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An operator C on a Hilbert space \\mathcal H dilates to an operator T on a Hilbert space \\mathcal K if there is an isometry V:\\mathcal H\\to \\mathcal K such that C= V^* TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor \\vartheta (d), expressed as a ratio of \\Gamma functions for d even, of all d\\times d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.

In the framework of sampled-data control, this paper deals with the lag synchronization of chaotic neural networks with time delay meanwhile taking the impulsive control into account. By constructing a proper Lyapunov function and employing the impulsive control theory, some sufficient conditions for lag synchronization of the addressed chaotic neural networks are derived in terms of linear matrix inequalities (LMIs). The hybrid controller including sampled-data controller and impulsive controller is designed based on the established LMIs. A numerical example is provided to demonstrate the effectiveness and advantage of the obtained results.

This paper discusses the synchronization problem for a class of reaction–diffusion neural networks with Dirichlet boundary conditions. Unlike other studies, a sampled-data controller with stochastic sampling is designed in order to synchronize the concerned neural networks with reaction–diffusion terms and time-varying delays, where m sampling periods are considered whose occurrence probabilities are given constants and satisfy the Bernoulli distribution. A novel discontinuous Lyapunov–Krasovskii functional with triple integral terms is introduced based on the extended Wirtinger’s inequality. Using Jensen’s inequality and reciprocally convex technique in deriving the upper bound for the derivative of the Lyapunov–Krasovskii functional, some new synchronization criteria are obtained in terms of linear matrix inequalities. Numerical examples are provided in order to show the effectiveness of the proposed theoretical results.

In this paper, we exploit the sliding mode control problem for a fluid power electrohydraulic actuator (EHA) system. To characterize the nonlinearity of the friction, the EHA system is modeled as a linear system with a system uncertainty. Practically, it is assumed that the system is also subject to the load disturbance and the external noise. An integral sliding mode controller is proposed to design. The advanced techniques such as the H ∞ control and the regional pole placement are employed to derive the optimal feedback gain which can be calculated by solving a necessary and sufficient condition in the form of linear matrix inequality. A sliding mode control law is developed such that the sliding mode reaching law is satisfied. Simulation and comparison results show the effectiveness of the proposed design method.

Addressed in this paper is the reachable set estimation (RSE) problem for fuzzy‐model‐based leader‐follower multi‐agent systems with time‐varying delays and false data injection attacks. First, the aperiodic sampled‐data control is designed for the follower agents with randomly occurring false data injection attacks. Then, using the Kronecker product, the error system between the leader and the follower is obtained in a compact general form. Next, a novel Lyapunov‐Krasovskii functional is constructed with the knowledge of sampling patterns and time‐varying delays. In the framework of linear matrix inequalities, sufficient consensus conditions are determined from the H∞$H_\\infty$ performance index and Lyapunov theory to guarantee that its reachable set is enclosed by an ellipsoid in the existence of bounded perturbations. In the end, the Duffing Van der Pol oscillator and the single‐link robot arm models are employed to validate the derived theoretical results. The reachable set estimation problem is considered for the first time for leader‐follower fuzzy MASs with time‐varying delays and bounded external disturbances. The aperiodic sampled‐data control is designed for all the follower agents with the information from the leader. Moreover, the Bernoulli distribution is used for modeling the randomly occurring false data injection attacks in the controller actuator channels. The proposed theoretical findings are validated by two practical examples, that is, Duffing Van der Pol oscillator and single‐link robot arm model.

Regenerative chatter deteriorates machining precision and accelerates tool wear, thereby limiting productivity. This paper presents the design, analysis, and verification of a novel finite-frequency band (FFB) H∞ state feedback control strategy, which is dedicated to chatter control of turning processes. In comparison with the available entire-frequency domain (EFD) controllers, one uniqueness of the proposed controller is that it achieves the finite-frequency band optimal control of chatter, and the user can specify the frequency band for optimization according to actual requirements. Dynamics of regenerative delay, cutting model uncertainty, and actuator output constraint are incorporated into the controller design. Utilizing the Lyapunov–Krasovskii functional (LKF) method and the generalized Kalman-Yakubovich-Popov (GKYP) lemma, a set of linear matrix inequalities (LMI) are derived and adopted to synthesize the FFB H∞ controller. The superiority of the developed controller versus EFD controllers is verified by carrying out both simulation and experimental studies. Results demonstrate that the chatter-free region can be substantially enlarged with the proposed method.

This research examines the Lyapunov-based criterion for global asymptotic stability of Bidirectional Associative Memory (BAM) neural networks that have mixed-interval time-varying delays. Using a second-order reciprocally convex approach, this paper introduces a novel stability criterion for BAM neural networks with time delays. The literature has recently incorporated a few triple integral expressions in the Lyapunov–Krasovskii functional to lessen conservatism in the analysis of system stability with interval time-varying delays using a second-order reciprocally convex combination strategy. This research work establishes the negative definiteness of the Lyapunov–Krasovskii functional derivative and is formulated using Linear Matrix Inequalities (LMIs). The effectiveness of the proposed result is demonstrated through numerical examples.

This work introduces a cutting plane algorithm to solve the maximization of the minimum frequency of truss structures subject to volume and compliance constraints. Multiple load cases and multiple scenarios of non-structural mass distributions are considered. This problem is formulated as a non-convex semi-definite programming problem with Bi-linear Matrix Inequality (BMI) constraints. The proposed algorithm consists of iteratively tightening a linear relaxation of that problem. A new family of linear constraints (cutting planes) is defined as a linearization of BMI constraints. It is proved that the algorithm can find a violated valid cut for any infeasible solution that could be found in any iteration. Implementation details of the algorithm are given. We show the robustness of the method with some numerical examples and compare its performance with other available solvers. The reported results indicate that the new method outperforms the previous ones when the number of non-structural mass scenarios is large.

This paper presents an improved equivalent-input-disturbance (EID) approach to deal with periodic disturbances. The approach has two degrees of freedom. One is an improved EID compensator, in which a repetitive controller is inserted in this study. The other is a conventional servo system for a reference input. The improved EID compensator estimates and compensates for periodic disturbances without steady-state error, and the servo system ensures a satisfactory tracking performance. The improved EID compensator is designed using the linear-matrix-inequality (LMI) method. Three parameters in an LMI are selected using the particle-swarm-optimization (PSO) algorithm. The state-feedback gain of the conventional servo system is designed using the linear-quadratic-regulator (LQR) method. Simulation results of a rotational control system demonstrate the validity of the approach and its advantage over others.

The presence of offshore wind farms (OWFs) reduces the inertia of the power system and jeopardizes its frequency stability. Considering virtual inertia control (VIC) for these farms improves the frequency stability and inertia in the power system. In this paper, the robust H∞ controller based on deep reinforcement learning (DRL) is designed to improve the frequency stability in the load–frequency control (LFC) of the power system by considering the effect of OWFs on frequency control. The proposed method is robust to disturbances (load and wind fluctuations) and uncertainties related to system parameters and can adapt to uncertainties. The robust H∞ controller is designed based on linear matrix inequality (LMI) and DRL optimizes the robust H∞ controller and will improve the overall performance of the system. To examine the performance of the proposed method ( H∞ –DRL), several scenarios have been considered and compared with DMPC and PID control methods. The results show the superiority of the proposed method, which has been able to reduce load and wind fluctuations, frequency deviations, and also power deviations of the tie‐line between the lines of the multi‐area power system.