Abstract
The stability of linear dynamic systems with hysteresis in feedback is considered. While the absolute stability for memoryless nonlinearities (known as Lure’s problem) can be proved by the well-known circle criterion, the multivalued rate-independent hysteresis poses significant challenges for feedback systems, especially for proof of convergence to an equilibrium state correspondingly set. The dissipative behavior of clockwise input-output hysteresis is considered with two boundary cases of energy losses at reversal cycles. For upper boundary cases of maximal (parallelogram shape) hysteresis loop, an equivalent transformation of the closed-loop system is provided. This allows for the application of the circle criterion of absolute stability. Invariant sets as a consequence of hysteresis are discussed. Several numerical examples are demonstrated, including a feedback-controlled double-mass harmonic oscillator with hysteresis and one stable and one unstable poles configuration.
Highlights
The stability of dynamic systems containing hysteresis has been investigated for some time, but still remains a challenging and appealing topic for systems and control theory
We have addressed the stability of linear dynamics with clockwise hysteresis in feedback
The lower boundary case, is shown to be non-dissipative and lossless from an energy viewpoint
Summary
The first requirement means that the hysteresis output trajectories, and the energy conservation or dissipation, do not depend on the input rate y This means a rate-independent ( called static) hysteresis map is invariant to affine transformations of time, i.e., a + bt ∀ a ∈ R, b ∈ R+. It is worth noting that for h → 0 the input-output map (2.7) approaches the lower boundary case of energy dissipation ∆E → 0, with a corresponding vanishing of hysteresis. It seems appropriate to assume that all clockwise hysteresis maps which lie between the lower and upper boundary cases, as defined above, will behave as dissipative upon the input cycles This is independent of the shape of the hysteresis loops. This is in the spirit of the Lure’s type systems [22] which have been thoroughly investigated and formalized for nonlinearities without memory
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