Suppose there are $11$ positions $P_1,P_2,\ldots,P_{11}$, and each substitution doesn't involve a position change. We relax the "condition" that each player can only appear in one place at a time, therefore we can assume that only one position is in the pitch at a time, and when position $P_1$ finishes, $P_2$ continues, ... , until $P_{11}$ finishes (so the game lasts $990$ minutes and always has one player on the pitch.) Now we can freely rearrange the stints of each players in each position $P_i$ without affecting the amount of substitutions. We'll rearrange as follows: For $k=1,2,\ldots,10$ : Since $55\nmid 90k$, after $P_1,P_2,\ldots,P_k$ some player $a_k$ must have played $>0$ but $<55$ minutes. Arrange the stints so that $a_k$ is not the last player playing at $P_k$. WLOG $a_k$ has a stint in $P_{k+1}$. Let $a_k$ start $P_{k+1}$. We now "fuse" the positions/games and look at the $990$-minute, one-position game. There are $18$ players, and for every $k$, $a_k$'s stint at the start of $P_{k+1}$ is discontinuous from all his/her previous stints. There are $10$ values of $k$, hence there are at least $18+10=28$ stints in the $990$-minute game. Now, from the $\geqslant 27$ change of stints, exactly $10$ is a change from $P_k$ to $P_{k+1}$ and are not substitutions. Therefore we're left with $\geqslant 17$ substitutions, and this is easily achievable (for example, see GGPiku's comment)