Continuous-time Algebraic Riccati Equations Research Articles

A general alternating-direction implicit Newton method for solving continuous-time algebraic Riccati equation

The complex continuous-time algebraic Riccati equation (CCARE) is quadratic, which is closely related to the analysis of the optimal control problem. In this paper, we apply Newton method as the outer iteration and an efficient general alternating-direction implicit (GADI) method as the inner iteration to solve CCARE. Meanwhile, we propose the inexact Newton-GADI method to further improve the efficiency of the algorithm. We give the convergence analysis of our proposed method and prove that its convergence rate is faster than the classical Newton-ADI method. Finally, some numerical examples are given to illustrate the effectiveness of our algorithms and the correctness of the theoretical analysis.

Continuous Time Algebraic Riccati Equation Solution for Second Order Systems

Abstract The continuous time algebraic Riccati equation (ARE) is often utilized in control, estimation, and optimization. For a linear system with a second order structure, the ARE required to be solved to get the control values in standard control problems results in complex sub-equations in terms of the second order system matrices. The computational costs of solving the algebraic Riccati equation through standard methods such as the Hamiltonian matrix pencil approach increase substantially as matrix sizes increase for a second order system, due to the eigendecomposition of the 2n x 2n system matrices involved. This work introduces a new solution that does not require the eigendecomposition of the 2n x 2n system matrices, while satisfying all of the requirements of the solution to the Riccati equation (e.g., detectability, stabilizability, positive semi-definite solution matrix).

Closed-Form Solutions to Continuous-Time Algebraic Riccati Equation for Second-Order Systems

Abstract Two closed-form methods to solve the continuous-time algebraic Riccati equation (CARE) for second-order systems in terms of the mass, damping, and stiffness matrices are presented. One method utilizes the modal transformation of mass and stiffness matrices, and the other does not require this transformation. Hundreds of high-dimensional second-order systems are used to show that these methods achieve similar or better accuracy compared to the state-of-the-art, while significantly reducing the computation time. Furthermore, advantages of these methods are illustrated in vibration control problems.

Memory-Efficient Interpolatory Projection Techniques for the Stabilization of Incompressible Navier-Stokes Flows

In this article, we are proposing an updated form of Krylov subspace-based interpolatory projection techniques for the stabilization of incompressible Navier-Stokes flows. In the proposed techniques, we utilize the reduced-order modelling approach implicitly, where reduced-order matrices need not be gathered explicitly. To estimate the aimed optimal feedback matrix, only the factored solution of the desired continuous-time algebraic Riccati equation (CARE) needs to be stored through the classical eigenvalue decomposition. The sparse structure of the target systems will remain invariant within the matrix-vector operations for generating the bases of the projector matrices through Krylov subspace techniques, where a cohesive projection scheme will be incorporated to ensure the potency of the projector matrices. So, the proposed techniques will be feasible for memory allocation and enhance the rapid convergence of the simulation. Analysis of the target systems’ transient characteristics, such as eigenvalues and step-responses, will be used to ascertain the competence and reliability of the proposed techniques. Necessary computation will be done numerically through MATLAB. Stabilization of the transient behaviors and minimization of the simulation time are the prime concerns in this work. Eventually, by comparing with the contemporary techniques the advancement of the proposed techniques will be confirmed.

Newton’s method for coupled continuous-time algebraic Riccati equations

Newton’s method for coupled continuous-time algebraic Riccati equations

A numerically stable high-order Chebyshev-Halley type multipoint iterative method for calculating matrix sign function

<abstract><p>A new eighth-order Chebyshev-Halley type iteration is proposed for solving nonlinear equations and matrix sign function. Basins of attraction show that several special cases of the new method are globally convergent. It is analytically proven that the new method is asymptotically stable and the new method has the order of convergence eight as well. The effectiveness of the theoretical results are illustrated by numerical experiments. In numerical experiments, the new method is applied to a random matrix, Wilson matrix and continuous-time algebraic Riccati equation. Numerical results show that, compared with some well-known methods, the new method achieves the accuracy requirement in the minimum computing time and the minimum number of iterations.</p></abstract>

A Flexible and Accurate Solver for Continuous-Time Algebraic Riccati Equations

A Flexible and Accurate Solver for Continuous-Time Algebraic Riccati Equations

Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems

In the context of infinite-horizon optimal control problems, the algebraic Riccati equations (ARE) arise when the stability of linear time-varying (LTV) systems is investigated. Using the zeroing neural network (ZNN) approach to solve the time-varying eigendecomposition-based ARE (TVE-ARE) problem, the ZNN model (ZNNTVE-ARE) for solving the TVE-ARE problem is introduced as a result of this research. Since the eigendecomposition approach is employed, the ZNNTVE-ARE model is designed to produce only the unique nonnegative definite solution of the time-varying ARE (TV-ARE) problem. It is worth mentioning that this model follows the principles of the ZNN method, which converges exponentially with time to a theoretical time-varying solution. The ZNNTVE-ARE model can also produce the eigenvector solution of the continuous-time Lyapunov equation (CLE) since the Lyapunov equation is a particular case of ARE. Moreover, this paper introduces a hybrid ZNN model for stabilizing LTV systems in which the ZNNTVE-ARE model is employed to solve the continuous-time ARE (CARE) related to the optimal control law. Experiments show that the ZNNTVE-ARE and HFTZNN-LTVSS models are both effective, and that the HFTZNN-LTVSS model always provides slightly better asymptotic stability than the models from which it is derived.

A low-rank solution method for Riccati equations with indefinite quadratic terms

Algebraic Riccati equations with indefinite quadratic terms play an important role in applications related to robust controller design. While there are many established approaches to solve these in case of small-scale dense coefficients, there is no approach available to compute solutions in the large-scale sparse setting. In this paper, we develop an iterative method to compute low-rank approximations of stabilizing solutions of large-scale sparse continuous-time algebraic Riccati equations with indefinite quadratic terms. We test the developed approach for dense examples in comparison to other established matrix equation solvers, and investigate the applicability and performance in large-scale sparse examples.

Distributed Optimization Design of Iterative Refinement Technique for Algebraic Riccati Equations

This article focuses on the problem of a distributed computation of continuous-time algebraic Riccati equations (CARE), where information of matrices is split and known by multiple agents. This article proposes a distributed optimization design of the iterative refinement technique (IRM), a well-established centralized method for CARE. By assuming that each agent only knows partial information of CARE, we reformulate IRM for CARE as three classes of distributed optimization subproblems with different formulations and constraints. Then, we propose distributed algorithms for obtained distributed optimization subproblems and prove convergence properties of proposed algorithms. Numerical results show the efficacy of the proposed distributed IRM.

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.

Effect of Shift Parameters in Rational Krylov Subspace Method for Solving Riccati Equations Arise from Power System Models

The simulation of continuous-time algebraic Riccati equations (CARE) derived from the very large power system models is a highly laborious task and most cases infeasible due to the sophisticated structural ingredients. The computation is very time costly and the rate of convergence can be severely affected in the direct solvers. To overcome those adversities, an iterative approach Rational Krylov Subspace Method (RKSM) is introduced to deal with those large-scale CAREs. The solutions of those CAREs and hence the optimal feedback matrices can be efficiently explored by the RKSM approach to stabilize the power system models of unstable categories. In this approach, shift parameters play a vital role in the convergence of the computations and size of the solution spaces. The goal of the work is to investigate the effect of different types of shift parameters on the stabilization process. To attain the mentioned objective, a modified version of the iterative RKSM algorithm is proposed by employing two types of shift parameters, namely, the adaptive ADI shifts and heuristic shifts. Qualitative discussions for the outcomes for those shift parameters are narrated by tabular and figurative methods.

Verified computation for the geometric mean of two matrices

An algorithm for numerically computing an interval matrix containing the geometric mean of two Hermitian positive definite (HPD) matrices is proposed. We consider a special continuous-time algebraic Riccati equation (CARE) where the geometric mean is the unique HPD solution, and compute an interval matrix containing a solution to the equation. We invent a change of variables designed specifically for the special CARE. By the aid of this special change of variables, the proposed algorithm gives smaller radii, and is more successful than previous approaches. Solutions to the equation are not necessarily Hermitian. We thus establish a theory for verifying that the contained solution is Hermitian. Finally, the positive definiteness of the solution is verified. Numerical results show effectiveness, efficiency, and robustness of the algorithm.

Modeling and Estimation of Multivariate Discrete and Continuous Time Stationary Processes

In this paper, we give an autoregressive model of order 1 type of characterization covering all multivariate strictly stationary processes indexed by the set of integers. Consequently, under square integrability, we derive continuous time algebraic Riccati equations for the parameter matrix of the characterization. This provides us with a natural way to define the corresponding estimator. In addition, we show that the estimator inherits consistency from autocovariances of the stationary process. Furthermore, the limiting distribution is given by a linear function of the limiting distribution of the autocovariances. We also present the corresponding existing results of the continuous time setting paralleling them to the discrete case treated in this paper.

Perturbation Analysis of Continuous-Time Linear Time-Invariant Systems

In this paper, we consider the perturbation analysis of linear time-invariant systems, which arise from the linear optimal control in continuous-time. We provide a method to compute condition numbers of continuous-time linear time-invariant systems. It solves the perturbed linear time-invariant systems via Riccati differential equations and continuous-time algebraic Riccati equations in finite and infinite time horizons. We derive the explicit expressions of measuring the perturbation bounds of condition numbers with respect to the solution of the linear time-invariant systems. Furthermore, condition numbers and their upper bounds of Riccati differential equations and continuous-time algebraic Riccati equations are also discussed. Numerical simulations show the sharpness of the perturbation bounds computed via the proposed methods.

A Numerical Comparison of Different Solvers for Large-Scale, Continuous-Time Algebraic Riccati Equations and LQR Problems

In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance.

On the structure of the solution of continuous-time algebraic Riccati equations with closed-loop eigenvalues on the imaginary axis.

This paper proposes a decomposition of the continuous-time algebraic Riccati equation aimed at eliminating the problem of the presence of closed-loop eigenvalues on the imaginary axis. In particular, we show that it is possible to parameterize the the entire set of solutions of the given Riccati equation in terms of the solutions of a reduced-order Riccati equation, which is associated to a Hamiltonian matrix with no eigenvalues on the imaginary axis, and some free parameters arising from the presence of imaginary eigenvalues of the Hamiltonian matrix.

Inheritance properties of Krylov subspace methods for continuous-time algebraic Riccati equations

We investigate the theory behind the Krylov subspace methods for large-scale continuous-time algebraic Riccati equations. We show that the solvability of the projected algebraic Riccati equation need not be assumed but can be inherited. This study of inheritance properties is the first of its kind. We study the stabilizability and detectability of the control system, the stability of the associated Hamiltonian matrix and perturbation in terms of residuals. Special attention is paid to the stabilizing and positive semi-definite properties of approximate solutions. Illustrative numerical examples for the inheritance properties are presented.

Low-rank updates and divide-and-conquer methods for quadratic matrix equations

In this work, we consider two types of large-scale quadratic matrix equations: continuous-time algebraic Riccati equations, which play a central role in optimal and robust control, and unilateral quadratic matrix equations, which arise from stochastic processes on 2D lattices and vibrating systems. We propose a simple and fast way to update the solution to such matrix equations under low-rank modifications of the coefficients. Based on this procedure, we develop a divide-and-conquer method for quadratic matrix equations with coefficients that feature a specific type of hierarchical low-rank structure, which includes banded matrices. This generalizes earlier work on linear matrix equations. Numerical experiments indicate the advantages of our newly proposed method versus iterative schemes combined with hierarchical low-rank arithmetic.

Nonlinear Least-Squares Approach for Large-Scale Algebraic Riccati Equations

We propose a new optimization approach for solving large-scale continuous-time algebraic Riccati equations with a low-rank right-hand side. First, we project the problem onto a Krylov-type low-dime...