Model Predictive Control (MPC) often requires that a new process model be developed from data collected while the process remains under feedback control. How can a good model be obtained for a process under constrained MPC, without excessive perturbation of the process? We propose a novel approach, model predictive control and identification (MPCI), that relies on augmenting the standard on-line MPC optimization with a series of persistent excitation (PE) constraints that current and future process inputs must satisfy over a finite horizon. For linear processes, the resulting on-line optimization problem is solved by solving a series of semi-definite programming (SP) problems, for which efficient numerical methods with guaranteed convergence exist. The addition of the PE constraint to standard MPC paves the way for a number of different MPCI variants. A variant of MPCI involving time-frequency process identification with wavelets is proposed in the paper.