Generalized predictive control (GPC) is a member of the MPC family of methods, where the mathematical model of the system is a “controlled auto-regressive and integrated moving-average” (CARIMA) model.

Long range predictive control (LRPC) seems (although I might be wrong since I am seeing this name for the first time) to be used interchangeably with MPC (especially around 1990–2000).

Model predictive control (MPC) is a family of control methods based on real-time repeated optimal control. These methods are intended for solving multivariable, constrained, infinite horizon, and possibly nonlinear, optimal control problems approximately via finite horizon solutions implemented in receding horizon fashion. These finite horizon solutions involve optimizing the objective function for the (finite) prediction horizon, where the predictions are based on a mathematical model of the dynamical system to be controlled.

There are many variants of MPC (there may be more):
> model algorithmic control: MPC using an impulse-response model
> dynamic matrix control: MPC using a step-response model
> generalized predictive control: MPC using a CARIMA model
> hybrid MPC: MPC using a hybrid system model (i.e., with binary state variables and/or control inputs in addition to continuous ones)
> with uncertainty treatment: robust MPC, stochastic MPC
> with real-time system identification: adaptive MPC (model is updated online)
> with parametric optimization: explicit MPC (control law is computed offline and implemented as a look-up table)