One submits "representations" to the council that served the PCN, and register an "appeal" at the adjudicators, London Tribunals. However like many things, the word 'appeal' is freely used by both OPs on here, and also by councils who should know better !
If you submit reps to the council within the discount period they usually re-offer the discount. However they, as with the other London councils, refuse virtually all representations no matter how valid, because they know most people then cough-up, being unsure of the law/regulations, and not wishing to risk the extra money by appealing to the adjudicators. And people who submit reps are in a small minority as well, as >90% of PCN recipents cough-up as soon as they get the PCN and never bother submitting reps, or seeking advice like this forum. This entitles them to Full Membership of the Mugged Club, (sorry, only joking !) This forum exists to make sure people know what the law actually is and says. (plus the regulations of course). In addition we will always give a view on whether the PCN has been lawfully served and is justified.