Employment and education reasons aren't accepted as valid reasons for failing to meet the residency requirement. So CIC won't consider your work as an engineer or your children's education. You can try using your mother's illness as a reason provided you have no other family in your home country (e.g. no siblings). You'll need to have hard proof (hospitalization records, doctor letters) to cover 15 years of failing to be in Canada. However even with evidence, I still think the chances of a PRTD being approved are effectively nil and an appeal will be refused as well. You've simply been outside of Canada for far too long.
Your entire family could also try flying to the US, re-entering Canada through a US/Canada land border using a private vehicle. If you are reported at the border for failing to meet the residency obligation (chances are high you will), you'll be called to a hearing and should expect to lose your PR status and be forced to leave Canada. If you are not reported at the border, you'll need to remain in Canada for two years straight without leaving before you'll be able to renew your PR cards and be sure you can keep your PR status. During this time, it may be difficult to get health care coverage or have your children attend school if all they have is the landing document / COPR from years ago.
The most realistic option is to formally renounce your PR status and apply again from scratch. Your children can be included as dependents in the application provided they are 21 years old or under.