Success and query on back payments

A big thank you first of all - help from your guides etc and advice on the forum were invaluable - Ive just been awarded PIP standard care high rate mobility and my ESA has been awarded to carry on too. Such a relief.
My only query is I was invited to claim PIP (previously on DLA) in June my award letter says it starts from 30th August.(letter is dated 3st JUly) Are you not allowed backdated PIP ? My award letter says my first payment will be 22nd Sept and this includes any arrears but it doesnt seem to include any difference as its too low. My next DLA payment is due 16th August (i think roughly)
So from what I make out my PIP award does not start until 31st Aug - 1st payment 22nd Sept for just three weeks money
A little confused .....
But thats not to say Im not over the moon and will make such a difference let alone stop the worry Ive felt since January with the re-assessment of ESA and then claiming PIP
Thank you once again you are all great - hopefully you can shed some light on the back payment senario

Your first PIP payment covers from the end of your DLA up until the start of your new four weekly payment cycle. It should be detailed in your decision letter along with the length of award