If I use the value of public equity to calculate the value of noncontrolling private equity, do I have to add lack of control discount?
I think, since the public equity and noncontrolling private equity are all minority interest (not controlling), we do not need to add lack of control discount.
Yes, you do. Private equity is far less accurately valued than public companies equity, so you should add the lack of control discount. Can you rely in the data provided from the private company you are valuing? I would not, sincerely.
Moreover, suppose you own a 15% stake in a private company. The day after the purchase the Mad King decides to kill your children, what can you do but just observe? You can’t leave, it is an illiquid position.
You could be overestimating the value of that position 
To be safe, better chop the price a little (or a lot).
Therefore,
Value of controlling private equity + lack of controlling discount = Value of public equity
Value of public equity + lack of controlling discount + lack of marketability discount = value of noncontrolling private equity
Value of controlling private equity + even larger lack of controlling discount + lack of marketability discount = value of noncontrolling private equity
Is logic correct?
With all due respect, that will lose you points on the Level II CFA exam.
If you use noncontrolling data, it already includes a discount for lack of control; to discount it again would give you too low a price, and will most likely be one of the wrong answers offered.
If you read well OP’s statement, you will realize he is implying that, if we use public equity data to emulate a private equity value, we do not need to add a lack of control discount since public equity is commonly largely shared (each investor possesses a little of the equity value). That is not correct, and that’s the point of the whole reading… to tells us that private equity value cannot just be valued using public companies data as comparables. We indeed need to adjusts the value applying discounts (lack of control, lack of marketeability, illiquidity).
What do you mean by “non-controlling data” exactly?
Data from stock sales that represent noncontrolling interests.
We can’t directly compare the data that non-controlling public equity shareholders receive, and the data a private equity investor receive from a private company. We need to make those discounts in order to arrive at a “true” risk-adjusted price, ergo equity return.