In this paper, the torus-event-based fault detection and isolation (FDI) problem is investigated for a class of time-varying multirate systems. An ellipsoidal constraint is first adopted to describe the fault in a more practical pattern, and a novel torus-event-triggering scheme is proposed to improve the unilateral triggering mechanism. The aim is to design the torus-event-based fault detection filter and fault isolation estimators such that both the prescribed variance constraint on the estimation error and the desired H _∞ performance on the disturbance are guaranteed over the finite horizon. Especially, the residual evaluation function is employed to detect the fault, and the residual matching function is developed to isolate the fault. Furthermore, three optimization problems are provided to seek separately the minimal parameters on the H _∞ performance level, the upper bound of the estimation error variance, and the triggering torus. Finally, two simulation examples are utilized to show the effectiveness of the FDI scheme proposed in this paper.

As one of the most significant applications of nanopositioning technology, the tremendous development of atomic force microscopes (AFMs) has been witnessed these years. Essentially, the scanning motions of AFMs are generally driven by piezoelectric tube scanners (PTSs), whose cross-coupling effect hinders their high speed and high-precision positioning. Therefore, it becomes an urgent yet challenging mission to establish a niche model for the nonlinear cross-coupling dynamics of the PTS. As an emergent pure data-driven learning approach, the Koopman operator sheds some light on the PTS modeling methodology, which is thus adopted in this article to approximate the nonlinear cross-coupling dynamics of PTSs in an infinite-dimensional space. Moreover, an online high-order extended dynamic mode decomposition algorithm is proposed for the finite-dimensional approximation of the Koopman operator online. The merit of the present model lies in updating the identified cross coupling upon the arrival of new data in an incremental way. Finally, experiments are conducted to approximate the X–Y axes cross coupling of PTSs of an NTMDT Prima AFM, which verifies the effectiveness and superiority of the proposed modeling algorithm. This article is expected to pave the way from the Koopman operator learning theory to real applications in dynamics modeling of abundant nanoscale measurement systems.

This article develops an identification algorithm for nonlinear systems. Specifically, the nonlinear system identification problem is formulated as a sparse recovery problem of a homogeneous variant searching for the sparsest vector in the null subspace. An augmented Lagrangian function is utilized to relax the nonconvex optimization. Thereafter, an algorithm based on the alternating direction method and a regularization technique is proposed to solve the sparse recovery problem. The convergence of the proposed algorithm can be guaranteed through theoretical analysis. Moreover, by the proposed sparse identification method, redundant terms in nonlinear functional forms are removed and the computational efficiency is thus substantially enhanced. Numerical simulations are presented to verify the effectiveness and superiority of the present algorithm.

This article presents a data-driven, Koopman operator-based modeling scheme for analyzing and predicting cross-coupling hysteresis effects of the piezoelectric tube scanners (PTSs) used in atomic force microscopes (AFMs). Such cross-coupling hysteresis effects between different PTS axes significantly reduce the positioning precision of AFMs. In contrast to most of the existing methods for PTS hysteresis, which involve complex nonlinear dynamics identification processes, the present study leverages the Koopman operator theory instead to treat the nonlinear hysteresis as a linear system. Therein, a Hankel extended dynamic mode decomposition (H-EDMD) algorithm is proposed to learn the finite-dimensional descriptions of the Koopman operator and the associated Koopman eigenspectrum. Moreover, the proposed H-EDMD even allows sparse sampling on the PTS systems, which is desirable in real industrial applications. Finally, extensive comparison experiments with a mainstream modified Prandtl-Ishlinskii model are conducted on an NTMDT Prima AFM to substantiate the effectiveness and superiority of the proposed H-EDMD method.