Global regulation with matrix-valued PID gains for higher-dimensional manipulators

Determine whether appropriately designed matrix-valued PID gains can achieve global regulation for higher-dimensional robot manipulators.

Background

The paper proves that scalar-gain classical PID control cannot globally regulate all multi-degree-of-freedom robot manipulators satisfying the stated Euler–Lagrange structural assumptions. Specifically, for every manipulator dimension n≥2, the authors construct a smooth robot manipulator for which no scalar gain triple (k_p,k_i,k_d) achieves global asymptotic regulation.

The unresolved issue is whether replacing the scalar gains with appropriately designed matrix-valued proportional, integral, and derivative gains can overcome this limitation and achieve global regulation for higher-dimensional robot manipulators. This question concerns a broader controller architecture than the scalar-gain PID class analyzed in the paper.

References

Determining whether appropriately designed matrix-valued PID gains can achieve global regulation for higher-dimensional robot manipulators remains an open problem and a promising direction for future research.

— On Global Regulatability of Robot Manipulators by Classical PID  (2609.01207 - Zhao et al., 1 Sep 2026) in Remark 3, Section 3.2, “Impossibility of global PID regulation for multi-DOF manipulators”