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This paper presents methods to find a positively invariant set (PIS) and derive conditions on the edge weights for a multirate Kuramoto oscillator network to achieve synchronization. These methods can be applied to a network with an arbitrary topology and nonidentical oscillators. The proposed methods are based on the construction of energy functions for this type of network. Two different conditions on the edge weights are provided by using graph spectral properties, or alternatively by analyzing a path set of a graph. These methods provide flexibility in checking the edge weight conditions. We further improve the estimate of the PIS for the multirate Kuramoto network when its damping coefficients are greater than a certain value. The effectiveness and conservativeness of the proposed methods are demonstrated by simulation studies. A comparison with another method from the existing literature shows that our method gives less conservative estimate of the PIS.

The corrosion of steel reinforcement in concrete often leads to huge unbudgeted expenses for maintaining, monitoring and renovating an infrastructure. This is mainly due to the presence of salts or chemical chlorides that pose a danger to the concrete structures. The determination of the existence of these corrosive salts is vital for defining the service life of concrete. This research looked at developing an electromagnetic induction (EMI) sensor for the detection of corrosive salts. The first design adopted a single-loop coil (SLC) concept, and the second was based on a multiple-loop coil (MLC) one using copper wire. Tests were conducted on these two techniques, and with the results obtained, the latter seemed more promising; thus, a prototype sensor was developed using the MLC concept. As this new prototype sensor was able to detect the manifestation of chemical contents in a concrete structure, it could be used as a non-destructive evaluation (NDE) technique for the detection of corrosive chemicals in concrete and has the further possibility of detecting corrosion in concrete.

This paper demonstrates a high-speed spiral imaging technique for an atomic force microscope (AFM). As an alternative to traditional raster scanning, an approach of gradient pulsing using a spiral line is implemented and spirals are generated by applying single-frequency cosine and sine waves of slowly varying amplitudes to the X and Y-axes of the AFM's piezoelectric tube scanner (PTS). Due to these single-frequency sinusoidal input signals, the scanning process can be faster than that of conventional raster scanning. A linear quadratic Gaussian controller is designed to track the reference sinusoid and a vibration compensator is combined to damp the resonant mode of the PTS. An internal model of the reference sinusoidal signal is included in the plant model and an integrator for the system error is introduced in the proposed control scheme. As a result, the phase error between the input and output sinusoids from the X and Y-PTSs is reduced. The spirals produced have particularly narrow-band frequency measures which change slowly over time, thereby making it possible for the scanner to achieve improved tracking and continuous high-speed scanning rather than being restricted to the back and forth motion of raster scanning. As part of the post-processing of the experimental data, a fifth-order Butterworth filter is used to filter noises in the signals emanating from the position sensors and a Gaussian image filter is used to filter the images. A comparison of images scanned using the proposed controller (spiral) and the AFM PI controller (raster) shows improvement in the scanning rate using the proposed method.