From public corporation to operating exec at PE portfolio company

I have 20+ years of broad operating experience in large corporate environments including general management, engineering, sales and marketing, strategy, international expansion, etc. Considering transition to run a mid market ($100-250M) PE backed firm. Will appreciate thoughts on making this transition. What differences in operating philosophy can I expect working for PE investors vs public if any? What's the best way to make the transition? How do I get noticed by the PE guys? What do they look for when seeking operating execs? By the way, my background is mostly in aerospace/defense but could be easily transferable to electronic, industrial, automotive or other similar manufacturing companies. Thanks in advance for your thoughts.