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In this article, the problem of finite‐horizon state estimation is studied for a class of time‐varying complex networks with sensor faults. The phenomenon of measurement quantization is considered such that the measurements are quantized probabilistically before transmitted to the state estimator. To deal with the unknown sensor fault, a neural network is introduced to appropriate the sensor fault whose weights are updated based on estimation error and the gradient descent method. Our aim is to design state estimators so that the state estimation errors are finite‐time bounded. First, sufficient conditions are established to ensure the existence of the desired state estimators. Then, the gains of the state estimators are derived in terms of the solutions to a set of recursive matrix inequalities. Finally, the usefulness of our estimation approach is confirmed by an illustrative example. [ABSTRACT FROM AUTHOR]

This study explores the problem of finite-time resilient control for periodic piecewise polynomial time-varying systems in the face of parameter uncertainties, time-varying state delays and external disturbances. Particularly, the considered system is characterized by dividing the fundamental period of periodic systems into numerous subintervals, each of which can be expressed by using matrix polynomial functions. The foremost intention of this work is to lay out a resilient controller such that the resulting closed-loop system is finite-time bounded and satisfies a mixed H ∞ and passivity performance index. Furthermore, by constructing a periodic piecewise time-varying Lyapunov–Krasovskii functional, a delay-dependent sufficient condition is established in line with Wiritinger's inequality and matrix polynomial lemma to guarantee the needed outcomes of the system under study. Following this, the gain matrix of the devised controller can be calculated by solving the established constraints. As a final step, we conclude with a numerical example that validates the potential and importance of the theoretical discoveries and the developed control scheme. [ABSTRACT FROM AUTHOR]

A novel non-backstepping output feedback control approach, called as the dual-gain scaling approach, is proposed by combining the extended Lyapunov matrix inequalities with dual-domination idea. This approach provides partially decentralised adaptive output feedback controllers with three event-triggered mechanisms to achieve the global practical tracking of a family of uncertain large-scale nonlinear systems subject to unknown control coefficients and strongly interconnected nonlinearities with output-polynomial growth rate. The dual gains are embedded in the observers and controllers, which can effectively compensate for time-varying control coefficients and unknown constant growth rate multiplied by output-polynomial function. Three different threshold schemes including fixed threshold, relative threshold and time-varying threshold are introduced to save communication resources. The stability analysis shows that the solutions of the closed-loop system are globally uniformly bounded and the tracking error converges to a preset compact set after a finite time; also, there will be no zeno behaviour in the closed-loop response. [ABSTRACT FROM AUTHOR]

This study deals with some basic inference problems about future observations in a general linear model (GLM) with linear parameter constraints, known as a constrained general linear model (CGLM). Combining the CGLM and its future observations, the author turns the model into a reparameterized form. Using some quadratic matrix optimization methods, the author derives analytical formulas for calculating the best linear unbiased predictors (BLUPs) of all unknown parameter matrices under a CGLM and its future observations. In particular, the author next gives a comprehensive search on the comparison of dispersion matrices of BLUPs of unknown vectors by establishing various equalities and inequalities for dispersion matrices of BLUPs under the model by using elementary block matrix operations and some formulas of rank and inertia of block matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study a system of nonlinear tempered fractional differential equations with multi-point coupled boundary conditions. By applying the properties of Green's function and the operator and combining the method of matrix analysis, we obtain the corresponding Lyapunov inequalities under two Banach spaces. And, we have compared two Lyapunov inequalities under certain conditions. An example is given to verify our results. [ABSTRACT FROM AUTHOR]

Considering the asymmetry of the system matrix caused by directed communication networks, a dynamic event-triggered control protocol for the robust H ∞ consensus of uncertain linear multi-agent systems accompanied by external disturbances on directed topologies is proposed. Through the introduction of the controlled output and generalized algebraic connectivity for directed graphs, the consensus problem is converted to the event-triggered stabilization of the uncertain linear closed-loop system with disturbance. On the basis of the relative state of neighbor agents at event-times and the introduced dynamic variables, a dynamic event-triggered control protocol is put forward, and the linear matrix inequality sufficient conditions for the system to achieve robust H ∞ consensus are obtained after a rigorous proof. Then, linear matrix inequalities are further decoupled to greatly reduce the computational complexity, and the parameters of the control protocol are determined by solving the decoupled linear matrix inequalities. Finally, simulations are applied for the verification of conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new characterizations of operator monotone functions using matrix mean inequalities. [ABSTRACT FROM AUTHOR]

This note offers some remarks on a norm version of the matrix arithmetic–geometric inequality. [ABSTRACT FROM AUTHOR]

We are concerned with log-majorization for matrices in connection with the multivariate Golden–Thompson trace inequality and the Karcher mean (i.e., a multivariate extension of the weighted geometric mean). We show an extension of Araki's log-majorization and apply it to the α -z-Rényi divergence in quantum information. We discuss the equality cases in the multivariate trace inequality of Golden–Thompson type and in the norm inequality for the Karcher mean. The paper includes an appendix to correct the proof of the author's old result on the equality case in the norm inequality for the weighted geometric mean. [ABSTRACT FROM AUTHOR]

Let J be a non trivial involutive Hermitian matrix. Consider C n equipped with the indefinite inner product induced by J, [ x , y ] = y ∗ J x for all x , y ∈ C n , which endows the matrix algebra C n × n with a partial order relation ≤ J between J-selfadjoint matrices. Inde-finite inequalities are given in this setup, involving the J-selfadjoint α -weighted geometric matrix mean. In particular, an indefinite version of Ando–Hiai inequality is proved to be equivalent to Furuta inequality of indefinite type. [ABSTRACT FROM AUTHOR]

This article is dedicated to event‐triggered anti‐disturbance control of networked control systems via a guaranteed cost triggering scheme. A distribution interconnected observer structure is built to estimate these unknown matched disturbances. A novel event triggered condition is constructed in terms of real‐time cost and averaged functions composed of observer states and the last successfully transmitted one. A composite control scheme is provided by an event‐triggered observer‐based state feedback controller and compensators for the matched disturbances. A sufficient condition in term of linear matrix inequality is supplied to ensure the asymptotic stability of the resultant closed‐loop system. A simulation is carried out to demonstrate the effectiveness of the proposed triggering control strategy. [ABSTRACT FROM AUTHOR]

In this paper, a generalized decentralized controller design is presented for robust stabilization and achieving certain L $ _2 $ 2 gain of a class of nonlinear interconnected systems via dynamic output feedback control within the LMI framework. This large-scale system is composed of linear subsystems coupled by nonlinear time-varying interconnections satisfying quadratic constraints. The proposed controller design uses the available neighbouring information to design a decentralized controller that operates at each subsystem. An LMI approach is sufficiently proposed as a feasibility problem of a BMI with respect to controller gains via dynamic output feedback structure. The application of power systems shows that the proposed controller can effectively stabilize the nonlinear interconnected systems and achieve desired performance, even in the presence of disturbances and uncertainties. [ABSTRACT FROM AUTHOR]

It is very difficult to control the distributed parameter system (DPS) for consistent spatial performance. In this paper, a dual scale spatio‐temporal control is designed for the DPS to achieve a consistent performance across the entire workspace under unknown exogenous disturbances. First, the generalized "spatial observer" should be designed under spectral decomposition/synthesis, to act as the nominal model of the DPS, through which the spatial mismatch between the model and the process could be estimated. Then a distributed disturbance compensator can be constructed to suppress all the undesirable spatial disturbance through the inner loop, and the compensated system will become closer to the nominal model and easier to control. Third, a dual controller will be designed in the outer loop for the final consistent spatial performance. Since the spatial performance is mainly affected by the dominant dynamics on the slow scale, a convex optimization algorithm can be effectively established in terms of nonlinear matrix inequalities for the dual controller. In this way, the nonlinear optimization problem is solved by converting it into the problem of a linear matrix inequalities with constraints. Besides, the spatio‐temporal state variable of the controlled plant is demonstrated to converge in Hilbert space. The feasibility and effectiveness of the proposed control method are verified by temperature control of a catalytic rod, a benchmark in chemical reactors. [ABSTRACT FROM AUTHOR]

In a celebrated paper of Marcus and Ree [M. Marcus, R. Ree. Diagonals of doubly stochastic matrices. Quart J Math Oxford Ser. 1959; 10(1):296–302], it was shown that if $ A=[a_{ij}] $ A = [ a ij ] is an $ n \times n $ n × n doubly stochastic matrix, then there is a permutation $ \sigma \in S_n $ σ ∈ S n such that $ \sum _{i,j=1}^{n} a_{i,j}^{2} \leq \sum _{i=1}^{n} a_{i,\sigma (i)} $ ∑ i , j = 1 n a i , j 2 ≤ ∑ i = 1 n a i , σ (i) . Erdős asked for which doubly stochastic matrices the inequality is saturated. Although Marcus and Ree provided some insight for the set of solutions, the question appears to have fallen into oblivion. Our goal is to provide a complete answer in the particular, yet non-trivial, case when n = 3. [ABSTRACT FROM AUTHOR]

This paper investigates the leader-following H ∞ consensus of fractional-order multi-agent systems (FOMASs) under input saturation via the output feedback. Based on the bounded real lemma for FOSs, the sufficient conditions of H ∞ consensus for FOMASs are provided in α ∈ 0 , 1 and 1 , 2 , respectively. Furthermore, the iterative linear matrix inequalities (ILMIs) approaches are applied for solving quadratic matrix inequalities (QMIs). The ILMI algorithms show a method to derive initial values and transform QMIs into LMIs. Mathematical tools are employed to transform the input saturation issue into optimal solutions of LMIs for estimating stable regions. The ILMI algorithms avoid the conditional constraints on matrix variables during the LMIs' construction and reduce conservatism. The approach does not disassemble the entire MASs by transformations to the Laplacian matrix, instead adopting a holistic analytical perspective to obtain gain matrices. Finally, numerical examples are conducted to validate the efficiency of the approach. [ABSTRACT FROM AUTHOR]

We establish new upper bounds for the Berezin number and Berezin norm of operator matrices, which are refinements of existing bounds. Among other bounds, we prove that if $ A=[A_{ij}] $ A = [ A i j ] is an $ n\times n $ n × n operator matrix with $ A_{ij}\in \mathbb {B}(\mathcal {H}) $ A i j ∈ B (H) for $ i,j=1,2,\dots, n $ i , j = 1 , 2 , ... , n , then $ \|A\|_{ber} \leq \left \|\left [\|A_{ij}\|_{ber}\right ]\right \| $ ‖ A ‖ b e r ≤ ‖ [ ‖ A i j ‖ b e r ] ‖ and $ \mathbf{ber}(A) \leq w([a_{ij}]), $ b e r (A) ≤ w ([ a i j ]) , where $ a_{ii}=\mathbf{ber}(A_{ii}), $ a i i = b e r (A i i) , $ a_{ij}=\big \||A_{ij}|+|A^*_{ji}|\big \|^{{1}/{2}}_{ber} \big \||A_{ji}|+|A^*_{ij}|\big \|^{{1}/{2}}_{ber} $ a i j = ‖ | A i j | + | A j i ∗ | ‖ b e r 1 / 2 ‖ | A j i | + | A i j ∗ | ‖ b e r 1 / 2 if ij. We also provide examples which illustrate these bounds for some concrete operators acting on the Hardy-Hilbert space. [ABSTRACT FROM AUTHOR]

We prove a reverse Lieb--Thirring inequality with a sharp constant for the matrix Schrödinger equation on the half-line. [ABSTRACT FROM AUTHOR]

This work presents a contribution to reduce the conservatism in nonlinear observer for TakagiSugeno fuzzy systems with unmeasurable premise variables based on the mean value theorem. The conservatism was reduced through the use of non-quadratic Lyapunov function (NQLF) rather than the quadratic form used in previous works. The stability conditions are given in the form of Bilinear Matrix Inequalities (BMI) whereby an effective solution to solve this type of problem is presented using linear solvers which are characterized by their computational efficiency. To show the improvement of the proposed work over the existing approach and to demonstrate its effectiveness in solving the Bilinear Matrix Inequalities, the feasibility area is illustrated by a numerical example, and a real-time implementation of the observer is applied to the induction motor (IM) where a big number of inequalities is present. • Using the TS fuzzy model representation which obtained by using the non linear sector transformation. • Reducing the conservatism of the classical Mean Value Theorem observer by using the polyquadratic Lyapunov function. • Presenting an iterative linear matrix inequalities method to solve bilinear matrix inequalities using linear optimization solvers. • Real-time implementation of the proposed observer on an induction motor. [ABSTRACT FROM AUTHOR]

BACKGROUND: Industrialization continues to increase in the 21st century, as industries start moving towards automation. Manufacturing of a component/item requires a lot of resources and energy. Usage of water in industries has increased in recent years, as it is used as both a resource and an energy generating source. Neutralization of a processed liquid is carried out by addition of solution through final control elements and most industries are utilizing pneumatic‐based control valves, which have a disadvantage in the regulation of additives. RESULTS: A motorized valve system is introduced, which is compared with the pneumatic system in terms of their performance. The proposed system is implemented with different controllers such as proportional–integral controller tuned using linear matrix inequalities, modified internal model controller, model predictive controller and nonlinear autoregressive (NAR) model, where the final control elements are tested for pH neutralization. On implementation of the proposed system the NAR model has proven to be better under the motorized control valve system for the neutralization process, with a peak value of 7.08 (pH), settling time of 75.16 s, rise time of 5.96 s, maximum overshoot of 10.1%, slew rate of 0.21 per second, integral square error of 0.95, integral absolute error of 0.48 and integral time absolute error of 1.88. CONCLUSION: Based on the results obtained it is observed that the NAR controller utilized with motorized control valve has better functionality in neutralizing a solution. © 2024 Society of Chemical Industry (SCI). [ABSTRACT FROM AUTHOR]

As a giant in energy consumption, buildings urgently need to optimize control strategies for the main energy consuming equipment inside buildings. It is of great significance to design advanced control algorithms to improve the efficiency of the main energy consuming equipment in buildings, namely air conditioning, in the current energy shortage. The model predictive control algorithms and strategies were used in this study to control the HVAC system to improve the energy utilization of the city. Then linear matrix inequality with robust model predictive feedback controller was used to optimize and get the model predictive control optimization algorithm. The research results showed that, under the influence of different factors, the three regions controlled by the model predictive control optimization algorithm showed a little overshooting in the initial state. But it was quickly corrected after adjustment. Meanwhile, the average tracking error of temperature and humidity in each region was 0.139°C and 0.13g/kg dry air, respectively. The average predicted mean vote was 0.32. In actual office buildings, the proposed algorithm controlled the temperature within the reference value range of 0.1°C throughout the entire process. The total electricity consumption and electricity price costs were reduced by 12.11% and 22.54%, respectively. In summary, the proposed method has good performance for HVAC system application, which can effectively realize energy saving and emission reduction and improve human comfort. This method makes important contributions to promote the construction of smart cities and the development of green buildings. [ABSTRACT FROM AUTHOR]

A conjecture from the distillability of quantum entanglement is that when A and B are 4 × 4 trace zero complex matrices and ‖ A ‖ 2 + ‖ B ‖ 2 = 1 / 4 (where ‖ ⋅ ‖ is the Frobenius norm), the sum of squares of the largest two singular values of A ⊗ I 4 + I 4 ⊗ B does not exceed 1/2. In this paper, the conjecture is proved when (i) A or B is unitarily similar to a direct sum of 2 × 2 trace zero matrices; (ii) A and B are unitarily similar to matrices, when partitioned into 2 × 2 blocks, having zero diagonal blocks. [ABSTRACT FROM AUTHOR]

This paper investigates the event‐triggered generalized extended state observer‐based estimation issue for a class of nonlinear networked control systems with unmodeled dynamics, external disturbances and multi‐channel attacks. In order to resist different forms of attack threats on multiple communication channels from sensors to the observer, Markov chain is introduced to describe the stochastic switching or jumping behavior between different attack modes. Considering the limited network resources, the measured outputs are transmitted to the observer only when the triggering conditions are met. Moreover, the parameters modeled by the Bernoulli process are adopted to help analyze potential random gain variations of the generalized extended state observer. By employing Lyapunov stability theory and stochastic systems analysis, sufficient conditions are derived to ensure that the augmented system is exponentially bounded in mean square, and the expected observer gains are further determined through linear matrix inequalities. Finally, a numerical example and a simulation related to the RLC series circuit system are conducted, illustrating the proposed method's effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, the global exponential stability of quaternion-valued neural networks with time-varying delays is discussed. On the basis of the matrix measure method and Halanay inequality, some sufficient criteria of exponential stability for quaternion-valued neural with delays are given. Different from the existing methods, the criteria of global exponential stability are obtained without constructing Lyapunov function. Moreover, the activation function is no longer assumed to be differentiable, which makes the analysis much easier. Finally, the numerical simulations are used to prove the validity of the main results. [ABSTRACT FROM AUTHOR]

This paper investigates the robust model predictive load frequency control problem for smart grids with wind power under cyber attacks. To accommodate intermittent power generation, the demand response of the system is considered by involving the air conditioning loads in the frequency regulation. In addition, the system uncertainties produced by the air conditioning load users and wind turbines when replacing traditional generator sets are considered. By using the cone complementary linearization algorithm and the linear matrix inequality technique, a resilience robust model predictive control strategy with mixed H2/H∞$H_2/H_\infty$ performance indexes is proposed. Furthermore, a rigorous derivation of the recursive feasibility of robust model predictive control is given. Finally, the simulation results of the two‐area load frequency control scheme show that the proposed model predictive control strategy is capable of realizing the load frequency control of the multi‐area smart grid and is robust to the parameter uncertainties and frequency regulation of the system. The results also show that the proposed model predictive control strategy has some resistance to cyber attacks and external disturbances. [ABSTRACT FROM AUTHOR]

This paper explores event-triggered model predictive control (MPC) for constrained switched systems, incorporating a buffer-based anti-attack (BBAA) approach to ensure network security. Since the system state is unknown and disturbed, an observer-based $ H_{\infty } $ H ∞ MPC approach is used to estimate the system state and suppress disturbance. Furthermore, the estimated state and control input can be optimised by minimising the performance function. Considering that the system state is subject to denial of service (DoS) attacks in the feedback channel, the BBAA approach is used to reconstruct the estimated state, thereby reducing the impact of attacks on system performance. To improve the utilisation of network resources, a set of event-triggering mechanisms (ETMs) is used. In addition, a set of matrix inequality conditions is given to constrain the gains of controllers and observers. Then, the system $ H_{\infty } $ H ∞ performance is demonstrated by applying the average dwell time (ADT) and the Lyapunov function method, and the complex coupling between model switching, DoS attacks and event-triggering is analysed. Simulation results validate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This study focuses on designing robust H ∞ controllers to mitigate actuator saturation-related issues of active suspension systems. Two design methods are presented: one utilizing full-state feedback and the other relying on measurable output feedback. The H ∞ state-feedback control incorporates actuator saturation directly into the design, representing the control input with the saturation function in the state-space model. The resulting optimization-based design problem, derived from theoretical stability analyses, poses a non-convex NP-hard problem under bilinear matrix inequality (BMI) constraints. To address this challenge, the study proposes a meta-heuristic optimization-based technique, providing a straightforward yet effective method to manage the BMI problem. The scope extends to practical implementation considerations, where suspension systems may only utilize measurable output variables. The H ∞ static output-feedback control case is explored by extending the design of a full-order state-feedback controller. Despite the BMI-related challenges entailed in this scenario, the proposed meta-heuristic optimization-based design technique demonstrates flexibility in managing BMI conditions, thus eliminating the need to modify the controller design methodology. A noteworthy contribution of this study is its consistent application of the same meta-heuristic optimization-based technique for designing both H ∞ state-feedback and static output-feedback controllers. Experimental results validate the effectiveness of the proposed robust H ∞ control strategies. [ABSTRACT FROM AUTHOR]

This article addresses the problem of designing an event‐triggered fault detector within a networked nonlinear system described by a Takagi–Sugeno (T–S) fuzzy model, accounting for time‐varying delays and denial‐of‐service (DoS) attacks characterized by a binary Markov chain. Our focus is on constructing an event‐triggered residual system, which aims to detect malfunctions in the original fuzzy system while optimizing communication resource allocation. Specifically, it is tailored to combat DoS attacks in the event generator‐to‐fault detector channel and mitigate network‐induced time‐varying delays. Utilizing the Lyapunov–Krasovskii function approach, we formulate linear matrix inequalities to establish the necessary conditions ensuring the asymptotic mean‐square stability of the residual system, meeting a predefined H∞$$ {H}_{\infty } $$ performance criterion. Finally, the simulation results are presented to validate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We establish a Pythagorean theorem for the absolute values of the blocks of a partitioned matrix. This leads to a series of remarkable operator inequalities. For instance, if the matrix \mathbb {A} is partitioned into three blocks A,B,C, then \begin{gather*} |\mathbb {A}|^3 \ge U|A|^3U^* + V|B|^3V^*+ W|C|^3W^*,\\ \sqrt {3} |\mathbb {A}| \ge U|A|U^* + V|B|V^*+ W|C|W^*, \end{gather*} for some isometries U,V,W, and \begin{equation*} \mu _4^2(\mathbb {A}) \le \mu _3^2(A) +\mu _2^2(B) + \mu _1^2(C) \end{equation*} where \mu _j stands for the j-th singular value. Our theorem may be used to extend a result by Bhatia and Kittaneh for the Schatten p-norms and to give a singular value version of Cauchy's Interlacing Theorem. [ABSTRACT FROM AUTHOR]

In this paper, we provide a formula to compute the permanent of block matrices depending on entries of each block. As a consequence, a generalized Lieb permanent inequality on positive semi-definite block matrices is given. [ABSTRACT FROM AUTHOR]

In this paper, a decentralised controller is extended for robust stabilisation and $ L_2 $ L 2 gain analysis of a class of nonlinear interconnected systems via dynamic output feedback control within the LMI framework. This large-scale system is composed of linear subsystems coupled by nonlinear time-varying interconnections satisfying quadratic constraints. An LMI approach is sufficiently proposed as a feasibility problem of a BMI with respect to controller gains via dynamic output feedback structure. An example of inverted pendulums is used to illustrate the applicability of extended approach. [ABSTRACT FROM AUTHOR]

This paper proposes a new condition for the root clustering of a real matrix A in a complex region D defined by a polynomial matrix inequality (PMI region). For a general case, a sufficient condition is given so that the eigenvalues of A lie in D . It was shown that this condition is necessary and sufficient for some particular PMI regions including linear matrix inequality (LMI) regions, quadratic matrix inequality (QMI) regions, polynomial regions and many others. This paper also provides an extension to the classical theorem of Cyparissos Stephanos, which formulates the relationship between the eigenvalues of two matrices and those of composite matrices of their Kronecker products. This extension turned out to be crucial for proving the main results obtained. Based on the analysis of the proposed condition, a guardian map function can be used for tackling the problem of robust D-stability of single-parameter uncertain linear systems, for which the exact and possibly disconnected domain of D-stability was determined. All the obtained results were illustrated by certain examples. [ABSTRACT FROM AUTHOR]

This paper discusses a synchronization issue of uncertain time-delay systems via flexible delayed impulsive control. A new Razumikhin-type inequality is presented, considering adjustable parameters the ϖ (t) , which relies on flexible impulsive gain. For the uncertain time-delay systems where delay magnitude is not constrained to impulsive intervals, sufficient conditions for global exponential synchronization (GES) are established. Furthermore, based on Lyapunov theory, a new differential inequality and linear matrix inequality design, and a flexible impulsive control method is introduced through using the variable impulsive gain and time-varying delays. It is interesting to find that uncertain time-delay systems can maintain GES by adjusting the impulsive gain and impulsive delay. Finally, two simulations are given to illustrate the effectiveness of the derived results. [ABSTRACT FROM AUTHOR]

For a matrix A which satisfies Crouzeix's conjecture, we construct several classes of matrices from A for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of 3 × 3 matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators. [ABSTRACT FROM AUTHOR]

Let A, B, X, and Y be n × n complex matrices such that A is self-adjoint, B ≥ 0 , ± A ≤ B , max (∥ X ∥ 2 , ∥ Y ∥ 2) ≤ 1 , and let f be a nonnegative increasing convex function on [ 0 , ∞) satisfying f (0) = 0 . Then 2 s j (f (| X A Y ∗ |)) ≤ max { ∥ X ∥ 2 , ∥ Y ∥ 2 } s j (f (B + A) ⊕ f (B − A)) for j = 1 , 2 , ... , n . This singular value inequality extends an inequality of Audeh and Kittaneh. Several generalizations for singular value and norm inequalities of matrices are also given. [ABSTRACT FROM AUTHOR]

In the paper, a design method of a static anti-windup compensator for systems with input saturations is proposed. First, an anti-windup controller is presented for system with cut-off saturations, and, secondly, the design problem of the compensator is presented to be a non-convex optimization problem easily solved using bilinear matrix inequalities formulation. This approach guarantees stability of the closed-loop system against saturation nonlinearities and optimizes the robust control performance while the saturation is active. [ABSTRACT FROM AUTHOR]

In this paper, we present some weighted Hölder-type inequalities for N −tuple of n × n complex matrices, with applications that include refined bounds for the norm of the sum of two matrices, the numerical radius of a matrix and the numerical radius for the product of two matrices. The relation with the existing literature will be discussed, where some results are proven to be improvements of celebrated results in the literature. Further, numerical examples are given to support the significance of other results. [ABSTRACT FROM AUTHOR]

This paper focuses on the containment control problem for a class of nonlinear multi-agent systems (MASs) with sampled-data. To enhance the efficiency of utilizing communication resources, a new distributed sampled-data-based event-triggered transmission strategy is proposed. This event-driven strategy requires that event detector works only at periodic sampling instants. By employing the techniques such as input delay method, inequality technique and the Lyapunov stability theory, sufficient criteria are derived to ensure that each follower converges to the convex hull spanned by the leaders. A convex optimization method is employed to search for the desired controller gain by maximizing the maximum-allowable bound of sampling intervals. Additionally, for some special scenarios, to reduce computational complexity, the matrix decoupling technique is utilized to reduce the dimension of matrix inequalities, thereby facilitating application to large-scale MASs. Finally, the effectiveness of the proposed theory is verified by a simulation example. [ABSTRACT FROM AUTHOR]

One of K. R. Parthasarathy's major and abiding interests was the theory of positive definite functions and especially its relation to classical and quantum probability. Here we recall his contribution in providing a seminal idea that stimulated a lot of work on matrix inequalities. [ABSTRACT FROM AUTHOR]

This paper presents the decentralized sampled‐data fuzzy observer design technique for nonlinear interconnected systems based on the Takagi–Sugeno fuzzy model. The interconnected system is considered that the subsystems include uncertain interconnections and are not needed to satisfy stable conditions. Based on the continuous‐time Lyapunov functional approach, the observer design technique is proposed to minimize the interconnection bound to attenuation degree ratio and formulated into the linear matrix inequality formats. Finally, by simulation example, it shows the validity and superiority of the proposed observer design technique. [ABSTRACT FROM AUTHOR]

Descriptor systems are more complex than normal systems, which are modeled by differential equations. This paper derives stability and stabilization criteria for uncertain fractional descriptor systems with neutral-type delay. Through the Lyapunov–Krasovskii functional approach, conditions subject to time-varying delay and parametric uncertainty are formulated as linear matrix inequalities. Based on the established criteria, static state- and output-feedback control laws are designed to ensure regularity and impulse-free properties, together with robust stability of the closed-loop system under permissible uncertainties. Numerical examples illustrate the effectiveness of the control methods and show that the results depend on the range of variation in the delays and on the fractional order, leading to stability analysis results that are less conservative than those reported in the literature. [ABSTRACT FROM AUTHOR]

By adding a scalar parameter to the classical Lyapunov matrix inequality, the underlying structure turns to be richer: Partial order of stability sets is introduced. This is then used to improve estimates of trajectories associated with differential inclusions. Technically, the spectrum of all matrices, satisfying a given Lyapunov inequality, lies within a special disk in the right-half plane: Under inversion such a disk is mapped onto itself. As a by-product, it is shown that these disks are a natural tool to understanding the Matrix-Sign-Function iteration scheme, used in matrix computations. Hyper-Lyapunov inclusions are formulated through Matrix-Quadratic-Form Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within sub -unit disks. [ABSTRACT FROM AUTHOR]

A real linear combination of products of minors which is nonnegative over all totally nonnegative (TN) matrices is called a determinantal inequality for these matrices. It is referred to as multiplicative when it compares two collections of products of minors and additive otherwise. Set theoretic operations preserving the class of TN matrices naturally translate into operations preserving determinantal inequalities in this class. We introduce index-row (and index-column) operations that act directly on all determinantal inequalities for TN matrices, and yield further inequalities for these matrices. These operations assist in revealing novel additive inequalities for TN matrices embedded in the classical identities due to Laplace [ Mem. Acad. Sciences Paris 1772] and Karlin (1968). In particular, for any square TN matrix A , these derived inequalities generalize – to every i th row of A and j th column of adj A – the classical Gantmacher–Krein fluctuating inequalities (1941) for i = j = 1. Furthermore, our index-row/column operations reveal additional undiscovered fluctuating inequalities for TN matrices. The introduced index-row/column operations naturally birth an algorithm that can detect certain determinantal expressions that do not form an inequality for TN matrices. However, the algorithm completely characterizes the multiplicative inequalities comparing products of pairs of minors. Moreover, the underlying index-row/column operations add that these inequalities are offshoots of certain "complementary/higher" ones. These novel results seem very natural, and in addition thoroughly describe and enrich the classification of these multiplicative inequalities due to Fallat–Gekhtman–Johnson [ Adv. Appl. Math. 2003] and later Skandera [ J. Algebraic Comb. 2004] (and revisited by Rhoades–Skandera [ Ann. Comb. 2005, J. Algebra 2006]). [ABSTRACT FROM AUTHOR]

This article deals with the stability and H∞$$ {H}_{\infty } $$ control synthesis for a class of linear sampled‐data systems. First, a novel integral inequality with the cubic term of integral interval length instead of reciprocal one of integral interval length is constructed by non‐orthogonal polynomials. This can make full use of slack matrix variables and additional information about sawtooth structural sampling pattern. Then, on the basis of the constructed integral inequality, a cubic‐term‐dependent discontinuous exponent Lyapunov–Krasovskii Functional (LKF) is developed containing additional information about system states to design an H∞$$ {H}_{\infty } $$ sampled‐data state feedback controller. To better estimate the upper bound of the derivative of LKF, a kind of multiple convex function approximation approach is developed, which can be used to deal with the two‐variable polynomial negative definite issue by dividing the variable interval into multiple subintervals. By the constructed integral inequality and multiple convex function approximation approach, sufficient conditions with less conservatism are deduced for the feasibility of H∞$$ {H}_{\infty } $$ sampled‐data state feedback controller. Moreover, by the inner convex approximation solution technique and proposed iterative algorithm, the bilinear matrix inequalities can be transformed into linear matrix ones, which can be easily tackled, to obtain the desired sampled‐data controller gain. Finally, by comparing with existing results, the advantages and superiority of our proposed approaches can be confirmed via two numerical examples. [ABSTRACT FROM AUTHOR]

We look at a stochastic time‐varying optimization problem and we formulate online algorithms to find and track its optimizers in expectation. The algorithms are derived from the intuition that standard prediction and correction steps can be seen as a nonlinear dynamical system and a measurement equation, respectively, yielding the notion of nonlinear filter design. The optimization algorithms are then based on an extended Kalman filter in the unconstrained case, and on a bilinear matrix inequality condition in the constrained case. Some special cases and variations are discussed, notably the case of parametric filters, yielding certificates based on LPV analysis and, if one wishes, matrix sum‐of‐squares relaxations. Supporting numerical results are presented from real data sets in ride‐hailing scenarios. The results are encouraging, especially when predictions are accurate, a case which is often encountered in practice when historical data is abundant. [ABSTRACT FROM AUTHOR]

This article focuses on the finite‐time mixed ℋ∞$$ {\mathscr{H}}_{\infty } $$ and passive filtering problem for discrete‐time singular systems with randomly occurring nonlinear perturbations. The main purpose is to design a filter, which ensures the singular error system is stochastically finite‐time bounded and satisfies mixed ℋ∞$$ {\mathscr{H}}_{\infty } $$ and passive performance. By constructing an augmented matrix, the properties of matrix determinant and rank are used to prove that the singular system is regular and causal, Finsler's lemma and Projection lemma are used in the design process to acquire additional slack variable matrices, thereby enhancing the solution space with extra degrees of freedom. Then, the unknown parameters of the filter are obtained by solving the less conservative linear matrix inequalities. Finally, the feasibility of the proposed method is verified through a common numerical example and by controlling a DC motor for an inverted pendulum. [ABSTRACT FROM AUTHOR]

Summary: An adaptive event‐triggered scheme is considered for networked control systems subject to deception and reply cyber attacks. In particular, the adaptive event‐triggered mechanism is used in the closed‐loop controller design to reduce signal transmission and communication burden. The attacker destroys the system's performance by employing deception attacks on sensor‐to‐controller communication channels and replay attacks on controller‐to‐actuator communication channels, respectively. By utilizing the Lyapunov stability approach, the closed‐loop system guarantees mean square stability and ensure security. The adaptive event‐triggered controller gains are presented by solving a set of matrix inequalities. Finally, a simulation result including the model of a batch reactor is presented to demonstrate the efficiency of the methods proposed. [ABSTRACT FROM AUTHOR]

• Memory sampled-data control is applied for the uncertain rigid spacecraft model. • Randomly missing measurement in the system output obeys the Bernoulli distribution. • It tolerates uncertainty and disturbances using the H ∞ technique. • A novel looped and delay-dependent LKF is constructed for the proposed systems. • It gives the minimum disturbance attenuation level compared with existing works. This paper focuses on stabilizing the attitude of the rigid spacecraft under the sampled-data control design technique with constant communication delay. The system dynamics with model uncertainty (perturbation) and missing measurements are considered. The aim is to design a desirable attitude control that ensures the stabilization of the rigid spacecraft with an optimal H ∞ performance level. A novel Lyapunov–Krasovskii functional is developed, incorporating looped characteristics, for the proposed study on the rigid spacecraft model by developing relevant new terms. Making use of the Lyapunov function as well as free-matrix-based integral inequality, sufficient stability criteria are established to assure the stability of the rigid spacecraft model. Furthermore, the desired sampled-data control gain matrices are acquired through the solution of linear matrix inequalities, which guarantees the asymptotic stability of the rigid spacecraft model. Finally, the numerical simulation demonstrates the efficacy of the theoretical study proposed for the rigid spacecraft model. [ABSTRACT FROM AUTHOR]