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We present a collection of different types of observation systems that work as differentiators. These observer-based differentiators can produce estimates for derivatives of a given signal, even though the given signal is prone to noise.

Image processing has become a critical technology in a variety of science and engineering disciplines. Although most image processing is performed digitally, optical analog processing has the advantages of being low-power and high-speed, but it requires a large volume. Here, we demonstrate flat optics for direct image differentiation, allowing us to significantly shrink the required optical system size. We first demonstrate how the differentiator can be combined with traditional imaging systems such as a commercial optical microscope and camera sensor for edge detection with a numerical aperture up to 0.32. We next demonstrate how the entire processing system can be realized as a monolithic compound flat optic by integrating the differentiator with a metalens. The compound nanophotonic system manifests the advantage of thin form factor as well as the ability to implement complex transfer functions, and could open new opportunities in applications such as biological imaging and computer vision.Vertical integration of a metalens to realize compound nanophotonic systems for optical analog image processing is realized, significantly reducing the size and complexity of conventional optical systems.

The attitude control problem is addressed for a quadrotor system subject to the modeling uncertainties and unknown disturbances. A novel attitude control scheme is proposed based on nonsingular fast terminal sliding mode (NFTSM) technique. First, the tracking differentiator (TD) is designed to obtain the smooth tracking signal and its derivative. Then, the extended state observer (ESO) is constructed to provide the estimate of the modeling uncertainties and unknown disturbances. With the designed TD and ESO, a novel NFTSM controller is developed such that tracking error converges to zero in finite time. The transient-state and the steady-state performances are both achieved with the new controller. Finally, the simulation and real experiment results verify the effectiveness and superiority of the proposed control method.

We present wave-based signal differentiation with unprecedented fidelity and flexibility by purposefully perturbing overmoded random scattering systems such that zeros of their scattering matrices lie exactly at the desired locations on the real frequency axis. Our technique overcomes limitations of hitherto existing approaches based on few-mode systems, both regarding their extreme vulnerability to fabrication inaccuracies or environmental perturbations and their inability to maintain high fidelity under in-situ adaptability. We demonstrate our technique experimentally by placing a programmable metasurface with hundreds of degrees of freedom inside a 3D disordered metallic box. Regarding the integrability of wave processors, such repurposing of existing enclosures is an enticing alternative to fabricating miniaturized devices. Our over-the-air differentiator can process in parallel multiple signals on distinct carriers and maintains high fidelity when reprogrammed to different carriers. We also perform programmable higher-order differentiation. Conceivable applications include segmentation or compression of communication or radar signals and machine vision. Here, the authors report purposefully perturbed wave chaos that enables analog signal processing with unprecedented fidelity and flexibility.

This paper presents a new high-order sliding mode tracking differentiator with backpropagation neural network based adaptive parameter estimation (SMF-BP) to obtain accurate filtering and differentiation estimates from signals that contain noise. It is improved based on the Levant’s high-order sliding mode tracking differentiator. First, SMF-BP incorporates a sigmoid function to reduce overshoot. Second, the BP neural network is introduced to adjust the parameters adaptively to balance response speed and filtering performance. Finally, the validity of SMF-BP has been demonstrated through numerical simulation examples.

Metasurface-enabled optical analog differentiation has garnered significant attention due to its inherent capacity of parallel operation, compactness, and low power consumption. Most previous works focused on the first- and second-order operations, while several significant works have also achieved higher-order differentiation in both real space and k-space. However, how to construct the desired optical transfer function in a practical system to realize scalable and multi-order-parallel high-order differentiation of images in real space, and particularly how to leverage it to tackle practical problems, have not been fully explored. Here, drawing on the basic mathematical feature of the Fourier transform, we theoretically propose universal phase-gradient functions of the Pancharatnam-Berry-phase-based meta-device for performing arbitrary order differentiation. The fifth-order optical differentiations for both intensity and phase images are realized in the experiment. More importantly, by exploring this elaborately designed spatial differentiator, we construct another scheme for optical super-resolution and achieve the estimation of the distance between two incoherent point sources within 0.015 of the Rayleigh distance, which thereby provides a potential toolkit for optical alignment in high-precision semiconductor nano-fabrication. Our findings hold promise for image processing, microscopy imaging, and optical super-resolution imaging. Here, the authors present a scalable approach for arbitrary high-order optical analog differentiations with metasurfaces. They realise fifth-order optical differentiations for both intensity and phase images, and demonstrate its use in optical super-resolution measurements for optical alignment in high-precision semiconductor nano-fabrication.

Analog spatial differentiation is used to realize edge-based enhancement, which plays an important role in data compression, microscopy, and computer vision applications. Here, a planar chip made from dielectric multilayers is proposed to operate as both first- and second-order spatial differentiator without any need to change the structural parameters. Third- and fourth-order differentiations that have never been realized before, are also experimentally demonstrated with this chip. A theoretical analysis is proposed to explain the experimental results, which furtherly reveals that more differentiations can be achieved. Taking advantages of its differentiation capability, when this chip is incorporated into conventional imaging systems as a substrate, it enhances the edges of features in optical amplitude and phase images, thus expanding the functions of standard microscopes. This planar chip offers the advantages of a thin form factor and a multifunctional wave-based analogue computing ability, which will bring opportunities in optical imaging and computing. The authors present a planar photonic chip, which operate as a multiple-order analog spatial differentiator. It provides a route for designing fast, power-efficient, compact and low-cost devices used in edge detection and optical image processing, thus expanding the functions of standard microscopes.

Filtering differentiators are capable both of rejecting large noises and exactly differentiating smooth signals. Even unbounded noises are rejected provided their local average is small. A special type constitute tracking filtering differentiators producing smooth outputs being derivatives one of another. Discrete filter versions are proposed, and the accuracy asymptotics are studied in the presence of noises and discrete sampling. Extensive computer simulation confirms the theoretical results.

This work is dedicated to the application of a semi-implicit homogeneous differentiator to estimate the angular velocity and the angular acceleration of each of the eight electric motors of the CRAFT cable-driven parallel robot from the recording of the angular position of their respective output shaft. These motors drive the winding or unwinding of eight cables to move the robot moving-platform. The results show that this differentiator, whose the definition is respectively based on two projectors, is an extremely efficient tool for estimating the angular velocities and accelerations of the eight motors. These estimated velocities and accelerations are much less noisy than their reference signals obtained by backward difference. Moreover, the estimated quantities obtained with this differentiator are successfully compared to those obtained by a non-linear observer based on the interpolation and numerical difference of the measured position variable. Those promising results will definitely contribute to a better control of the CRAFT robot.

In this paper, a novel and robust backstepping controller is introduced for a quadrotor system by combining an integral linear extended state observer (ILESO), an auxiliary dynamic system, and a sliding mode differentiator. The ILESO is specifically designed to observe unmeasured states and address the challenges posed by the ‘lump disturbance’, encompassing unknown external disturbances and modeling uncertainties, while also mitigating measurement noise effects. To contend with input saturation in the quadrotor, an auxiliary dynamic system is implemented. Additionally, a sliding mode differentiator is employed to counteract the differential explosion issue inherent in traditional backstepping controllers. The controller’s efficacy is established through the assurance of the quadrotor’s capability to accurately track desired attitudes. The uniform ultimate boundedness of all signals in the closed-loop system is demonstrated using the Lyapunov theory. Simulation results further illustrate the effectiveness and superiority of the proposed controller.