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Journal articles on the topic 'Linear quadratic theory'

Author: Grafiati

Published: 6 September 2023

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1

Bakushev, S. V. "LINEAR THEORY OF ELASTICITY WITH QUADRATIC SUMMAND." STRUCTURAL MECHANICS AND ANALYSIS OF CONSTRUCTIONS 303, no. 4 (February 28, 2022): 29–36. http://dx.doi.org/10.37538/0039-2383.2022.1.29.36.

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We suggest a linear theory version based on Taylor decompositions for stresses and power-series for quadratic summand deformations. Thus, static equations of equilibrium in stresses are written in the form of the second-order partial derivatives differential equations. The resolving equations of equilibrium in displacements are represented in the form of the third order partial derivatives differential equations. The physical equations in this version of the linear theory of elasticity are written in the same way as in the classical linear theory of elasticity. Equilibrium equations, along with other parameters – physical constants of the medium – contain minor parameters dx, dy, dz, the value of which, as shown by numerical modelling, has little effect on the nature of the stress-strain state. It is suggested to use experimental data to determine them. Along with the formulating of the basic equations of the three-dimensional theory of elasticity, particular cases of the stress-strain state of elastic continuous medium are considered: uniaxial stressed state; uniaxial deformed state; flat deformation; generalized plane stress state. Determination of the stressed and deformed state of a thin elastic bar by integrating the resolving equations in stresses and displacements is considered as examples. The suggested version of the linear theory of elasticity, due to the quadratic summand in Taylor decompositions for stresses and in power-series for deformations, expands the classical linear theory of elasticity and, with an appropriate experimental justification, can lead to new qualitative effects in the calculation of elastic deformable bodies.

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2

Moir, T. J., and J. F. Barrett. "Wiener theory of digital linear-quadratic control." International Journal of Control 49, no. 6 (June 1989): 2123–55. http://dx.doi.org/10.1080/00207178908559766.

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3

Alford, R., and E. Lee. "Sampled data hereditary systems: Linear quadratic theory." IEEE Transactions on Automatic Control 31, no. 1 (January 1986): 60–65. http://dx.doi.org/10.1109/tac.1986.1104106.

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4

van den Broek, W. A., J. C. Engwerda, and J. M. Schumacher. "An equivalence result in linear-quadratic theory." Automatica 39, no. 2 (February 2003): 355–59. http://dx.doi.org/10.1016/s0005-1098(02)00228-5.

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5

Răsvan, Vladimir. "Linear quadratic problems (On “linear” approaches in nonlinear system theory)." Journal of Physics: Conference Series 1864, no. 1 (May 1, 2021): 012003. http://dx.doi.org/10.1088/1742-6596/1864/1/012003.

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6

Lapierre, Helene, and Germain Ostiguy. "Structural model verification with linear quadratic optimization theory." AIAA Journal 28, no. 8 (August 1990): 1497–503. http://dx.doi.org/10.2514/3.25244.

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7

Rasina, Irina Viktorovna, and Oles Vladimirovich Fesko. "Approximate optimal control synthesis for nonuniform discrete systems with linear-quadratic state." Program Systems: Theory and Applications 10, no. 2 (2019): 67–77. http://dx.doi.org/10.25209/2079-3316-2019-10-2-67-77.

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Nonuniform discrete systems linear-quadratic over its state are the subject of intense study in optimal control theory. This work presents an approximate optimal control synthesis method in this class based on Krotov’s sufficient optimality conditions and illustrates it with a simple example.

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8

Horwitz, Noam. "Linear resolutions of quadratic monomial ideals." Journal of Algebra 318, no. 2 (December 2007): 981–1001. http://dx.doi.org/10.1016/j.jalgebra.2007.06.006.

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9

CIARLET, PHILIPPE G., and LILIANA GRATIE. "A NEW APPROACH TO LINEAR SHELL THEORY." Mathematical Models and Methods in Applied Sciences 15, no. 08 (August 2005): 1181–202. http://dx.doi.org/10.1142/s0218202505000704.

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We propose a new approach to the existence theory for quadratic minimization problems that arise in linear shell theory. The novelty consists in considering the linearized change of metric and change of curvature tensors as the new unknowns, instead of the displacement vector field as is customary. Such an approach naturally yields a constrained minimization problem, the constraints being ad hoc compatibility relations that these new unknowns must satisfy in order that they indeed correspond to a displacement vector field. Our major objective is thus to specify and justify such compatibility relations in appropriate function spaces. Interestingly, this result provides as a corollary a new proof of Korn's inequality on a surface. While the classical proof of this fundamental inequality essentially relies on a basic lemma of J. L. Lions, the keystone in the proposed approach is instead an appropriate weak version of a classical theorem of Poincaré. The existence of a solution to the above constrained minimization problem is then established, also providing as a simple corollary a new existence proof for the original quadratic minimization problem.

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10

AHMED, N. U., and P. LI. "Quadratic Regulator Theory and Linear Filtering Under System Constraints." IMA Journal of Mathematical Control and Information 8, no. 1 (1991): 93–107. http://dx.doi.org/10.1093/imamci/8.1.93.

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11

Kohlmann, Michael, and Shanjian Tang. "Minimization of Risk and Linear Quadratic Optimal Control Theory." SIAM Journal on Control and Optimization 42, no. 3 (January 2003): 1118–42. http://dx.doi.org/10.1137/s0363012900372465.

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12

Rotea, Mario A., and Jacinto L. Marchetti. "Internal model control using the linear quadratic regulator theory." Industrial & Engineering Chemistry Research 26, no. 3 (March 1987): 577–81. http://dx.doi.org/10.1021/ie00063a026.

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13

Jonckheere, Edmond A., Jyh-Ching Juang, and Leonard M. Silverman. "Spectral theory of the linear-quadratic and H∞ problems." Linear Algebra and its Applications 122-124 (September 1989): 273–300. http://dx.doi.org/10.1016/0024-3795(89)90656-3.

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14

Likaj, Rame, and Ahmet Shala. "Optimisation and Control of Vehicle Suspension Using Linear Quadratic Gaussian Control." Strojnícky casopis – Journal of Mechanical Engineering 68, no. 1 (April 1, 2018): 61–68. http://dx.doi.org/10.2478/scjme-2018-0006.

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Abstract The paper deals with the optimal design and analysis of quarter car vehicle suspension system based on the theory of linear optimal control because Linear Quadratic Gaussian (LQG) offers the possibility to emphasize quantifiable issues like ride comfort or road holding very easily by altering the weighting factor of a quadratic criterion. The theory used assumes that the plant (vehicle model + road unevenness model) is excited by white noise with Gaussian distribution. The term quadratic is related to a quadratic goal function. The goal function is chosen to provide the possibility to emphasize three main objectives of vehicle suspensions; ride comfort, suspension travel and road holding. Minimization of this quadratic goal function results in a law of feedback control. For optimal designs are used the optimal parameters which have been derived by comparison of two optimisation algorithms: Sequential Quadratic Program (SQP) and Genetic Algorithms (GA's), for a five chosen design parameters. LQG control is considered to control active suspension for the optimal parameters derived by GA's, while the main focus is to minimise the vertical vehicle body acceleration

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15

Wu, Shuo Mei, Jian Wei Song, and Wen Qing Zhang. "Optimal Control Theory Research on Inverted Pendulum System." Applied Mechanics and Materials 494-495 (February 2014): 1118–21. http://dx.doi.org/10.4028/www.scientific.net/amm.494-495.1118.

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The state space expression can be deduced by establishing the mathematical model of inverted pendulum system. In this paper, linear quadratic regulator (LQR) is used to control the inverted pendulum system, providing better balance between system robustness stability and rapidity. The simulation structure shows that the better the system anti-interference capability is, the shorter its recovery time is. Good control effect can be achieved by applying linear quadratic optimal control in the control of double inverted pendulum balancing system.

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16

Bibby, Christin, and Justin Hilburn. "Quadratic-linear duality and rational homotopy theory of chordal arrangements." Algebraic & Geometric Topology 16, no. 5 (November 7, 2016): 2637–61. http://dx.doi.org/10.2140/agt.2016.16.2637.

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17

Gibson, J. S., and A. Adamian. "Approximation Theory for Linear-Quadratic-Gaussian Control of Flexible Structures." SIAM Journal on Control and Optimization 29, no. 1 (January 1991): 1–37. http://dx.doi.org/10.1137/0329001.

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18

Balogun, O. S., M. Hubbard, and J. J. DeVries. "Automatic Control of Canal Flow Using Linear Quadratic Regulator Theory." Journal of Hydraulic Engineering 114, no. 1 (January 1988): 75–102. http://dx.doi.org/10.1061/(asce)0733-9429(1988)114:1(75).

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19

Bhansali, R. J., L. Giraitis, and P. S. Kokoszka. "Approximations and limit theory for quadratic forms of linear processes." Stochastic Processes and their Applications 117, no. 1 (January 2007): 71–95. http://dx.doi.org/10.1016/j.spa.2006.05.015.

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20

Bittanti, Sergio, and Giuseppe de Nicolao. "The autonomous linear quadratic control problem: Theory and numerical solutions." Automatica 30, no. 3 (March 1994): 555–56. http://dx.doi.org/10.1016/0005-1098(94)90138-4.

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21

LEE, B. K., B. S. CHEN, and Y. P. LIN. "Extension of linear quadratic optimal control theory for mixed backgrounds." International Journal of Control 54, no. 4 (October 1991): 943–72. http://dx.doi.org/10.1080/00207179108934194.

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22

Napp, D., and H. L. Trentelman. "Linear-quadratic control and quadratic differential forms for multidimensional behaviors." Linear Algebra and its Applications 434, no. 1 (January 2011): 117–30. http://dx.doi.org/10.1016/j.laa.2010.08.031.

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23

Zhou, Xue-Gang, Bing-Yuan Cao, and Seyed Hadi Nasseri. "Optimality Conditions for Fuzzy Number Quadratic Programming with Fuzzy Coefficients." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/489893.

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The purpose of the present paper is to investigate optimality conditions and duality theory in fuzzy number quadratic programming (FNQP) in which the objective function is fuzzy quadratic function with fuzzy number coefficients and the constraint set is fuzzy linear functions with fuzzy number coefficients. Firstly, the equivalent quadratic programming of FNQP is presented by utilizing a linear ranking function and the dual of fuzzy number quadratic programming primal problems is introduced. Secondly, we present optimality conditions for fuzzy number quadratic programming. We then prove several duality results for fuzzy number quadratic programming problems with fuzzy coefficients.

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24

Jin, Jun, Xiaohong Wang, Lamin Zhan, and Hongping Hu. "Strong quadratic acousto-optic coupling in 1D multilayer phoxonic crystal cavity." Nanotechnology Reviews 10, no. 1 (January 1, 2021): 443–52. http://dx.doi.org/10.1515/ntrev-2021-0034.

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Abstract Four methods are applied to calculate the acousto-optic (AO) coupling in one-dimensional (1D) phoxonic crystal (PXC) cavity: transfer matrix method (TMM), finite element method (FEM), perturbation theory, and Born approximation. Two types of mechanisms, the photoelastic effect (PE) and the moving interface effect (MI), are investigated. Whether the AO coupling belongs to linear or quadratic, the results obtained by the perturbation theory are in good agreement with the numerical results. We show that the combination method of FEM and perturbation theory has some advantages over Born approximation. The dependence of linear and quadratic couplings on the symmetry of acoustic and optical modes has been discussed in detail. The linear coupling will vanish if the defect acoustic mode is even symmetry, but the quadratic effect may be enhanced. Based on second-order perturbation theory, the contribution of each optical eigenfrequency to quadratic coupling is clarified. Finally, the quadratic coupling is greatly enhanced by tuning the thickness of the defect layer, which is an order of magnitude larger than that of normal defect thickness. The enhancement mechanism of quadratic coupling is illustrated. The symmetry of the acoustic defect mode is transformed from odd to even, and two optical defect modes are modulated to be quasi-degenerated modes. This study opens up a possibility to achieve tunable phoxonic crystals on the basis of nonlinear AO effects.

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25

Mack, Thomas. "A proof of quadratic reciprocity via linear recurrences." Acta Arithmetica 199, no. 4 (2021): 433–40. http://dx.doi.org/10.4064/aa210213-21-3.

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26

Fukshansky, Lenny. "Small zeros of quadratic forms with linear conditions." Journal of Number Theory 108, no. 1 (September 2004): 29–43. http://dx.doi.org/10.1016/j.jnt.2004.05.001.

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27

Johnson, Ch R., J. M. Pena, and T. Szulc. "Rank of Linear and Quadratic Combinations of Matrices." Electronic Journal of Linear Algebra 36, no. 36 (April 1, 2020): 169–76. http://dx.doi.org/10.13001/ela.2020.4949.

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In this paper, the rank of some combinations of matrices is analysed. In particular, the rank of all matrices on the line joining two rank $1$ matrices is characterized, and the rank of convex combinations of two matrices and quadratic combinations of three matrices is studied. Presented results concern the problem of robustness of rank under certain kinds of perturbations of a matrix.

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28

Zhang, Si Qi, Tian Xia Zhang, and Shu Wen Zhou. "Vehicle Dynamics Control Based on Linear Quadratic Regulator." Applied Mechanics and Materials 16-19 (October 2009): 876–80. http://dx.doi.org/10.4028/www.scientific.net/amm.16-19.876.

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The paper presents a vehicle dynamics control strategy devoted to prevent vehicles from spinning and drifting out. With vehicle dynamics control system, counter braking are applied at individual wheels as needed to generate an additional yaw moment until steering control and vehicle stability were regained. The Linear Quadratic Regulator (LQR) theory was designed to produce demanded yaw moment according to the error between the measured yaw rate and desired yaw rate. The results indicate the proposed system can significantly improve vehicle stability for active safety.

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29

Liu, Zixin, Yuanan Liu, and Lianglin Xiong. "Robust Linear Neural Network for Constrained Quadratic Optimization." Discrete Dynamics in Nature and Society 2017 (2017): 1–10. http://dx.doi.org/10.1155/2017/5073640.

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Based on the feature of projection operator under box constraint, by using convex analysis method, this paper proposed three robust linear systems to solve a class of quadratic optimization problems. Utilizing linear matrix inequality (LMI) technique, eigenvalue perturbation theory, Lyapunov-Razumikhin method, and LaSalle’s invariance principle, some stable criteria for the related models are also established. Compared with previous criteria derived in the literature cited herein, the stable criteria established in this paper are less conservative and more practicable. Finally, a numerical simulation example and an application example in compressed sensing problem are also given to illustrate the validity of the criteria established in this paper.

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30

Zhao, Yanjuan, and Yuanguo Zhu. "Fuzzy optimal control of linear quadratic models." Computers & Mathematics with Applications 60, no. 1 (July 2010): 67–73. http://dx.doi.org/10.1016/j.camwa.2010.04.030.

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31

Meyer, D., and G. Franklin. "A connection between normalized coprime factorizations and linear quadratic regulator theory." IEEE Transactions on Automatic Control 32, no. 3 (March 1987): 227–28. http://dx.doi.org/10.1109/tac.1987.1104569.

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32

Caravani, P. "On Extending Linear Quadratic Control Theory to Non-Symmetric Risky Objectives." IFAC Proceedings Volumes 19, no. 10 (June 1986): 367–72. http://dx.doi.org/10.1016/s1474-6670(17)59694-9.

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33

Yoshida, Taketoshi, and Kenneth A. Loparo. "Quadratic regulatory theory for analytic non-linear systems with additive controls." Automatica 25, no. 4 (July 1989): 531–44. http://dx.doi.org/10.1016/0005-1098(89)90096-4.

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34

Caravani, P. "On extending linear quadratic control theory to non-symmetric risky objectives." Journal of Economic Dynamics and Control 10, no. 1-2 (June 1986): 83–88. http://dx.doi.org/10.1016/0165-1889(86)90022-9.

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35

Srivastava, Hari M., Waseem Z. Lone, Firdous A. Shah, and Ahmed I. Zayed. "Discrete Quadratic-Phase Fourier Transform: Theory and Convolution Structures." Entropy 24, no. 10 (September 23, 2022): 1340. http://dx.doi.org/10.3390/e24101340.

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The discrete Fourier transform is considered as one of the most powerful tools in digital signal processing, which enable us to find the spectrum of finite-duration signals. In this article, we introduce the notion of discrete quadratic-phase Fourier transform, which encompasses a wider class of discrete Fourier transforms, including classical discrete Fourier transform, discrete fractional Fourier transform, discrete linear canonical transform, discrete Fresnal transform, and so on. To begin with, we examine the fundamental aspects of the discrete quadratic-phase Fourier transform, including the formulation of Parseval’s and reconstruction formulae. To extend the scope of the present study, we establish weighted and non-weighted convolution and correlation structures associated with the discrete quadratic-phase Fourier transform.

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36

Zhao, Huihong, and Chenghui Zhang. "Finite-TimeH∞Filtering for Linear Continuous Time-Varying Systems with Uncertain Observations." Journal of Applied Mathematics 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/710904.

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This paper is concerned with the finite-timeH∞filtering problem for linear continuous time-varying systems with uncertain observations andℒ2-norm bounded noise. The design of finite-timeH∞filter is equivalent to the problem that a certain indefinite quadratic form has a minimum and the filter is such that the minimum is positive. The quadratic form is related to a Krein state-space model according to the Krein space linear estimation theory. By using the projection theory in Krein space, the finite-timeH∞filtering problem is solved. A numerical example is given to illustrate the performance of theH∞filter.

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37

Aliane, Mohamed, Mohand Bentobache, Nacima Moussouni, and Philippe Marthon. "Direct method to solve linear-quadratic optimal control problems." Numerical Algebra, Control & Optimization 11, no. 4 (2021): 645. http://dx.doi.org/10.3934/naco.2021002.

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<p style='text-indent:20px;'>In this work, we have proposed a new approach for solving the linear-quadratic optimal control problem, where the quality criterion is a quadratic function, which can be convex or non-convex. In this approach, we transform the continuous optimal control problem into a quadratic optimization problem using the Cauchy discretization technique, then we solve it with the active-set method. In order to study the efficiency and the accuracy of the proposed approach, we developed an implementation with MATLAB, and we performed numerical experiments on several convex and non-convex linear-quadratic optimal control problems. The obtained simulation results show that our method is more accurate and more efficient than the method using the classical Euler discretization technique. Furthermore, it was shown that our method fastly converges to the optimal control of the continuous problem found analytically using the Pontryagin's maximum principle.</p>

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38

Sargent, Oliver. "Density of values of linear maps on quadratic surfaces." Journal of Number Theory 143 (October 2014): 363–84. http://dx.doi.org/10.1016/j.jnt.2014.04.020.

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39

Uç, M., H. Özdemir, and A. Y. Özban. "On the quadraticity of linear combinations of quadratic matrices." Linear and Multilinear Algebra 63, no. 6 (July 3, 2014): 1125–37. http://dx.doi.org/10.1080/03081087.2014.922967.

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40

Todd, Michael J. "Linear and quadratic programming in oriented matroids." Journal of Combinatorial Theory, Series B 39, no. 2 (October 1985): 105–33. http://dx.doi.org/10.1016/0095-8956(85)90042-5.

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41

Khanbaghi, Maryam, and Aleksandar Zecevic. "Jump Linear Quadratic Control for Microgrids with Commercial Loads." Energies 13, no. 19 (September 23, 2020): 4997. http://dx.doi.org/10.3390/en13194997.

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Due to the aging power-grid infrastructure and increased usage of renewable energies, microgrids (μGrids) have emerged as a promising paradigm. It is reasonable to expect that they will become one of the fundamental building blocks of a smart grid, since effective energy transfer and coordination of μGrids could help maintain the stability and reliability of the regional large-scale power-grid. From the control perspective, one of the key objectives of μGrids is load management using local generation and storage for optimized performance. Accomplishing this task can be challenging, however, particularly in situations where local generation is unpredictable both in quality and in availability. This paper proposes to address that problem by developing a new optimal energy management scheme, which meets the requirements of supply and demand. The method that will be described in the following models μGrids as a stochastic hybrid dynamic system. Jump linear theory is used to maximize storage and renewable energy usage, and Markov chain theory is applied to model the intermittent generation of renewable energy based on real data. Although the model itself is quite general, we will focus exclusively on solar energy, and will define the performance measure accordingly. We will demonstrate that the optimal solution in this case is a state feedback law with a piecewise constant gain. Simulation results are provided to illustrate the effectiveness of such an approach.

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42

Tamsaouete, Karima, and Baha Alzalg. "An Algebraic-Based Primal–Dual Interior-Point Algorithm for Rotated Quadratic Cone Optimization." Computation 11, no. 3 (March 2, 2023): 50. http://dx.doi.org/10.3390/computation11030050.

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In rotated quadratic cone programming problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of rotated quadratic cones. In this paper, we introduce the rotated quadratic cone programming problems as a “self-made” class of optimization problems. Based on our own Euclidean Jordan algebra, we present a glimpse of the duality theory associated with these problems and develop a special-purpose primal–dual interior-point algorithm for solving them. The efficiency of the proposed algorithm is shown by providing some numerical examples.

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43

G. Dani, Shrikrishna. "Simultaneous diophantine approximation with quadratic and linear forms." Journal of Modern Dynamics 2, no. 1 (2008): 129–38. http://dx.doi.org/10.3934/jmd.2008.2.129.

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44

SINGH, PARAMPREET, and NARESH DADHICH. "FIELD THEORIES FROM THE RELATIVISTIC LAW OF MOTION." Modern Physics Letters A 16, no. 02 (January 20, 2001): 83–90. http://dx.doi.org/10.1142/s0217732301002900.

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From the relativistic law of motion we attempt to deduce the field theories corresponding to the force law being linear and quadratic in four-velocity of the particle. The linear law leads to the vector gauge theory which could be the Abelian Maxwell electrodynamics or the non-Abelian Yang–Mills theory. On the other hand, the quadratic law demands space–time metric as its potential which is equivalent to demanding the principle of equivalence. It leads to the tensor theory of gravitational field — general relativity. It is remarkable that a purely dynamical property of the force law leads uniquely to the corresponding field theories.

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45

Astapenko, Valery, Andrei Letunov, and Valery Lisitsa. "From the Vector to Scalar Perturbations Addition in the Stark Broadening Theory of Spectral Lines." Universe 7, no. 6 (June 2, 2021): 176. http://dx.doi.org/10.3390/universe7060176.

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The effect of plasma Coulomb microfied dynamics on spectral line shapes is under consideration. The analytical solution of the problem is unachievable with famous Chandrasekhar–Von-Neumann results up to the present time. The alternative methods are connected with modeling of a real ion Coulomb field dynamics by approximate models. One of the most accurate theories of ions dynamics effect on line shapes in plasmas is the Frequency Fluctuation Model (FFM) tested by the comparison with plasma microfield numerical simulations. The goal of the present paper is to make a detailed comparison of the FFM results with analytical ones for the linear and quadratic Stark effects in different limiting cases. The main problem is connected with perturbation additions laws known to be vector for small particle velocities (static line shapes) and scalar for large velocities (the impact limit). The general solutions for line shapes known in the frame of scalar perturbation additions are used to test the FFM procedure. The difference between “scalar” and “vector” models is demonstrated both for linear and quadratic Stark effects. It is shown that correct transition from static to impact limits for linear Stark-effect needs in account of the dependence of electric field jumping frequency in FFM on the field strengths. However, the constant jumping frequency is quite satisfactory for description of the quadratic Stark-effect. The detailed numerical comparison for spectral line shapes in the frame of both scalar and vector perturbation additions with and without jumping frequency field dependence for the linear and quadratic Stark effects is presented.

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46

Fabien, B. C., R. W. Longman, and F. Freudenstein. "The Design of High-Speed Dwell-Rise-Dwell Cams Using Linear Quadratic Optimal Control Theory." Journal of Mechanical Design 116, no. 3 (September 1, 1994): 867–74. http://dx.doi.org/10.1115/1.2919462.

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This paper uses linear quadratic optimal control theory to design high-speed Dwell-Rise-Dwell (D-R-D) cams. Three approaches to D-R-D cam design are compared. In the first approach the cam is designed to be optimal at a fixed operating speed, i.e., a tuned cam design is obtained. In the second approach the cam profile is determined by minimizing a sum of quadratic cost functions over a range of discrete speeds, thus producing a cam-follower system which is optimal over a range of speeds. The third technique uses trajectory sensitivity minimization to design a cam which is insensitive to speed variations. All design methods are formulated as linear quadratic optimal control problems and solved using an efficient numerical procedure. It is shown that the design techniques developed can lead to cams that have significantly lower peak contact stress, contact force and energy loss when compared to a polydyne cam design. Furthermore, the trajectory sensitivity minimization approach is shown to yield cams that have lower residual vibration, over a range of speeds, when compared to a polydyne cam design.

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47

Chen, Yuefen, and Bo Li. "Multi-Dimension Uncertain Linear Quadratic Optimal Control with Cross Term." Journal of Advanced Computational Intelligence and Intelligent Informatics 19, no. 5 (September 20, 2015): 670–75. http://dx.doi.org/10.20965/jaciii.2015.p0670.

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In this paper, we consider a multi-dimension uncertain linear quadratic (LQ) optimal control with cross term. With the aid of the equation of optimality of a general multi-dimension uncertain optimal control, we present a necessary and sufficient condition for the existence of optimal linear feedback optimal control which is associated with a Riccati differential equation. Moreover, some properties of the solution for the Riccati differential equation are discussed. Furthermore, the uniqueness of the feedback optimal control for the uncertain linear quadratic optimal control with cross term is proved. Finally, as an application, an example is presented to illustrate the theory obtained.

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48

Fan, Zhixian, Yang Yan, Xiangyu Wang, and Haizhu Xu. "Path Tracking Control of Commercial Vehicle Considering Roll Stability Based on Fuzzy Linear Quadratic Theory." Machines 11, no. 3 (March 13, 2023): 382. http://dx.doi.org/10.3390/machines11030382.

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Commercial vehicles generally drive at a higher speed on structured expressways, and their higher center of mass leads to a lower rollover threshold and a greater rollover risk while steering. Therefore, the design of a lateral trajectory-tracking control strategy for commercial vehicles should not only consider the accuracy of trajectory tracking but also consider roll stability. Based on this control objective, a fuzzy linear quadratic controller was designed in this study to ensure rolling stability in the path-tracking control process and improve the adaptability of the strategy to the driving scenario. Firstly, a steering and braking cooperative control model based on the four-degree-of-freedom model and the multi-point preview model was established. Then, a path tracking controller considering roll stability was designed based on the linear quadratic theory. On this basis, a fuzzy linear quadratic controller was designed to realize the online optimization of cost function weights. Finally, the effectiveness of the control strategy was verified using co-simulation and hardware-in-loop experiments. The results show that the designed controller can effectively adjust the weight of path-tracking and stability according to the vehicle’s state. This effectively improves the vehicle’s control distribution problem.

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49

Xu, Ce, Yingyue Yang, and Jianwen Zhang. "Explicit evaluation of quadratic Euler sums." International Journal of Number Theory 13, no. 03 (February 9, 2017): 655–72. http://dx.doi.org/10.1142/s1793042117500336.

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Abstract:

In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.

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50

Wilson, David A. "Tutorial solution to the linear quadratic regulator problem using H2optimal control theory." International Journal of Control 49, no. 3 (March 1989): 1073–77. http://dx.doi.org/10.1080/00207178908559686.

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