Showing papers by "John B. Moore published in 1968"

Algebraic Structure of Generalized Positive Real Matrices

Abstract: Square matrices $Z( \cdot )$ of real rational functions of a complex variable are considered with two properties : (1) $Z(\infty )$ has finite elements; (2) $Z(j\omega ) + Z'( - j\Omega )$ is nonnegative definite Hermitian for all real $\omega $, other than those for which $j\omega $ is a pole of an element of $Z( \cdot )$. Necessary and sufficient conditions for the nonnegativity property are derived which involve the existence of constant. matrices satisfying several algebraic equations. The work thereby extends earlier results on the structure of rational positive real matrices.

A generalization of the Popov criterion

Abstract: A criterion for the stability of control systems which contain an arbitrary finite number of memoryless nonlinearities is considered. The criterion is such that the degree of stability may be specified, and such that for the case when the absolute stability of a control system having one memoryless nonlinearity is considered, it reduces to the original Popov criterion.

Extensions of quadratic minimization theory I. Finite time results

Abstract: Necessary and sufficient conditions are developed for the existence and calculation of well-defined Riccati differential equation solutions associated with quadratic loss minimization problems. Of particular interest is the fact that a. covariance condition is involved. The disclosure of this condition not only extends the range of optimal control problems for which a solution, guaranteed to be well defined, may be calculated, but also introduces an approach for establishing the existence of well-defined solutions in other problems involving covariance conditions, as for example, in a time-varying spectral factorization procedure. This paper is concerned with finite time results, whilo a companion paper considers tho infinite time case.

Simulation of stationary stochastic processes

Abstract: A ‘spectral-factorisation’ procedure involving the solution of a Riccati matrix differential equation is considered to determine systems which, with white-noise input signals, may be used in the simulation of stochastic processes having prescribed stationary covariances. More spe$itically, the specification of a system is made so that the covariance of the system output is a prescribed stationary covariance R(t – ~) for all t and -r greater than or equal to the ‘switch-on’ time of the system. The advantage of the ‘spectralfactorisation’ procedure described compared with those previously given is that, assuming an initial-state mean of zero, a suitable initial-state covariance is calculated as an intermediate result in the procedure. The calculation of an appropriate initial-state covariance is of interest since, if zero initial conditions are used in an attempted simulation, an undesirable time lapse may be necessary for the output covariance to be acceptable as a simulation of the prescribed stationary covariance. For the case when the system is given or is determined using alternative procedures to those described in the paper, the initial-state covariance is calculated from the solution of a linear matrix equation. The problem considered in the paper is the simulation of stationary stochastic processes with prescribed covariances using linear, finite-dimensional, time-invariant systems with white-noise input. Of particular interest is the selection of an initial-state covariance, so that the covariance of the outputs will be indistinguishable from that observed over the same time interval for the hypothetical limiting case as the initial time approaches – m. Systems which may be used in the simulation of stationary stochastic processes with prescribed covariances may be determined from any of a number of spectral-factorisation procedures, ] z,* With regard to the initial conditions, Cttrrent practice is to set these to zero and ignore the outputs for a period corresponding to a few time constants of the system. The inadequacy of this procedure has been recognised.3 In the paper two results are presented. The first is a method for selecting an initial-state covariance for a given system, so that the application of white noise at the input results in outputs that may be considered, after the switch-on, as sample functions of a stationary stochastic process; this is the best possible real-time simulation for a stationary stochastic process. All that is required in order to obtain the result is the solution of a linear matrix equation. The second result of the paper is a spectral factorisation of a specified covariance matrix using theorems from Anderson. t The procedure gives a system having a stable transferfunction matrix with a stable inverse (often required in certain optimisation problems), together with the initial-state covariance; the advantage of the particular approach presented is that all the information necessary for the simulation is given in one procedure. The key step in the procedure is the solution of a quadratic matrix equation which satisfies certain constraints. This solution, which is unique, may be found using algebraic means similar to those of Reference 4 or by determining the steadystate solution of a Riccati matrix differential equation. t The method avoids the need to carry out any of the procedures in References 1, 2 or *, which prove very complex in cases where the covariance is a matrix rather than scalar.

Convergence properties of Riccati equation solutions

Abstract: Existence results are developed for Riccati equations. In particular, it is shown that the existence of one solution to a Riccati equation implies the existence of a whole family of solutions whose initial condition lies in a cone determined by the initial condition associated with the known solution.

A circle criterion generalization for relative stability

Abstract: It is shown that the circle criterion for linear systems containing a single time-varying element corresponds to a specialization of the Popov stability theory. The circle criteria are generalized using the Popov theory to give some measure of the degree of the system stability.

'Circle criteria' in the parameter plane

Abstract: The `circle criteria? for giving stability information of linear systems containing one time-varying element are shown to have useful graphical interpretations for design purposes on a parameter-plane diagram. The significance of the parameter-plane approach is that, in a system design, the adjustable parameters of either the time-varying element or the time-invariant subsystem may be selected to satisfy the system stability constraints directly from the diagram. This means that for some design problems the parameter-plane approach is more efficient than the application of the well known complex-plane methods.

Optimal State Estimation in High Noise

Abstract: The problem is examined of estimating the state of a linear dynamical system in the presence of high measurement noise. It is concluded that optimal filter design may be simplified to the extent that it need not depend on the solution of a matrix Riccati differential equation, but only on the solution of a matrix linear differential equation. For a related problem, that of estimating a signal s(t) given noisy measurements s(t) + w(t) where the noise is large and the covariance of s(t) is known, optimal filter design is immediate.

Liapunov function generation for a class of time-varying systems

Abstract: Finite dimensional systems with time-varying feedback, which satisfy the conditions of the circle criterion for stability, are considered. A recent result giving a system theory description of positive real matrices is used to generate Liapunov functions.

Optimum differentiation using Kalman filter theory

Abstract: Consideration is given to the construction of an optimum differentiator to give the minimum-variance unbiased estimate of the first derivatives of random signals corrupted by white noise It is assumed that the signals are differentiable and are the outputs of a known linear finite-dimensional (possibly time-varying) system excited by white noise Extension of the results to consider higher-order differentiation is straightforward

Structural stability of linear time-varying systems

Abstract: An application of Popov criterion generalizations for time-varying systems is considered in regard to the tolerance of small amounts of memoryless sector nonlinearities existent in any practical realization of a linear system. It is shown that such nonlinearities can be tolerated if they are sufficiently small, without disturbing system stability.

Optimal and suboptimal performance of stochastic linear tracking systems

Abstract: The derivation of performance indices for both optimal and suboptimal linear tracking systems is organised so that the expressions for the indices consist of terms that may be calculated from the system input and its parameters and which represent separately various identifiable components of the overall cost. The derivation of the performance index for optimal linear tracking systems is necessary to give a complete solution to the optimal linear tracking problem. A knowledge of the various cost terms in the performance index of suboptimal linear tracking systems means that the performance of existing or approximate designs, or of optimal designs either modified or having different noise inputs or performance indices, may be calculated. Further, the improvement of existing or approximate designs may be carried out by a systematic trial-and-error approach, which aims at reducing perhaps only the largest cost terms for each iteration.