I don't think that analytic power series can be controlled by algebraic power series in an algebraic way, because they are often transcendental over the former. 

> Suppose an automorphism $\phi$ of $R$ is identity on the subring of algebraic power series. Is $\phi$ identity on the analytic power series?

The answer is no. 

For a counter-example let $k$ be any field of characteristic zero and let $K \subseteq k((X))$ be the field of elements that are algebraic over $k(X)$.  By the accepted answer of http://mathoverflow.net/questions/21290/whats-an-example-of-a-transcendental-power-series, the exponental function $\text{exp}$ is transcendental over $k(X)$ and by transitivity of algebraic extensions, $\text{exp}$ is also transcendental over $K$. Extend the identity on $K$ to the automorphism $$\phi: K(\text{exp}) \to K(\text{exp}),\,\,\text{exp}\mapsto \text{exp} + X$$
(the inverse sends $\text{exp}\mapsto \text{exp} - X)$. 

Let $R$ be an algebraic closure of $k((X))$. Then $\phi$ extends to an automorphism of $R$ (cf. https://en.wikipedia.org/wiki/Transcendence_degree, "Applications"). This $\phi$ is the identity on algebraic power series but not on analytic power series. 

> At least, does $\phi$ send analytic to analytic ?

I *think* the answer is no, but I have no example. To produce an example one could look for a non-analytic power series $f$ such that $\text{exp}, f$ are algebraically independent over $k(X)$. Then modify the $\phi$ above by mapping $\text{exp} \mapsto f,\,\, f \mapsto \text{exp}$.