The molecular scattering of light from certain organic liquids

An attempt is made to discover in the molecular scattering of light a method by which atomic linkages in molecules may be recognized. The essential features of the method are described. There are rather serious difficulties in its application to complex molecules, two of which are suggested. Nevertheless, it may be claimed that the work is not without result, because frequencies attributable to linkages which hold externally substituted atoms or groups seem to be relatively undisturbed by positional effects.In conclusion, the authors wish to express their indebtedness to Mr. G. M. J. Mackay whose interest in this problem made the present work possible.     

Stability analysis of stochastic switching singular systems with jumps

This paper deals with the stability problem of a class of stochastic switching singular systems with jumps, where all the continuous subsystems can be unstable. By introducing multiple time-varying discretized Lyapunov functions and designing novel switching signals via mode-dependent dwell times, sufficient conditions ensuring mean square and almost sure stabilities of the considered systems are obtained. On the basis of these findings, some existing results in the literature are further generalized. We finally provide a numerical example to verify the applicability of the proposed theoretical results.     

Observability of singular Boolean control networks with state delays

This paper investigates the observability of singular Boolean control networks (SBCNs) with state delays. First, via Cheng product, the considered system is converted into its algebraic form. Then, under the uniqueness of the solution, two methods are proposed to transform the system into a general Boolean control network (BCN). Therefore, the considered system’s observability is equivalent to the BCN’s observability. Subsequently, the restricted input-state incidence matrix is introduced and some necessary and sufficient conditions are presented for the observability of BCNs with state delays. Finally, two illustrative examples are presented to show that the main results obtained are effective in analyzing the observability of SBCNs with state delays.     

Homomorphic image watermarking with a singular value decomposition algorithm

In this paper, a new homomorphic image watermarking method implementing the Singular Value Decomposition (SVD) algorithm is presented. The idea of the proposed method is based on embedding the watermark with the SVD algorithm in the reflectance component after applying the homomorphic transform. The reflectance component contains most of the image features but with low energy, and hence watermarks embedded in this component will be invisible. A block-by-block implementation of the proposed method is also introduced. The watermark embedding on a block-by-block basis makes the watermark more robust to attacks. A comparison study between the proposed method and the traditional SVD watermarking method is presented in the presence of attacks. The proposed method is more robust to various attacks. The embedding of chaotic encrypted watermarks is also investigated in this paper to increase the level of security.     

Stabilization of switched linear singular systems with state reset

This paper investigates the stability and stabilization of switched linear singular systems with state reset at switching instants. Based on the dynamics decomposition of singular subsystems, a sufficient stability condition for the system with the given state reset is obtained. Then, the stabilization problem by state reset is investigated and an algorithm for computing the reset matrices is presented. The obtained results extend some previous works on both singular switched systems and reset control for normal switched systems. Finally, a numerical example is presented to illustrate the effectiveness of the proposed approach.     

Accurate approximate solutions to oscillatory problems with perturbing singular damping

In this paper we attempt to obtain approximate solutions of improved accuracy for a class of differential equations of the form

$\text{d}{}^2\text{y}\text{d}\text{x}{}^2\text{+εμ(x)}\text{d}\text{y}\text{d}\text{x}\text{+ω}{}^2{}_{\mathrm{c}}\text{y = 0}$

, where ε is a real parameter less than unity, ω_c is a positive real constant of order unity and μ(x) is a singular function of x in the region of interest. It does not appear to be possible to find a general analytic expression for the error estimate of the approximate solution. For the case μ(x) = x^?2, however, it is shown that the approximate solution is accurate to 0(ε²), as x → 0^? from negative values, by comparing it with the numerically integrated solution. For the same case, the approximate solution is orders of magnitude more accurate than Poincaré's first-order perturbation solution, which is accurate to $\text{0(}\text{ε}{}^2\text{ln}\text{|x|}\text{|x|}\text{)}$ as x → 0^?. This work arose in search of analytic solutions to a linearized form of the restricted three-body problem.     

Embedding singular linear system with initial condition into distribution space

For inconsistent initial value problem of singular linear system, passing from classical solution to distributional solution is formulated as a process of embedding the system with initial condition into distribution space. Three technical notions, function space on R₊ with given initial value, consistent extension of the pair of the function space and input, and order exchange of the two operations of embedding and differentiating, result in a rigorous definition of distributional solution.     

Robust control for discrete-time singular large-scale systems with parameter uncertainty

This paper addresses the problem of robust stabilization for uncertain discrete-time singular large-scale systems with parameter uncertainties. The system under consideration is not necessarily regular. The parameter uncertainties are assumed to be time invariant, but norm-bounded. The purpose of the robust stabilization problem is to design state feedback controllers such that, for all admissible uncertainties, the closed-loop system is regular, causal and stable. In terms of strict LMIs, sufficient conditions for the solvability of the problem is presented, and the parameterization of desired state feedback controllers is also given. A numerical example is given to demonstrate the applicability of the proposed design approach.     

Robust finite-time stability of singular nonlinear systems with interval time-varying delay

The problem of robust finite-time stability (RFTS) for singular nonlinear systems with interval time-varying delay is studied in this paper. Some delay-dependent sufficient conditions of RFTS are derived in the form of the linear matrix inequalities (LMIs) by using Lyapunov–Krasovskii functional (LKF) method and singular analysis technique. Two examples are provided to show the applications of the proposed criteria.     

Exponentially admissibility of neutral singular systems with mixed interval time-varying delays

The chief aim of this paper is to analyze the exponentially admissibility of neutral singular systems (NSSs) with mixed interval time-varying delays (TVDs). A new Lyapunov-Krasovskii functional (LKF) is presented based on state decomposition method. Via adopting linear matrix inequality (LMI) method and zero-value equation technology, a novel delay-dependent exponentially admissibility criterion is acquired. Furthermore, the exponentially admissibility analysis of the NSSs with equal mixed TVDs is discussed. Three numerical examples show that our method is effective and superior. In particular, the reasonable analysis and design for two practical application models (direct current (DC) motor model and oil catalytic cracking process (OCCP) model) illustrate that our research is very meaningful.     

Observer-based control for singular nonhomogeneous Markov jump systems with packet losses

This paper is concerned with the observer-based H_∞ control for a class of singular Markov jump systems over a finite-time interval, where the transition probability (TP) is time-varying and is limited to a convex hull. Due to the limited capacity of network medium, packet losses are presented in the underlying systems. Firstly, using a stochastic Lyapunov functional, a sufficient condition on singular stochastic H_∞ finite-time boundedness for the corresponding closed-loop error systems is provided. Subsequently, a linear matrix inequality (LMI) condition on the existence of the H_∞ observer-based controller is developed from a new perspective. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed controller design method, wherein it is shown that the proposed method yields less conservative results than those in the literature.     

State bounding estimation of positive singular discrete-time systems with unbounded time-varying delays

This paper investigates the problem of state bounding for linear positive singular discrete-time systems with unbounded time-varying delay. Our approach is based on the comparison principle and the spectral analysis of matrices. Using the proposed approach, we provide new sufficient conditions for the regularity, causality, positivity and the existence of the smallest ultimate upper bound of such systems. The conditions are given in terms of linear programming problems, which can be solved by LP optimal toolbox. Numerical examples with simulation are given to demonstrate the advantage and validity of the proposed theoretical results. The proposed method is the first trial in the state bounding problem of linear singular discrete-time systems with unbounded time-varying delays.     

Resilient sampled-data control design for singular networked systems with random missing data

This paper proposes an active resilient control strategy for singular networked control systems with external disturbances and missing data scenario based on sampled-data scheme. To characterize the missing data scenario, a stochastic variable satisfying Bernoulli distributed white sequence is introduced. Based on this scenario, in this paper, two different models are proposed. For both the models, by using Lyapunov–Krasovskii functional approach, which fully uses the available information about the actual sampling pattern, some sufficient conditions in terms of linear matrix inequalities (LMIs) are separately obtained to guarantee that the resulting closed-loop system is admissible and strictly dissipative with a prescribed performance index. In particular, Jensen’s and Wirtinger based integral inequalities are employed to simplify the integral terms which appeared in the derivation of stabilization results. Then, if the obtained LMIs are feasible, the corresponding parameters of the designed resilient sampled-data controller are determined. Finally, two numerical examples are presented to demonstrate the effectiveness of the proposed control design technique.     

Reachable set control for singular systems with disturbance via sliding mode control

The problem of the reachable set (RS) control of sliding mode control (SMC) for a class of singular systems with or without time-varying delay under zero initial conditions is studied. The purpose is to get an RS boundary containing all states of the system by designing an SMC. Firstly, singular systems with or without time-varying delay are decomposed into slow and fast subsystems by using the decomposition approach. Then, the augmented Lyapunov functional is built utilizing the decomposed state vector. The SMC is designed based on the exponential reaching criterion, resulting in the corresponding closed-loop control system (CLCS) construction. As a consequence, an RS criterion is constructed by employing the inequality scaling approach and the free-weighting matrix in conjunction with the linear matrix inequality (LMI). Finally, the validity and primacy of the results are provided by two numerical and practical examples.     

Stability of mode-dependent linear switched singular systems with stable and unstable subsystems

This paper studies the E-exponential stability of mode-dependent linear switched singular systems with stable and unstable subsystems. First, by constructing an appropriate multiple discontinuous Lyapunov function, new sufficient conditions of E-exponential stability for linear switched singular systems are established. Considering the feature of mode-dependent average dwell time switching, we adopt the switching strategy where fast switching and slowing switching are respectively applied to unstable and stable subsystems. Compared with the existing results, our approach is more flexible and tighter bounds can be obtained. Finally, a numerical example is provided to illustrate the effectiveness of the proposed criteria.     

Improved results on admissibility analysis for singular systems with periodically time-varying delay

This paper is concerned with the admissibility analysis for the singular system with a periodically time-varying delay. By dividing the periodic delay interval into monotonically decreasing and increasing intervals, a novel Lyapunov–Krasovskii functional (LKF) is developed in virtue of the looped functional philosophy, relaxing the positive definition of the LKF and making full use of the system state and the time-varying delay function information. Then, a new admissibility condition is derived by combining the newly constructed LKF and the second-order canonical Bessel–Legendre integral inequality. Finally, two numerical examples are given to demonstrate the superiority of the developed method.     

Two dimensional quaternion valued singular spectrum analysis with application to image denoising

Compared to the traditional single color plane based image denoising methods, the quaternion valued singular value decomposition (QSVD) exploits the relationship among different color planes. Hence, it has been applied to the color image denoising. On the other hand, compared to the non-overlapping based image denoising methods, the two dimensional real valued singular spectrum analysis (2DSSA) constructs the trajectory matrix with many elements in the matrix being overlapped. Since the 2DSSA exploits the local information within each color plane, it has also been applied to the single color plane based image denoising. However, neither these two image denoising methods can exploit the relationship among the color planes and the local information within each color plane simultaneously. Therefore, this paper proposes a two dimensional quaternion valued singular spectrum analysis (2DQSSA) based method for performing the color image denoising. Our proposed method can enjoy the advantages of both methods. However, the most critical issue for the 2DQSSA is on the selection of these 2DQSSA components. This paper finds that the optimal total number of the 2DQSSA components used for performing the reconstruction is monotonic decreasing with respect to the power of the noise in the image. Therefore, the polynomial fitting approach is proposed to model this relationship. Since the 2DQSSA based denoising method exploits the relationship among the red color plane, the green color plane and the blue color plane, the 2DQSSA based denoising method outperforms the conventional single color plane based denoising methods. Moreover, since the 2DQSSA based denoising method also exploits the local relationship within each color plane, the 2DQSSA based denoising method outperforms the non-overlapping based methods.     

Robust finite-time sliding mode control for discrete-time singular system with time-varying delays

This paper explores the finite-time bounded issue for discrete-time singular time-varying delay system via sliding mode control method. A suitable discrete-time sliding mode control law is constructed to drive the state trajectories onto the specified sliding surface in a given finite time interval. Meanwhile, sufficient conditions for finite-time bounded to the closed-loop delayed system are provided in both reaching phase and sliding motion phase. In addition, the finite-time sliding mode controller gain matrix can be solved by using the linear matrix inequalities approach. Finally, three numerical examples are illustrated to demonstrate the superiority and practicability of presented results.     

Event-based sliding mode control for fuzzy singular systems with semi-Markovian switching parameters

This paper addresses the observer-based dynamic event-triggered (DET) sliding mode control (SMC) problem for fuzzy singular semi-Markovian jump systems (FSS-MJSs) subject to generalized dissipative performance, in which a novel double-quantized structure is reasonably merged into a unified model. The main aim of this paper is to develop a mode-dependent adaptive sliding mode control (ASMC) law through the DET rule, which not only makes the closed-loop systems mean-square admissibility and generalized dissipative, but also the finite-time reachability around the predefined sliding mode surface (SMS) can be achieved. Firstly, in order to improve the data transmission efficiency and save network bandwidth resources, DET and doubled-quantized-based control protocol are introduced, in which the event-based threshold function is dynamically regulated and the data of input and output are both quantized; Secondly, due to the sensor information constraints, system state information is not always obtained in practice, hence, a suitable observer design can make up for this defect. Meantime, in terms of elegant linearization technique and implicit function theorem, the uniqueness of the solution for FSS-MJSs is also established; Additionally, by making use of the Lyapunov functional and linear matrix inequality (LMI) technique, both the desired SMC gains, observer gains and triggering parameter matrices are co-designed, more than that the derivative singular matrix is also integrated into the whole design process such that the derived conditions are much more easily to be checked; Finally, a numerical example and a practical application example are co-given to verify the effectiveness of our design mentality.