A monodromy operator-based interpretation for repetitive control of robot manipulators with controller synthesis and stability analysis

Geun Il Song; Jung Hoon Kim; Geun Il Song; Jung Hoon Kim

doi:10.3934/math.20261207

AIMS Mathematics

2026, Volume 11, Issue 9: 30460-30483. doi: 10.3934/math.20261207

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A monodromy operator-based interpretation for repetitive control of robot manipulators with controller synthesis and stability analysis

Geun Il Song ,
Jung Hoon Kim ^,

Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea

Received: 21 July 2026 Revised: 05 September 2026 Accepted: 15 September 2026 Published: 18 September 2026
MSC : 93C85

A monodromy operator-based interpretation for repetitive control of robot manipulators subject to periodic reference and disturbance signals is established in this paper. An inverse dynamics strategy is first employed to convert the coupled nonlinear input–output relation of a robot manipulator into a decoupled linear time-invariant (LTI) representation, for which the conventional repetitive control approach can be applied. Such an application leads to a unified expression, i.e., a delay-feedback system, suitable for both controller synthesis and stability analysis. Two linear matrix inequality (LMI) formulations are established to obtain repetitive controllers that minimize the $ H_\infty $ and generalized $ H_2 $ norms from the disturbance to the tracking error, respectively. The resulting closed-loop systems are shown to be exponentially stable if and only if the spectrum radii of their monodromy operators are less than $ 1 $. We further show that the exponential stability also implies the input-to-state stability (ISS) under bounded disturbances. Finally, simulation comparisons are presented to demonstrate the effectiveness of the overall arguments.
- robot manipulators,
- controller synthesis,
- stability analysis
Citation: Geun Il Song, Jung Hoon Kim. A monodromy operator-based interpretation for repetitive control of robot manipulators with controller synthesis and stability analysis[J]. AIMS Mathematics, 2026, 11(9): 30460-30483. doi: 10.3934/math.20261207

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Abstract

A monodromy operator-based interpretation for repetitive control of robot manipulators subject to periodic reference and disturbance signals is established in this paper. An inverse dynamics strategy is first employed to convert the coupled nonlinear input–output relation of a robot manipulator into a decoupled linear time-invariant (LTI) representation, for which the conventional repetitive control approach can be applied. Such an application leads to a unified expression, i.e., a delay-feedback system, suitable for both controller synthesis and stability analysis. Two linear matrix inequality (LMI) formulations are established to obtain repetitive controllers that minimize the $ H_\infty $ and generalized $ H_2 $ norms from the disturbance to the tracking error, respectively. The resulting closed-loop systems are shown to be exponentially stable if and only if the spectrum radii of their monodromy operators are less than $ 1 $. We further show that the exponential stability also implies the input-to-state stability (ISS) under bounded disturbances. Finally, simulation comparisons are presented to demonstrate the effectiveness of the overall arguments.

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