Showing papers on "Riccati equation published in 2022"

New perturbed conformable Boussinesq-like equation: Soliton and other solutions

Abstract: In recent years, finding exact solutions to nonlinear differential equations has become a fascinating study topic. In the present study, using the modified Kudryashov method and the improved generalized Riccati equation mapping method in the conformable sense, we acquired the exact soliton-type solutions for the new time-fractional perturbed Boussinesq-like equation. This equation has been successfully obtained by the use of asymptotic methods on the defocusing Camassa–Holm nonlinear Schrödinger equation. We obtain the exponential, trigonometric, hyperbolic, and rational type solutions comprising dark solitons, kink solitons, bisymmetry solitons, singular solitons, combined complex solitons, and periodic solutions. These solutions are of great importance in various fields of applied sciences, coastal, and ocean engineering. Furthermore, all acquired solutions have been validated by putting them back into the original equation with the help of the Mathematica package software.

Nonlinear Differential Equations with Distributed Delay: Some New Oscillatory Solutions

Abstract: The oscillation of a class of fourth-order nonlinear damped delay differential equations with distributed deviating arguments is the subject of this research. We propose a new explanation of the fourth-order equation oscillation in terms of the oscillation of a similar well-studied second-order linear differential equation without damping. The extended Riccati transformation, integral averaging approach, and comparison principles are used to provide some additional oscillatory criteria. An example demonstrates the efficacy of the acquired criteria.

<i>H</i>∞ Pinning Control of Complex Dynamical Networks Under Dynamic Quantization Effects: A Coupled Backward Riccati Equation Approach

Abstract: In this article, a pinning control strategy is developed for the finite-horizon H∞ synchronization problem for a kind of discrete time-varying nonlinear complex dynamical network in a digital communication circumstance. For the sake of complying with the digitized data exchange, a feedback-type dynamic quantizer is introduced to reflect the transformation from the raw signals into the discrete-valued ones. Then, a quantized pinning control scheme takes place on a small fraction of the network nodes with the hope of cutting down the control expenses while achieving the expected global synchronization objective. Subsequently, by resorting to the completing-the-square technique, a sufficient condition is established to ensure the finite-horizon H∞ index of the synchronization error dynamics against both quantization errors and external noises. Moreover, a controller design algorithm is put forward via an auxiliary H2 -type criterion, and the desired controller gains are acquired in terms of two coupled backward Riccati equations. Finally, the validity of the presented results is verified via a simulation example.

Decentralized Control for Networked Control Systems With Asymmetric Information

Abstract: This article considers the decentralized control for networked control systems (NCSs) with asymmetric information. In this NCSs model, the controller 2 (C2) shares its observations and part of its historical control inputs with the controller 1 (C1), whereas C2 cannot obtain the information of C1 due to network constraints. Under the linear control strategies assumption, we present the optimal estimators for C1 and C2 respectively based on asymmetric observations. Since the information for C1 and C2 are asymmetric, the estimation error covariance (EEC) is coupled with the controller which means that the classical separation principle fails. By applying the Pontryagin’s maximum principle, we obtain a solution to the forward and backward stochastic difference equations. Based on this solution, we derive the optimal controllers to minimize a quadratic cost function. Combining the linear optimal controllers with the EEC, the controller C1 is decoupled from the EEC. It should be emphasized that the control gain is dependent on the estimation gain. What is more, the estimation gain satisfies the forward Riccati equation and the control gain satisfies the backward Riccati equation which makes the problem more challenging. We propose iterative solutions to the Riccati equations and give a suboptimal solution to the optimal decentralized control problem.

Gradient-augmented Supervised Learning of Optimal Feedback Laws Using State-Dependent Riccati Equations

Abstract: A supervised learning approach for the solution of large-scale nonlinear stabilization problems is presented. A stabilizing feedback law is trained from a dataset generated from State-dependent Riccati Equation solvers. The training phase is enriched by the use of gradient information in the loss function, which is weighted through the use of hyperparameters. High-dimensional nonlinear stabilization tests demonstrate that real-time sequential large-scale Algebraic Riccati Equation solvers can be substituted by a suitably trained feedforward neural network.

Fuzzy State-Dependent Riccati Equation (FSDRE) Control of the Reverse Osmosis Desalination System With Photovoltaic Power Supply

Abstract: The two challenges facing human life are water and energy. Reverse osmosis (RO) desalination systems are popular owning to their unique advantages. However, robust performance and power supply are the two main challenges in this desalination system. This power is used to drive an induction motor that rotates a centrifugal pump to apply the required back pressure to the RO membrane. To solve these two challenges, a complete RO system powered by a photovoltaic (PV) system was considered, and for each subsystem, a robust controller was designed based on their dynamic models. A fuzzy controller optimized by the invasive weed algorithm (IWA) was designed to track the maximum power in the photovoltaic subsystem under different environmental conditions. A fuzzy-PID controller was used to control the motor-pump subsystem. Furthermore, it is focused on designing a robust controller with the ability to compensate for large set-point changes, reject external disturbances, and cope with parametric uncertainties, such as variations in feed water salinity. Hence, state-dependent Riccati equation control (SDRE) was used to control the reverse osmosis system. The simulation results for different scenarios show that the proposed controller performs well under different operating conditions and can remove the effects of disturbances on the system.

Output Consensus of Heterogeneous Multiagent Systems: A Distributed Observer-Based Approach

Abstract: As the control tasks become complex, fulfilling such tasks cooperatively is the first choice in practice. In this article, the output consensus problem of heterogeneous multiagent systems is studied by deploying distributed observers in follower agents. Each observer in the follower only measures part of the leader’s output, which relieves the burden of a simple agent when the leader’s output is of large-scale dimensions. Then, all followers in the system work cooperatively to estimate the full state of the leader. By using parameterized Riccati equation and output regulation theory, sufficient conditions are given to design the distributed observers and the output consensus protocol. Finally, a numerical example is conducted to verify the obtained result.

A New Approach to Solving Fuzzy Quadratic Riccati Differential Equations

Abstract: Moa’ath N. Oqielat, Ahmad El-Ajou, Zeyad Al-Zhour, Tareq Eriqat, and Mohammed Al-Smadi;. International Journal of Fuzzy Logic and Intelligent Systems 2022;22:23-47. https://doi.org/10.5391/IJFIS.2022.22.1.23

Integrability, similarity reductions and solutions for a (3+1)-dimensional modified Kadomtsev–Petviashvili system with variable coefficients

Abstract: In this study, a (3+1)-dimensional modified Kadomtsev–Petviashvili system with variable-coefficients (vcmKP) arising in many fields of science like plasma physics, optics, and fluid mechanics is studied. The consistent Riccati expansion solvability technique is applied to the vcmKP system to obtain necessary integrability conditions between the variable coefficients. Moreover, many novel different types of travelling wave solutions are given. Additionally, the vcmKP system is reduced to a nonlinear third order ordinary differential equation and many other new solutions are obtained. Finally, some plots are given to illustrate how the arbitrary function choices affect the propagation of the wave solutions.

New solitary wave solutions for fractional Jaulent–Miodek hierarchy equation

Abstract: In this paper, the generalized rational function procedure was used to produce new solitary solutions to the fractional Jaulent–Miodek hierarchy equation. The equation is shortly called the Jaulent–Miodek equation, which was first derived by Jaulent and Miodek and associated with energy-dependent Schrödinger potentials. The equation is converted into a fourth-order partial differential equation using a transformation. In order, we find some solitary solutions such as soliton, rational and hyperbolic function solutions of Jaulent–Miodek hierarchy equation by the help of generalized exponential rational function procedure. Then, for some parameters, we draw three-dimensional graphics of real values of some exact solutions that we acquired by using this procedure.

Quaternion-based state-dependent differential Riccati equation for quadrotor drones: Regulation control problem in aerobatic flight

Abstract: The quaternion is a powerful and common tool to avoid singularity in rotational dynamics in three-dimensional (3D) space. Here it has been particularly used as an alternative to Euler angles and rotation matrix. The application of the quaternion is exercised in quadrotor modeling and control. It changes the dynamics and represents a singularity-free attitude model. Here for the first time (for the best knowledge of authors), the state-dependent differential Riccati equation (SDDRE) control has been implemented on the quaternion-based model of a quadcopter. The proposed control structure is capable of aerobatic flight, and the Pugachev’s Cobra maneuver is chosen to assess the capability of the quaternion-based SDDRE approach. The introduced control simulator is validated by comparison with conventional dynamics based on Euler angles, controlled using a proportional-derivative (PD) controller on a normal regulation flight. The simulator successfully performed the Cobra maneuver and also validated the proposed structure. The more precision in regulation along with lower energy consumption demonstrated the superiority of the introduced approach.

Piecewise parameterization for multifactor uncertain system and uncertain inventory-promotion optimization

Abstract: The analytic solution provides a great deal of convenience for linear quadratic (LQ) model in industrial implementation. But, it usually dependents on the solution of Riccati differential equation. In many cases, this will increase the complexity and design cost of controller because we are not able to find the analytical solution of Riccati differential equation. In addition, the dynamic systems may be disturbed by more than one uncertain factor. Here, we discuss the piecewise parameterization for multifactor uncertain system and an uncertain inventory promotion optimization problem. First, we obtain an analytic optimal control of multifactor uncertain LQ model. For simplifying the expression of optimal control, a parametric multifactor uncertain LQ model is formulated. Moreover, a control variable piecewise parameterization method is presented for obtaining the optimal control parameters. Finally, an inventory promotion optimization problem under uncertain environment is considered for demonstrating the effectiveness of the formulated multifactor uncertain model and presented method.

On several specific cases of the simple equations method (SEsM): Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method, general projective Riccati equations method, and first intergal method

Abstract: We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that the Jacobi Elliptic Function Expansion Method, F-Expansion method, Modified Simple Equation method, Trial Function Method, General Projective Riccati Equations Method and the First Integral Method are specific cases of SEsM.

Gradient-augmented Supervised Learning of Optimal Feedback Laws Using State-Dependent Riccati Equations

Abstract: A supervised learning approach for the solution of large-scale nonlinear stabilization problems is presented. A stabilizing feedback law is trained from a dataset generated from State-dependent Riccati Equation solvers. The training phase is enriched by the use of gradient information in the loss function, which is weighted through the use of hyperparameters. High-dimensional nonlinear stabilization tests demonstrate that real-time sequential large-scale Algebraic Riccati Equation solvers can be substituted by a suitably trained feedforward neural network.

Symbolic Regulator Sets for a Weakly Nonlinear Discrete Control System with a Small Step

Abstract: For a class of discrete weakly nonlinear state-dependent coefficient (SDC) control systems, a suboptimal synthesis is constructed over a finite interval with a large number of steps. A one-point matrix Padé approximation (PA) of the solution of the initial problem for the discrete matrix Riccati equation is constructed based on the state-dependent Riccati equation (SDRE) approach and the asymptotics by the small-step of the boundary layer functions method. The symmetric gain coefficients matrix for Padé control synthesis is constructed based on the one-point PA. As a result, the parametric closed-loop control is obtained. The results of numerical experiments illustrate, in particular, the improved extrapolation properties of the constructed regulator, which makes the algorithm applicable in control systems for a wider range of parameter variation.

Simple equations method (SEsM): Review and new results

Abstract: We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that specific case of SEsM can be used in order to reproduce the methodology of the Inverse Scattering Transform Method for the case of the Burgers equation and Korteweg - de Vries equation. This specific case is connected to use of a specific case of Step. 2 of SEsM: representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a ”small” parameter ε, solving the differential equations occurring from this representation by means of Fourier series and transition from the obtained solution for small values of ε to solution for arbitrary finite values of ε. Next, we discuss the application of composite functions in SEsM. We proof two propositions connected to obtaining solutions of nonlinear differential equations with polynomial nonlinearities by means of use of composite functions. We present several examples of applications of this methodology and obtain exact solutions of the generalized Korteweg - deVries equation, Olver equation, and several other equations. Next we discuss the most simple version of SEsM: the Modified Method of Simplest Equation (MMSE). We start with the role of the simplest equation and discuss the several cases of simplest equations such as nonlinear ordinary differential equations called Riccati equation and Bernoulli equation. The theory is illustrated by obtaining exact solution of various nonlinear partial differential equations such as Newel-Whitehead equation, FitzHugh-Nagumo equation, etc. MMSE is further illustrated by obtaining exact solutions of many equations such as Swift-Hohenberg equation, Rayleigh equation, Huxley equation. Special attention is given to the process of obtaining of balance equations in the MMSE. This process is illustrated by obtaining balance equations for several model nonlinear differential equations from the area of ecology and population dynamics. Among the discussed examples are the reaction-diffusion equation with density-dependent diffusion as well as the reaction-telegraph equation. Finally we obtain exact solution of two nonlinear model differential equations connected to the water wave propagation. These are the extended Korteweg-de Vries equation and the generalized Camassa-Holm equation. We close the discussion by several remark on the methodology and about the future plans connected to our research in this area.

New Solitary-Wave Solutions of the Van der Waals Normal Form for Granular Materials via New Auxiliary Equation Method

Abstract: In this paper, we use general Riccati equation to construct new solitary wave solutions of the Van der Waals normal form, which is one of the most famous models for natural and industrial granular materials. It is very important to understand the static and dynamic characteristics of this models in many application fields. We solve the general Riccati equation through different function transformation, and many new hyperbolic function solutions are obtained. Then, it is substituted into the Van der Waals normal form as an auxiliary equation. Abundant types of solitary-wave solutions are obtained by choosing different coefficient in the general Riccati equation, and some of them have not been found in other documents. The results show that the analysis method we used is very simple and effective for dealing with nonlinear models.