Relay self-oscillations for second order, stable, nonminimum phase plants
We study a relay feedback system (RFS) having an ideal relay element and a linear, time-invariant, second order plant. We model the relay element using an ideal on-off switch. And we model the second order plant with a transfer function that: (i) is Hurwitz stable, (ii) is proper, (iii) has a positi...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
26.10.2018
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Subjects | |
Online Access | Get full text |
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Summary: | We study a relay feedback system (RFS) having an ideal relay element and a
linear, time-invariant, second order plant. We model the relay element using an
ideal on-off switch. And we model the second order plant with a transfer
function that: (i) is Hurwitz stable, (ii) is proper, (iii) has a positive real
zero, and (iv) has a positive DC gain.
We analyze this RFS using a state space description, with closed form
expressions for the state trajectory from one switching time to the next. We
prove that the state transformation from one switching time to the next: (a)
has a Schur stable linearization, (b) is a contraction mapping, and (c) maps
points of large magnitudes to points with lesser magnitudes. Then using the
Banach contraction mapping theorem, we prove that every trajectory of this RFS
converges asymptotically to an unique limit cycle. This limit cycle is
symmetric, and is unimodal as it has exactly two relay switches per period.
This result helps understand the behaviour of the relay autotuning method, when
applied to second order plants with no time delay.
We also treat cases where the plant either has no finite zero, or has exactly
one zero and that is negative. |
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DOI: | 10.48550/arxiv.1810.11371 |