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The Complex Adaptive Delta-Modulator in Sliding Mode Theory.

Dhafer Almakhles^1,2

¹College of Engineering, Communications & Networks Engineering, Prince Sultan University, Riyadh 11586, Saudi Arabia.

Entropy (Basel, Switzerland)

|December 8, 2020

Summary

This study analyzes the stability and periodic behaviors of discrete-time delta modulators (Δ-M) and adaptive Δ-M. Quasi-sliding mode theory confirms stability conditions and reveals periodic dynamics in these delta modulation systems.

Keywords:

adaptive delta-modulator hitting-time periodicity quasi-sliding mode single-bit two-level quantizer

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Area of Science:

Control Systems Engineering
Signal Processing
Nonlinear Dynamics

Background:

Discrete-time delta modulators (Δ-M) are crucial in signal processing.
Adaptive Δ-M offers enhanced performance but requires thorough stability analysis.
Understanding dynamical behaviors is key to optimizing modulator design.

Purpose of the Study:

To investigate the stability constraints and conditions for both discrete-time Δ-M and adaptive Δ-M.
To explore the periodic behaviors exhibited by these systems under specific input and parameter conditions.
To validate the theoretical findings through simulation.

Main Methods:

Application of quasi-sliding mode theory to derive stability conditions.
Analysis of system dynamics for steady-state inputs.
Exploration of parameter-dependent behaviors.
Validation using simulated examples.

Main Results:

Derived stability constraints and conditions for discrete-time Δ-M and adaptive Δ-M.
Identified conditions leading to periodic behaviors in both systems.
Confirmed the theoretical findings through simulation, validating stability and periodicity.

Conclusions:

The study successfully established stability conditions for Δ-M and adaptive Δ-M using quasi-sliding mode theory.
Periodic dynamical behaviors were confirmed to exist under specific conditions.
Simulations validated the theoretical derivations, providing confidence in the results for practical applications.