I have crossed a problem, and it took me about 3 days. I wonder how I can find the Taylor series of "function of functions". To be explicit, I am looking for the Taylor series of
$$y(s)=f(u_1(s),u_2(s),s)$$
I think the Taylor series at t in the T neighborhood of t-T (Where T is small enough) is as follows. Am I right?
$$y(t)=y(t-T)+\left.\frac{\partial f}{\partial u_1}\frac{\partial u_1}{\partial s}\right|_{s=t-T}u_1(t-T)+\left.\frac{\partial f}{\partial u_2}\frac{\partial u_2}{\partial s}\right|_{s=t-T}u_2(t-T)+\left.\frac{\partial f}{\partial s}\right|_{s=t-T}(t-T)+\cdots $$ I want it to become as follows: $$y(t)=\theta_1(t) u_1(t)+\theta_2(t) u_2(t)+\theta_3(t) $$