Matrix Riccati Differential Equation Research Articles

Explicit Symplectic-Precise Iteration Algorithms for Linear Quadratic Regulator and Matrix Differential Riccati Equation

Efficient, robust and precise algorithms for linear quadratic regulator (LQR) and matrix differential Riccati equation (MDRE) are essential in optimal control. However, there are lack of good algorithms for time-varying LQR problem because of the difficulty of solving the nonlinear time-varying MDRE. In this paper, we proved that the n-th order LQR problem is equivalent to n parallel 1-dim Hamiltonian systems and proposed the explicit symplectic-precise iteration method (SPIM) for solving LQR and MDRE. The explicit symplectic-precise iteration algorithms (ESPIA) designed with SPIM have three typical merits: firstly, there are no accumulative errors in the sense of long-term time which inherits from symplectic difference scheme; secondly the stiffness problem due to the inverse of matrix is avoided by the precise iteration method; and finally the algorithmic structure of ESPIA is simple and no extra assumptions are required. Systematic analysis shows that the time complexity of the symplectic algorithms for the n-th order LQR and MDRE is <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) where k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">max</sub> is the iteration times specified by the time duration. Numerical examples and simulations are provided to validate the performance of the ESPIA.

Output Control by Linear Time-Varying Systems using Parametric Identification Methods

In this paper the problem of control for time-varying linear systems by the output (i.e. without measuring the vector of state variables or derivatives of the output signal) was considered. For the control design, the well-known online procedure for solving the Riccati matrix differential equation is chosen. This procedure involves the synthesis of linear static feedbacks on state variables in the case of known parameters of the plant. If state variables are not measured, then for the observer design using the matrix Riccati differential equation, using the dual scheme, which provides for the transposition of the state matrix and the replacement of the input matrix by the output matrix. It is well known that an observer of state variables built on the basis of a solution of the Riccati matrix differential equation ensures the exponential stability of a closed loop system in the case of uniform observability. Despite the fact that this type of observer can be classified as universal, its have a number of significant drawbacks. The main problem of such observers is the need for accurate knowledge of the parameters and the requirement for uniform observability, which in practice cannot always be realized. Thus, the problem of the new methods design for constructing observers of state variables of linear non-stationary systems is still relevant. Some time ago, a number of methods for the adaptive observers design of state variables for nonlinear systems were proposed. The main idea of the synthesis of observers was based on the transformation of the original dynamic system to a linear regression model containing unknown parameters, which in turn were functions of the initial conditions of the state variables of the control object. This approach in the English language literature is called PEBO. This paper, based on the PEBO method, proposes a new approach for the observers design of state for non-stationary systems. This approach provides the possibility of obtaining monotonic convergence estimates with transient time tuning.

Backstepping Method for Stabilizing System of 2×2 Riccati Matrix Differential Equations

In this research paper, the backstepping method (BSM) will be proposed for stabilizing and solving system of 2×2 Riccati matrix ordinary differential equations. Such equations have many difficulties in the studying their solutions and stability. The basic idea behind of this approach is to use the BSM as a transformation method for transforming the original system into an equivalent one which is stabilizable and solvable based on constructing the Lyapunov function.

Time-asymptotic Dynamics of Hermitian Riccati Differential Equations

The matrix Riccati differential equation (RDE) raises in a wide variety of applications for science and applied mathematics. We are particularly interested in the Hermitian Riccati Differential Equation (HRDE). Radon's lemma gives a solution representation to HRDE. Although solutions of HRDE may show the finite escape time phenomenon, we can investigate the time asymptotic dynamical behavior of HRDE by its extended solutions. In this paper, we adapt the Hamiltonian Jordan canonical form to characterize the time asymptotic phenomena of the extended solutions for HRDE in four elementary cases. The extended solutions of HRDE exhibit the dynamics of heteroclinic, homoclinic and periodic orbits in the elementary cases under some conditions.

A Memory-Efficient Formulation of Precise-Integration Time-Domain Method With Riccati Matrix Differential Equations

A novel numerical method, referred to as the Riccati precise-integration time-domain (Riccati-PITD) method, is proposed to reduce the memory requirement of the conventional PITD method. In the Riccati-PITD method, the electromagnetic field components of the transverse electric/transverse magnetic (TE/TM) wave are arranged in a matrix for constructing the Riccati matrix differential equations (RDEs) about Maxwell's curl equations. Then the precise integration technique is adopted for the numerical integration of the RDEs. Theoretical analyses about the memory requirements of the conventional and the Riccati-PITD methods are presented. It is found that the memory requirement can be more greatly reduced by solving the electromagnetic-field components in the form of the matrices in the Riccati-PITD method rather than in the form of the vectors in the conventional PITD method. Numerical examples are also given to confirm that the proposed method has significant advantages over the conventional PITD method with respect to the execution time and the memory requirement.

Order Reduction Methods for Solving Large-Scale Differential Matrix Riccati Equations

We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising cl...

Finite-Time Stability of Switched Linear Time-Delay Systems Based on Time-Dependent Lyapunov Functions

In this paper, finite-time stability of switched linear time-delay systems has been addressed. By constructing a class of time-dependent common (multiple) Lyapunov functions, new explicit conditions for finite-time stability of the system under arbitrary switching and average dwell-time switching are established respectively. Compared with most of existing results in the literature, our results are easily verifiable by solving several linear matrix inequalities rather than complex matrix Riccati differential equations. The effectiveness of the proposed method is demonstrated by numerical examples.

State synchronization of controlled nodes via the dynamics of links for complex dynamical networks

A continuous time-varying complex dynamical network with the graph model may be regarded to be composed of two interconnected subsystems, one of which is the nodes subsystem (NS) and the other is the links subsystem (LS). The two subsystems can be modeled mathematically by the state differential equations, in which the weighted values of links are regarded as the state variables of LS. This paper mainly focuses on the dynamics of LS, which is modeled as the Riccati matrix differential equation, and investigates the state synchronization of NS associated with the synthesized state feedback controller for NS and the constructed coupling relation in LS. Firstly, this paper proposes the equivalent condition of state synchronization by using the matrix algebra method. Then, the state feedback controller for NS is proposed and the coupling relation in LS is also constructed based on the Lyapunov stability theory, by which the asymptotic state synchronization of NS is sure to be realized. Finally, numerical simulations are given to verify the effectiveness of the theoretical results in this paper.

An extended block Golub–Kahan algorithm for large algebraic and differential matrix Riccati equations

In this paper we propose a new projection method to solve both large-scale continuous-time matrix Riccati equations and differential matrix Riccati equations. The new approach projects the problem onto an extended block Krylov subspace and gets a low-dimensional equation. We use the block Golub–Kahan procedure to construct the orthonormal bases for the extended Krylov subspaces. For matrix Riccati equations, the reduced problem is then solved by means of a direct Riccati scheme such as the Schur method. When we solve differential matrix Riccati equations, the reduced problem is solved by the Backward Differentiation Formula (BDF) method and the obtained solution is used to build the low rank approximate solution of the original problem. Finally, we give some theoretical results and present numerical experiments.

On one method of finding optimal control in construction industry

In this paper we consider options for solving optimal control problems with a quadratic quality criterion. There are optimal management tasks in the construction industry also. For example, in optimizing the construction technology of large complexes, construction work cycles at the facilities. A quadratic quality criterion leads to Riccati matrix differential equations that are generally not solvable in quadratures. In this paper we consider solution options in the case of a stationary system. We find a symmetric solution of algebraic matrix Riccati equation. We offer some new analytical methods for solving this nonlinear equation in various cases.

Adaptive control for complex dynamical networks with structural balance via external stimulus signals

This paper investigates the adaptive structural balance control of complex dynamical networks by employing the controlled external stimulus signals which are coupled and transmitted to the dynamics of complex dynamical network. The control objective is to assure the asymptotical convergence of the dynamical links to the structural balance by the controlled external stimulus signals. The dynamical links of complex dynamical network are represented in this paper mathematically as the Riccati matrix differential equation with the controlled external stimulus signals which are coupled approximately in the form of Hebb rule. Compared with the existing results which are mainly concerned with the dynamical characteristics of nodes such as synchronization, this paper is mainly focused on the dynamical characteristic of links so named as the structural balance which is asymptotically obtained by the adaptive control scheme of external stimulus signals. Finally, a simulation example is given to show the validity of result proposed in this paper.

Genera of Conjoined Bases for (Non)oscillatory Linear Hamiltonian Systems: Extended Theory

In this paper we study the properties of conjoined bases of a general linear Hamiltonian system without any controllability condition. When the Legendre condition holds and the system is nonoscillatory, it is known from our previous work that conjoined bases with eventually the same image form a special structure called a genus. In this work we extend the theory of genera of conjoined bases to arbitrary systems, for which the Legendre condition is not assumed and/or the system may be oscillatory. We derive a classification of all genera of conjoined bases and show that they form a complete lattice. These results are based on the relationship between subspaces of solutions of a linear control system and orthogonal projectors satisfying a certain Riccati type differential equation. The presented theory is applied in our paper (Sepitka in Discrete Contin Dyn Syst 39(4):1685–1730, 2019) to general Riccati matrix differential equations for possibly uncontrollable linear Hamiltonian systems.

Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

SummaryIn this paper, we consider large‐scale nonsymmetric differential matrix Riccati equations with low‐rank right‐hand sides. These matrix equations appear in many applications such as control theory, transport theory, applied probability, and others. We show how to apply Krylov‐type methods such as the extended block Arnoldi algorithm to get low‐rank approximate solutions. The initial problem is projected onto small subspaces to get low dimensional nonsymmetric differential equations that are solved using the exponential approximation or via other integration schemes such as backward differentiation formula (BDF) or Rosenbrock method. We also show how these techniques can be easily used to solve some problems from the well‐known transport equation. Some numerical examples are given to illustrate the application of the proposed methods to large‐scale problems.

Reduction of the optimal control problem for a magnetoelectric power drive

A linear-quadratic optimal control problem is considered for a singularly perturbed differential system, which describes the dynamics of a magnetoelectric. The optimal control law is constructed as feedback. Using the method of integral manifolds allows us to reduce the order of the matrix differential Riccati equation.

An explicit Floquet-type representation of Riccati aperiodic exponential semigroups

The article presents a rather surprising Floquet-type representation of time-varying transition matrices associated with a class of nonlinear matrix differential Riccati equations. The main difference with conventional Floquet theory comes from the fact that the underlying flow of the solution matrix is aperiodic. The monodromy matrix associated with this Floquet representation coincides with the exponential (fundamental) matrix associated with the stabilising fixed point of the Riccati equation. The second part of this article is dedicated to the application of this representation to the stability of matrix differential Riccati equations. We provide refined global and local contraction inequalities for the Riccati exponential semigroup that depend linearly on the spectral norm of the initial condition. These refinements improve upon existing results and are a direct consequence of the Floquet-type representation, yielding what seems to be the first results of this type for this class of models.

Exponential synchronization of the high-dimensional Kuramoto model with identical oscillators under digraphs

For the Kuramoto model and its variations, it is difficult to analyze the exponential synchronization under the general digraphs due to the lack of symmetry. In this paper, for the high-dimensional Kuramoto model of identical oscillators, a matrix Riccati differential equation (MRDE) is proposed to describe the error dynamics. Based on the MRDE, the exponential synchronization is proved by constructing a total error function for the case of digraphs admitting directed spanning trees.

Robust State Observer Design for Dynamic Connection Relationships in Complex Dynamical Networks

The complex dynamical network with the time-varying links may be regarded to be composed of the two mutually coupled subsystems, which are called the nodes subsystem and the connection relationships subsystem respectively. Designing the state observer for the nodes subsystem has been discussed in some existing researches by employing the known connection relationships. However, the state observer design for the connection relationships subsystem is not shown in the existing researches. In practical applications, the connection relationships subsystem possesses also the state variables such as the relationship strength in social network, the size of web tension in the web winding system. Therefore, designing the state observer for the connection relationships subsystem is also of practical significance. In this paper, a novel state observer is proposed for the connection relationships subsystem modeled mathematically by the Riccati matrix differential equation. The observer utilizes the output measurements of the connection relationships subsystem and the state of nodes subsystem. Compared with the state observer of nodes subsystems which employs the known connection relationships, the state observer for the connection relationships subsystem employs the state of nodes subsystem and is represented in form of the matrix differential equation. By using Lyapunov stability theory, it is proved in this paper that the state observer for the connection relationships subsystem is asymptotical under certain mathematic conditions. Finally, the illustrative simulation is given to show the efficiency and validity of the proposed method in this paper.

Riccati equations for linear Hamiltonian systems without controllability condition

In this paper we develop new theory of Riccati matrix differential equations for linear Hamiltonian systems, which do not require any controllability assumption. When the system is nonoscillatory, it is known from our previous work that conjoined bases of the system with eventually the same image form a special structure called a genus. We show that for every such a genus there is an associated Riccati equation. We study the properties of symmetric solutions of these Riccati equations and their connection with conjoined bases of the system. For a given genus, we pay a special attention to distinguished solutions at infinity of the associated Riccati equation and their relationship with the principal solutions at infinity of the system in the considered genus. We show the uniqueness of the distinguished solution at infinity of the Riccati equation corresponding to the minimal genus. This study essentially extends and completes the work of W. T. Reid (1964, 1972), W. A. Coppel (1971), P. Hartman (1964), W. Kratz (1995), and other authors who considered the Riccati equation and its distinguished solution at infinity for invertible conjoined bases, i.e., for the maximal genus in our setting.

A periodic solution of the coupled matrix Riccati differential equations

A periodic solution of the coupled matrix Riccati differential equations

Solution of matrix differential Riccati equations in the medium of modeling system of MVTU 3.7

Purpose. For the design of systems under the action of random perturbations, a method of stochastic dynamic programming has been developed. The method is very effective because it allows one to find the optimal control of the parameters of elastic-dissipative connections of the crew suspension as a function of estimating the phase coordinates. The main reason that hinders the widespread use of this method for linear dynamic systems with many degrees of freedom is related to the lack of software for solving the matrix non-linear differential equation of Riccati type, which is included in the algorithm of the method. The aim of the work is to develop modeling schemes for solving differential equations of Riccati type in the system of modeling MVTU 3.7 (modeling in technical devices). The technique. The simulation system of MVTU 3.7 allows studying transient processes in complex dynamic systems and analyzing the stability and stability of oscillatory processes along a phase trajectory. A mathematical model of a dynamic system is created - differential equations, after - a structural diagram of the model and in the environment of MVTU 3.7 a modeling scheme is constructed. MVTU 3.7 has a graphics editor, so the modeling scheme is assembled from library blocks, which is located on the working field. The block parameters, initial conditions, integration method, simulation time, solution accuracy, etc. are set. The simulation results are given in the form of graphs. In addition, there is a text editor that records the numerical data of the simulation results. To solve differential equations like Riccati, a structural scheme is not needed. Results. A simulation scheme was created for solving differential equations of Riccati type and correlation functions of the matrix of parameters of the optimal filter were obtained. The simulation of the oscillation process of the body of the 2TE10L locomotive and the filter was performed. It has been established that for the 2TE10L locomotive it is not required to create devices for actively controlling the parameters of elastic-dissipative links. Naukova novelty. A method for solving differential equations in the simulation system of MVTU 3.7 Practical value. The main reason that prevented the widespread use of the method of stochastic dynamic programming has been eliminated and has hampered the development of the theory of creating active suspensions of transport crews.