Consider the nonlinear state equation (see pic for equation) with nominal state (see pic for equation)
and constant nominal input ?u(t) = 1. Verify that ?x(t) is a solution to ? ?x(t) = f( ?x, ?u). Show that the nominal output is ?y(t) = ?2/3. Linearize the state equation about the nominal solution.

Question

$Consider the nonlinear state equation \[ \begin{array}{l} \dot{\boldsymbol{x}}(t)=\boldsymbol{f}(\boldsymbol{x}(t), u(t))=\le$

Consider the nonlinear state equation

\overset{?}{x} (t) = f (x (t), u (t)) = u (t) x_{1} (t) + x_{3} (t) 3 x_{2} (t) ? 2 u (t) x_{3} (t) y (t) = x_{1} (t) + x_{3} (t)

with nominal state

\tilde{x} (t) = t ? \frac{2}{3} t ? \frac{7}{9} ? t ? \frac{2}{3}

and constant nominal input

\tilde{u} (t) = 1

. Verify that

\tilde{x} (t)

is a solution to

\overset{?}{\tilde{x}} (t) = f (\tilde{x}, \tilde{u})

. Show that the nominal output is

\tilde{y} (t) = ? \frac{2}{3}

. Linearize the state equation about the nominal solution.

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