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Journal articles on the topic 'Lyapunov-based approach'

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Published: 15 June 2024

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1

Hoang, H., F. Couenne, C. Jallut, and Y. Le Gorrec. "Thermodynamic approach for Lyapunov based control." IFAC Proceedings Volumes 42, no. 11 (2009): 357–62. http://dx.doi.org/10.3182/20090712-4-tr-2008.00056.

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2

Dixon, W. E., E. Zergeroglu, D. M. Dawson, and B. T. Costic. "Repetitive learning control: a Lyapunov-based approach." IEEE Transactions on Systems, Man and Cybernetics, Part B (Cybernetics) 32, no. 4 (August 2002): 538–45. http://dx.doi.org/10.1109/tsmcb.2002.1018772.

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3

Kansha, Yasuki, Li Jia, and Min-Sen Chiu. "Self-tuning PID controllers based on the Lyapunov approach." Chemical Engineering Science 63, no. 10 (May 2008): 2732–40. http://dx.doi.org/10.1016/j.ces.2008.02.026.

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4

Senthilkumar, L., M. Meenakshi, and J. Vasantha Kumar. "Lyapunov Optimization Based Cross Layer Approach for Green Cellular Network." Journal of Green Engineering 5, no. 2 (2016): 129–50. http://dx.doi.org/10.13052/jge1904-4720.523.

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5

Margaliot, Michael, and Gideon Langholz. "Fuzzy Lyapunov-based approach to the design of fuzzy controllers." Fuzzy Sets and Systems 106, no. 1 (August 1999): 49–59. http://dx.doi.org/10.1016/s0165-0114(98)00356-x.

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6

Hoang, N. Ha, Denis Dochain, and Nicolas Hudon. "A thermodynamic approach towards Lyapunov based control of reaction rate." IFAC Proceedings Volumes 47, no. 3 (2014): 9117–22. http://dx.doi.org/10.3182/20140824-6-za-1003.01958.

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7

Mutlu, Ilhan, Frank Schrödel, Naim Bajcinca, Dirk Abel, and M. Turan Söylemez. "Lyapunov Equation Based Stability Mapping Approach: A MIMO Case Study." IFAC-PapersOnLine 49, no. 9 (2016): 130–35. http://dx.doi.org/10.1016/j.ifacol.2016.07.512.

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8

Abdelmalek, Ibtissem, Noureddine Goléa, and Mohamed Hadjili. "A New Fuzzy Lyapunov Approach to Non-Quadratic Stabilization of Takagi-Sugeno Fuzzy Models." International Journal of Applied Mathematics and Computer Science 17, no. 1 (March 1, 2007): 39–51. http://dx.doi.org/10.2478/v10006-007-0005-4.

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A New Fuzzy Lyapunov Approach to Non-Quadratic Stabilization of Takagi-Sugeno Fuzzy ModelsIn this paper, new non-quadratic stability conditions are derived based on the parallel distributed compensation scheme to stabilize Takagi-Sugeno (T-S) fuzzy systems. We use a non-quadratic Lyapunov function as a fuzzy mixture of multiple quadratic Lyapunov functions. The quadratic Lyapunov functions share the same membership functions with the T-S fuzzy model. The stability conditions we propose are less conservative and stabilize also fuzzy systems which do not admit a quadratic stabilization. The proposed approach is based on two assumptions. The first one relates to a proportional relation between multiple Lyapunov functions and the second one considers an upper bound to the time derivative of the premise membership functions. To illustrate the advantages of our proposal, four examples are given.

9

Kuzmych, Olena, Abdel Aitouche, Ahmed El Hajjaji, and Jerome Bosche. "Nonlinear control for a diesel engine: A CLF-based approach." International Journal of Applied Mathematics and Computer Science 24, no. 4 (December 1, 2014): 821–35. http://dx.doi.org/10.2478/amcs-2014-0061.

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Abstract In this paper, we propose a control Lyapunov function based on a nonlinear controller for a turbocharged diesel engine. A model-based approach is used which predicts the experimentally observed engine performance for a biodiesel. The basic idea is to develop an inverse optimal control and to employ a Lyapunov function in order to achieve good performances. The obtained controller gain guarantees the global convergence of the system and regulates the flows for the variable geometry turbocharger as well as exhaust gas recirculation systems in order to minimize the NOx emission and the smoke of a biodiesel engine. Simulation of the control performances based on professional software and experimental results show the effectiveness of this approach.

10

Masoumnezhad, Mojtaba, Maziar Rajabi, Amirahmad Chapnevis, Aleksei Dorofeev, Stanford Shateyi, Narges Shayegh Kargar, and Hassan Saberi Nik. "An Approach for the Global Stability of Mathematical Model of an Infectious Disease." Symmetry 12, no. 11 (October 27, 2020): 1778. http://dx.doi.org/10.3390/sym12111778.

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The global stability analysis for the mathematical model of an infectious disease is discussed here. The endemic equilibrium is shown to be globally stable by using a modification of the Volterra–Lyapunov matrix method. The basis of the method is the combination of Lyapunov functions and the Volterra–Lyapunov matrices. By reducing the dimensions of the matrices and under some conditions, we can easily show the global stability of the endemic equilibrium. To prove the stability based on Volterra–Lyapunov matrices, we use matrices with the symmetry properties (symmetric positive definite). The results developed in this paper can be applied in more complex systems with nonlinear incidence rates. Numerical simulations are presented to illustrate the analytical results.

11

Azhmyakov, Vadim. "Stability of differential inclusions: A computational approach." Mathematical Problems in Engineering 2006 (2006): 1–15. http://dx.doi.org/10.1155/mpe/2006/17837.

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We present a technique for analysis of asymptotic stability for a class of differential inclusions. This technique is based on the Lyapunov-type theorems. The construction of the Lyapunov functions for differential inclusions is reduced to an auxiliary problem of mathematical programming, namely, to the problem of searching saddle points of a suitable function. The computational approach to the auxiliary problem contains a gradient-type algorithm for saddle-point problems. We also extend our main results to systems described by difference inclusions. The obtained numerical schemes are applied to some illustrative examples.

12

SHEN, Tielong, and Katsutoshi TAMURA. "Nonlinear Robust H^|^infin; Control-An Approach Based on Lyapunov Function-." Transactions of the Society of Instrument and Control Engineers 34, no. 9 (1998): 1191–97. http://dx.doi.org/10.9746/sicetr1965.34.1191.

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13

Hong, Yiguang, Lixin Gao, Daizhan Cheng, and Jiangping Hu. "Lyapunov-Based Approach to Multiagent Systems With Switching Jointly Connected Interconnection." IEEE Transactions on Automatic Control 52, no. 5 (May 2007): 943–48. http://dx.doi.org/10.1109/tac.2007.895860.

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14

Xu, Xiang, Lu Liu, and Gang Feng. "Stability and Stabilization of Infinite Delay Systems: A Lyapunov-Based Approach." IEEE Transactions on Automatic Control 65, no. 11 (November 2020): 4509–24. http://dx.doi.org/10.1109/tac.2019.2958557.

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15

Xu, Jun, Daoying Pi, Yong-Yan Cao, and Shouming Zhong. "On Stability of Neural Networks by a Lyapunov Functional-Based Approach." IEEE Transactions on Circuits and Systems I: Regular Papers 54, no. 4 (April 2007): 912–24. http://dx.doi.org/10.1109/tcsi.2007.890604.

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16

Fridman, Emilia. "Stability of linear descriptor systems with delay: a Lyapunov-based approach." Journal of Mathematical Analysis and Applications 273, no. 1 (September 2002): 24–44. http://dx.doi.org/10.1016/s0022-247x(02)00202-0.

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17

Benaoumeur, Ibari, Benchikh Laredj, Haniﬁ Elhachimi Amar Reda, and Ahmed-foitih Zoubir. "Backstepping Approach for Autonomous Mobile Robot Trajectory Tracking." Indonesian Journal of Electrical Engineering and Computer Science 2, no. 3 (June 1, 2016): 478. http://dx.doi.org/10.11591/ijeecs.v2.i3.pp478-485.

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This paper proposes a backstepping controller design for trajectory tracking of unicycle-type mobile robots. The main object of the control algorithms developed is to design a robust output tracking controller. The design of the controller is based on the lyapunov theorem, kinematic tracking controller of an unicycle-like mobile robot is used to provides the desired values of the linear and angular velocities for the given trajectory. A Lyapunov-based stability analysis is presented to guarantee the robot stability of the tracking errors. Simulation and experimental results show the effectiveness of the proposed robust controller in term of accuracy and stability under different load conditions.

18

Zhao, Yongchi, Shengxian Zhuang, Weiming Xiang, and Lin Du. "Discretized Lyapunov Function Approach for Switched Linear Systems under Dwell Time Constraint." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/905968.

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This paper is concerned with the stability and disturbance attenuation properties of switched linear system with dwell time constraint. A novel time-scheduled Lyapunov function is introduced to deal with the problems studied in this paper. To numerically check the existence of such time-scheduled Lyapunov function, the discretized Lyapunov function technique usually used in time-delay system is developed in the context of switched system in continuous-time cases. Based on discretized Lyapunov function, sufficient conditions ensuring dwell-time constrained switched system global uniformly asymptotically stable are established, then the disturbance attenuation properties in the sense ofL2gain are studied. The main advantage of discretized Lyapunov function approach is that the derived sufficient conditions are convex in subsystem matrices, which makes the analysis results easily used and generalized. Thus, theH∞control synthesis problem is considered. On the basis of analysis results in hand, the control synthesis procedures including both controller and switching law design are unified into one-step method which explicitly facilitates the control synthesis process. Several numerical examples are provided to illustrate the results within our paper.

19

Hoang, Nguyen Quang, and Soon Geul Lee. "Energy-Based Approach for Controller Design of Overhead Cranes: A Comparative Study." Applied Mechanics and Materials 365-366 (August 2013): 784–87. http://dx.doi.org/10.4028/www.scientific.net/amm.365-366.784.

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In this paper, five controllers including linear and nonlinear ones for an underactuated overhead crane are derived based on the passivity of the system. The total energy of the system and its square are used in Lyapunov candidate function to design controllers. The equilibrium point of the closed loop is proven to be asymptotically stable by the Lyapunov technique and LaSalle invariance theorem. In addition, the optimal linear controller is also combined to force the swing angle to converge fast to zero by reaching destination of the trolley. Numerical simulations are carried out to evaluate the controllers.

20

Vafamand, Navid. "Global non-quadratic Lyapunov-based stabilization of T–S fuzzy systems: A descriptor approach." Journal of Vibration and Control 26, no. 19-20 (February 13, 2020): 1765–78. http://dx.doi.org/10.1177/1077546320904817.

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This article studies the problem of global stability of the Takagi–Sugeno fuzzy systems based on a novel descriptor-based non-quadratic Lyapunov function. A modified non-quadratic Lyapunov function, which comprises an integral term of the membership functions, and a modified non-parallel distributed controller constructed by constant delayed premise variables are considered that assure the global stability of the closed-loop T–S fuzzy system. The special structure of the used non-quadratic Lyapunov function results in time-delayed terms of the membership functions, instead of appearing their time derivatives, which is the well-known issue of the common non-quadratic Lyapunov functions in the literature. Also, the memory fuzzy controller is chosen such that the artificial constant delay-dependent stability analysis conditions for a non-delayed closed-loop T–S fuzzy system are formulated in terms of linear matrix inequalities. To further reduce the conservatives, some slack matrices are introduced by deploying the descriptor representation and decoupling lemmas. Moreover, the design of the robust fuzzy controller is studied through the [Formula: see text] performance criteria. The main advantages of the proposed approach are its small conservatives and the global stability analysis, which distinguish it from the state-of-the-art methods. To show the merits of the proposed approach, comparison results are provided, and two numerical case studies, namely, flexible joint robot and two-link joint robot are considered.

21

Martin, Christoph, Nahal Sharafi, and Sarah Hallerberg. "Estimating covariant Lyapunov vectors from data." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 3 (March 2022): 033105. http://dx.doi.org/10.1063/5.0078112.

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Covariant Lyapunov vectors characterize the directions along which perturbations in dynamical systems grow. They have also been studied as predictors of critical transitions and extreme events. For many applications, it is necessary to estimate these vectors from data since model equations are unknown for many interesting phenomena. We propose an approach for estimating covariant Lyapunov vectors based on data records without knowing the underlying equations of the system. In contrast to previous approaches, our approach can be applied to high-dimensional datasets. We demonstrate that this purely data-driven approach can accurately estimate covariant Lyapunov vectors from data records generated by several low- and high-dimensional dynamical systems. The highest dimension of a time series from which covariant Lyapunov vectors are estimated in this contribution is 128.

22

HAN, R., M. LEMM, and W. SCHLAG. "Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base." Ergodic Theory and Dynamical Systems 40, no. 10 (April 17, 2019): 2788–853. http://dx.doi.org/10.1017/etds.2019.19.

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We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew-shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman’s subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large-deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite-volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large-deviation estimates and the avalanche principle, effective.

23

Ates, Muzaffer, and Nezir Kadah. "Novel stability and passivity analysis for three types of nonlinear LRC circuits." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 11, no. 2 (July 31, 2021): 227–37. http://dx.doi.org/10.11121/ijocta.01.2021.001073.

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In this paper, the global asymptotic stability and strict passivity of three types of nonlinear RLC circuits are investigated by utilizing the Lyapunov direct method. The stability conditions are obtained by constructing appropriate Lyapunov function, which demonstrates the practical application of the Lyapunov theory with a clear perspective. The meaning of Lyapunov functions is not clear by many specialists whose studies based on Lyapunov theory. They construct Lyapunov functions by using some properties of Lyapunov functions with much trial and errors or for a system choose candidate Lyapunov functions. So, for a given system Lyapunov function is not unique. But we insist that Lyapunov (energy) function is unique for a given physical system. In this study we highly simplified Lyapunov’s direct method with suitable tools. Our approach constructing energy function based on power-energy relationship that also enable us to take the derivative of integration of energy function. These aspects have not been addressed in the literature. This paper is an attempt towards filling this gap. The results are provided within and are of central importance for the analysis of nonlinear electrical, mechanical, and neural systems which based on the system energy perspective. The simulation results given from Matlab successfully verifies the theoretical predictions.

24

Zhu, Hailing, Khmaies Ouahada, and Suvendi Rimer. "Real Time Energy Storage Sharing With Load Scheduling: A Lyapunov-Based Approach." IEEE Access 9 (2021): 46626–40. http://dx.doi.org/10.1109/access.2021.3067788.

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25

冷, 轩. "Global Dynamics for a Plant Disease Model Based on Generalized Lyapunov Approach." Advances in Applied Mathematics 10, no. 04 (2021): 1086–95. http://dx.doi.org/10.12677/aam.2021.104117.

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26

Dubljević, Stevan, and Nikolaos Kazantzis. "A new Lyapunov design approach for nonlinear systems based on Zubov's method." Automatica 38, no. 11 (November 2002): 1999–2007. http://dx.doi.org/10.1016/s0005-1098(02)00110-3.

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27

Wang, Ruigang, and Jie Bao. "A differential Lyapunov-based tube MPC approach for continuous-time nonlinear processes." Journal of Process Control 83 (November 2019): 155–63. http://dx.doi.org/10.1016/j.jprocont.2018.11.006.

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28

Faieghi, Mohammadreza, Seyed Kamal-e.-ddin Mousavi Mashhadi, and Dumitru Baleanu. "Sampled-data nonlinear observer design for chaos synchronization: A Lyapunov-based approach." Communications in Nonlinear Science and Numerical Simulation 19, no. 7 (July 2014): 2444–53. http://dx.doi.org/10.1016/j.cnsns.2013.11.021.

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29

Wang, Yaonan, Zhiqiang Miao, Hang Zhong, and Qi Pan. "Simultaneous Stabilization and Tracking of Nonholonomic Mobile Robots: A Lyapunov-Based Approach." IEEE Transactions on Control Systems Technology 23, no. 4 (July 2015): 1440–50. http://dx.doi.org/10.1109/tcst.2014.2375812.

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30

Montoya, Oscar Danilo, and Walter Gil-González. "Nonlinear analysis and control of a reaction wheel pendulum: Lyapunov-based approach." Engineering Science and Technology, an International Journal 23, no. 1 (February 2020): 21–29. http://dx.doi.org/10.1016/j.jestch.2019.03.004.

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31

Namachchivaya, N. Sri, and H. J. Van Roessel. "Stochastic Stability of Coupled Oscillators in Resonance: A Perturbation Approach." Journal of Applied Mechanics 71, no. 6 (November 1, 2004): 759–68. http://dx.doi.org/10.1115/1.1795813.

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A perturbation approach is used to obtain an approximation for the moment Lyapunov exponent of two coupled oscillators with commensurable frequencies driven by a small intensity real noise with dissipation. The generator for the eigenvalue problem associated with the moment Lyapunov exponent is derived without any restriction on the size of pth moment. An orthogonal expansion for the eigenvalue problem based on the Galerkin method is used to derive the stability results in terms of spectral densities. These results can be applied to study the moment and almost-sure stability of structural and mechanical systems subjected to stochastic excitation.

32

TayebiHaghighi, Shahnaz, and Insoo Koo. "Sensor Fault Diagnosis Using a Machine Fuzzy Lyapunov-Based Computed Ratio Algorithm." Sensors 22, no. 8 (April 13, 2022): 2974. http://dx.doi.org/10.3390/s22082974.

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Anomaly identification for internal combustion engine (ICE) sensors has become an important research area in recent years. In this work, a proposed indirect fuzzy Lyapunov-based computed ratio observer integrated with a support vector machine (SVM) was designed for sensor fault classification. The proposed fuzzy Lyapunov-based computed ratio observer integrated with SVM has three main layers. In the preprocessing (first) layer, the resampled root mean square (RMS) signals are extracted from the original signals to the designed indirect observer. The second (observation) layer is the principal part with the proposed indirect fuzzy sensor-fault-classification technique. This layer has two sub-layers: signal modeling and estimation. The Gaussian autoregressive-Laguerre approach integrated with the fuzzy approach is designed for resampled RMS fuel-to-air-ratio normal signal modeling, while the subsequent sub-layer is used for resampled RMS fuel-to-air-ratio signal estimation using the proposed fuzzy Lyapunov-based computed ratio observer. The third layer, for residual signal generation and classification, is used to identify ICE sensor anomalies, where residual signals are generated by the difference between the original and estimated resampled RMS fuel-to-air-ratio signals. Moreover, SVM is suggested for residual signal classification. To test the effectiveness of the proposed method, the results are compared with two approaches: a Lyapunov-based computed ratio observer and a computed ratio observer. The results show that the accuracy of sensor anomaly classification by the proposed fuzzy Lyapunov-based computed ratio observer is 98.17%. Furthermore, the proposed scheme improves the accuracy of sensor fault classification by 8.37%, 2.17%, 6.17%, 4.57%, and 5.37% compared to other existing methods such as the computed ratio observer, the Lyapunov-based computed ratio observer, fuzzy feedback linearization observation, self-tuning fuzzy robust multi-integral observer, and Kalman filter technique, respectively.

33

Aly, Islam A., and Kadriye Merve Dogan. "Discrete-Time Adaptive Control for Uncertain Scalar Multiagent Systems with Coupled Dynamics: A Lyapunov-Based Approach." Electronics 13, no. 3 (January 27, 2024): 524. http://dx.doi.org/10.3390/electronics13030524.

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Discrete-time architectures offer a distinct advantage over their continuous counterparts, as they can be seamlessly implemented on embedded hardware without the necessity for discretization processes. Yet, because of the difficulty of ensuring Lyapunov difference expressions, their designs, which are based on quadratic Lyapunov-based frameworks, are highly complex. As a result, various existing continuous-time results using adaptive control methods to deal with system uncertainties and coupled dynamics in agents of a multiagent system cannot be directly applied to the discrete-time context. Furthermore, compared to their continuous-time equivalent, discrete-time information exchange based on periodic time intervals is more practical in the control of multiagent systems. Motivated by these standpoints, in this paper, we first introduce a discrete-time adaptive control architecture designed for uncertain scalar multiagent systems without coupled dynamics as a preliminary result. We then introduce another discrete-time adaptive control approach for uncertain multiagent systems in the presence of coupled dynamics. Our approach incorporates observer dynamics to manage unmeasurable coupled dynamics, along with a user-assigned Laplacian matrix to induce cooperative behaviors among multiple agents. Our solution includes Lyapunov analysis with logarithmic and quadratic Lyapunov functions for guaranteeing asymptotic stability with both controllers. To demonstrate the effectiveness of the proposed control architectures, we provide an illustrative example. The illustrative numerical example shows that the standard discrete-time adaptive control in the absence of observer dynamics cannot guarantee the reference state vector tracking, while the proposed discrete-time adaptive control can ensure the tracking objective.

34

Huang, Xi Chang, Zhi Jian Zong, and Zhong Tu Liu. "Robust Nonlinear Control of Electric Power Steering System Based on Lyapunov Redesign." Advanced Materials Research 588-589 (November 2012): 1619–23. http://dx.doi.org/10.4028/www.scientific.net/amr.588-589.1619.

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Based on the dynamic characteristics of EPS, whole vehicle and tire, the dynamic equation of the EPS system was established. The Krasovk’s method was used to obtian the candidate Lyapunov function for the EPS system. A nonlinear control for EPS system considering parameter uncertainty was presented by Lyapunov redesign approach. The results were obtained from simulation based on Dymola, where is shown that Control law of EPS have the robust performance.

35

Rakhshan, Mohsen, Navid Vafamand, Mohammad Mehdi Mardani, Mohammad-Hassan Khooban, and Tomislav Dragičević. "Polynomial control design for polynomial systems: A non-iterative sum of squares approach." Transactions of the Institute of Measurement and Control 41, no. 7 (October 1, 2018): 1993–2004. http://dx.doi.org/10.1177/0142331218793476.

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This paper proposes a non-iterative state feedback design approach for polynomial systems using polynomial Lyapunov function based on the sum of squares (SOS) decomposition. The polynomial Lyapunov matrix consists of states of the system leading to the non-convex problem. A lower bound on the time derivative of the Lyapunov matrix is considered to turn the non-convex problem into a convex one; and hence, the solutions are computed through semi-definite programming methods in a non-iterative fashion. Furthermore, we show that the proposed approach can be applied to a wide range of practical and industrial systems that their controller design is challenging, such as different chaotic systems, chemical continuous stirred tank reactor, and power permanent magnet synchronous machine. Finally, software-in-the-loop (SiL) real-time simulations are presented to prove the practical application of the proposed approach.

36

Liao, Wenqi, Hongbing Zeng, and Huichao Lin. "Stability Analysis of Linear Time-Varying Delay Systems via a Novel Augmented Variable Approach." Mathematics 12, no. 11 (May 23, 2024): 1638. http://dx.doi.org/10.3390/math12111638.

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This paper investigates the stability issues of time-varying delay systems. Firstly, a novel augmented Lyapunov functional is constructed for a class of bounded time-varying delays by introducing new double integral terms. Subsequently, a time-varying matrix-dependent zero equation is introduced to relax the constraints of traditional constant matrix-dependent zero equations. Secondly, for a class of periodic time-varying delays, considering the monotonicity of the delay and combining it with an augmented variable approach, Lyapunov functionals are constructed for monotonically increasing and monotonically decreasing delay intervals, respectively. Based on the constructed augmented Lyapunov functionals and the employed time-varying zero equation, less conservative stability criteria are obtained separately for bounded and periodic time-varying delays. Lastly, three examples are used to verify the superiority of the stability conditions obtained in this paper.

37

Ivanov, Danil, Goncalo Amaro, Yaroslav Mashtakov, Mikhail Ovchinnikov, and Anna Guerman. "Formation Flying Lyapunov-Based Control Using Lorentz Forces." Aerospace 10, no. 1 (January 1, 2023): 39. http://dx.doi.org/10.3390/aerospace10010039.

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A formation flying control algorithm using the Lorentz force for Low Earth Orbits to achieve a trajectory with required shape and size is proposed in the paper. The Lorentz force is produced as a result of interaction between the Earth’s magnetic field and an electrically charged spacecraft. Achievement of the required trajectories represents a challenge since the control in three-dimensional space is a scalar value of the satellite’s charge. A Lyapunov-based control algorithm is developed for elimination of the initial relative drift after the launch. It also aims at reaching a required amplitudes for close relative trajectories for in-plane and out-of-plane motion. Due to the absence of full controllability, the algorithm is incapable of correcting all the parameters of the relative trajectory such as in-plane and out-of-plane phase angles. The proposed control allows to converge to the trajectory with required shape and size, though with some oscillating errors in the vicinity of the required trajectory parameters. Numerical simulation of the relative motion is used to study performance of the control algorithm for one case of one controlled satellite and two cases of five controlled satellites forming a nested ellipses and train formations. The convergence time and final trajectory accuracy are evaluated for different control parameters and orbits using Monte Carlo approach.

38

Beisenbi, Мamyrbek, and Samal Kaliyeva. "Approach to the synthesis of an aperiodic robust automatic control system based on the gradient-speed method of Lyapunov vector functions." Eastern-European Journal of Enterprise Technologies 1, no. 3 (121) (February 28, 2023): 6–14. http://dx.doi.org/10.15587/1729-4061.2023.274063.

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One of the actual problems for the theory and practice of control of dynamic objects is the development of methods for research and synthesis of control systems of multidimensional objects. The paper proposes a universal approach to construct Lyapunov vector functions directly from the equation of state of control system and a new gradient-speed method of Lyapunov vector functions to study aperiodic robust stability of linear control system with m inputs and n outputs. The study of aperiodic robust stability of automatic control systems is based on the construction of Lyapunov vector functions and gradient-speed dynamic control systems. The basic statements of Lyapunov's theorem about asymptotic stability and notions of stability of dynamic systems are used. The representation of control systems as gradient systems and Lyapunov functions as potential functions of gradient systems from the catastrophe theory allow to construct the full-time derivative of Lyapunov vector functions always as a sign-negative function equal to the scalar product of the velocity vector on the gradient vector. The conditions of aperiodic robust stability are obtained as a system of inequalities on the uncertain parameters of the automatic control system, which are a condition for the existence of the Lyapunov vector-function. A numerical example of synthesis of aperiodic robustness of a multidimensional control object is given. The example shows the main stages of the developed synthesis method, the study of the system stability at different values of the coefficients k, confirming the consistency of the proposed method. Transients in the system satisfy all requirements

39

Uskenbayeva, Gulzhan. "Research of the Robust Stability of Control Systems Using a New Approach to the Lyapunov Functions Construction." Modern Applied Science 9, no. 11 (September 30, 2015): 176. http://dx.doi.org/10.5539/mas.v9n11p176.

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<p class="22">We investigate a new approach to the construction of vector Lyapunov functions. An approach to the construction of Lyapunov functions as vector functions is developed based on a geometrical interpretation of the second method of Lyapunov. The negative of the gradient is determined from the components of the time derivative of the state vector (i.e., the right-hand side of the state equation). The region of stability of a closed-loop linear, stationary system with uncertain parameters is governed by inequalities in the matrix elements of the closed-loop system. This study developed a method for analysing the robust stability of SISO and MIMO linear systems in canonical forms.</p>

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Arrifano, Natache S. D., Vilma A. Oliveira, and Lúcia V. Cossi. "Synthesis of an LMI-based fuzzy control system with guaranteed cost performance: a piecewise Lyapunov approach." Sba: Controle & Automação Sociedade Brasileira de Automatica 17, no. 2 (June 2006): 213–25. http://dx.doi.org/10.1590/s0103-17592006000200009.

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A new stability analysis and design of a fuzzy switching control based on uncertain Takagi-Sugeno fuzzy systems are proposed. The fuzzy system adopted is composed by a family of local linear uncertain systems with aggregation. The control design proposed uses local state feedback gains obtained from an optimization problem with guaranteed cost performance formulated in the context of linear matrix inequalities and a fuzzy switching scheme built from local Lyapunov functions. The global stability is guaranteed by considering a class of piecewise quadratic Lyapunov functions. Examples are given to illustrate the applicability of the proposed approach.

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Agung, Hendi Wicaksono, and Fransisco Jordan. "Semi-decentralized Lyapunov-based formation control of multiple omnidirectional mobile robots." Indonesian Journal of Electrical Engineering and Computer Science 35, no. 2 (August 1, 2024): 823. http://dx.doi.org/10.11591/ijeecs.v35.i2.pp823-833.

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<p>This paper introduces an advanced formation control algorithm based on a Lyapunov approach for coordinating multiple omnidirectional mobile robots in collaborative object transport tasks. The semi-decentralized strategy ensures that the robots maintain a predefined geometric formation, crucial for stability during material transportation, and dynamically adapt to avoid collisions using onboard sensors. Experimental with a physical robot simulator demonstrates successful maintenance of line and triangle formations achieving an average side length maintenance of 1.00 meters with minimal deviation. Quantitative analysis across 30 experimental runs reveals consistent performance, with a maximum side length fluctuation of only 2 centimeters, validating the effectiveness of maintaining formation within a multi-robot system (MRS) framework. The Lyapunov-based approach proves to be an efficient method for cooperative object transport, achieving consistent performance with minimal deviation.</p>

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Tao, Zhao, Yi-hua Wang, Guang-qi Li, and Gang Hou. "Lyapunov based global trajectory tracking control of wheeled mobile robot." Journal of Physics: Conference Series 2478, no. 10 (June 1, 2023): 102018. http://dx.doi.org/10.1088/1742-6596/2478/10/102018.

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Abstract This paper considers the global trajectory tracking control problem of wheeled mobile robot. A novel tracking controller for the kinematic model of wheeled mobile robot is presented based on the cascaded approach theory. Meanwhile, by using the Lyapunov stability theory, the stability of the controller is proved. Finally, the proposed scheme is implemented on the wheeled mobile robot. The simulation results show the effectiveness of the controller.

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Barreira, Luis, Davor Dragičević, and Claudia Valls. "Tempered exponential dichotomies and Lyapunov exponents for perturbations." Communications in Contemporary Mathematics 18, no. 05 (July 18, 2016): 1550058. http://dx.doi.org/10.1142/s0219199715500583.

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We establish a Perron-type result for the perturbations of a linear cocycle in the context of ergodic theory. More precisely, we show that the Lyapunov exponents of a linear cocycle are preserved under sufficiently small nonautonomous perturbations. Our approach is based on the Lyapunov theory of regularity.

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Fiedler, Robert, and Hartmut Hetzler. "Numerical Approximation of LYAPUNOV-Exponents for Quasiperiodic Motions." MATEC Web of Conferences 241 (2018): 01009. http://dx.doi.org/10.1051/matecconf/201824101009.

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This paper proposes an approach to approximate the LYAPUNOV -spectrum of quasiperiodic flows on isolated invariant manifolds numerically. Once the invariant manifold has been determined, integrations over the infinite, one dimensional time interval – as calculating the LYAPUNOV -spectrum for instance – can be transformed into an integral over a finite, p-dimensional domain, where p is the dimension of the manifold. The application of the proposed approach is demonstrated by calculating the LYAPUNOV -spectrum of periodic and quasiperiodic motions of a forced VAN-DER-POL equation. The results are compared to results from a classical time integration based method using a continuous GRAM-SCHMIDT orthonormalization.

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Leeghim, Henzeh, Dong-Hyun Cho, Su-Jang Jo, and Donghoon Kim. "Generalized Guidance Scheme for Low-Thrust Orbit Transfer." Mathematical Problems in Engineering 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/407087.

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The authors present an orbital guidance scheme for the satellite with an electrical propulsion system using a Lyapunov feedback control. The construction of a Lyapunov candidate is based on orbital elements, which consist of angular momentum and eccentricity vectors. This approach performs orbit transfers between any two arbitrary elliptic or circular orbits without any singularity issues. These orbital elements uniquely describe a non degenerate Keplerian orbit. The authors improve the reliability of the existing Lyapunov orbital guidance scheme by considering the energy term. Additional improvement is achieved by adding the penalty function. Furthermore, it is shown that the final suggested approach is suitable for the satellite passing the earth’s shadow area.

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Choi, Hyungjin, Umesh Vaidya, and Yongxin Chen. "A Convex Data-Driven Approach for Nonlinear Control Synthesis." Mathematics 9, no. 19 (October 1, 2021): 2445. http://dx.doi.org/10.3390/math9192445.

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We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.

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LAM, H. K., and F. H. F. LEUNG. "SYNCHRONIZATION OF UNCERTAIN CHAOTIC SYSTEMS BASED ON THE FUZZY-MODEL-BASED APPROACH." International Journal of Bifurcation and Chaos 16, no. 05 (May 2006): 1435–44. http://dx.doi.org/10.1142/s0218127406015404.

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This paper investigates the synchronization of chaotic systems subject to parameter uncertainties. Based on the fuzzy-model-based approach, a switching controller will be proposed to deal with the synchronization problem. The stability conditions will be derived based on the Lyapunov approach. The tracking performance and parameter design of the proposed switching controller will be formulated as a generalized eigenvalue minimization problem which can be solved numerically using some convex programming techniques. Simulation examples will be given to show the effectiveness of the proposed approach.

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Amato, F., M. Ariola, F. Calabrese, and R. Ambrosino. "Finite-time stability of linear systems: an approach based on polyhedral Lyapunov functions." IET Control Theory & Applications 4, no. 9 (September 1, 2010): 1767–74. http://dx.doi.org/10.1049/iet-cta.2009.0182.

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Rashidi, Mehran, and Ebrahim Farjah. "Lyapunov exponent-based optimal PMU placement approach with application to transient stability assessment." IET Science, Measurement & Technology 10, no. 5 (August 1, 2016): 492–97. http://dx.doi.org/10.1049/iet-smt.2015.0232.

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Caccia, M., A. Carta Colombo, G. Casalin, M. Decia, and G. Veruggio. "Closed-Loop Approach Algorithm Based on Lyapunov Techniques for an Autonomous Underwater Vehicle." IFAC Proceedings Volumes 28, no. 2 (May 1995): 101–6. http://dx.doi.org/10.1016/s1474-6670(17)51658-4.

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