Time-Frequency Representation of MIMO Dynamical Systems

Abstract

Many deterministic and random physical signals can be modeled as the output of a multi-input-multi-output (MIMO) dynamical system. Since physical signals are typically nonstationary, their frequency content changes with time. To understand this time variation, we transform the MIMO system to the time-frequency domain. The result is a time-frequency MIMO dynamical system, whose input and output are the time-frequency spectra of the original input and output signals in the time domain. The time-frequency system reveals the spectral mechanisms involved in the generation of nonstationary signals. We apply our method to the case of a MIMO system with two vibrational modes and a nonstationary noise at the input. We obtain the time-frequency spectrum of the output, which shows how the spectrum of the modes changes with time. This result cannot be achieved with classical spectral techniques, because they require the input random process to be wide sense stationary.

The eigenvalue problem in phase space

“…The above is a special case of methods that have been developed for transforming ordinary and partial differential equations into phase space equations. [8][9][10]…”

“…The Weyl operator, G(q,p), is defined by the substitution of the operator for position, q, and the operator for momentum, p for q and p in eq. (7), Gðq; pÞ 5 ð ðĝ ðu; sÞ e iuq1isp du ds (8) Using the fact that [7] e iuq1isp 5e ius h=2 e iuq e isp (9) we also have Gðq; pÞ 5 ð ðĝ ðu; sÞ e ius h=2 e iuq e isp du ds…”

Advanced Time-Frequency Signal and System Analysis

“…The MIMO system in Eq. (4.9.26) can be transformed to the time-frequency domain, and the result of this transformation is the time-frequency system [93]…”

Solution method for time-frequency responds of servo control systems based on frequency domain characteristics

“…18,19 The Wigner distribution was successfully used for tracking error control, demonstrating the feasibility and promise of time-frequency analysis for motion system analysis. Galleani 20 used the Wegener distribution to transform the motion control system from the time domain to the time-frequency domain so that the time-frequency spectrum of the system output can be calculated. This method achieves the solution of the time-frequency response by time-frequency transformation of the system, but the calculation of the time-frequency transformation of the system is very complicated.…”