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.     

Discrete J-spectral factorization of possibly singular polynomial matrices

We develop two state-space algorithms for discrete-time J-spectral factorization of possibly singular, possibly non-column-reduced, and with possible zeros at the origin para-hermitian matrices, by assignment of matrices in optimal LQ return difference equality. The column degrees of the spectral factor equal the column degrees of the given matrix and the dimension of a discrete-time algebraic Riccati system equals the sum of the column degrees. Necessary and sufficient condition for the least-order property of our state-space realizations is column reduction of the given matrix.     

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.     

(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.     

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.     

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.     

On the discrete and continuous time infinite-dimensional algebraic Riccati equations

The standard state space solution of the finite-dimensional continuous time quadratic cost minimization problem has a straightforward extension to infinite-dimensional problems with bounded or moderately unbounded control and observation operators. However, if these operators are allowed to be sufficiently unbounded, then a strange change takes place in one of the coefficients of the algebraic Riccati equation, and the continuous time Riccati equation begins to resemble the discrete time Riccati equation. To explain why this phenomenon must occur we discuss a particular hyperbolic PDE in one space dimension with boundary control and observation (a transmission line) that can be formulated both as a discrete time system and as a continuous time system, and show that in this example the continuous time Riccati equation can be recovered from the discrete time Riccati equation. A particular feature of this example is that the Riccati operator does not map the domain of the generator into the domain of the adjoint generator, as it does in the standard case.     

Fractional transformation group induced byQR factorization and linear prediction problem

A new fractional transformation group is found which acts transitively on the space of linear predictors for nonstationary processes by using the QR factorization of nonsingular matrices.     

Stable Hermitian solutions of discrete algebraic Riccati equations

Stable and Lipschitz stable hermitian solutions of the discrete algebraic Riccati equations are characterized, in the complex as well as in the real case.This paper was written while the first author visited The College of William and Mary.Partially supported by NSF Grant DMS-8802836 and by the Binational United States-Israel Science Foundation.     

Lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations

In this paper, we propose lower matrix bounds for the continuous algebraic Riccati and Lyapunov matrix equations. We give comparisons between the parallel estimates. Finally, we give examples showing that our bounds can be better than the previous results for some cases.     

Synthesis of stable H controller via the chain scattering framework

A sufficient condition for the existence of suboptimal stable stabilizing H^∞ controllers is given. By exploiting the free parameter in the parameterization of stabilizing controllers and using the chain scattering framework, we reformulate the H^∞ strong stabilization problem as an equivalent H^∞ optimization problem which can be solved via only one algebraic Riccati equation. A parameterization of all suboptimal stable stabilizing H^∞ controllers is also given.     

On the H fixed-lag smoothing: how to exploit the information preview

The problem of the continuous-time H^∞ fixed-lag smoothing over the infinite horizon is studied. The first solution to the problem is derived in terms of one algebraic Riccati equation of the same dimension as in the filtering case and the mechanism by which the performance improvements with respect to the H^∞ filtering occur is clarified. It is shown that the H^∞ smoother exploits the information preview in an “H² manner”.     

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.     

Product of nonnegative operators and infinite-dimensional H∞ Riccati equations

In this paper we collect some useful properties of the product of nonnegative operators in a Hilbert space. We then apply them to the standard H_∞-control problem for infinite-dimensional time-varying systems and give necessary and sufficient conditions for the existence of a suboptimal controller by three conditions involving two independent Riccati equations with a coupling inequality.     

-Insensitizing controls for pointwise observations of the heat equation

In this paper, we study the -insensitizing controllability for the functional given by integrating in time the square of the solution of the heat equation in a finite number of points of the domain , i.e., when the observation set is reduced to a finite set of points. We reduce the controllability problem to a unique continuation property for a cascade system of linear heat equations and we obtain a positive result in the case N3.     

Closed formulae for a parametric-mixed-sensitivity problem for Pritchard-Salamon systems

The parametric-mixed-sensitivity problem is posed for the class of irrational transfer matrices with a realization as a Pritchard-Salamon system. An exact solution is obtained in terms of two uncoupled, standard Riccati equations.     

Finite Integer Computations: An Algebraic Foundation for Their Correctness

Even though computations on finite integer representations are as old as computers themselves, there is one problem that has been inexcusably neglected: integer computations in programming languages do not operate on the ring of integer numbers but only on finite subsets of it. Changing the size of integer representations may change the results of operations performed on them in unexpected ways. In particular, increasing the representation size of intermediate results during computation may lead to incorrect final results. In this paper, we develop an algebraic foundation for integer arithmetic under changing representation sizes and, in particular, a criterion for safely replacing one finite integer arithmetic by another. This safety criterion has also been verified in the theorem prover Isabelle/HOL. Based on this formal development, we not only reveal and explain an inconsistency in the Java Card integer arithmetic but also propose an optimization for Java Card integer expressions and their safe transformation into Java Card bytecode. We also discuss the application of this safety criterion to constant folding, a standard compiler optimization, for Java and C compilers. Received August 2004 Revised January 2006 Accepted February 2006 by C. B. Jones     

Controller design and reduction of bullwhip for a model supply chain system using z-transform analysis

In this work, a discrete time series model of a supply chain system is derived using material balances and information flow. Transfer functions for each unit in the supply chain are obtained by z-transform. The entire chain can be modeled by combining these transfer functions into a close loop transfer function for the network. The model proves to be very useful in revealing the dynamics characteristic of the system. The system can be viewed as a linear discrete system with lead time and operating constraints. The stability of the system can be analyzed using the characteristic equation. Controllers are designed using frequency analysis. The bullwhip effect, i.e. magnification of amplitudes of demand perturbations from the tail to upstream levels of the supply chain, is a very important phenomenon for supply chain systems. We proved that intuitive operation of a supply chain system with demand forecasting will cause bullwhip. Moreover, lead time alone would not cause bullwhip. It does so only when accompanied by demand forecasting. Furthermore, we show that by implementing a proportional intergral or a cascade inventory position control and properly synthesizing the controller parameters, we can effectively suppress the bullwhip effect. Moreover, the cascade control structure is superior in meeting customer demand due to its better tracking of long term trends of customer demand.