A New Approach to Linearly Perturbed Riccati Equations Arising in Stochastic Control

In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed Riccati differential equation and deriving new convergence results for systems not necessarily observable. Accepted 30 July 1996     

Some predictor–corrector‐type iterative schemes for solving nonsymmetric algebraic Riccati equations arising in transport theory

It is as well known that nonsymmetric algebraic Riccati equations arising in transport theory can be translated to vector equations. In this paper, we propose six predictor–corrector‐type iterative schemes to solve the vector equations. And we give the convergence of these schemes. Unlike the previous work, we prove that all of them converge to the minimal positive solution of the vector equations by the initial vector (e,e), where e = (1,1, ? ,1)^T. Moreover, we prove that all the sequences generated by the iterative schemes are strictly and monotonically increasing and bounded above. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach. Copyright © 2014 John Wiley & Sons, Ltd.     

On the direct solution of Riccati equations arising in boundary control theory

Two classes of Riccati equations arising in the boundary control of parabolic systems are studied by direct methods. The new feature with respect to previous works on this subject is the low regularity of the final data. The classes considered here generalize those of [7]and [5]on one side, and of [14]on the other one. Completely new methods are used to obtain the solution of the Riccati equations, in both cases. The central theme is the dependence of the solutions on a «symmetric» norm of the final data, yielding these new results as well as a new proof of existence for the related algebraic Riccati equation under more general assumptions. The synthesis of the associated linear-quadratic-regulator problems is easily solved using these results.     

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.     

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.     

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.     

代数Riccati方程可稳解的条件数

１．引言 关于代数Ｒｉｃｃａｔｉ方程（ＡＲＥ）的研究是大量的．从数值角度看,有关数值方法,扰动理论的研究已比较深入．而关于条件数理论的研究则还不多［３］,［６］． Ｒｙｅｒｓ［１］研究了时连续代数 Ｒｉｃｃａｔｉ方程可稳解的条件数; Ｋｅｎｎｅｙ和 Ｈｅｗｅｒ［３］讨论了时连续代数 Ｒｉｃｃａｔｉ方程（以下简称 ＣＴＡＲＥ）可稳解的敏度分析,给出了一阶扰动界,引进了条件数; Ｓｕｎ［６］从最佳向后扰动理论角度研究了时离散代数Ｒｉｃｃａｔｉ方程（以下简称ＤＴＡＲＥ）可稳解的条件数;徐树方［８］针对 ＣＴＡ…     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方法,即改进的广义射影Ricccati方程方法, 求解(2+1)维色散长波方程, 得到该方程的新的更一般形式的行波解, 包括扭状孤波解, 钟状解,孤子解和周期解. 并对部分新形式孤波解画图示意.     

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.     

Transformations between discrete-time and continuous-time algebraic Riccati equations

We introduce a transformation between the discrete-time and continuous-time algebraic Riccati equations. We show that under mild conditions the two algebraic Riccati equations can be transformed from one to another, and both algebraic Riccati equations share common Hermitian solutions. The transformation also sets up the relations about the properties, commonly in system and control setting, that are imposed in parallel to the coefficient matrices and Hermitian solutions of two algebraic Riccati equations. The transformation is simple and all the relations can be easily derived. We also introduce a generalized transformation that requires weaker conditions. The proposed transformations may provide a unified tool to develop the theories and numerical methods for the algebraic Riccati equations and the associated system and control problems.     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方 法,即改进的广义射影Ricccati方程方法,求解(2 1)维色散长波方程,得到该方程的新的 更一般形式的行波解,包括扭状孤波解,钟状解,孤子解和周期解．并对部分新形式孤波解画 图示意．     

A modified Newton method for solving non-symmetric algebraic Riccati equations arising in transport theory

Liang Bao ^{The non-symmetric algebraic Riccati equation arising in transporttheory can be rewritten as a vector equation and the minimalpositive solution of the non-symmetric algebraic Riccati equationcan be obtained by solving the vector equation. In this paper,we apply the modified Newton method to solve the vector equation.Some convergence results are presented. Numerical tests showthat the modified Newton method is feasible and effective, andoutperforms the Newton method.}     

Solutions for the Linear-Quadratic Control Problem of Markov Jump Linear Systems

The paper is concerned with recursive methods for obtaining the stabilizing solution of coupled algebraic Riccati equations arising in the linear-quadratic control of Markovian jump linear systems by solving at each iteration uncoupled algebraic Riccati equations. It is shown that the new updates carried out at each iteration represent approximations of the original control problem by control problems with receding horizon, for which some sequences of stopping times define the terminal time. Under this approach, unlike previous results, no initialization conditions are required to guarantee the convergence of the algorithms. The methods can be ordered in terms of number of iterations to reach convergence, and comparisons with existing methods in the current literature are also presented. Also, we extend and generalize current results in the literature for the existence of the mean-square stabilizing solution of coupled algebraic Riccati equations.     

Optimal feedback control for constrained mechanical systems

An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we want to use it to control constrained dynamical systems (differential algebraic equations of Index 3). To describe their discrete dynamics, a constrained variational integrators [1] is used. Using a discrete version of the Lagrange-d’Alembert principle yields a forced constrained discrete Euler-Lagrange equation in a position-momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (q_opt, p_opt) and according control input u_opt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input u_R. Adding u_opt and u_R to the discrete Euler-Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

求解非对称代数Riccati 方程几个新的预估-校正法

来源于输运理论的非对称代数Riccati 方程可等价地转化成向量方程组来求解. 本文提出了求解该向量方程组的几个预估-校正迭代格式,证明了这些迭代格式所产生的序列是严格单调递增且有上界,并收敛于向量方程 组的最小正解. 最后,给出了一些数值实验,实验结果表明,本文所提出的算法是有效的.     

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Summary This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.Research partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

The matrix sign function and computations in systems

The matrix sign function has several interesting properties which form the basis of new solution algorithms for problems which occur frequently in systems and control theory applications. Presented in this paper are new algorithms, based on the matrix sign function, for the solution of algebraic matrix Riccati equations, Lyapunov equations, coupled Riccati equations, spectral factorization, matrix square roots, pole assignment, and the algebraic eigenvalue-eigenvector problem. Examples of the application of each algorithm are also presented.     

Second-order nonsmooth optimization for <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> synthesis

The standard way to compute H _∞ feedback controllers uses algebraic Riccati equations and is therefore of limited applicability. Here we present a new approach to the H _∞ output feedback control design problem, which is based on nonlinear and nonsmooth mathematical programming techniques. Our approach avoids the use of Lyapunov variables, and is therefore flexible in many practical situations.     

Solving Differential Matrix Equations using Parareal

Differential matrix equations appear in many applications like optimal control of partial differential equations, balanced truncation model order reduction of linear time varying systems and many more. Here, we will focus on differential Riccati equations (DRE). Solving such matrix-valued ordinary differential equations (ODE) is a highly time consuming process. We present a Parareal based algorithm applied to Rosenbrock methods for the solution of the matrix-valued differential Riccati equations. Considering problems of moderate size, direct matrix equation solvers for the solution of the algebraic Lyapunov equations arising inside the time intgration methods are used. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)