Definition of General Operator Space and The s-gap Metric for Measuring Robust Stability of Control Systems with Nonlinear Dynamics

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Saki Saman; Bolandi Hossein

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Journal Paper

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Journal: Journal of Control Year:2022 | Volume:16 | Issue:2 Page(s): 41-54

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Information Journal Paper

Title

Definition of General Operator Space and The s-gap Metric for Measuring Robust Stability of Control Systems with Nonlinear Dynamics

Author(s)

Saki Saman | Bolandi Hossein | Issue Writer Certificate

Keywords

The Gain BoundsQ4

The Generalized Stability MarginQ4

Nonlinear MetricsQ4

Operator Space and NonlinearityQ4

Abstract

In the recent decades, metrics have been introduced as mathematical tools to determine the robust stability of the closed loop control systems. However, the metrics drawback is their limited applications in the closed loop control systems with nonlinear dynamics. As a solution in the literature, applying the metric theories to the linearized models is suggested. In this paper, we show that using the linear model is not adequate to analyze the robust stability. To this end, the definition of general operator space is proposed as the important novelty to determine the weakest topology between two nonlinear dynamic systems. In this space, all nonlinear operators (nonlinear dynamic systems) with differentiable manifold topology can be considered as isometric isomorphism. The result of this definition is possibility of the nonlinear gap metric solution which leads to definition of the s-gap metric. In fact, we show that the calculation of the nonlinear gap metric leads to the most conservative tangent spaces in the defined space. Also, based on the new results, the gain bound of the closed-loop system is determined, which together with the s-gap offers a new theory for robust stability analysis. Advanced operator theory and simulation results confirm the correctness of the claims.

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