• East Asian mathematical journal
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• v.36 no.1
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• pp.25-34
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• 2020

The classical limit theorems like strong law of large numbers, central limit theorems and law of iterated logarithms are fundamental theories in probability and statistics. These limit theorems are proved under additivity of probabilities and expectations. In this paper, we investigate strong law of large numbers under sub-linear expectation which generalize the classical ones. We give strong law of large numbers under sub-linear expectation with respect to the partial sums and some conditions similar to Petrov's. It is an extension of the classical Chung type strong law of large numbers of Jardas et al.'s result. As an application, we obtain Chung's strong law of large number and Marcinkiewicz's strong law of large number for independent and identically distributed random variables under the sub-linear expectation. Here the sub-linear expectation and its related capacity are not additive.

• Journal of applied mathematics & informatics
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• v.29 no.1_2
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• pp.201-210
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• 2011

In this paper, we obtain the H$\`{a}$jeck-R$\`{e}$nyi type inequality and the strong law of large numbers for asymptotically linear negative quadrant dependent random variables by using this inequality. We also give the strong law of large numbers for the linear process under asymptotically linear negative quadrant dependence assumption.

• Coupled systems mechanics
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• v.12 no.4
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• pp.297-313
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• 2023

This paper deals with delamination analysis of non-linear viscoelastic multilayered beam subjected to bending in the plane of the layers. For this purpose, first, a non-linear viscoelastic model is presented. In order to take into account the non-linear viscoelastic behaviour, a non-linear spring and a non-linear dashpot are assembled in series with a linear spring connected in parallel to a linear dashpot. The behaviours of the non-linear spring and dashpot are described by applying non-linear stress-strain and stress-rate of strain relationships, respectively. The constitutive law of the model is derived. Due to the non-linear spring and dashpot, the constitutive law is non-linear. This law is used for describing the time-dependent mechanical behaviour of the beam under consideration. The material properties involved in the constitutive law vary along the beam length due to the continuous material inhomogeneity of the layers. Solution of the strain energy release rate for the delamination is obtained by analyzing the balance of the energy with considering of the non-linear viscoelastic behaviour. The strain energy release rate is found also by using the complementary strain energy for verification. A parametric study is carried-out by using the solution obtained. The solutions derived and the results obtained help to understand the time-dependent delamination of non-linear viscoelastic beams under loading in the plane of layers.

• Communications of the Korean Mathematical Society
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• v.20 no.1
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• pp.161-168
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• 2005

In this paper, we obtain strong law of large numbers for linear process generated by asymptotically linear negative quadrant dependence processes.

• Journal of the Ergonomics Society of Korea
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• v.31 no.5
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• pp.647-655
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• 2012

Objective: The purpose of this study was to confirm difference between angular foot movement time and existing foot Fitts' law predicting times, and to develop the angular foot Fitts' law in the foot tapping task. Background: Existing studies of foot Fitts' law focused on horizontal movement to predict the movement time. However, when driving a car, humans move their foot from the accelerator to the brake with a fixed heel. Therefore, we examined the experiment to measure angular foot movement time in reciprocal foot tapping task and compared to conventional foot Fitts' law predicting time. And, we developed the angular foot Fitts' law. Method: In this study, we compared the angular foot movement time in foot tapping task and the predicted time of four conventional linear foot Fitts' law models - Drury's foot Fitts' law, Drury's ballistic, Hoffmann's ballistic, Hoffmann's visually-controlled. 11 subjects participated in this experiment to get a movement time and three target degrees of 20, 40, and 60 were used. And, conventional models were calculated for the prediction time. To analyze the movement time, linear and arc distance between targets were used for variables of model. Finally, the angular foot Fitts' law was developed from experimental data. Results: The average movement times for each experiment were 412.2ms, 474.9ms, and 526.6ms for the 89mm, 172mm, and 253mm linear distance conditions. The results also showed significant differences in performance time between different angle level. However, all of conventional linear foot Fitts' laws ranged 135.6ms to 401.2ms. On the other hand, the angular foot Fitts' law predicted the angular movement time well. Conclusion: Conventional linear foot Fitts' laws were underestimated and have a limitation to predict the foot movement time in the real task related angular foot movement. Application: This study is useful when considering the human behavior of angular foot movement such as driving or foot input device.

• Honam Mathematical Journal
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• v.32 no.1
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• pp.91-99
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• 2010

In this paper we introduce the concept of asymptotically almost negatively associated random variables in a Hilbert space and obtain the strong law of large numbers for a strictly stationary asymptotically almost negatively associated sequence of H-valued random variables with zero means and finite second moments. As an application we prove a strong law of large numbers for a linear process generated by asymptotically almost negatively random variables in a Hilbert space with this result.

• Journal of Institute of Control, Robotics and Systems
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• v.13 no.1
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• pp.15-18
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• 2007

We consider the problem of designing a static output feedback sliding mode control law for linear dynamical systems with mismatched uncertainties in the state matrix. We assume that an output dependent sliding surface guaranteeing the quadratic stability of the sliding mode dynamics is given, the reachability condition is not required to be satisfied globally, and the output feedback sliding mode control law complises both linear and discontinuous parts. We reduce the problem of designing the linear part of the sliding mode control law to a simple LMI problem which offers design flexibility for combining various useful convex design specifications. Our approach does not require state transformation and it can be applied to mismatched uncertain systems.

• International Journal of Ocean System Engineering
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• v.1 no.3
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• pp.120-135
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• 2011

This paper proposes a model based trajectory tracking control scheme for under-actuated underwater robotic vehicles. The difficulty in stabilizing a non-linear system using smooth static state feedback law means that the design of a feedback controller for an under-actuated system is somewhat challenging. A necessary condition for the asymptotic stability of an under-actuated vehicle about a single equilibrium is that its gravitational field has nonzero elements corresponding to non-actuated dynamics. To overcome this condition, we propose a continuous time-varying control law based on the direct estimation of vehicle dynamic variables such as inertia, damping and Coriolis & centripetal terms. This can work satisfactorily under commonly encountered uncertainties such as an ocean current and parameter variations. The proposed control law cancels the non-linearities in the vehicle dynamics by introducing non-linear elements in the input side. Knowledge of the bounds on uncertain terms is not required and it is conceptually simple and easy to implement. The controller parameter values are designed using the Taguchi robust design approach and the control law is verified analytically to be robust under uncertainties, including external disturbances and current. A comparison of the controller performance with that of a linear proportional-integral-derivative (PID) controller and sliding mode controller are also provided.

• Honam Mathematical Journal
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• v.33 no.4
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• pp.591-602
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• 2011

In this paper we establish the central limit theorem and the strong law of large numbers for linear random fields generated by identically distributed linear negative quadrant dependent random variables on $\mathbb{Z}^2$.

• Proceedings of the KIEE Conference
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• 2002.11c
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• pp.54-57
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• 2002

In this paper model reference adaptive control (MRAC) of linear time-varying(LTV) systems is considered. MRAC for a linear time invariant(LTI) system does not assure the boundedness of the output and parameter estimation errors in the presence of time variations of the parameters. However, changing the adaptive laws such as use of $\sigma$-modification can result in the boundedness of the output and parameter estimation errors[5]. Together with the $\sigma$-modification in the adaptive law, we also modify the control law by adding an additional term to the standard control law. The additional term leads to smaller bounds of the output and parameter estimation errors when compared to the case where only the standard control law is applied.