Analysis of small oscillations of complex electrical systems by Allaev K.R., Makhmudov T.F.

“…A joint solution of the equations of nodal voltages and the matrix Lyapunov equation is given, which makes it possible to determine the stability conditions for the electrical system and identify the generator first approaching the stability limit. …”
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Delay dependent stability of linear time-delay systems by Stojanović Sreten B., Debeljković Dragutin Lj.

“…The delay-dependent criteria are derived by Lyapunov's direct method and are exclusively based on the solvents of particular matrix equation and Lyapunov equation for non-delay systems. Obtained stability conditions do not possess conservatism. …”
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Damping of Power Systems Oscillations by using Genetic Algorithm-Based Optimal Controller by Akram F. Bat

“…It requires solving first order or second order linear matrix Lyapunov equation for shifting dominant poles to much better location that guaranteed less overshoot and less settling time of system transient response following a disturbance.…”
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Optimal nonlinear polynomial control of a quarter-vehicle suspension system under harmonic and random road excitations by Xudong Gu, Bingxin Zhao, Yuhao Liu

“…The linear feedback matrix and nonlinear feedback matrix in the NPC force are determined by the Riccati equation and Lyapunov equation, respectively. The root mean square (RMS) responses of the sprung mass acceleration, suspension deformation and road load of driving vehicles under deterministic and random road surfaces are calculated. …”
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Event-triggered coordination of multi-agent systems via a Lyapunov-based approach for leaderless consensus by Chen, Ci, Lewis, Frank L., Li, Xiaolei

“…We study such a problem by using a Lyapunov equation for leaderless consensus. It is shown that a Lyapunov-based approach can complete stability analysis of event-triggered consensus of multi-agent systems over general directed graphs.…”
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Energy amplification in channel flow over riblets. by Duncan, S, Papachristodoulou, A

“…The energy amplification is calculated for the streamwise constant component of the flow from the controllability grammian and the associated Lyapunov equation is solved efficiently by exploiting the structure of the problem. …”

Periodically refreshed quantum thermal machines by Archak Purkayastha, Giacomo Guarnieri, Steve Campbell, Javier Prior, John Goold

“…We also introduce an elegant description of the PReB process for Gaussian systems in terms of a discrete-time Lyapunov equation. Further, our analysis also reveals interesting connections with Zeno and anti-Zeno effects.…”
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The Online Parameter Identification Method of Permanent Magnet Synchronous Machine under Low-Speed Region Considering the Inverter Nonlinearity by Qiushi Zhang, Ying Fan

“…Moreover, the <i>d</i>-axis voltage equation is established considering the nonlinearity of the inverter and an online adaptive observer was proposed. Based on the Lyapunov equation, the adaptive laws are derived. Further, the decoupling of the deadtime voltage and resistance voltage is realized by using the result of neural network identification. …”
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Synchronization Control of Dynamic Positioning Ships Using Model Predictive Control by Cheng Liu, Ting Sun, Qizhi Hu

“…Thirdly, stability, frequently neglected by traditional MPC, is ensured by the means of adding a terminal cost function exported from the Lyapunov equation into the objective function. Finally, the effectiveness and advantages of the proposed control design are illustrated by extensive simulations.…”
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Time Delay Estimation Control of Permanent Magnet Spherical Actuator Based on Gradient Compensation by Xiuqin Wang, Rui Zhang, Guoli Li, Qunjing Wang, Yan Wen

“…Furthermore, the stability of the designed controller is proved by the Lyapunov equation. Finally, the correctness and effectiveness of the controller are validated by comparison with other controllers through simulation. …”
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Distributed Fault Estimation of Multi–Agent Systems Using a Proportional–Integral Observer: A Leader–Following Application by Farrera Beltrán, López-Estrada Francisco-Ronay, Chadli Mohammed, Valencia-Palomo Guillermo, Gómez-Peñate Samuel

“…Second, based on the proposed quadratic Lyapunov equation, sufficient conditions for the asymptotic convergence of the observer are obtained as a set of linear matrix inequalities. …”
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Designing Active Disturbance Rejection Control of Permanent Magnetic Spherical Actuator Based on Nonlinear Extended State Observer by Xiwen Guo, Xianda Liao, Qunjing Wang, Yan Wen, Tao Wu

“…Finally, the nonlinear control law is used to compensate for the observed disturbance, and the stability is proved based on the Lyapunov equation. Simulation and experimental results show that the proposed control scheme can achieve decoupling and tracking control of the complex system in the presence of model error, random disturbance, and other uncertain factors, has good control accuracy and response speed, and has strong robustness to uncertain disturbances.…”
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Solving the differential biochemical Jacobian from metabolomics covariance data. by Thomas Nägele, Andrea Mair, Xiaoliang Sun, Lena Fragner, Markus Teige, Wolfram Weckwerth

“…A differential biochemical Jacobian was calculated using an approach which links this metabolic interaction matrix and the covariance of metabolomics data satisfying a Lyapunov equation. The predictions of the differential Jacobian from real metabolomic data were found to be correct by testing the corresponding enzymatic activities. …”
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Centralized fusion robust filtering for networked uncertain systems with colored noises, one-step random delay, and packet dropouts by Shuang Li, Wenqiang Liu, Guili Tao

“…To demonstrate the robustness of the proposed CF estimators, the non-negative definite matrix decomposition method and Lyapunov equation approach are employed. It is proved that the robust accuracy of CF estimator is higher than that of each local estimator. …”
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Energy Dissipation and Performance Assessment of the Connected Structure with a One-Side Damping Layer by Liangkun Liu, Yuze Zheng, Zhaodong Pan, Qing Lyu

“…Based on the Kanai-Tajimi filtered spectrum model, the seismic response variances are calculated using the Lyapunov equation. To investigate the seismic energy distribution and mitigation performance, three models of the connected structure with a damping layer are analyzed using the index of the mean kinetic energy. …”
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Saturated Nonsingular Fast Sliding Mode Control for the Crane-Form Pipeline System by Baigeng Wang, Shurong Li

“…Owing to the existence of saturation functions in the controller, the stability analysis using the Lyapunov equation needs to be discussed in different cases. …”
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Modelling channel flow over riblets: Calculating the energy amplification by Kasliwal, A, Duncan, S, Papachristodoulou, A

“…However, solving the associated Lyapunov equation can be computationally prohibitive, which limits the density of the discretized grid. …”

Trajectory tracking control strategy of the gantry welding robot under the influence of uncertain factors by Hongwei Sun, Haonan Wang, Yicun Xu, Jiong Yang, Yujie Ji, Lei Cheng, Chao Liu, Benshun Zhang

“…Based on the computed torque control, the adaptive fuzzy feedback control strategy is adopted to compensate for the modeling errors and external disturbances in the gantry welding robot system, and the dead zone adaptive fuzzy control strategy is designed to compensate for the nonlinear dead zone structure with unknown parameters. The Lyapunov equation is introduced to prove the stability of the controller. …”
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Hybrid synchronization and parameter estimation of a complex chaotic network of permanent magnet synchronous motors using adaptive integral sliding mode control by Nazam Siddique, Fazal U. Rehman

“…The compensator controller, as well as the adopted laws, are designed to get the first derivative of the Lyapunov equation strictly negative. To give an illustration, the proposed technique is applied to 4-coupled motor systems yielding the convergence of error dynamics to zero, estimation of uncertain parameters, and hybrid synchronization of system states. …”
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Dynamical Optimal Values of Parameters in the SSOR, AOR, and SAOR Testing Using Poisson Linear Equations by Chein-Shan Liu, Essam R. El-Zahar, Chih-Wen Chang

“…Numerical testings of the linear Poisson equation discretized to a matrix–vector form and a Lyapunov equation form were used to assess the performance of the DOSSOR, DOAOR, and DOSAOR dynamical optimal methods.…”
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