Spectrogram-MPC: Enforcing hard constraints on systems' output spectra

A novel model predictive control (MPC) scheme that allows one to enforce hard constraints on the spectrum of a constrained system's output signal is presented. The approach is based on a time-local analysis of the spectrum of output signals by means of the short-time Fourier transform (STFT), and its squared magnitude, called the spectrogram. It is shown that an MPC problem with spectrogram constraints can be formulated as a quadratically constrained quadratic program (QCQP). We prove recursive feasibility and stability of the proposed spectrogram MPC scheme via an ellipsoidal invariant set, including spectrogram constraints. Moreover, it is pointed out how the proposed spectrogram MPC approach can be extended to MPC for tracking while ensuring recursive feasibility. Finally, we present simulation results of spectrogram MPC applied to a resonant system. Our simulations show that, by employing the proposed spectrogram MPC approach, oscillations can be attenuated in the system output, tracking a reference signal, by explicitly enforcing hard constraints on its spectrum.