Convergence analysis of the Newton–Shamanskii method for a nonsymmetric algebraic Riccati equation

In this paper, we consider the nonsymmetric algebraic Riccati equation arising in transport theory. An important feature of this equation is that its minimal positive solution can be obtained via computing the minimal positive solution of a vector equation. We apply the Newton–Shamanskii method to solve the vector equation. Convergence analysis shows that the sequence of vectors generated by the Newton–Shamanskii method is monotonically increasing and converges to the minimal positive solution of the vector equation. Numerical experiments show that the Newton–Shamanskii method is feasible and effective, and outperforms the Newton method. Copyright © 2008 John Wiley & Sons, Ltd.     

Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations

For the algebraic Riccati equation whose four coefficient matrices form a nonsingular M-matrix or an irreducible singular M-matrix K, the minimal nonnegative solution can be found by Newton’s method and the doubling algorithm. When the two diagonal blocks of the matrix K have both large and small diagonal entries, the doubling algorithm often requires many more iterations than Newton’s method. In those cases, Newton’s method may be more efficient than the doubling algorithm. This has motivated us to study Newton-like methods that have higher-order convergence and are not much more expensive each iteration. We find that the Chebyshev method of order three and a two-step modified Chebyshev method of order four can be more efficient than Newton’s method. For the Riccati equation, these two Newton-like methods are actually special cases of the Newton–Shamanskii method. We show that, starting with zero initial guess or some other suitable initial guess, the sequence generated by the Newton–Shamanskii method converges monotonically to the minimal nonnegative solution.We also explain that the Newton-like methods can be used to great advantage when solving some Riccati equations involving a parameter.     

Versions of inexact Kleinman-Newton methods for Riccati equations

Optimal control problems involving PDEs often lead in practice to the numerical computation of feedback laws for an optimal control. This is achieved through the solution of a Riccati equation which can be large scale, since the discretized problems are large scale and require special attention in their numerical solution. The Kleinman-Newton method is a classical way to solve an algebraic Riccati equation. We look at two versions of an extension of this method to an inexact Newton method. It can be shown that these two implementable versions of Newton's method are identical in the exact case, but differ substantially for the inexact Newton method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis and modificaton of Newton's method for algebraic Riccati equations

When Newton's method is applied to find the maximal symmetric solution of an algebraic Riccati equation, convergence can be guaranteed under moderate conditions. In particular, the initial guess need not be close to the solution. The convergence is quadratic if the Fréchet derivative is invertible at the solution. In this paper we examine the behaviour of the Newton iteration when the derivative is not invertible at the solution. We find that a simple modification can improve the performance of the Newton iteration dramatically.

     

On the convergence of inexact Newton methods for discrete-time algebraic Riccati equations

In this paper, we present a convergence analysis of the inexact Newton method for solving Discrete-time algebraic Riccati equations (DAREs) for large and sparse systems. The inexact Newton method requires, at each iteration, the solution of a symmetric Stein matrix equation. These linear matrix equations are solved approximatively by the alternating directions implicit (ADI) or Smith?s methods. We give some new matrix identities that will allow us to derive new theoretical convergence results for the obtained inexact Newton sequences. We show that under some necessary conditions the approximate solutions satisfy some desired properties such as the d-stability. The theoretical results developed in this paper are an extension to the discrete case of the analysis performed by Feitzinger et al. (2009) [8] for the continuous-time algebraic Riccati equations. In the last section, we give some numerical experiments.     

On inexact Newton methods based on doubling iteration scheme for non‐symmetric algebraic Riccati equations

Newton iteration method can be used to find the minimal non‐negative solution of a certain class of non‐symmetric algebraic Riccati equations. However, a serious bottleneck exists in efficiency and storage for the implementation of the Newton iteration method, which comes from the use of some direct methods in exactly solving the involved Sylvester equations. In this paper, instead of direct methods, we apply a fast doubling iteration scheme to inexactly solve the Sylvester equations. Hence, a class of inexact Newton iteration methods that uses the Newton iteration method as the outer iteration and the doubling iteration scheme as the inner iteration is obtained. The corresponding procedure is precisely described and two practical methods of monotone convergence are algorithmically presented. In addition, the convergence property of these new methods is studied and numerical results are given to show their feasibility and effectiveness for solving the non‐symmetric algebraic Riccati equations. Copyright © 2010 John Wiley & Sons, Ltd.     

The King-Werner method for solving nonsymmetric algebraic Riccati equation

For the nonsymmetric algebraic Riccati equation arising from transport theory, we concern about solving its minimal positive solution. In [1], Lu transferred the equation into a vector form and pointed out that the minimal positive solution of the matrix equation could be obtained via computing that of the vector equation. In this paper, we use the King-Werner method to solve the minimal positive solution of the vector equation and give the convergence and error analysis of the method. Numerical tests show that the King-Werner method is feasible to determine the minimal positive solution of the vector equation.     

A preconditioned block Arnoldi method for large scale Lyapunov and algebraic Riccati equations

In the present paper, we propose a preconditioned Newton–Block Arnoldi method for solving large continuous time algebraic Riccati equations. Such equations appear in control theory, model reduction, circuit simulation amongst other problems. At each step of the Newton process, we solve a large Lyapunov matrix equation with a low rank right hand side. These equations are solved by using the block Arnoldi process associated with a preconditioner based on the alternating direction implicit iteration method. We give some theoretical results and report numerical tests to show the effectiveness of the proposed approach.     

Newton additive and multiplicative Schwarz iterative methods

Daniel B. Szyld^¶ Department of Mathematics, Temple University, Philadelphia, PA 19122, USA ^{Convergence properties are presented for Newton additive andmultiplicative Schwarz (AS and MS) iterative methods for thesolution of nonlinear systems in several variables. These methodsconsist of approximate solutions of the linear Newton step usingeither AS or MS iterations, where overlap between subdomainscan be used. Restricted versions of these methods are also considered.These Schwarz methods can also be used to precondition a Krylovsubspace method for the solution of the linear Newton steps.Numerical experiments on parallel computers are presented, indicatingthe effectiveness of these methods.}     

On multivariable encryption schemes based on simultaneous algebraic Riccati equations over finite fields

New multivariable asymmetric public-key encryption schemes based on the NP-complete problem of simultaneous algebraic Riccati equations over finite fields are suggested. We also provide a systematic way to describe any set of quadratic equations over any field, as a set of algebraic Riccati equations. This has the benefit of systematic algebraic crypt-analyzing any encryption scheme based on quadratic equations, to any possible vulnerable hidden structure, in view of the fact that the set of all solutions to any given single algebraic Riccati equation is fully described in terms of all the T-invariant subspaces of some restricted dimension, where T is the matrix of coefficients of the related algebraic Riccati equation.     

On the Minimal Nonnegative Solution of Nonsymmetric Algebraic Riccati Equation

We study perturbation bound and structured condition number about the minimal nonnegative solution of nonsymmetric algebraic Riccati equation, obtaining a sharp perturbation bound and an accurate condition number. By using the matrix sign function method we present a new method for finding the minimal nonnegative solution of this algebraic Riccati equation. Based on this new method, we show how to compute the desired M-matrix solution of the quadratic matrix equation X^2 - EX - F = 0 by connecting it with the nonsymmetric algebraic Riccati equation, where E is a diagonal matrix and F is an M-matrix.     

Quadratic optimal control of stable well-posed linear systems

We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.

     

A new equation for the linear regulator problem

In this article, a new equation is derived for the optimal feedback gain matrix characterizing the solution of the standard linear regulator problem. It will be seen that, in contrast to the usual algebraic Riccati equation which requires the solution ofn(n + 1)/2 quadratically nonlinear algebraic equations, the new equation requires the solution of onlynm such equations, wherem is the number of system input terminals andn is the dimension of the state vector of the system. Utilizing the new equation, results are presented for the inverse problem of linear control theory.     

Approximation of low rank solutions for linear quadratic control of partial differential equations

Algebraic Riccati equations (ARE) of large dimension arise when using approximations to design controllers for systems modeled by partial differential equations. We use a modified Newton method to solve the ARE that takes advantage of several special features of these problems. The modified Newton method leads to a right-hand side of rank equal to the number of inputs regardless of the weights. Thus, the resulting Lyapunov equation can be more efficiently solved. The Cholesky-ADI algorithm is used to solve the Lyapunov equation resulting at each step. The algorithm is straightforward to code. Performance is illustrated with a number of standard examples. An example on controlling the deflection of the Euler-Bernoulli beam indicates that for weakly damped problems a low rank solution to the ARE may not exist. Further analysis supports this point.     

On the numerical solution of large-scale sparse discrete-time Riccati equations

We discuss the numerical solution of large-scale discrete-time algebraic Riccati equations (DAREs) as they arise, e.g., in fully discretized linear-quadratic optimal control problems for parabolic partial differential equations (PDEs). We employ variants of Newton??s method that allow to compute an approximate low-rank factor of the solution of the DARE. The principal computation in the Newton iteration is the numerical solution of a Stein (aka discrete Lyapunov) equation in each step. For this purpose, we present a low-rank Smith method as well as a low-rank alternating-direction-implicit (ADI) iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.     

Alternately linearized implicit iteration methods for the minimal nonnegative solutions of the nonsymmetric algebraic Riccati equations

For the non‐symmetric algebraic Riccati equations, we establish a class of alternately linearized implicit (ALI) iteration methods for computing its minimal non‐negative solutions by technical combination of alternate splitting and successive approximating of the algebraic Riccati operators. These methods include one iteration parameter, and suitable choices of this parameter may result in fast convergent iteration methods. Under suitable conditions, we prove the monotone convergence and estimate the asymptotic convergence factor of the ALI iteration matrix sequences. Numerical experiments show that the ALI iteration methods are feasible and effective, and can outperform the Newton iteration method and the fixed‐point iteration methods. Besides, we further generalize the known fixed‐point iterations, obtaining an extensive class of relaxed splitting iteration methods for solving the non‐symmetric algebraic Riccati equations. Copyright © 2006 John Wiley & Sons, Ltd.     

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

Delta算子Riccati方程研究的新结果

基于Delta算子描述，统一研究连续时间代数Riccati方程(CARE)和离散时间代数Riccati方程(DARE)的定界估计问题，提出了统一代数Riccati方程(UARE)解矩阵的上下界，给出UARE中P与R和Q的几个基本关系．     

Computation of the stabilizing solution of game theoretic Riccati equation arising in stochastic <Emphasis Type="Italic">H</Emphasis><Subscript> ∞ </Subscript> control problems

In this paper, the problem of the numerical computation of the stabilizing solution of the game theoretic algebraic Riccati equation is investigated. The Riccati equation under consideration occurs in connection with the solution of the H _∞ control problem for a class of stochastic systems affected by state dependent and control dependent white noise. The stabilizing solution of the considered game theoretic Riccati equation is obtained as a limit of a sequence of approximations constructed based on stabilizing solutions of a sequence of algebraic Riccati equations of stochastic control with definite sign of the quadratic part. The efficiency of the proposed algorithm is demonstrated by several numerical experiments.     

ON THE MINIMAL NONNEGATIVE SOLUTION OF ONSYMMETRIC ALGEBRAIC RICCATI EQUATION

We study perturbation bound and structured condition number about the minimalnonnegative solution of nonsymmetric algebraic Riccati equation,obtaining a sharp per-turbation bound and an accurate condition number.By using the matrix sign functionmethod we present a new method for finding the minimal nonnegative solution of this al-gebraic Riccati equation.Based on this new method,we show how to compute the desiredM-matrix solution of the quadratic matrix equation X~2-EX-F=0 by connecting itwith the nonsymmetric algebraic Riccati equation,where E is a diagonal matrix and F isan M-matrix.