To control 17 dimmable led strips you need a 17 PWM outputs, but the Uno has only six.
You can you use PCA9685 16 channel PWM controller for 16 strips and control the last strip by Uno itself
You don't need to be limited by the Arduino's restricted number of PWM channels. To drive lights, you can use a much larger number of software-generated channels. But then you'd have to set up a shift-register chain to get the data out of the Arduino, because it hasn't got enough pins to drive 17 channels. Or maybe you could set up a 4x5 matrix somehow? But that's probably too complicated to be worth doing.
Does it really have to be 17 channels? 16 would be much simpler to deal with.
Oops, I didn't stop to think about how many pins the basic Arduino actually has. Yes, it could be done with 1 pin per channel. But it would mean there's not room for much else. Question for the originator of this thread--how is this device to be controlled, via a serial link perhaps? That needs some port pins too.
@veqlargh wants to be able to dim each LED strip separately, so if standard GPIO pins are used that would involve using software PWM to control each of them. At best this would be inconvenient
@UKHeliBob
Using regular pins as software PWM is not my suggestion, but @Beminetonight
I was just reacting to the statement that the Arduino Uno does not have enough pins to connect 17 outputs