why ios 11.1 make my iphone 5s become lagging

The iPhone 5's processor has a lot more to do under iOS 11.1, thus the lag you may be seeing. Sorry to say that you may need to update the hardware (the iPhone) to get the software (iOS) to perform adequately.

The best thing to try is to:

Backup the iPhone under iTunes (link #1 below)

Use iTunes to restore the phone "as new" (link #2 below)

Test performance with a minimal installation

If performance is the same under "as new", don't bother testing any more and restore the backup you created above (link #3 below)

If performance is improved under "as new", you may need to rebuild the iPhone's contents from scratch. Admittedly, not fun.

#1 How to back up your iPhone, iPad, and iPod touch - Apple Support

#2 Restore your iPhone, iPad, or iPod to factory settings - Apple Support

#3 Restore your iPhone, iPad, or iPod touch from a backup - Apple Support

So far, I have not yet updated my iPhone 5s to iOS 11.x and am still at iOS 9.3.5.

Same observation on my iPhone 5S : keyboard lag (while prediction and correction are disabled), slow wake up, slow unlock, ... foreseen obsolescence or bugs ?