hard to answer; for instance: is the system stable?

In this and the following chapters, we consider a few analytic and numerical techniques for dealing with nonlinear systems.

A state-space model has the general form $$\begin{align} \frac{\diff\bm{x}}{\diff t} &= \bm{f}(\bm{x},\bm{u},t) \label{eq:state_nonlinear2} \\ \bm{y} &= \cloze{\bm{g}(\bm{x},\bm{u},t)} \label{eq:output_nonlinear2} \end{align}$$ where \(\bm{f}\) and \(\bm{g}\) are vector-valued functions that depend on the system. Nonlinear state-space models are those for which \(\bm{f}\) is a nonlinear functional of either \(\bm{x}\) or \(\bm{u}\). For instance, a state variable \(x_1\) might appear as \(x_1^2\) or two state variables might combine as \(x_1 x_2\) or an input \(u_1\) might enter the equations as \(\log u_1\).