Abstract
In this paper, an analytic condition is given for determining delayed positive feedback controller for stabilizing an oscillatory system. The $\tau $ -decomposition and $D$ -decomposition methods are employed in deriving this condition. The obtained results are then used to stabilize second order delay systems by a proportional controller. Under-damped and over-damped systems are treated separately, where the Smith-predictor structure is used in the over-damped case to obtain the stability conditions. Illustrative examples are given to show the effectiveness of the proposed approach.
Highlights
It was shown in [1] that delayed positive feedback can stabilize oscillatory systems
It was proven that a single integrator is stabilizable by a single delay and that a chain of integrators can be stabilized by multiple delays [2]
Based on the obtained results, an extension was formulated to a delayed second order system
Summary
It was shown in [1] that delayed positive feedback can stabilize oscillatory systems. S. Elmadssia et al.: Stabilization Domains for Second Order Delay Systems approach to study this problem by concentrating on concluding stability or instability of the system with respect to a single or multiple system parameters. The D-decomposition method was largely used in robust stability problems and design of low fixed order controllers for time delay systems, as it permits to decompose the controller’s parameter space and determine stability boundaries using a set of parametric equations. These equations explicitly depend on the control parameters, which results in finding the stability regions and can help in obtaining analytical expressions for tuning the controllers.
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