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The number of seniors and persons facing a loss of autonomy in need of daily assistance increases and the human resources available in the upcoming years will certainly be insufficient. To remedy this situation, smart habitats are considered by many researchers as an innovative avenue to help support the needs of elders. It aims at providing cognitive assistance in taking decisions by giving hints, suggestions and reminders, with different kinds of effectors to residents. To implement such technology, the first challenge we need to overcome is the recognition of the ongoing activity. In the literature, some researchers have proposed solutions based on binary sensors or cameras, but these types of approaches infringed the privacy of residents and showed inappropriate for smart homes. Therefore, a new affordable activity recognition system based on passive RFID technology and able to detect errors related to cognitive impairment is presented in this paper. The entire system relies on an innovative model of elliptical trilateration with several filters as well as on an ingenious representation of activities with spatial zones. This system has been implemented and deployed in a real smart-home prototype. This paper also renders the results of a complete set of experiments conducted on this new activity-recognition system with real cases scenarios.