Riccati Differential Equation Research Articles

Algorithm for Finding Feedback in a Problem with Constraints for One Class of Nonlinear Control Systems

For a continuous nonlinear control system on a finite time interval with control constraints, where the right-hand side of the dynamics equations is linear in control and linearizable in the vicinity of the zero equilibrium position, we consider the construction of a feedback according to the Kalman algorithm. For this, the solution of an auxiliary optimal control problem with a quadratic functional is used by analogy with the SDRE approach.Since this approach is used in the literature to find suboptimal synthesis in optimal control problems with a quadratic functional with formally linear systems, where all coefficient matrices in differential equations and criteria can contain state variables, then on a finite time interval it becomes necessary to solve a complicated matrix differential Riccati equations, with state-dependent coefficient matrices. This circumstance, due to the nonlinearity of the system, in comparison with the Kalman algorithm for linear-quadratic problems, significantly increases the number of calculations for obtaining the coefficients of the gain matrix in the feedback and for obtaining synthesis with a given accuracy. The proposed synthesis construction algorithm is constructed using the extension principle proposed by V. F. Krotov and developed by V. I. Gurman and allows not only to expand the scope of the SDRE approach to nonlinear control problems with control constraints in the form of closed inequalities, but also to propose a more efficient computational algorithm for finding the matrix of feedback gains in control problems on a finite interval. The article establishes the correctness of the application of the extension principle by introducing analogs of the Lagrange multipliers, depending on the state and time, and also derives a formula for the suboptimal value of the quality criterion. The presented theoretical results are illustrated by calculating suboptimal feedbacks in the problems of managing three-sector economic systems.

Optimal Control and Stabilization for Itô Systems with Input Delay

The paper considers the linear quadratic regulation (LQR) and stabilization problems for Ito stochastic systems with two input channels of which one has input delay. The underlying problem actually falls into the field of asymmetric information control because of the nonidentical measurability induced by the input delay. In contrast with single-channel single-delay problems, the challenge of the problems under study lies in the interaction between the two channels which are measurable with respect to different filtrations. The key techniques conquering such difficulty are the stochastic maximum principle and the orthogonal decomposition and reorganization technique proposed in a companion paper. The authors provide a way to solve the delayed forward backward stochastic differential equation (D-FBSDE) arising from the maximum principle. The necessary and sufficient solvability condition and the optimal controller for the LQR problem are given in terms of a new Riccati differential equation established herein. Further, the necessary and sufficient stabilization condition in the mean square sense is provided and the optimal controller is given. The idea proposed in the paper can be extended to solve related control problems for stochastic systems with multiple input channels and multiple delays.

다중 레이더 배치를 위한 폐쇄형 표적추적 성능지표 산출

This paper suggests the closed-form target tracking performance measure which is necessary for formulating a multi-radar placement problem. It is well-known that there is no analytic solution to the DRE(differential Riccati equation) of a nonlinear target tracking filter, which makes the problem intractable. As a fundamental resolution to this technical issue, the line-of-sight frame equivalents of the given nonlinear filter are exploited. For such case, unlike the existing numerical method, the analytic solutions to the corresponding DREs can be readily derived. This idea was based on the fact that the Fisher information is preserved regardless of the coordinates system in which a tracking filter is designed. The proposed approach allows us to clearly describe the tracking performance measure as a function of the radars’ positions as well as their specification. Simulations confirm that our scheme is beneficial for reducing the inherent complexity of a multi-radar placement problem.

Optimal regulators for a class of nonlinear stochastic systems

We consider a class of nonlinear stochastic systems with square-root nonlinearities appearing in the diffusion terms. The optimal control problems with indefinite quadratic criteria in both finite and infinite horizon are formulated and solved in an explicit closed form. It turns out that all optimal controls are of an affine state-feedback form, despite the fact that the system is nonlinear. We use the method of completion of squares and new types of Riccati differential and algebraic equations to find the solutions. An application to the problem of optimal investment in a market with a stochastic interest rate is given.

Adaptive boundary sine cosine optimizer with population reduction for robustness analysis of finite time horizon systems

This paper proposes an adaptive boundary sine cosine optimizer with population reduction. It is designed specifically to calculate an upper bound on the worst-case gain of a known finite horizon Linear Time Varying (LTV) system and a perturbation. The input/output behavior of the perturbation is described by a time domain Integral Quadratic Constraint (IQC). The analysis condition is formulated as a parametric Riccati differential equation which depends on the IQC representation. A nonlinear optimization problem is posed that minimizes the upper bound on the worst-case gain over the IQC parameterization. For industrial size applications like a space launcher, the number of decision variables can grow arbitrarily large depending on the number of considered perturbations as well as the type and representation of the IQC. This is aggravated by nonlinear constraints on the decision variables imposed by the Riccati differential equation making it challenging to solve. Several established Meta-Heuristics (MHs) along with the proposed algorithm are applied to an industry size worst case analysis of a space launcher during its atmospheric ascend. Their respective performances are evaluated to emphasize the advantages of the developed optimizer. This work builds the foundation of applying MHs to IQC based robustness analysis of finite horizon LTV systems.

Optimal Attack Strategy Against Fault Detectors for Linear Cyber-Physical Systems

This paper is concerned with the design of false data injection attacks (FDIAs) against fault detectors (FD) for linear cyber-physical systems (CPSs). The purpose of the attacker is to design an stealthy attack scheme such that the defenders cannot detect the faults in time or fail to detect the faults, and at the same time the robustness of the CPS is deteriorated. Unlike the existing optimal attack strategies (OASs) that are established based on sufficient conditions, necessary and sufficient conditions (NASCs) are established to maximize the degradation of the FD performance and robustness of CPS while maintaining stealth. Subsequently, an OAS is constructed by solving coupled backward recursive Riccati difference equations (RDEs). Finally, two simulation examples are employed to show the effectiveness of the designed attack scheme.

Data-Driven Virtual Inertia Control Method of Doubly Fed Wind Turbine

This paper presents a data-driven virtual inertia control method for doubly fed induction generator (DFIG)-based wind turbine to provide inertia support in the presence of frequency events. The Markov parameters of the system are first obtained by monitoring the grid frequency and system operation state. Then, a data-driven state observer is developed to evaluate the state vector of the optimal controller. Furthermore, the optimal controller of the inertia emulation system is developed through the closed solution of the differential Riccati equation. Moreover, a differential Riccati equation with self-correction capability is developed to enhance the anti-noise ability to reject noise interference in frequency measurement process. Finally, the simulation verification was performed in Matlab/Simulink to validate the effectiveness of the proposed control strategy. Simulation results showed that the proposed virtual inertia controller can adaptively tune control parameters online to provide transient inertia supports for the power grid by releasing the kinetic energy, so as to improve the robustness and anti-interference ability of the control system of the wind power system.

Nonlinear Optimal Feedback Control of the Two-Level Open Non-Markovian Stochastic Quantum System

In this paper, a nonlinear optimal feedback control based on the nonlinear state estimator for a two-level open non-Markovian stochastic quantum system is proposed. The proposed nonlinear state estimator is designed by using the state-dependent differential Riccati equation and constructed to optimally estimate the state. The estimated state is updated by the output data of continuous weak measurement of the controlled quantum system and used to design the nonlinear state feedback controller to track the output of the reference model. The output state of reference model is chosen as the desired performance. The numerical simulation results verify the achievability of the proposed state feedback control strategy, and the capability to steer the state of system from any arbitrary initial state to the final reference target state with high state transfer success rate.

Approximate-analytical solutions of some classical Riccati differential equation using the Daftardar-Gejji Jafari method

This present paper considers the approximate-analytical solution of some classical Riccati Differential Equations (RDEs). Here, an efficient numerical method referred to as Daftardar-Gejji Jafari Method (DJM) for solving the functional differential equations is applied. Three numerical examples are considered to show the accuracy of the proposed method.

Numerical investigation for Caputo-Fabrizio fractional Riccati and Bernoulli equations using iterative reproducing kernel method

The pivotal aim of present work is to design and implement an effective numerical approach, the iterative reproducing-kernel algorithm, to provide numerical solutions to the fractional Riccati and Bernoulli differential equations in the Caputo-Fabrizio sense. To achieve this, we develop an advanced operational algorithm based on creating a complete orthonormal function system utilizing the reproducing-kernel function in order to formulate the solution in the form of uniformly convergent Fourier series. Additionally, error and stability analysis of the proposed method are studied in the appropriate space W2,2(Δ). On the other hand, some meaningful numerical applications are included to demonstrate the feasibility and reliability of this approach under the qualitative effect of the Caputo-Fabrizio fractional derivative. In numerical viewpoints, the achieved results using the proposed approach indicate a high level of accuracy and rare technical skills, which qualifies it to deal with many fractional models emerging in various mathematical and physical disciplines within the frame of Caputo-Fabrizio derivatives.

Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

Analysis of spatially extended excitable Izhikevich neuron model near instability

The article focuses on the issue of a spatiotemporal excitable biophysical model that describes the propagation of electrical potential called spikes to model the diffusion-induced dynamics based on an analytical development of amplitude equations. Considering the Izhikevich neuron model consisting of coupled systems of ODEs, we demonstrate various results of spatiotemporal architecture (PDEs) using a suitable parameter regime. We analytically perform the saddle-node bifurcation and Hopf bifurcation analysis with bifurcating periodic solutions that show the transition phases in the system dynamics. We study different firing patterns both analytically and numerically by the formation of Riccati differential equation. To examine the characteristics of diffusive instabilities, we use Turing amplitude equations by multiscaling method. The instabilities and Turing bifurcation are established using theoretical analysis and numerical simulations. The spatial dynamics has potential effects on the deterministic system as a result of the diffusive matrices with various couplings, and the coupled oscillators with this nearest-neighbor coupling show synchronization measured by the synchronization factor analysis. Our results qualitatively reproduce different phenomena of the extended excitable system based with an efficient analytical scheme.

Construction of optical solitons for conformable generalized model in nonlinear media

In the current analysis, the conformable generalized Kudryashov equation of pulses propagation with power non-linearity is processed. The considered higher order equation represents a generalized mathematical model of many well-known ones in nonlinear media. A variety of multiple kinks, bi-symmetry, periodic, singular, bright and dark optical solitons are extracted via the generalized Riccati equation mapping method. Basing on the Riccati differential equation, the theoretical algorithm extracts a number of empirical solutions that do not exist in the literature. The obtained results showed that the present technique is an effective and strong tool for solving nonlinear fractional partial differential equations and produces a very large number of solutions.

Optical magnetic helicity with binormal electromagnetic vortex filament flows in MHD

In this paper, we construct new efficient insight into the understanding of the configurations of the binormal electromagneto-vortex formation and their consequences in the energy dissipation and magnetic helicity in magnetohydrodynamics (MHD). Firstly, we introduce new formulations of the magnetic helicity of the Lorentz force of the Serret–Frenet frame vectors and associated electromagnetic vector fields of the binormal electromagnetic field lines E − B. Then, we investigate the Godbillon–Vey helicity model. We also compute the time evolution of the binormal electromagnetic field lines of the Frenet–Serret vectors and associated geometric quantities in MHD. Then, we determine a set of partial differential equation systems describing the Hall effect, which establishes the time evolution relation between the binormal electromagnetic field lines and magnetic vector potential in MHD. Finally, for a better comprehension of the time evolution of the binormal electromagnetic field lines, we consider the traveling wave transformation and the Riccati differential equation system to evaluate the evolution of the magnetic helicity and energy dissipation based on the calculation of the theoretical models.

On the lack of monotonicity of Newton–Hewer updates for Riccati equations

We provide a set of counterexamples for the monotonicity of the Newton–Hewer method (Hewer, 1971) for solving the discrete-time algebraic Riccati equation in dynamic settings, drawing a contrast with the Riccati difference equation (Caines and Mayne, 1970).

Galerkin trial spaces and Davison-Maki methods for the numerical solution of differential Riccati equations

The differential Riccati equation appears in different fields of applied mathematics like control and system theory. Recently, Galerkin methods based on Krylov subspaces were developed for the autonomous differential Riccati equation. These methods overcome the prohibitively large storage requirements and computational costs of the numerical solution. Known solution formulas are reviewed and extended. Because of memory-efficient approximations, invariant subspaces for a possibly low-dimensional solution representation are identified. A Galerkin projection onto a trial space related to a low-rank approximation of the solution of the algebraic Riccati equation is proposed. The modified Davison-Maki method is used for time discretization. Known stability issues of the Davison-Maki method are discussed. Numerical experiments for large-scale autonomous differential Riccati equations and a comparison with high-order splitting schemes are presented.

A semi-analytical solutions of fractional Riccati's differential equation via singular and non-singular operators

Looking for offer a solution of the Riccati's differential equations of fractional order (FRDEs) involving Caputo derivative (CD), Caputo-Fabrizio derivative (CFD) or Atangana-Baleanu derivative (ABD) in this comparative research is based on a semi-analytical iterative approach. Temimi and Ansari introduced this method and called it TAM. The comparison of the time used in minutes is given for three derivatives CD, CFD and ABD. Meanwhile, the comparison of the approximate solutions with CD, CFD and ABD are presented. Regarding the help of the software Mathematica, all the results have been obtained and the calculations have been done.

New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

Fuzzy Homotopy Analysis Method For Solving Fuzzy Riccati Differential Equation

In this work, we have used fuzzy homotopy analysis method to find the fuzzy series solution (fuzzy semi-analytical solution) of the first order fuzzy Riccati differential equation. The fuzzy approximate-analytical solutions that we obtained during this paper are accurate solutions and very close to the fuzzy exact-analytical solutions. Some numerical results are given to illustrate the method. The obtained numerical results are compared with the exact solutions.

On the dynamics of the singularly perturbed Riccati differential equation with two different delays

On the dynamics of the singularly perturbed Riccati differential equation with two different delays