共查询到20条相似文献,搜索用时 62 毫秒
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The standard algebraic stability condition for general linear methods (GLMs) is considered in a modified form, and connected
to a branch of Control Theory concerned with the discrete algebraic Riccati equation (DARE). The DARE theory shows that, for
an algebraically stable method, there is a minimal G-matrix, G
*, satisfying an equation, rather than an inequality. This result, and another alternative reformulation of algebraic stability,
are applied to construct new GLMs with 2 steps and 2 stages, one of which has order p=4 and stage order q=3. The construction process is simplified by method-equivalence, and Butcher’s simplified order conditions for the case p≤q+1.
相似文献
3.
In this paper, we propose a class of special Krylov subspace methods to solve continuous algebraic Riccati equation (CARE), i.e., the Hessenberg-based methods. The presented approaches can obtain efficiently the solution of algebraic Riccati equation to some extent. The main idea is to apply Kleinman-Newton"s method to transform the process of solving algebraic Riccati equation into Lyapunov equation at every inner iteration. Further, the Hessenberg process of pivoting strategy combined with Petrov-Galerkin condition and minimal norm condition is discussed for solving the Lyapunov equation in detail, then we get two methods, namely global generalized Hessenberg (GHESS) and changing minimal residual methods based on the Hessenberg process (CMRH) for solving CARE, respectively. Numerical experiments illustrate the efficiency of the provided methods. 相似文献
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1.引言若不特别说明,以下记号都是常规的,可参见山.在最优控制中占有核心地位的代数Xiccati方程(ARE)有两种基本形式:连续型的ARE(CARE):离散型的**E(**RE):其中只兄见NE贮””,GIE皿””m,GZE*m”m,in<2,*T二K>风*T二N>几*2二*2>凡从应用角度看,主要关心**E的对称半正定解.对于**RE,已有大量的研究工作.特别,陈春晖门、徐树方问、Ghwimi-Laub同等研究了扰动理论.对于DARE,它与***E的一个明显区别是:**M是二次矩阵方程而***E具有高度非线性.这种区别使得DARE远为复杂… 相似文献
5.
Vejdi I. Hasanov 《Numerical Functional Analysis & Optimization》2013,34(12):1532-1559
The perturbation results for the solutions of two linearly perturbed algebraic Riccati equations are derived. We generalize the results of Sun [SIAM J. Matrix Anal. Appl., 19 (1998):39–65] for continuous (CARE) and discrete (DARE) algebraic Riccati equations, respectively. The results are illustrated by numerical examples. 相似文献
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Xiao-xiaGuo Zhong-zhiBai 《计算数学(英文版)》2005,23(3):305-320
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. 相似文献
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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. 相似文献
8.
Augusto Ferrante 《Linear and Multilinear Algebra》2013,61(11):1460-1474
This paper proposes a reduction technique for the generalized Riccati difference equation arising in optimal control and optimal filtering. This technique relies on a study on the generalized discrete algebraic Riccati equation. In particular, an analysis on the eigenstructure of the corresponding extended symplectic pencil enables to identify a subspace in which all the solutions of the generalized discrete algebraic Riccati equation are coincident. This subspace is the key to derive a decomposition technique for the generalized Riccati difference equation. This decomposition isolates a “nilpotent” part, which converges to a steady-state solution in a finite number of steps, from another part that can be computed by iterating a reduced-order generalized Riccati difference equation. 相似文献
9.
ON SIGN FUNCTION AND SQUARE ROOT METHODS FOR SOLUTION OF REAL ALGEBRAIC RICCATI EQUATIONS 总被引:1,自引:0,他引:1
卢琳璋 《高等学校计算数学学报(英文版)》1992,(1)
In this paper, we first justify the theory of the square root algorithm presented in [1], [2] by simple application of the matrix sign function, then we present an iterative square root algorithm for the continuous time algebraic Riccati equation. 相似文献
10.
Chun-Hua Guo 《Numerical Functional Analysis & Optimization》2013,34(5):516-529
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. 相似文献
11.
Olof J. Staffans 《Transactions of the American Mathematical Society》1997,349(9):3679-3715
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.
12.
A noniterative algebraic solution for Riccati equations satisfying two-point boundary-value problems
A noniterative algebraic method is presented for solving differential Riccati equations which satisfy two-point boundary-value problems. This class of numerical problems arises in quadratic optimization problems where the cost functionals are composed of both continuous and discrete state penalties, leading to piecewise periodic feedback gains. The necessary condition defining the solution for the two-point boundary value problem is cast in the form of a discrete-time algebraic Riccati equation, by using a formal representation for the solution of the differential Riccati equation. A numerical example is presented which demonstrates the validity of the approach.The authors would like to thank Dr. Fernando Incertis, IBM Madrid Scientific Center, who reviewed this paper and pointed out that the two-point boundary-value necessary condition could be manipulated into the form of a discrete-time Riccati equation. His novel approach proved to be superior to the authors' previously proposed iterative continuation method. 相似文献
13.
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. 相似文献
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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. 相似文献
16.
As is known, Alternating-Directional Doubling Algorithm (ADDA) is quadratically convergent for computing the minimal nonnegative solution of an irreducible singular M-matrix algebraic Riccati equation (MARE) in the noncritical case or a nonsingular MARE, but ADDA is only linearly convergent in the critical case. The drawback can be overcome by deflating techniques for an irreducible singular MARE so that the speed of quadratic convergence is still preserved in the critical case and accelerated in the noncritical case. In this paper, we proposed an improved deflating technique to accelerate further the convergence speed – the double deflating technique for an irreducible singular MARE in the critical case. We proved that ADDA is quadratically convergent instead of linearly when it is applied to the deflated algebraic Riccati equation (ARE) obtained by a double deflating technique. We also showed that the double deflating technique is better than the deflating technique from the perspective of dimension of the deflated ARE. Numerical experiments are provided to illustrate that our double deflating technique is effective. 相似文献
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In this paper we restudy, by a radically different approach, the optimal quadratic cost problem for an abstract dynamics,
which models a special class of second-order partial differential equations subject to high internal damping and acted upon
by boundary control. A theory for this problem was recently derived in [LLP] and [T1] (see also [T2]) by a change of variable
method and by a direct approach, respectively. Unlike [LLP] and [T1], the approach of the present paper is based on singular
control theory, combined with regularity theory of the optimal pair from [T1]. This way, not only do we rederive the basic
control-theoretic results of [LLP] and [T1]—the (first) synthesis of the optimal pair, and the (first) nonstandard algebraic
Riccati equation for the (unique) Riccati operator which enters into the gain operator of the synthesis—but in addition, this
method also yields new results—a second form of the synthesis of the optimal pair, and a second (still nonstandard) algebraic
Riccati equation for the (still unique) Riccati operator of the synthesis. These results, which show new pathologies in the
solution of the problem, are new even in the finite-dimensional case.
This research was made possible by NATO Collaborative Research Grant SA.5-2-05 (CRG.940161) 274/94/JARC-501, whose support
is gratefully acknowledged. The research of I. Lasiecka and R. Triggiani was supported also by the National Science Foundation
under Grant NSF-DMS-92-04338. The research of L. Pandolfi was written with the programs of GNAFA-CNR. The main results of
the present paper were announced in [LPT]. 相似文献
18.
To factorize a spectral density matrix of a vector moving average process, we propose a state space representation. Although this state space is not necessarily of minimal dimension, its associated system matrices are simple and most matrix multiplications involved are nothing but index shifting. This greatly reduces the complexity of computation. Moreover, in this article we stack every q consecutive observations of the original process MA(q) and generate a vector MA(1) process. We consider a similar state space representation for the stacked process. Consequently, the solution hinges on a surprisingly compact discrete algebraic Riccati equation (DARE), which involves only one Toeplitz and one Hankel block matrix composed of autocovariance functions. One solution to this equation is given by the so-called iterative projection algorithm. Each iteration of the stacked version is equivalent to q iterations of the unstacked one. We show that the convergence behavior of the iterative projection algorithm is characterized by the decreasing rate of the partial correlation coefficients for the stacked process. In fact, the calculation of the partial correlation coefficients via the Whittle algorithm, which takes a very simple form in this case, offers another solution to the problem. To achieve computational efficiency, we apply the general Newton procedure given by Lancaster and Rodman to the DARE and obtain an algorithm of quadratic convergence rate. One immediate application of the new algorithms is polynomial stabilization. We also discuss various issues such as check of positivity and numerical implementation. 相似文献
19.
Cristian Oara 《Numerical Algorithms》1994,7(2):355-373
An algorithm for computing proper deflating subspaces with specified spectrum for an arbitrary matrix pencil is presented. The method uses refined algorithms for computing the generalized Schur form of a matrix pencil and enlightens the connection that exists between reducing and proper deflating subspaces. The proposed algorithm can be applied for computing the stabilizing solution of the generalized algebraic Riccati equation, a recently introduced concept which extends the usual algebraic Riccati equation. 相似文献
20.
By using symmetries associated to Riccati equation in standard form (SRE), we obtain a family which can be integrated by quadratures. As a consequence, we get a new integrability condition for the generalized Riccati equation (GRE). We illustrate the result with some examples and we give some applications in the solitons theory. 相似文献