This paper examines the ability of a multivariable PID controller rejecting measurement noise without the use of any external filter. The work first provides a framework for the design of the PID gains comprising of necessary and sufficient conditions for boundedness of trajectories and zero-error convergence in presence of measurement noise. It turns out that such convergence requires time-varying gains. Subsequently, novel recursive algorithms providing optimal and sub-optimal time-varying PID gains are proposed for discrete-time varying linear multiple-input multiple-output (MIMO) systems. The development of the proposed optimal algorithm is based on minimising a stochastic performance index in presence of erroneous initial conditions, white measurement noise, and white process noise. The proposed algorithms are shown to reject measurement noise provided that the system is asymptotically stable and the product of the input-output coupling matrices is full-column rank. In addition, convergence results are presented for discretised continuous-time plants. Simulation results are included to illustrate the performance capabilities of the proposed algorithms.