Showing papers by "Brian D. O. Anderson published in 1972"
TL;DR: Two small-gain theorems and a passivity theorem yielding stability results for general feedback systems were shown to be equivalent in this article, and two small gain theorem and a small gain theorem were also shown to yield stability.
Abstract: Two small-gain theorems and a passivity theorem yielding stability results for general feedback systems are shown to be equivalent.
109 citations
01 Sep 1972
TL;DR: In this paper, it was shown that the positive-definite solution of the algebraic equation PF + F?P?PGR?1G?P + S = 0, provided that it exists and is unique, can be obtained as the limiting solution of a quadratic matrix difference equation.
Abstract: The paper describes an iterative algorithm for computing the limiting, or steady-state, solution of the matrix Riccati differential equation associated with quadratic minimisation problems in linear systems. It is shown that the positive-definite solution of the algebraic equation PF + F?P?PGR?1G?P + S = 0, provided that it exists and is unique, can be obtained as the limiting solution of a quadratic matrix difference equation that converges from any nonnegative definite initial condition. The algorithm is simple, and, at least for moderate dimensions of the solution matrix, competitive in computational effort with other current techniques for obtaining the limiting solution of the Riccati equation.
39 citations
TL;DR: In this article, the Cauchy index of a real rational function can be computed by evaluating the signature of a certain Hankel matrix, and alternative procedures for its computation are presented, one of which offers greater computational simplicity.
Abstract: The Cauchy index of a real rational function can be computed by evaluating the signature of a certain Hankel matrix. Alternative procedures for its computation are presented here, one of which offers greater computational simplicity.
27 citations
TL;DR: The conditional Fokker Planck equation yielding the probability density of the state of a nonlinear dynamical system, conditioned on measurements over a fixed interval, is derived in a novel way.
Abstract: An equation is derived for the probability density of the state of a nonlinear dynamical system, conditioned on measurements over a fixed interval. In deriving the equation, the conditional Fokker Planck equation yielding the probability density of the filtering problem is used several times in a novel way.
24 citations
20 citations
TL;DR: In this article, it was shown that exponential asymptotic stability of the homogeneous part of the system is equivalent to the system mapping inputs with finite L 2 norm into outputs with finite l 2 norm.
Abstract: Finite-dimensional linear systems are considered with the properties of uniform complete controllability and observability. It is shown that exponential asymptotic stability of the homogeneous part of the system is equivalent to the system mapping inputs with finite L 2 norm into outputs with finite L 2 norm.
13 citations
12 citations
TL;DR: In this article, the smoothed estimate of the state of a noisy linear system in terms of filtered estimates of the system's state, for both continuous and discrete time, is given.
Abstract: Formulas are given expressing the smoothed estimate of the state of a noisy linear system in terms of filtered estimates of the state, for both continuous and discrete time.
11 citations
10 citations
TL;DR: In this article, three basic methods for constructing suboptimal continuous-time linear fixed-lag smoothers are described, and the degree of complexity of these smooths determines how closely they perform to the limit obtainable from the theoretical optimal linear fixedlag smoother.
Abstract: Three basic methods for constructing suboptimal continuous-time linear fixed-lag smoothers are described. In contrast to the optimal smoother, all the suboptimal smoothers are stable. The degree of complexity allowed in the suboptimal smoothers determines how closely they perform to the limit obtainable from the theoretical optimal linear fixed-lag smoother.
5 citations
TL;DR: In this article, the classical results connecting Hurwitz polynomials, LC positive real functions, and RC real functions are all viewed as being equivalent to the positive definiteness of a symmetric matrix.
Abstract: Classical results connecting Hurwitz polynomials, LC positive real functions, and RC positive real functions are all viewed as being equivalent to the positive definiteness of a symmetric matrix.
TL;DR: In this article, the question is asked: When does there exist a family of coordinate transformations defined by a set of nonsingular matrices T(\lambda), continuously dependent on \lambda, with T(0) = I and with mapping the state vector associated with one minimal realization into the one associated with the other?
Abstract: Suppose one is given two minimal realizations of the same transfer function matrix. The question is asked: When does there exist a family of coordinate transformations defined by a set of nonsingular matrices T(\lambda) , continuously dependent on \lambda , with T(0) = I and with (1) mapping the state vector associated with one minimal realization into the state vector associated with the other? The quesion is answered, and a procedure is given for constructing the family when it exists.