Settling Time Decomposition and Gain Selection for Fixed-Time Underactuated Euler–Lagrange Systems

Source abstract

A fixed-time stabiliser for underactuated Euler–Lagrange systems was established in Part 1 of this study. This paper examines what its fixed-time certificate costs and what is achieved by tightening it. We decompose the full-state settling time into a reaching phase governed by the certificate and an internal transient governed by the plant, and we show that the two obey different constraints: the settling bound depends on the reaching gains alone and tightens in exact inverse proportion to their scale, whereas the unactuated coordinate obeys a rate limit set by the coupling authority. We prove that this rate limit floors the full-state settling time at every gain level, so that the ratio of settling time to certificate grows without bound as the gains do, and we give a gain-free estimate of the internal transient that lies within 13–18% of the measured settling time and accounts for the whole gain effect as a tracking lag. On the TORA benchmark, a 13.3-fold gain range tightens the bound 13.3-fold, while full-state settling improves by 6.0% and control energy rises by 53%. Fourteen controls, including actuator saturation, a second benchmark, parametric mismatch, bounded disturbances, a search over all gains, a sampled-data implementation, a gain range extended to κ=64, and a linear-quadratic anchor, establish that the floor is structural and that the designs that buy a tight certificate are also the ones that chatter and amplify noise. A matched-performance protocol places the fixed-time law and three linear counterparts at energy parity. The result is a gain selection rule with a closed-form saturation-aware form: choose the reaching gains against the guarantee that must be certified and treat the response speed as a question about coupling authority rather than feedback gain.