Solution Of Riccati Equation Research Articles

Fuzzy Form of Euler Method to Solve Fuzzy Differential Equations

Background: The Euler method is a elementary method to solve fuzzy differential equations numerically. Several authors have explored the Euler method by applying various approaches and derivative concepts. Methods: This paper proposes a fuzzy form of the Euler method to solve fuzzy initial value problems. Novelty of this approach is that the method developed based on fuzzy arithmetic. The solution by this method is readily available in the form of fuzzy-valued function. The method does not require to re-write fuzzy differential equation into a system of two crisp ordinary differential equations. Results: The algorithm of proposed method and local error expression are discussed. An illustration and solution of the fuzzy Riccati equation are provided for the applicability of the method. Conclusion: The proposed Euler method is a natural generalization of crisp Euler method. It is very efficient in solving linear and nonlinear fuzzy differential equations. One of the advantage of this method is that the solution obtained by the method is always a fuzzy-valued function due to well-defined fuzzy arithmetic.

Moment-Driven Predictive Control of Mean-Field Collective Dynamics

The synthesis of control laws for interacting agent-based dynamics and their mean-field limit is studied. A linearization-based approach is used for the computation of sub-optimal feedback laws obtained from the solution of differential matrix Riccati equations. Quantification of dynamic performance of such control laws leads to theoretical estimates on suitable linearization points of the nonlinear dynamics. Subsequently, the feedback laws are embedded into nonlinear model predictive control framework where the control is updated adaptively in time according to dynamic information on moments of linear mean-field dynamics. The performance and robustness of the proposed methodology is assessed through different numerical experiments in collective dynamics.

Closed-form solution of non-symmetric algebraic Riccati matrix equation

In this work we obtain closed-form expression for the solution of non-symmetric algebraic Riccati equation (NARE), AX+XB+XCX=D,A,B,C,D∈ℂn×n, under some conditions on the matrices A,B,C and D. We show that the solution (real and complex) of this NARE is X=(A2+DC−A)C−1, where M is square root of matrix M. Finally this formula applied on some NAREs.

An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problems

Navier-Stokes equations are well known in modeling of an incompressible Newtonian fluid, such as air or water. This system of equations is very complex due to the non-linearity term that characterizes it. After the linearization and the discretization parts, we get a descriptor system of index-2 described by a set of differential algebraic equations (DAEs). The two main parts we develop through this paper are focused firstly on constructing an efficient algorithm based on a projection technique onto an extended block Krylov subspace, that appropriately allows us to construct a reduced system of the original DAE system. Secondly, we solve a Linear Quadratic Regulator (LQR) problem based on a Riccati feedback approach. This approach uses numerical solutions of large-scale algebraic Riccati equations. To this end, we use the extended Krylov subspace method that allows us to project the initial large matrix problem onto a low order one that is solved by some direct methods. These numerical solutions are used to obtain a feedback matrix that will be used to stabilize the original system. We conclude by providing some numerical results to confirm the performance of our proposed method compared to other known methods.

Properties of solutions of quaternionic Riccati equations

In this paper we study properties of regular solutions of quaternionic Riccati equations. The obtained results we use for study of the asymptotic behavior of solutions of two first-order linear quaternionic ordinary differential equations.

Leader-Following Cluster Consensus as a Graphical Differential Game With a Nash Equilibrium Solution

This paper formulates the leader-following cluster consensus problem of a homogeneous linear time-invariant multi-agent system over a directed graph as an N-player graphical differential game. The agents utilize distributed state feedback control laws based on the available information to calculate their best response. In general, the absence of centralized information prevents the best responses of agents from constituting a Nash equilibrium solution. We show that the best responses depending on the unique solution of an algebraic Riccati equation constitute a Nash equilibrium for the proposed graphical differential game with a modified cost function. A numerical example illustrates the Nash equilibrium solution by showing that the agents can not improve their performance by altering their responses while solving the leader-following cluster consensus problem.

A fundamental solution concept for a family of state constrained regulator problems

A family of continuous-time regulator problems, subject to linear time-varying dynamics and convex state constraints, is parameterized with respect to a space of semiconcave terminal costs. By embedding the state constraints into an extended real-valued running cost, a family of relaxations is employed to develop a min-plus convolution-based fundamental solution concept for the family of regulator problems. A representation of the fundamental solution concept is provided in terms of solutions of associated differential Riccati and related equations, and possible avenues for efficient numerical computation are explored.

Kernel representation of Kalman observer and associated H-matrix based discretization

In deterministic estimation, applying a Kalman filter to a dynamical model based on partial differential equations is theoretically seducing but solving the associated Riccati equation leads to a so-called curse of dimensionality for its numerical implementation. In this work, we propose to entirely revisit the theory of Kalman filters for parabolic problems where additional regularity results proves that the Riccati equation solution belongs to the class of Hilbert-Schmidt operators. The regularity of the associated kernel then allows to proceed to the numerical analysis of the Kalman full space-time discretization in adapted norms, hence justifying the implementation of the related Kalman filter numerical algorithm with H-matrices typically developed for integral equations discretization.

Investigation of Finite-Difference Schemes for the Numerical Solution of a Fractional Nonlinear Equation

The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a nonlocal implicit finite difference scheme (IFDS) are proved. For IFDS, it is shown that the scheme converges with the order corresponding to the estimate for approximating the Gerasimov-Caputo fractional operator. The IFDS scheme is solved by the modified Newton’s method (MNM), for which it is shown that the method is locally stable and converges with the first order of accuracy. In the case of the fractional Riccati equation, approximation, stability, and convergence theorems are proved for a nonlocal explicit finite difference scheme (EFDS). It is shown that EFDS conditionally converges with the first order of accuracy. On specific test examples, the computational accuracy of numerical methods was estimated according to Runge’s rule and compared with the exact solution. It is shown that the order of computational accuracy of numerical methods tends to the theoretical order of accuracy with increasing nodes of the computational grid.

Computationally Efficient Optimal Control for Unstable Power System Models

In this article, the focus is mainly on gaining the optimal control for the unstable power system models and stabilizing them through the Riccati-based feedback stabilization process with sparsity-preserving techniques. We are to find the solution of the Continuous-time Algebraic Riccati Equations (CAREs) governed from the unstable power system models derived from the Brazilian Inter-Connected Power System (BIPS) models, which are large-scale sparse index-1 descriptor systems. We propose the projection-based Rational Krylov Subspace Method (RKSM) for the iterative computation of the solution of the CAREs. The novelties of RKSM are sparsity-preserving computations and the implementation of time-convenient adaptive shift parameters. We modify the Low-Rank Cholesky-Factor integrated Alternating Direction Implicit (LRCF-ADI) technique-based nested iterative Kleinman–Newton (KN) method to a sparse form and adjust this to solve the desired CAREs. We compare the results achieved by the Kleinman–Newton method with that of using the RKSM. The applicability and adaptability of the proposed techniques are justified numerically with MATLAB simulations. Transient behaviors of the target models are investigated for comparative analysis through the tabular and graphical approaches.

An approximate solution of fractional order Riccati equations based on controlled Picard’s method with Atangana–Baleanu fractional derivative

In this paper, a new computationally efficient approach to solve fractional differential equations with Atangana–Baleanu operator is introduced. Controlled Picard’s method is employed for solving a class of fractional differential equations with order 0<α<1. The proposed approach can cover wide range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for nonlinear differential equations. Two models of fractional Riccati equation are solved to validate and illustrate the accuracy of the new approach. Figures has been used to construct the results obtained from the presented approach. It is shown that the proposed method is efficient, credible, and easy to implement for various related problems in science and engineering.

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.

SVD-Krylov based Sparsity-preserving Techniques for Riccati-based Feedback Stabilization of Unstable Power System Models

We propose an efficient sparsity-preserving reduced-order modelling approach for index-1 descriptor systems extracted from large-scale power system models through two-sided projection techniques. The projectors are configured by utilizing Gramian based singular value decomposition (SVD) and Krylov subspace-based reduced-order modelling. The left projector is attained from the observability Gramian of the system by the low-rank alternating direction implicit (LR-ADI) technique and the right projector is attained by the iterative rational Krylov algorithm (IRKA). The classical LR-ADI technique is not suitable for solving Riccati equations and it demands high computation time for convergence. Besides, in most of the cases, reduced-order models achieved by the basic IRKA are not stable and the Riccati equations connected to them have no finite solution. Moreover, the conventional LR-ADI and IRKA approach not preserves the sparse form of the index-1 descriptor systems, which is an essential requirement for feasible simulations. To overcome those drawbacks, the fitting of LR-ADI and IRKA based projectors from left and right sides, respectively, desired reduced-order systems attained. So that, finite solution of low-rank Riccati equations, and corresponding feedback matrix can be executed. Using the mechanism of inverse projection, the Riccati-based optimal feedback matrix can be computed to stabilize the unstable power system models. The proposed approach will maintain minimized ℌ2 -norm of the error system for reduced-order models of the target models.

The Laplace transform of the integrated Volterra Wishart process

AbstractWe establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore, we link the characteristic exponents to a system of non‐standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear‐quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.

Nonlinear optimal control for multi‐DOF robotic manipulators with flexible joints

AbstractA main problem in the control of flexible‐joint robots and in the precise positioning of their end‐effector, is synchronization between the turn angles of the robots' joints and the turn angles of the motors that provide actuation to these joints. In this article, a nonlinear optimal (H‐infinity) control method is proposed for multi‐DOF robotic manipulators with flexible joints. The dynamic model of these robotic manipulators is shown to be underactuated and to satisfy differential flatness properties. The state‐space description of these robots undergoes approximate linearization around a temporary operating point which is recomputed at each iteration of the control method. The linearization procedure makes use of first‐order Taylor‐series expansion and relies on the computation of Jacobian matrices. For the approximately linearized model of these multi‐DOF robotic systems an H‐infinity feedback controller is designed. This controller stands for the solution of the robots' optimal control problem under model uncertainty and external disturbances. The computation of the controller's feedback gain requires the solution of an algebraic Riccati equation, which is performed again at each time‐step of the control algorithm. The stability properties of the control scheme are proven trough Lyapunov analysis. First, it is confirmed that the controller satisfies the H‐infinity tracking performance criterion which ascertains its robustness. Moreover, it is proven that the control loop is globally asymptotically stable. Finally, to implement sensorless control for such robotic systems the H‐infinity Kalman Filter is used as a robust state estimator. The proposed control method for multi‐DOF robotic manipulators with flexible joints retains the advantages of linear optimal control, that is fast and accurate tracking of its reference setpoints under moderate variations of the control inputs.

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.

Extended Kalman Filter Design for Semilinear Distributed Parameter Systems with Application to Anaerobic Digestion

This paper is concerned with the online state estimation of a class of one-dimensional semilinear partial differential equation (PDE) systems considering piecewise measurements over the spatial domain. In the context of infinite-dimensional linear systems, it is well-known that the Kalman filter minimizes the mean square estimation error. For semilinear infinite-dimensional systems, the extended Kalman filter (EKF) is a widely used extension relying on successive linearizations of the estimation error dynamics. In this paper, we propose a computationally tractable implementation of the EKF using a sample-and-hold approach for which the optimal output injection operator associated to the proposed estimator is computed at each sampling time via the approximate solution of the infinite-dimensional Riccati equation. The performance of the observer is exemplified through numerical experiments which demonstrate the efficiency of the proposed approach.

Riccati-Type Equations Associated with Higher Order Ordinary Differential Equations

It is shown, similarly as the solution of the Riccati equation can be given via quotients of two linear independent solutions of the second order ordinary differential equation, the solution of the Riccati-Abel differential equation is presented by three linear independent solutions of the third order ordinary differential equation. The method is extended to the class of generalized Riccati-type equations governed by the characteristic polynomials of the associated higher order ordinary differential equations. An explicit functional dependence between solutions of the higher degree Riccati equations with the linear independent solutions of the associated ordinary differential equations established.

Hereditary Riccati Equation with Fractional Derivative of Variable Order

The Riccati differential equation with a fractional derivative of variable order is considered. A derivative of variable fractional order in the original equation implies the hereditary property of the medium, i.e., the dependence of the current state of a dynamic system on its previous states. A software called Numerical Solution of a Fractional-Differential Riccati Equation (briefly NSFDRE) is created; it allows one to compute a numerical solution of the Cauchy problem for the Riccati differential equation with a derivative of variable fractional order. The numerical algorithm implemented in the software is based on the approximation of the variable-order derivative by finite differences and the subsequent solution of the corresponding nonlinear algebraic system. New distribution modes depending on the specific type of variable order of the fractional derivative were obtained. We also show that some distribution curves are specific for other hereditary dynamic systems.

Symbolic Padé representation of stabilizing regulators for a class of nonlinear control systems with a parameter

An approach for the construction of a parametric set of stabilizing controls for a class of pseudo-linear control systems with a parameter that varies on a certain interval of the positive semi-axis is described. The algorithm is based on a combination of the State-dependent Riccati equation approach (SDRE) for feedback control design, matrix Padé approximations and intellectual choice theory. Firstly, asymptotic expansions for the solutions of certain matrix algebraic state-dependent Riccati equations in the vicinity of the boundary points of the interval of the parameter variation are constructed. The obtained asymptotic expansions are matched into one symbolic construction using two-point diagonal Padé approximations and as a result a parametric set of solutions for the entire domain of parameter variation is obtained. The resulting controllers have the form of the Kalman regulators and can be used for systems with different control gains coefficients. The symbolic and parametric representation of the regulators allows to increase the efficiency of calculations and to propose algorithms for the intellectual selection of the parameters taking into account the changes in the characteristics of motion in real time.