Example 5 It is known that for any $x$,
$$
a \cos x + b \cos 2x \geqslant -1
$$

always holds. Find the minimum value of $a + b$.
(2009, Peking University Independent Admission Examination)
When $x=0$, $a+b \geqslant-1$. Taking $a=-\frac{4}{5}, b=-\frac{1}{5}$, then
$$
\begin{array}{l}
a \cos x+b \cos 2 x \\
=-\frac{2}{5}(\cos x+1)^{2}+\frac{3}{5} \\
\geqslant-\frac{2}{5}(1+1)^{2}+\frac{3}{5} \\
=-1 .
\end{array}
$$

Therefore, the minimum value of $a+b$ is -1.