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

Source abstract

A fixed-time stabilizer for underactuated Euler–Lagrange systems was established in Part 1 of this study , which certified fixed-time reaching on the actuated manifold and supplied a separate certificate for the internal dynamics. This paper asks what that certificate costs and what it buys. We decompose the total settling time into the reaching time governed by the certificate and an internal transient governed by the plant, and show that the two obey different constraints: the settling bound depends on the reaching gains alone and tightens exactly in inverse proportion to their scale, whereas the unactuated coordinate obeys a rate limit fixed by the coupling authority and is untouched by those gains. We prove that this forces a strictly positive floor on total settling time as the gains grow without bound, and we measure that floor to be far more binding in practice than the proof alone guarantees. On the translational oscillator with rotational actuator, scaling the reaching gains over a range of 13.3 tightens the analytical bound by a factor of 13.3 while the full-state settling time improves by 6.0% and the control energy rises by 53%. The floor is not an artifact of actuator saturation: with the actuation limit raised a hundredfold the same gain range still moves settling by only 11.4%. It is not an artifact of the plant either, since the cart-pole shows the separation more sharply, nor of the reaching-law gain family specifically, since settling varies by only 7.0% across a 49-design two-dimensional sweep over which the bound varies by a factor of 53.3. The floor is likewise unchanged at 1% and 5% settling tolerances, across a sixteenfold range of parametric mismatch including its sign, and under bounded disturbances, which the certified channel absorbs completely while the internal channel does not. A search over all eight controller parameters, a sampled-data implementation with measurement noise and torque quantization, three further cost metrics, and a fourfold finer sweep leave the result and the parity finding intact, and show that the designs that buy a tight certificate are also the ones that chatter and amplify noise. The bound itself is valid throughout, honoured by every design with margins between 62% and 95% once the event it governs is identified correctly. A matched-performance protocol puts the fixed-time law and its linear counterpart at energy parity, with ratios between 0.934 and 1.128. By contrast, increasing the coupling authority eightfold makes the response 7.4 times faster while reducing the energy integral by a factor of 14.6. The resulting design rule is to select gains against the guarantee that must be certified and to treat achievable response speed as a question about coupling authority rather than about feedback gain.