A new framework to closed-loop process identification is proposed. It relies on simultaneous constrained model predictive control (MPC) and identification (MPCI). MPCI obtains sufficient model information on a process under constrained MPC, while minimally disturbing that process. To select a process input at each time step, MPCI solves a constrained optimization problem on-line with respect to process input values over a finite moving horizon. These process inputs should satisfy all conventional MPC constraints, as well as additional constraints that assure persistent excitation of the process by these inputs. The persistent excitation constraints can be enabled or disabled, according to process identification needs of the closed loop. An iterative scheme is proposed for the numerical solution of the on-line optimization problem. At each iteration, that iterative scheme finds a suboptimal feasible solution of the on-line optimization problem by solving a semidefinite programming problem whose global convergence is guaranteed. The effectiveness of the proposed new methodology is illustrated through simulations on a linear time-varying process.