We study the problem of optimally generating quantum gates in a logical subspace embedded in a larger Hilbert space, where the dynamics is also affected by unknown static imperfections. This general problem is widespread across various emergent quantum technology architectures. We derive the fidelity susceptibility in the computational subspace as a measure of robustness to perturbations, and define a cost function that quantifies leakage out of the subspace. We tackle both effects using a two-stage optimization where two cost functions are minimized in series. Specifically, we apply this framework to the generation of single-qubit gates in a superconducting transmon system, and find high-fidelity solutions robust to detuning and amplitude errors across various parameter regimes. We also show control pulses which maximize fidelity while minimizing leakage at all times during the evolution. However, finding control solutions that address both effects simultaneously is shown to be much more challenging, indicating the presence of a trade-off relation.

Nonlinear friction has long been, and continues to be, one of the major challenges for precision motion control systems. A linear asymptotic observer of the motion state variables with nonlinear friction uses a dedicated state-space representation of the dynamic friction force (including pre-sliding) (Ruderman, 2023b), which is robust to the noisy input signals. The observer implements the reduced-order Luenberger observation law, while assuming the output displacement is the only available measurement. The latter is relevant for multiple motion control applications with use of encoder-type sensors. The uniform asymptotic stability and convergence analysis of the proposed observer are elaborated in the present work by using the Lyapunov function-based stability criterion by Ignatyev and imposing parametric constraints on the time-dependent eigenvalues of the system matrix to be always negative real. A design procedure for assigning a dominant, and thus slowest, real pole of the observer is proposed. Explanative numerical examples accompany the developed analysis. In addition, a thorough experimental evaluation is given for the proposed observer-based friction compensation which is performed for positioning and tracking tasks. The observer-based compensation, which can serve as a plug-in to a standard feedback controller, extends a PID feedback control that is optimally tuned for disturbance suppression. The experimental results are compared with and without the plugged-in observer-based compensator.