Linear Constant Systems Research Articles

Data assimilation and uncertainty analysis of environmental assessment problems—an application of Stochastic Transfer Function and Generalised Likelihood Uncertainty Estimation techniques

Abstract Stochastic Transfer Function (STF) and Generalised Likelihood Uncertainty Estimation (GLUE) techniques are outlined and applied to an environmental problem concerned with marine dose assessment. The goal of both methods in this application is the estimation and prediction of the environmental variables, together with their associated probability distributions. In particular, they are used to estimate the amount of radionuclides transferred to marine biota from a given source: the British Nuclear Fuel Ltd (BNFL) repository plant in Sellafield, UK. The complexity of the processes involved, together with the large dispersion and scarcity of observations regarding radionuclide concentrations in the marine environment, require efficient data assimilation techniques. In this regard, the basic STF methods search for identifiable, linear model structures that capture the maximum amount of information contained in the data with a minimal parameterisation. They can be extended for on-line use, based on recursively updated Bayesian estimation and, although applicable to only constant or time-variable parameter (non-stationary) linear systems in the form used in this paper, they have the potential for application to non-linear systems using recently developed State Dependent Parameter (SDP) non-linear STF models. The GLUE based-methods, on the other hand, formulate the problem of estimation using a more general Bayesian approach, usually without prior statistical identification of the model structure. As a result, they are applicable to almost any linear or non-linear stochastic model, although they are much less efficient both computationally and in their use of the information contained in the observations. As expected in this particular environmental application, it is shown that the STF methods give much narrower confidence limits for the estimates due to their more efficient use of the information contained in the data. Exploiting Monte Carlo Simulation (MCS) analysis, the GLUE technique is used to estimate how the errors involved in the STF model structure and observations influence the model outputs and errors. The discussion on updating information originating from different locations using GLUE procedure is also given. A final aim of the paper is to use the results obtained in this particular example to explore the differences between the GLUE and STF approaches.

An algebraic framework for linear identification

A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.

LINEAR PROGRAMMING BASED GAIN SCHEDULING FOR LPV AND PL SYSTEMS

Parametric linear constant and gain-scheduled control systems design are studied for a class of linear parameter varying (LPV) systems and piecewise linear (PL) systems based on stabilization via convexity restrictions. Specifically, the maximum row sum (matrix) norm is utilized to obtain a computationally attractive state space method for the scheduling of simple linear controllers under incomplete modelling information. Furthermore, certain close connections of the studied framework with fuzzy gain scheduling are discussed.

Correcteurs proportionnels-intégraux généralisés

For constant linear systems we are introducing integral reconstructors and generalized proportional-integral controllers , which permit to bypass the derivative term in the classic PID controllers and more generally the usual asymptotic observers. Our approach, which is mainly of algebraic flavour, is based on the module-theoretic framework for linear systems and on operational calculus in Mikusinski's setting. Several examples are discussed.

The unconditional stability for the hyperneutral type constant linear control system with delays (I)

By means of the frequency domain method and the inequality analysis, we discuss the unconditional stability problem for the hyperneutral type constant linear control system with delays, and obtain some precise sufficient, sufficient and necessary conditions.

The unconditional stability for the hyperneutral type constant linear control system with delays

In this paper, by means of the frequency domain method and the inequality analysis, unconditional stability problem for the hyperneutral type constant linear control system with delays are discussed, and some precise sufficient, sufficient and necessary conditions are obtained.

Characteristic modeling and the control of flexible structure

Appropriate modeling for a controlled plant has been a remarkable problem in the control field. A new modeling theory, i.e. characteristic modeling, is roundly demonstrated. It is deduced in detail that a general linear constant high-order system can be equivalently described with a two-order time-varying difference equation. The application of the characteristic modeling method to the control of flexible structure is also introduced. Especially, as an example, the Hubble Space Telescope is used to illustrate the application of the characteristic modeling and adaptive control method proposed in this paper.

Basic theory of linear singular discrete systems with delay

This paper discusses existence and uniqueness of solutions of constant coefficient linear singular discrete systems with delay, presents the results that the systems have a unique solution in some cases, and have no solution or have infinite solutions in other cases.

Stability of constant coefficient linear singular systems with delay

In this paper, the general class of singular systems with delay and linear constant coefficient singular systems with delay are discussed. First, several definitions of stability are presented for singular systems with delay, and general sufficient stability conditions and instability conditions are obtained. Second, stability and instability are analyzed for linear constant coefficient singular systems with delay.

On strict system equivalence for multidimensional systems

Strict system equivalence describes the connection between least-order realizations of a transfer function in terms of a general linear constant system (Rosenbrock, H. H., 1970, State-space and Multivariable Theory (London: Nelson)). The paper generalizes this and related results to the case of multidimensional systems, i.e. systems of linear partial difference or differential equations with constant coefficients.

The unconditional stability for the multigroup multidelays neutral type linear constant control system

Applying the frequency domain method and the inequality method, we discussed the unconditional stability problem of the multigroup multidelays neutral type linear constant continuous control system, and obtained some sufficient conditions.

LFT representations of parametrized polynomial systems

The paper focuses on general linear constant differential systems in which the coefficients depend polynomially on several parameters. It is shown how the system matrix can be written in terms of a linear fractional transformation (LFT), which is a representation that extracts the parametric uncertainty. The LFT form yields lower bounds for the robust stability radius of the system via /spl mu/-analysis tools. The method is applied to the linearized model of a transistor amplifier.

Bifurcation on stability of singular systems with delay

In this paper, the general class of singular systems with delay and second order linear constant coefficients singular systems with delay, is considered. First, for singular systems with delay, new definitions of stability are presented, and general sufficient stability conditions and instability conditions are derived. Secondly, these results are applied to the analysis of stability and instability of second-order linear constant coefficients singular systems with delay, the location of a possible bifurcation is discovered.

Continuous time-varying linear systems

We discuss implicit systems of ordinary linear differential equations with (time-) variable coefficients, their solutions in the signal space of hyperfunctions according to Sato and their solution spaces, called time-varying linear systems or behaviours, from the system theoretic point of view. The basic result, inspired by an analogous one for multidimensional constant linear systems, is a duality theorem which establishes a categorical one–one correspondence between time-varying linear systems or behaviours and finitely generated modules over a suitable skew-polynomial ring of differential operators. This theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic (hyper-)functions or of distributions on R . It is used to obtain various results on key notions of linear system theory. Several new algorithms for modules over rings of differential operators and, in particular, new Gröbner basis algorithms due to Insa and Pauer make the system theoretic results effective.

On the Notion of Duality for Linear Delay Systems

The notion of the dual (or adjoint) ofa system is extended to the case of linear (constant or time-varying) systems with constant pointwise delays. It results that the dual of a retarded state representation needs not be retarded. The duality between several notions of controllability and observability is exhibited.

Separated Dirac operators and asymptotically constant linear systems

Levinson's theorem with all its ramifications is a well-established tool in the spectral analysis of ordinary differential operators (see in particular §§3·10, 11 and 4·3, 4 of Eastham's book [5]). In this note we should like to draw attention to a result that describes the solutions of asymptotically constant linear systems under weaker assumptions less precisely than the Levinson theorem. This result can be called the Perron–Lettenmeyer–Hartman–Wintner theorem after the contributions of these authors in [11, 10, 6, 8]. (Note that the Hartman–Wintner theorem that is discussed in [5, p. 17] is a substantially different version of this theorem.)

Turbulence and separation induced pressure fluctuations on a finite circular cylinder — application of a linear unsteady strip theory

Abstract In a turbulent stream, the fluctuating pressures on bluff bodies are mainly caused by the oncoming turbulence. Further unsteadiness is engendered by the wake turbulence and by the vortex shedding. Although the gross properties of the flow around a body and in particular the separated flow are affected by the intensity and the scale of the incident turbulence, these sources of pressure fluctuations are thought to be statistically independent. On this basis, the physical system of a turbulent flow around a body is approximated by an artificial black box model. This model corresponds to a constant parameter linear system with multiple stationary random inputs, i.e. turbulent velocity fluctuations far upstream the body. The model output is equal to an instantaneous surface pressure, which is subdivided into a turbulence induced contribution and an extraneous noise term to account for the residual fluctuations. By means of velocity/pressure cross spectral densities (i) significant velocity fluctuations, (ii) the associated admittance functions and (iii) the conditioned and residual pressure spectra are identified experimentally for the flow around a surface mounted circular cylinder in an atmospheric boundary layer flow. In a first approximation, the longitudinal u′(t) and lateral velocity fluctuation ν′(t) of a single stream filament are thought to be the only decisive inputs to the multivariate system (linear unsteady strip theory). The latter induces maximum spectral amplitudes in the mid-frequency range, whereas the longitudinal component achieves maximum effectiveness at the lowest frequencies. However, significant deviations between the (u, ν)-conditioned and the factual pressure spectrum remain. Depending on the circumferential angle and in a certain frequency range, vortex shedding contributes up to 45% to the total pressure RMS-values.

Optimal pole placement in time-dependent linear systems

In this paper a problem of optimal modal control for general linear, time-dependent systems is posed and solved. We review the recent modal decomposition of the general linear system. We then pose the problem of controlling one unstable mode, with a specified new Lyapunov exponent, but minimizing the mean-square gain function. The method is applied to the shallow-angle re-entry of a Delta Clipper-like vehicle, and the modal decomposition leads to both stability information and characterization of how the trajectory is unstable. The unstable Lyapunov exponent is moved so that the trajectory is stable, as verified by closed-loop Lyapunov exponent analysis. When implemented within the original nonlinear system, the controller can handle initial errors at the kilometer level. Only 4% modulation of the original drag factor is required for control. I. Introduction T HE problem of control of constant coefficient linear systems is well understood, forming the basis of most of control theory. Control of time-dependent systems is not yet at the same level of understanding. Fairly comprehensive discussion of the time-periodic case have appeared: see Breakwell et al. 1 for the linear quadratic regulator for periodic systems and Calico and Wiesel2 for pole placement in periodic systems. Recently, the current author has successfully produced the modal decomposition for a general timedependent linear system3 and introduced the first pole placement algorithm for such systems. In this paper we extend the method to optimal pole placement and will successfully control a shallowangle re-entry trajectory by this new technique.

The solution of unsymmetric tridiagonal Toeplitz systems by the strides reduction algorithm

A cyclic reduction method is described for the fast numerical solution of constant tridiagonal Toeplitz linear systems which occur repeatedly in the solution of the implicit finite difference equations derived from linear first order hyperbolic equations, i.e. the Transport equation, under a variety of boundary conditions. In this paper, we show that the linear systems can be solved efficiently by the Stride of 3 reduction algorithm.

Index of an implicit time-varying linear differential equation: a noncommutative linear algebraic approach

An intrinsic definition of the index for implicit linear time-varying differential systems is proposed. This definition relies on linear algebraic techniques over the noncommutative principal ideal ring of nonconstant linear differential operators (noncommutative transfer function or state variable representation). Accordance with the existing definition for constant coefficient linear systems is demonstrated.