Free Matrices Research Articles

Discrete event-triggered security control for Markovian CVNNs with additive time-varying delays under random deception attacks

This paper is concerned with the security stabilization problem for a class of Complex-valued Neural Networks (CVNNs) with Markov Jump Parameters (MJPs) and Additive Time-varying Delays (ATVDs) under Random Deception Attacks (RDAs). Different from the existing literature, the instant and strength of RDAs considered in this paper is both random, which is more in line with the real situation. Secondly, a general Lyapunov–Krasovskii Functional (LKF) contains more information about MJPs and ATVDs is constructed, and a new Complex-valued Reciprocally Convex Inequality (CVRCI) containing more free matrices and ATVDs parameters is proposed, which play a key role in reducing the conservativeness of security stabilization criteria. Thirdly, a Discrete Event-triggered Mechanism (DETM) is introduced to mitigate the transmission burden of communication networks, in which the triggering condition of DETM mainly relies on the current sampled state and the last triggered state. Then, by combining with the LKF, CVRCI, DETM, and other analysis techniques, some less conservative security stabilization criteria for the underlying systems are provided in terms of Linear Matrix Inequalities (LMIs). Finally, the effectiveness of our results are verified by two numerical examples and a practical example.

[formula omitted] performance analysis for aperiodic sampled-data systems via a matrix-injection method

This study proposes a matrix-injection method to complete the H∞ performance analysis of linear systems under aperiodic sampled data. First of all, an augmented looped-functional is constructed by combining the looped-functional method with state vector augmentation. To make use of sampling period-related information in this augmented functional, sampling period-dependent quadratic terms are kept in the derivative of the functional. Then, to deal with the aforementioned quadratic terms, a matrix-injection method is developed. This method introduces two free matrices, which allows for a reduction in the conservativeness of the results. A sufficient condition with less conservatism is provided using these techniques. Finally, three examples are given to demonstrate the superiority of the presented approach.

A metaplectic perspective of uncertainty principles in the linear canonical transform domain

We derive Heisenberg uncertainty principles for pairs of Linear Canonical Transforms of a given function, by resorting to the fact that these transforms are just metaplectic operators associated with free symplectic matrices. The results obtained synthesize and generalize previous results found in the literature, because they apply to all signals, in arbitrary dimension and any metaplectic operator (which includes Linear Canonical Transforms as particular cases). Moreover, we also obtain a generalization of the Robertson-Schrödinger uncertainty principle for Linear Canonical Transforms. We also propose a new quadratic phase-space distribution, which represents a signal along two intermediate directions in the time-frequency plane. The marginal distributions are always non-negative and permit a simple interpretation in terms of the Radon transform. We also give a geometric interpretation of this quadratic phase-space representation as a Wigner distribution obtained upon Weyl quantization on a non-standard symplectic vector space. Finally, we derive the multidimensional version of the Hardy uncertainty principle for metaplectic operators and the Paley-Wiener theorem for Linear Canonical Transforms.

Stability analysis of discrete‐time systems with a time‐varying delay via improved methods

AbstractThis paper is concerned with the stability analysis of discrete‐time systems with a time‐varying delay. The conservatism and computation burden are two important factors to evaluate a stability condition. By taking the relationship of two reciprocally convex parts into consideration, a new combined matrix‐separation‐based inequality is proposed that involves only a few free matrices. Moreover, an improved matrix‐injection‐based transformation lemma with the parameter varying within a closed interval is proposed by introducing only one free matrix. By constructing an appropriate Lyapunov–Krasovskii functional and applying the improved methods, a relaxed stability condition is consequently obtained with a small number of decision variables. Two numerical examples are given to show the merits of the proposed methods.

New results on stability analysis for a class of generalized delayed neural networks

This paper addresses the stability analysis of neural networks with a time-varying delay, where the delay is periodically varying bounded function with constrained derivatives. In order to obtain less conservative stability criteria, two novel Lyapunov-Krasovskii functionals (LKFs) with delay-derivative-variation-dependent matrices are constructed. In one LKF, two delay-product terms are double utilized and more free matrices are incorporated. Additionally, a new type of the LKF with delay-derivative-variation-dependent matrices, namely, the neuronal-activation-function-based looped functional, is developed by accounting for the periodic nature of the delay function and the coupling information of the system state and the neural activation function. Then, several stability conditions of delayed neural networks are developed by combining some integral inequalities. Finally, the superiority and validity of developed stability conditions are validated on two benchmark examples.

Stability Analysis of Sampled-Data Systems Based on Sawtooth-Characteristic-Based Hierarchical Integral Inequality.

The aim of this article is to investigate the stability of sampled-data systems (SDSs) by introducing a sawtooth-characteristic-based hierarchical integral inequality (SCBHII) and to obtain the maximum allowable sampling period that maintains the stability of the system. First, by associating the sawtooth characteristics of the input delay in SDSs with free matrices, an SCBHII is proposed; its accuracy improves as the hierarchy increases. Subsequently, a high-order two-sided looped-functional, which considers both the sampling multi-integral states and the sawtooth pattern, is introduced to cater to the aforementioned inequality. In addition, the system variables are augmented by sawtooth pattern-related terms, which eliminates the need for additional secondary processing when determining the negative-definiteness of derivatives with high-order terms. By combining the high-order two-sided looped-functional with the proposed SCBHII, a stability criterion for SDSs with reduced conservatism is achieved, presented in the form of linear matrix inequalities. The proposed inequality technique and the stability criterion are shown to be effective and superior through three numerical examples and a real-world simplified power market model.

Nonfragile power and frequency control in islanded microgrids under abnormal asynchronous stochastic cyber attacks

In this paper, we focus on the real power sharing and frequency regulation of distributed generators in islanded microgrids with abnormal asynchronous stochastic cyber attacks, which is of great significance to the information security and stable operation of microgrids. Firstly, considering the possible cyber attacks in the communication network, a distributed non-fragile controller with coupled memory delay is proposed according to the nonperiodic sampled-data control. Then, the construction of delay-dependent two-sided looped-functional makes the Lyapunov–Krasovskii functional contain more delay and sampling information and relaxes constraints on free matrices. In addition, based on the enhanced integral inequality technique and the linear convex combination method, a sampling-based consensus protocol is presented to solve the issues of real power sharing and frequency regulation in the islanded microgrids. Finally, to verify the effectiveness and feasibility of the designed control strategy, a modified IEEE 34-bus test system is used for the experiment.

Magnetic-Responsive Liposomal Hydrogel Membranes for Controlled Release of Small Bioactive Molecules-An Insight into the Release Kinetics.

This work explores the unique features of magnetic-responsive hydrogels to obtain liposomal hydrogel delivery platforms capable of precise magnetically modulated drug release based on the mechanical responses of these hydrogels when exposed to an external magnetic field. Magnetic-responsive liposomal hydrogel delivery systems were prepared by encapsulation of 1,2-dipalmitoyl-sn-glycero-3-phosphocoline (DPPC) multilayered vesicles (MLVs) loaded with ferulic acid (FA), i.e., DPPC:FA liposomes, into gelatin hydrogel membranes containing dispersed iron oxide nanoparticles (MNPs), i.e., magnetic-responsive gelatin. The FA release mechanisms and kinetics from magnetic-responsive liposomal gelatin were studied and compared with those obtained with conventional drug delivery systems, e.g., free liposomal suspensions and hydrogel matrices, to access the effect of liposome entrapment and magnetic field on FA delivery. FA release from liposomal gelatin membranes was well described by the Korsmeyer-Peppas model, indicating that FA release occurred under a controlled diffusional regime, with or without magnetic stimulation. DPPC:FA liposomal gelatin systems provided smoother controlled FA release, relative to that obtained with the liposome suspensions and with the hydrogel platforms, suggesting the promising application of liposomal hydrogel systems in longer-term therapeutics. The magnetic field, with low intensity (0.08 T), was found to stimulate the FA release from magnetic-responsive liposomal gelatin systems, increasing the release rates while shifting the FA release to a quasi-Fickian mechanism. The magnetic-responsive liposomal hydrogels developed in this work offer the possibility to magnetically activate drug release from these liposomal platforms based on a non-thermal related delivery strategy, paving the way for the development of novel and more efficient applications of MLVs and liposomal delivery systems in biomedicine.

The free field: Realization via unbounded operators and Atiyah property

Let X1,…,Xn be operators in a finite von Neumann algebra and consider their division closure in the affiliated unbounded operators. We address the question when this division closure is a skew field (aka division ring) and when it is the free skew field. We show that the first property is equivalent to the strong Atiyah property and that the second property can be characterized in terms of the noncommutative distribution of X1,…,Xn. More precisely, X1,…,Xn generate the free skew field if and only if there exist no non-zero finite rank operators T1,…,Tn such that ∑i[Ti,Xi]=0. Sufficient conditions for this are the maximality of the free entropy dimension or the existence of a dual system of X1,…,Xn. Our general theory is not restricted to selfadjoint operators and thus does also include and recover the result of Linnell that the generators of the free group give the free skew field.We give also consequences of our result for the question of atoms in the distribution of rational functions in free variables or in the asymptotic eigenvalue distribution of matrices over polynomials in asymptotically free random matrices. This solves in particular a conjecture of Charlesworth and Shlyakhtenko.

Zero-Sum Squares in $\{-1, 1\}$-Matrices with Low Discrepancy

Given a matrix $M = (a_{i,j})$ a square is a $2 \times 2$ submatrix with entries $a_{i,j}$, $a_{i, j+s}$, $a_{i+s, j}$, $a_{i+s, j +s}$ for some $s \geq 0$, and a zero-sum square is a square where the entries sum to $0$. Recently, Arévalo, Montejano and Roldán-Pensado proved that all large $n \times n$ $\{-1,1\}$-matrices $M$ with discrepancy $|\sum a_{i,j}| \leq n$ contain a zero-sum square unless they are split. We improve this bound by showing that all large $n \times n$ $\{-1,1\}$-matrices $M$ with discrepancy at most $n^2/4$ are either split or contain a zero-sum square. Since zero-sum square free matrices with discrepancy at most $n^2/2$ are already known, this bound is asymptotically optimal.

A novel predictive control based management strategy considering smart PHEV in digital twin simulation

A combined layout predictive control and H2 / H∞ control is proposed using gain-scheduled control rules for DC-DC boost converters (DDBCs) fed through constant power load (CPL) for improving the stability of the DC bus voltage during extensive disturbances considering the DC loads such as plug-in hybrid electric vehicle (PHEV) demands. Accordingly, to track the reference signal, linear parameter variables with bounded disturbances are applied to express its kinematic error. In this method, the linear matrix inequality (LMI) statuses for the error following layout are solved for the purpose of stabilizing the controller. Afterward, stability constraints have been defined so that the LMI auxiliary variables may be regarded as upper bounds. Furthermore, by allowing more free matrices, conservatism could be reduced. Lastly, the suggested controller has been validated theoretically by a simulation test with a variety of operational conditions, and the performance and efficiency of the controller has been proven. The suggested method can track reference signals with a low steady-state error in different scenarios such as; alternation in the CPL’s desired power; alternation in reference voltage signal; alternation in input voltage. This has been done in solar-based energy systems using digital twin simulation.

Finite time stability of linear time varying delay systems using free matrix based integral inequalities

This paper discusses the problem of finite time stability (FTS) for linear delayed systems where the time-varying delay is bounded. First, based on a Lyapunov–Krasovskii Functional (LKF) which contains terms of triple integrals, delay-dependent FTS conditions are provided by introducing some free-weighting matrices. Then, new approximations of the single and multiple quadratic integrals that appear in the LKF derivative are established using integral inequalities, called free-matrix-based integral inequalities. The dimensions of these free matrices integral in our results are less than those obtained in the literature. This reduction in the number of variables does not mean that our method is a particular case but simply that our approach is completely different from the others and therefore more effective. Finally, less conservative design conditions in terms of Linear Matrix Inequalities (LMIs) are proposed and solved by the LMI Tools of MATLAB to show the advantage and effectiveness of the proposed approach.

Improved Integral Inequality Based on Free Matrices and Its Application to Stability Analysis of Delayed Neural Networks via Matrix-valued Cubic Polynomial Inequality

This paper deals with the stability analysis of delayed neural networks (DNNs). Firstly, an improved integral inequality based on free matrices (I3BFM) is newly proposed. The I3BFM gives the upper bound of the integral quadratic form including the state-related vector stacked by the state, its derivative, and its integration by introducing some free matrices. The upper bound of the I3BFM fully reflects the information not only on each term in the state-related vector but also on the cross-terms between the three terms. Secondly, the double integral Lyapunov-Krasovskii functional (LKF) of the integral quadratic form including the state-related vector is established to utilize the proposed I3BFM in the stability analysis of DNN. From the derivative of the double integral LKF, a matrix-valued cubic polynomial on a time-varying delay is generated in the process of stability analysis. Therefore, a matrix-valued cubic polynomial inequality is applied to make the matrix-valued cubic polynomial numerically tractable. Finally, the relaxed stability criteria utilizing the I3BFM and the double integral inequality are derived, and the superiority of the derived stability criteria is demonstrated by two well-known numerical examples.

Enhanced Fuzzy State Estimation of Discrete-Time Nonlinear Circuits via Two Relaxed Measures

This brief gives an enhanced result for fuzzy state estimation of discrete-time nonlinear circuits by developing two new technical measures of relaxations. Firstly, a novel balanced matrix method is proposed in which all the involved free matrices are with arbitrary variable orders, and thus a lot of freedom can be introduced into different switching modes for reducing the conservatism caused by existing those free matrices with fixed orders in the recent result. Secondly, an important property is employed in order to further relax the conservatism caused by the mismatch of adjacent normalized fuzzy weighting functions. Therefore, the scope of application of fuzzy state estimation can be enlarged by a large margin and much better estimation performance is obtained than before. Finally, the superiority of the developed approach is tested and verified based on the common benchmark example applied in the literature.

An extended generalized integral inequality based on free matrices and its application to stability analysis of neural networks with time-varying delays

This paper proposes an extended generalized integral inequality based on free matrices (EGIIFM) and applies it to the stability analysis of neural networks with time-varying delays. The EGIIFM estimates an upper bound for a quadratic form of a positive definite matrix with an augmented vector staked not only with the state and its derivative but also with the nonlinear activation function. By reflecting the correlated cross-information among the terms in the augmented vector as free matrices, the EGIIFM provides a tighter upper bound and encompasses various existing single integral inequalities as special cases. In addition, by establishing a new double integral Lyapunov–Krasovskii functional including the correlated cross-information, a less conservative stability criterion is obtained. Through three well-known numerical examples, the effectiveness of the EGIIFM is evaluated.

Bound states of a one-dimensional Dirac equation with multiple delta-potentials

Two approaches are developed for the study of the bound states of a one-dimensional Dirac equation with the potential consisting of N δ-function centers. One of these uses Green’s function method. This method is applicable to a finite number N of δ-point centers, reducing the bound state problem to finding the energy eigenvalues from the determinant of a 2 N × 2 N matrix. The second approach starts with the matrix for a single delta-center that connects the two-sided boundary conditions for this center. This connection matrix is obtained from the squeezing limit of a piecewise constant approximation of the delta-function. Having then the connection matrices for each center, the transmission matrix for the whole system is obtained by multiplying the one-center connection matrices and the free transfer matrices between neighbor centers. An equation for bound state energies is derived in terms of the elements of the total transfer matrix. Within both approaches, the transcendental equations for bound state energies are derived, the solutions to which depend on the strength of delta-centers and the distance between them, and this dependence is illustrated by numerical calculations. The bound state energies for the potentials composed of one, two, and three delta-centers (N = 1, 2, 3) are computed explicitly. The principle of strength additivity is analyzed in the limits as the delta-centers merge at a single point or diverge to infinity.

Development of Vaginal Carriers Based on Chitosan-Grafted-PNIPAAm for Progesterone Administration.

Chitosan-based hydrogels possess numerous advantages, such as biocompatibility and non-toxicity, and it is considered a proper material to be used in biomedical and pharmaceutical applications. Vaginal administration of progesterone represents a viable alternative for maintaining pregnancy and reducing the risk of miscarriage and in supporting the corpus luteum during fertilization cycles. This study aimed to develop new formulations for vaginal administration of progesterone (PGT). A previously synthesized responsive chitosan-grafted-poly (N-isopropylacrylamide) (CS-g-PNIPAAm) was formulated in various compositions with polyvinyl alcohol (PVA) as external crosslinking agent to obtain pH- and temperature-dependent hydrogels; the hydrogels had the capacity to withstand shear forces encountered in the vagina due to its mechanism of swelling once in contact with vaginal fluids. Three different hydrogels based on grafted chitosan were analyzed via Fourier-transform infrared spectroscopy (FTIR), swelling tests, in vitro drug release, and bioadhesion properties by TA.XTplus texture analysis. A higher amount of PVA decreased the swelling and the bioadhesion capacities of the hydrogel. All hydrogels showed sensitivity to temperature and pH in terms of swelling and in vitro delivery characteristics. By loading progesterone, the studied hydrogels seemed to possess even higher sensitivity than drug–free matrices. The release profile of the active substance and the bioadhesion characteristics recommended the CS-g-PNIPAAm/PVA 80/20 +PGT (P1) hydrogel as a proper constituent for the vaginal formulation for progesterone administration.

Double Asynchronous Quasi-Time-Dependent $H_{\infty}$ Filtering for Nonlinear Switched Systems via Event-Triggered Communication Mechanism

Based on an interval type-2 fuzzy model, an event-triggered communication mechanism, and the mode-dependent persistent dwell time, the quasi-time-dependent <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty}$</tex-math></inline-formula> filtering problem is investigated for nonlinear switched systems in the presence of double asynchronous phenomenon. The filter and the system’s premise variables and modes are usually mismatched in practical environment, i.e., the double asynchronous phenomenon. The mode asynchronous interval is allowed to exceed the dwell time. The free matrices are introduced to deal with the mismatched membership function. By the quasi-time-dependent Lyapunov function and the event-triggered communication mechanism, the sufficient conditions are given to make the filtering error system exponentially stable and satisfy an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_{\infty}$</tex-math></inline-formula> performance. Then, the double asynchronous quasi-time-dependent filter is determined by solving linear matrix inequalities. Finally, a tunnel diode circuit simulation example is given to illustrate the validity of the results.

Relaxed Fault Estimation of Discrete-Time Nonlinear System Based on a New Multi-Instant Real-Time Scheduling Fuzzy Observer

Relaxed fault estimation (FE) criteria of a class of discrete-time nonlinear plants are proposed based on a new multi-instant real-time scheduling fuzzy FE observer. Compared with the existing methods given in the recent literature, the newly developed fuzzy observer is capable of providing much freedom for reducing the conservatism of designing feasible FE criteria. At each sampling point, the information of adjacent groups of normalized fuzzy weighting functions can be updated and utilized for deciding the activated mode of our developed fuzzy FE observer. Based on this, an exclusive pair of gain matrices can be given by introducing a set of well-designed free matrices into the designed FE criteria for every activated mode. Moreover, it can be found that the proposed result will include the previous one as a special case if all of those free matrices are removed. Finally, the well-known nonlinear truck–trailer plant with actuator fault is utilized to validate the superiority of our approach over the existing ones reported in the literature.

Relaxed Conditions of Observer Design of Discrete-Time Takagi–Sugeno Fuzzy Systems via a New Multi-Instant Gain-Scheduling Scheme

A novel multi-instant gain-scheduling scheme is proposed in order to further improve the state estimation performance of discrete-time Takagi&#x2013;Sugeno fuzzy systems in this article. Different from those existing methods, the enabled working pattern at each sampling instant can be determined in cycle according to utilization of the updated information of adjacent normalized fuzzy weighting functions in depth, and then the featured time-varying free matrices are introduced for each working pattern, which can remove those existing constraints on traditional free matrices. Consequently, the conservatism can be reduced significantly and it does broaden the range of application for observer-based fuzzy state estimation. Despite the fact that the quantity of the free matrices becomes much bigger than before, it needs to be pointed out that the actual computational complexity only increases by a small amount through depressing the number of cycles. Given that the developed designing process belongs to an offline computation, it is an acceptable tradeoff between the increasement of computational burdens and the reduction of conservatism. In the end, the effectiveness of the developed method is illustrated through the detailed analysis of numerical simulations.