In this paper a new algorithm for solving algebraic Riccati equations (both continuous-time and discrete-time versions) is presented. The method studied is a variant of the classical eigenvector approach and uses instead an appropriate set of Schur vectors thereby gaining substantial numerical advantages. Complete proofs of the Schur approach are given as well as considerable discussion of numerical issues. The method is apparently quite numerically stable and performs reliably on systems with dense matrices up to order 100 or so, storage being the main limiting factor. The description given below is a considerably abridged version of a complete report given in [0].

A Schur method for solving algebraic Riccati equations

Control Theory for Linear Systems deals with the mathematical theory of feedback control of linear systems. It treats a wide range of control synthesis problems for linear state space systems with inputs and outputs. The book provides a treatment of these problems using state space methods, often with a geometric flavour. Its subject matter ranges from controllability and observability, stabilization, disturbance decoupling, and tracking and regulation, to linear quadratic regulation, $H_2$ and $H_\infty$ control, and robust stabilization. Each chapter of the book contains a series of exercises, intended to increase the reader's understanding of the material. Often, these exercises generalize and extend the material treated in the regular text.

/pdf/control-theory-for-linear-systems-24kzsm66fq.pdf

Control theory for linear systems

Exponential dichotomies and transversal homoclinic points

The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert's 16th problem? Section 2: The first part of Hilbert's 16th problem. Section 3: The second part of Hilbert's 16th problem: introduction. Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop. Section 5: Finiteness problem. Section 6: The weakened Hilbert's 16th problem. Section 7: Global and local bifurcations of Zq–equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n.

Hilbert's 16th problem and bifurcations of planar polynomial vector fields

This paper represents a first attempt to derive a closed-form (Hankel-norm) optimal solution for multivariable system reduction problems. The basic idea is to extend the scalar ease approach in [5] to deal with the multivariable systems. The major contribution lies in the development of a minimal degree approximation (MDA) theorem and a computation algorithm. The main theorem describes a closed-form formulation for the optimal approximants, with the optimality verified by a complete error analysis. In deriving the main theorem, some useful singular value/vector properties associated with block-Hankel matrices are explored and a key extension theorem is also developed. Imbedded in the polynomial-theoretic derivation of the extension theorem is an efficient approximation algorithm. This algorithm consists of three steps: i) compute the minimal basis solution of a polynomial matrix equation; ii) solve an algebraic Riccati equation; and iii) find the partial fraction expansion of a rational matrix.

Optimal Hankel-norm model reductions: Multivariable systems

Matrix quadratic equations have found the most diverse applications. The present article gives a connected account of their theory, and contains some new results and new proofs of known results.

/pdf/matrix-quadratic-equations-2vn284iqg2.pdf

Matrix quadratic equations

The properties of the degree of a matrix of rational functions are obtained in a simplified way, which enables them to be generalised to matrices whose elements are not necessarily rational functions. On the basis of these results a theory of realisations is developed, which similarly generalises the theory of state space realisations of a matrix of rational functions.

/pdf/matrices-of-rational-functions-4lbeojfob6.pdf

Matrices of rational functions

An integral manifold for a system of differential equations is a manifold such that any solution of the equations which has a point on it is entirely contained on it. The method of averaging establishes the existence of such a manifold for a system which is a perturbation of an autonomous system with a periodic orbit. The existence of the manifold is established here under more general hypotheses, namely for perturbations which are ‘integrally small’. The method differs from the original method of Bogolyubov and Mitropolskii and operates directly with the individual solutions. This is made possibly by the use of an appropriate norm, and is equivalent to solving the partial differential equation which occurs in work by Moser and Sacker by the method of characteristics rather than by the introduction of an artificial viscosity term. Moreover, detailed smoothness properties of the manifold are obtained. For periodic perturbations the integral manifold is a torus and these smoothness properties are just sufficient to permit the application of Denjoy's theorem.

/pdf/averaging-and-integral-manifolds-ii-4r6gu3zmph.pdf

Averaging and integral manifolds (II)

It is shown that a quadratic system with a degenerate critical point has at most one limit cycle.

/pdf/quadratic-systems-with-a-degenerate-critical-point-1q0wai07wa.pdf

Quadratic systems with a degenerate critical point

The basic elementary results about convex sets are derived successively from various properties of segments. The complete set of properties is shown to form a natural set of axioms characterising the convex sets in a real vector space

W.A. Coppel

Papers

Matrix quadratic equations

Matrices of rational functions

Averaging and integral manifolds (II)

Quadratic systems with a degenerate critical point

Axioms for convexity