On some iterations for optimal control of jump linear equations

We consider different iterative methods for computing Hermitian solutions of the coupled Riccati equations of the optimal control problem for jump linear systems. We have constructed a sequence of perturbed Lyapunov algebraic equations whose solutions define matrix sequences with special properties proved under proper initial conditions. Several numerical examples are included to illustrate the effectiveness of the considered iterations.     

Stein iterations for the coupled discrete-time Riccati equations

We consider a set of discrete-time coupled algebraic Riccati equations that arise in quadratic optimal control of Markovian jump linear systems. Two iterations for computing a symmetric (maximal) solution of this system are investigated. We construct sequences of the solutions of the decoupled Stein equations and show that these sequences converge to a solution of the considered system. Numerical experiments are given.     

Sensitivity analysis of the algebraic Riccati equations

Summary. In this paper, some sharp perturbation bounds for the Hermitian positive semi-definite solution to an algebraic Riccati equation are developed. A further analysis for these bounds is done. This analysis shows that there is, presumably, some intrinsic relation between the sensitivity of the solution to the algebraic Riccati equation and the distance of the spectrum of the closed-loop matrix from the imaginary axis. Received December 16, 1994     

On equivalence of pencils from discrete-time and continuous-time control

We study two matrix pencils that arise, respectively, in discrete-time and continuous-time optimal and robust control. We introduce a one-to-one transformation between these two pencils. We show that for the pencils under the transformation, their regularity is preserved and their eigenvalues and deflating subspaces are equivalently related. The eigen-structures of the pencils under consideration have strong connections with the associated control problems. Our result may be applied to connect the discrete-time and continuous-time control problems and eventually lead to a unified treatment of these two types of control problems.     

Smooth factorizations of matrix valued functions and their derivatives

Summary In this paper we study the numerical factorization of matrix valued functions in order to apply them in the numerical solution of differential algebraic equations with time varying coefficients. The main difficulty is to obtain smoothness of the factors and a numerically accessible form of their derivatives. We show how this can be achieved without numerical differentiation if the derivative of the given matrix valued function is known. These results are then applied in the numerical solution of differential algebraic Riccati equations. For this a numerical algorithm is given and its properties are demonstrated by a numerical example.     

Cayley transform and the Kronecker product of Hermitian matrices

We consider the conditions under which the Cayley transform of the Kronecker product of two Hermitian matrices can be again presented as a Kronecker product of two matrices and, if so, if it is a product of the Cayley transforms of the two Hermitian matrices. We also study the related question: given two matrices, which matrix under the Cayley transform yields the Kronecker product of their Cayley transforms.     

The convergence rate of continued fractions representing solutions of a Riccati equation

The aim of this note is to generalize and apply results on matrix continued fractions representing the solution of discrete matrix Riccati equations. Assuming uniform bounds for the norm of the matrix coefficients of the continued fraction, the minimal and maximal solutions of the corresponding algebraic Riccati equation can be accurately enclosed.     

Perturbation theory for nonlinear feedback control systems and spencer-goldschmidt integrability of linear partial differential equations

This paper aims to develop the differential-geometric and Lie-theoretic foundations of perturbation theory for control systems, extending the classical methods of Poincaré from the differential equation-dynamical system level where they are traditionally considered, to the situation where the element of control is added. It will be guided by general geometric principles of the theory of differential systems, seeking approximate solutions of the feedback linearization equations for nonlinear affine control systems. In this study, certain algebraic problems of compatibility of prolonged differential systems are encountered. The methods developed by D. C. Spencer and H. Goldschmidt for studying over-determined systems of partial differential equations are needed. Work in the direction of applying theio theory is presented.Supported by grants from the Ames Research Center of NASA and the Applied Mathematics and Systems Research Programs of the National Science Foundation     

Residual bounds of approximate solutions of the discrete-time algebraic Riccati equation

Let be a Hermitian matrix which approximates the unique Hermitian positive semi-definite solution to the discrete-time algebraic Riccati equation (DARE) where , is Hermitian positive definite, , the pair is stabilizable, and the pair is detectable. Assume that is nonsingular, and is stable. Let , and let be the residual of the DARE with respect to . Define the linear operator by The main result of this paper is: If where denotes any unitarily invariant norm, and then Received June 7, 1995 / Revised version received February 28, 1996     

The time-varying discrete four block Nehari problem: A generalized Popov-Yakubovich type approach

The paper provides necessary and sufficient solvability conditions for the time-variant discrete four block Nehari problem in terms of the existence of the stabilizing solutions to two coupled Riccati equations. A parametrization of the class of all solutions is also given. The results are easily obtained from a signature condition — a generalized Popov Yakubovich type argument-imposed on an appropiate rational node. The present development may be seen as an alternative of the theory developed by Gohberg, Kaashoek and Woerdeman [15].     

Invariants for feedback equivalence and Cauchy characteristic multifoliations of nonlinear control systems

Linearization of a nonlinear feedback control system under nonlinear feedback is treated as a problem of equivalence-under the Lie pseudogroup of feedback transformations-of distributions on the product manifold of the state and control variables. The new feature of this paper is that it introduces the Cauchy characteristic sub-distributions of these distributions and their derived distributions. These Cauchy characteristic distributions are involutive and nested, hence define a Multifoliate Structure. A necessary condition for feedback equivalence of two nonlinear control systems is that these multifoliations be transformed under the feedback pseudogroup. For linear systems, this Cauchy characteristic multifoliate structuee is readily computed in terms of the (A, B)-matrix that defines the linear system. Assuming that the conditions for local feedback linearization are satisfied, the existence of a global feedback linearizing transformation is dependent on computing an element of the first cohomology group of the space with coefficients in the sheaf of groupoid of infinitesimal feedback automorphisms of the linear system. The theorem quoted above about the Cauchy characteristic multifoliations provides some information about this groupoid. It is computed explicitly and directly for control systems with one- or two-state dimensions. Finally, these Cauchy characteristic sub-distributions must inevitably play a role in the numerical or symbolic computational analysis of the Hunt-Su partial differential equations for the feedback-linearizing transformation.Senior Research Associate of the National Research Council at the Ames Research Center of NASA.     

A note on the regularity of solutions of infinite dimensional Riccati equations

This note is concerned with the regularity of solutions of algebraic Riccati equations arising from infinite dimensional LQR control problems. We show that distributed parameter systems described by certain parabolic partial differential equations often have a special structure that smooths solutions of the corresponding Riccati equation. This analysis is motivated by the need to find specific representations for Riccati operators that can be used in the development of computational schemes for problems where the input and output operators are not Hilbert-Schmidt. This situation occurs in many boundary control problems and in certain distributed control problems associated with optimal sensor/actuator placement.     

A numerical method for computing the Hamiltonian Schur form

We derive a new numerical method for computing the Hamiltonian Schur form of a Hamiltonian matrix that has no purely imaginary eigenvalues. We demonstrate the properties of the new method by showing its performance for the benchmark collection of continuous-time algebraic Riccati equations. Despite the fact that no complete error analysis for the method is yet available, the numerical results indicate that if no eigenvalues of are close to the imaginary axis then the method computes the exact Hamiltonian Schur form of a nearby Hamiltonian matrix and thus is numerically strongly backward stable. The new method is of complexity and hence it solves a long-standing open problem in numerical analysis. Volker Mehrmann was supported by Deutsche Forschungsgemeinschaft, Research Grant Me 790/11-3.     

Instability indices for matrix polynomials

There is a well-established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are Hermitian and skew-Hermitian. This theory relates the number of negative directions for the matrix coefficients which are Hermitian to the total number of unstable eigenvalues for the polynomial. Herein we extend the theory to ?-even matrix polynomials of any finite degree. In particular, unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two. We also consider Hermitian matrix polynomials, and derive an index which counts the number of eigenvalues with nonpositive imaginary part. The results are refined if we consider the Hermitian matrix polynomial to be a perturbation of a ?-even polynomials; however, this refinement requires additional assumptions on the matrix coefficients.     

Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory

We determine and compare the convergence rates of various fixed-point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic Riccati equations arising in transport theory.     

Exponential Stability for Discrete Time Linear Equations Defined by Positive Operators

In this paper the problem of exponential stability of the zero state equilibrium of a discrete-time time-varying linear equation described by a sequence of linear positive operators acting on an ordered finite dimensional Hilbert space is investigated. The class of linear equations considered in this paper contains as particular cases linear equations described by Lyapunov operators or symmetric Stein operators as well as nonsymmetric Stein operators. Such equations occur in connection with the problem of mean square exponential stability for a class of difference stochastic equations affected by independent random perturbations and Markovian jumping as well us in connection with some iterative procedures which allow us to compute global solutions of discrete time generalized symmetric or nonsymmetric Riccati equations. The exponential stability is characterized in terms of the existence of some globally defined and bounded solutions of some suitable backward affine equations (inequalities) or forward affine equations (inequalities).     

Join-semidistributive lattices and convex geometries

We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for the different embedding results in the class of join-semidistributive lattices. In particular, we prove that every finite join-semidistributive lattice can be embedded into a lattice S_P(A) of algebraic subsets of a suitable algebraic lattice A. This latter construction, S_P(A), is a key example of a convex geometry that plays an analogous role in hierarchy of join-semidistributive lattices as a lattice of equivalence relations does in the class of modular lattices. We give numerous examples of convex geometries that emerge in different branches of mathematics from geometry to graph theory. We also discuss the introduced notion of a strong convex geometry that might promise the development of rich structural theory of convex geometries.     

Note on the Riccati method for differential eigenvalue problems showing algebraic growth

The Riccati transformation method is used to pick out eigenvalues of a system of linear ordinary differential equations on a semi-infinite interval. The eigenvalues are determined by the condition which ensures algebraic rather than exponential growth of the solutions as the independent variable tends to infinity. Numerical results which demonstrate the viability of the Riccati transformation method for this type of problem are included.     

Condition numbers of algebraic Riccati equations in the Frobenius norm

Consider the continuous-time algebraic Riccati equation (CARE) and the discrete-time algebraic Riccati equation (DARE) which arise in linear control and system theory. It is known that appropriate assumptions on the coefficient matrices guarantee the existence and uniqueness of Hermitian positive semidefinite stabilizing solutions. In this note, we apply the theory of condition developed by Rice to define condition numbers of the CARE and DARE in the Frobenius norm, and derive explicit expressions of the condition numbers in a uniform manner. Both the complex case and real case are considered, and connections to certain existing condition numbers of the CARE and DARE are discussed.