Frequency Domain Analysis of Uncertain Time-Varying Discrete-Time Systems

The paper develops tools and methods for the analysis of uncertain linear time-varying, discrete-time systems. Our considerations are concerned with the analysis of a system in the frequency domain, in particular with obtaining approximate Bode amplitude diagrams by means of two methods: one based on singular value decomposition and the other on the discrete Fourier transform (SVD-DFT). Starting with the literature review of frequency domain methods for time-varying systems, we introduce the state space model with its equivalent operator form description and the most essential definitions for the Bode diagram approximations using two methods called the spectral and the averaged transfer function, respectively. Next we analyze a system with an additive uncertainty, in particular, given as a linear combination of perturbations. Theoretical considerations are summed up and illustrated by numerical examples of applications to robot manipulators. The obtained bounds (computed by the proposed method) are compared with the results obtained by means of the MATLAB wcgain function.