Necessary and sufficient conditions for existence of J-spectral factorization for para-Hermitian rational matrix functions

For a para-Hermitian rational matrix function G(λ)=J+C(λI_p−A)⁻¹B, where J=J∗ is invertible, and which has no poles and zeros on the imaginary line, we give necessary and sufficient conditions in terms of A,B,C and J for the existence of a J-spectral factorization, as well as an algorithm to obtain the J-spectral factor in case it exists.     

Polynomial J-spectral factorization in minimal state space

By a specific choice of matrices in optimal LQ return difference equality, we develop a simple state-space algorithm for polynomial J-spectral factorization. For this purpose we solve a minimal-order algebraic Riccati equation, the order obtained by solving an Integer Linear Programming (ILP) problem. An example that cannot be solved by the existing algorithms illustrates our algorithm.     

Polynomial solutions to H∞ problems

The paper presents a polynomial solution to the standard H_∞-optimal control problem. Based on two polynomial J-spectral factorization problems, a parameterization of all suboptimal compensators is obtained. A bound on the McMillan degree of suboptimal compensators is derived and an algorithm is formulated that may be used to solve polynomial J-spectral factorization problems.     

A numerical study of the Navier–Stokes-αβ model

We present a numerical study of the NS-αβ model, which is a recently proposed multiscale variation of the NS-α model that attempts to recapture scales lost through over-regularization by separately modeling dissipation-range scales. We develop a similarity theory for the new model which shows that it is better equipped than the NS-α model to capture smaller-scale behavior. Next, we propose and study an unconditionally stable, optimally accurate, and efficient finite-element implementation for the NS-αβ model; rigorous proofs for stability and convergence are provided. Finally, we present results from two numerical experiments that demonstrate the advantages of the NS-αβ model over the NS-α model.     

Sub-optimal Hankel norm approximation problem: a frequency-domain approach

We obtain a simple solution for the sub-optimal Hankel norm approximation problem for the Wiener class of matrix-valued functions. The approach is via J-spectral factorization and frequency-domain techniques.     

New Algorithms for Polynomial J-Spectral Factorization

In this paper new algorithms are developed for J-spectral factorization of polynomial matrices. These algorithms are based on the calculus of two-variable polynomial matrices and associated quadratic differential forms, and share the common feature that the problem is lifted from the original one-variable polynomial context to a two-variable polynomial context. The problem of polynomial J-spectral factorization is thus reduced to a problem of factoring a constant matrix obtained from the coefficient matrices of the polynomial matrix to be factored. In the second part of the paper, we specifically address the problem of computing polynomial J-spectral factors in the context of H _∞ control. For this, we propose an algorithm that uses the notion of a Pick matrix associated with a given two-variable polynomial matrix. Date received: January 1, 1998. Date revised: October 15, 1998.     

Nehari problems and equalizing vectors for infinite-dimensional systems

For a class of infinite-dimensional systems we obtain a simple frequency domain solution for the suboptimal Nehari extension problem. The approach is via J-spectral factorization, and it uses the concept of an equalizing vector. Moreover, the connection between the equalizing vectors and the Nehari extension problem is given.     

Algebraic Riccati equation and J-spectral factorization for estimation

In this paper we investigate on the existence of the stabilizing solution of the algebraic Riccati equation (ARE) related to the filtering problem with a prescribed attenuation level γ. It is well known that such a solution exists and is positive definite for γ larger than a certain γ_F and it does not exist for γ smaller than a certain γ₀. We consider the intermediate case γ(γ₀,γ_F] and show that in this interval the stabilizing solution does exist, except for a finite number of values of γ. We show how the solution of the ARE may be employed to obtain a minimum-phase J-spectral factor of the J-spectrum associated with the filtering problem.     

(J, J′)-lossless factorization based on conjugation

This paper is concerned with a derivation of the state-space form of the (J, J′)-lossless factorization which contains both the inner-outer factorization and the spectral factorization of positive matrices as special cases. Also, the (J, J′)-lossless factorization gives a unified framework of H^∞ control theory. We use the method of conjugation which makes the derivation much simpler than the previous literature, most of which used the technique of (J, J′)-spectral factorization. A necessary and sufficient condition is represented in terms of two Riccati equations one of which is degenerated.     

Discrete J-spectral factorization

This paper deals with the J-spectral factorization for general discrete rational matrices. A simple approach based on the Kalman filtering in Krein space is proposed. The main idea is to construct a stochastic state space filtering model in Krein space such that the spectral matrix of the output is equal to the rational matrix to be factorized. The spectral factor is then easily derived by using the generalized Kalman filtering in Krein space, which is similar to the H₂ spectral factorization. Our approach unifies the treatment of the H₂ spectral factorization and the J-spectral factorization. The applications of the derived results in H_∞ and risk-sensitive estimation for both nonsingular and singular systems are demonstrated.     

J-spectral factorization and equalizing vectors

For the Wiener class of matrix-valued functions we provide necessary and sufficient conditions for the existence of a J-spectral factorization. One of these conditions is in terms of equalizing vectors. The second one states that the existence of a J-spectral factorization is equivalent to the invertibility of the Toeplitz operator associated to the matrix to be factorized. Our proofs are simple and only use standard results of general factorization theory (Clancey and Gohberg, Factorization of Matrix Functions and Singular Integral Operators, Operator Theory: Advances and Applications, Vol. 3, Birkhäuser, Basel, 1981). Note that we do not use a state space representation of the system. However, we make the connection with the known results for the Pritchard–Salamon class of systems where an equivalent condition with the solvability of an algebraic Riccati equation is given.     

Model-free Q-learning designs for linear discrete-time zero-sum games with application to H-infinity control

In this paper, the optimal strategies for discrete-time linear system quadratic zero-sum games related to the H-infinity optimal control problem are solved in forward time without knowing the system dynamical matrices. The idea is to solve for an action dependent value function Q(x,u,w) of the zero-sum game instead of solving for the state dependent value function V(x) which satisfies a corresponding game algebraic Riccati equation (GARE). Since the state and actions spaces are continuous, two action networks and one critic network are used that are adaptively tuned in forward time using adaptive critic methods. The result is a Q-learning approximate dynamic programming (ADP) model-free approach that solves the zero-sum game forward in time. It is shown that the critic converges to the game value function and the action networks converge to the Nash equilibrium of the game. Proofs of convergence of the algorithm are shown. It is proven that the algorithm ends up to be a model-free iterative algorithm to solve the GARE of the linear quadratic discrete-time zero-sum game. The effectiveness of this method is shown by performing an H-infinity control autopilot design for an F-16 aircraft.     

A Toeplitz algorithm for polynomial J-spectral factorization

A block Toeplitz algorithm is proposed to perform the J-spectral factorization of a para-Hermitian polynomial matrix. The input matrix can be singular or indefinite, and it can have zeros along the imaginary axis. The key assumption is that the finite zeros of the input polynomial matrix are given as input data. The algorithm is based on numerically reliable operations only, namely computation of the null-spaces of related block Toeplitz matrices, polynomial matrix factor extraction and linear polynomial matrix equations solving.     

Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon

This paper is concerned with a stochastic linear-quadratic (LQ) problem in an infinite time horizon with multiplicative noises both in the state and the control. A distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite. A new type of algebraic Riccati equation – called a generalized algebraic Riccati equation (GARE) – is introduced which involves a matrix pseudo-inverse and two additional algebraic equality/inequality constraints. It is then shown that the well-posedness of the indefinite LQ problem is equivalent to a linear matrix inequality (LMI) condition, whereas the attainability of the LQ problem is equivalent to the existence of a “stabilizing solution” to the GARE. Moreover, all possible optimal controls are identified via the solution to the GARE. Finally, it is proved that the solution to the GARE can be obtained via solving a convex optimization problem called semidefinite programming.     

Yager’s new class of implications Jf and some classical tautologies

Recently, Yager [R. Yager, On some new classes of implication operators and their role in approximate reasoning, Information Sciences 167 (2004) 193-216] has introduced a new class of fuzzy implications, denoted J_f, called the f-generated implications and has discussed some of their desirable properties, such as neutrality, exchange principle, etc. In this work, we discuss the class of J_f implications with respect to three classical logic tautologies, viz., distributivity, law of importation and contrapositive symmetry. Necessary and sufficient conditions under which J_f implications are distributive over t-norms and t-conorms and satisfy the law of importation with respect to a t-norm have been presented. Since the natural negations of J_f implications, given by N_Jf(x)=J_f(x,0), in general, are not strong, we give sufficient conditions under which they become strong and possess contrapositive symmetry with respect to their natural negations. When the natural negations of J_f are not strong, we discuss the contrapositivisation of J_f. Along the lines of J_f implications, a new class of implications called h-generated implications, J_h, has been proposed and the interplay between these two types of implications has been discussed. Notably, it is shown that while the natural negations of J_f are non-filling those of J_h are non-vanishing, properties which determine the compatibility of a contrapositivisation technique.     

The extended generalized distance problem in discrete time

We obtain the class of all solutions to the extended (two block) generalized distance problem for discrete-time systems by employing the so-called ‘signature condition’—a generalized Popov theory type argument which parallels the J-spectral factorization approach. The novelty is that we derive explicit state-space formulae in terms of one Riccati and one Lyapunov equation while we remove the usual assumption in the discrete case on the time reversibility (invertibility of the state matrix) of the system to be approximated. © 1998 John Wiley & Sons, Ltd.     

Efficient algorithms for volterra integral equations of the second kind

Spline function of degreem, deficiencyJ?1, i. e. inC ^m?J , are used in conjunction with (Gaussian) quadrature rules to construct algorthms for the numerical solution of a general Volterra integral equation of the second kind. For a givenm, the method is of order (m+1) and, in general, requires 0(N) evaluations of the kernel. This is in sharp contrast to the 0(N ²) evaluations required by hitherto known methods. It is shown that the method for spline functions with full continuity (J=1) is numerically unstable for allm>2. However, stability is established forJ=m, m?1, for allm. Furthermore, form=3,J=1, it is demonstrated that by appropriately modifying the original method, a whole family of stable methods is obtained.     

广义代数Riccati方程和最优调节器的研究

利用能稳性和精确能观性, 对广义代数Riccati方程和相关的随机最优调节器问题进行了深入的研究. 对广义代数Riccati方程得到了下列结果: 如果随机系统既是能稳定的又是精确能观的, 则广义代数Riccati方程有一个最大解, 同时也是一个反馈镇定解. 在精确能观性的假设下, 广义代数Riccati方程的所有非负定解(如果存在的话)必是正的反馈镇定解. 作为应用, 最优调节器问题, 广义代数Riccati方程的最大解, 反馈镇定解三者之间的关系获得了澄清. 所有这些结果在随机控制和随机稳定性理论中是有