Linear functional state bounding for perturbed time-delay systems and its application

This paper considers time-delay systems with bounded disturbances. We study a new problem of finding an upper bound of an absolute value function of any given linear functional of the state vector starting from the origin of the system. Based on the Lyapunov-Krasovskii method combining with the recent Wirtinger-based integral inequality that has just been proposed by Seuret & Gouaisbaut (2013. Wirtinger-based integral inequality: application to time-delay systems. Automatica, 49, 2860-2866), sufficient conditions for the existence of an upper bound of the function are derived. The obtained results are shown to be more effective than those adapted from the existing works on reachable set bounding. Furthermore, the obtained results are applied to refine existing ellipsoidal bounds of the reachable sets. The effectiveness of the obtained results is illustrated by two numerical examples.

A reduced-order linear functional observer for linear stochastic systems

This paper presents necessary and sufficient conditions for the existence of a reduced-order linear functional state observer for linear stochastic systems. The order of the observer is the same as the dimension of the vectors to be estimated. A simple design algorithm is given together with a numerical example to illustrate the simplicity of the design procedure.

A low-order linear functional observer for time delay systems

This paper presents an efficient technique to design low order state function observers for linear time-delay systems. Assuming the existence of a linear state feedback controller to achieve stability or some control performance criteria of the time-delay system, a design procedure is proposed for
reconstruction of the state feedback control action. The procedure involves solving an optimisation problem with the objective to generate a matrix that is as close as possible to the given feedback gain of the required feedback controller. A condition for robust stability of the time-delay system using the observer-based control scheme is given. The attractive features of the proposed design procedure are that the resulted linear functional state observer is of a very low order and it requires information of a small number of outputs. Numerical examples are given to demonstrate the design procedure and its merits.

New results in common functional state estimation for two linear systems with unknown inputs

This paper presents new developments in common functional observers for two systems. We improve an existing common functional observer scheme by reducing its order, and then investigate its existence conditions in terms of the original system matrices. These conditions have never been explored and they enable the users to know at the outset the class of systems for which the scheme is applicable. They also show that both observers can be designed independently of each other which significantly simplifies the design process. A numerical simulation verifies the findings.

Partial state bounding with a pre-specified time of non-linear discrete systems with time-varying delays

In this study, the authors address a new problem of finding, with a pre-specified time, bounds of partial states of non-linear discrete systems with a time-varying delay. A novel computational method for deriving the smallest bounds is presented. The method is based on a new comparison principle, a new algorithm for finding the infimum of a fractal function, and linear programming. The effectiveness of our obtained results is illustrated through a numerical example.

Design of reduced-order functional observers for linear systems with unknown inputs

This brief paper presents new conditions for the existence and design of reduced-order linear functional state observers for linear systems with unknown inputs. Systematic procedures for the synthesis of reduced-order functional observers are given. Numerical examples are given to illustrate the attractiveness and simplicity of the new design procedures.

Design of linear functional observers for time-delay systems of the neutral-type

This paper presents the design of reduced-order linear functional observers for a class of linear time-delay systems of the neutral-type. The type of the observer proposed in this paper is without internal delay and its order is the same as the number of linear functions to be estimated. First, conditions for the existence of the reduced-order functional observers that are capable of asymptotically estimating any given function of the state vector are derived. Then, based on the newly derived existence conditions, a procedure is given for the determination of the observer parameters. The results derived in this paper include a range of linear systems and extend some existing results of linear functional observers to linear neutral delay systems. A numerical example is given to illustrate the design procedure.

Linear functional observers with guaranteed ε-convergence for discrete time-delay systems with input/output disturbances

The problem of designing linear functional observers for discrete time-delay systems with unknown-but-bounded disturbances in both the plant and the output is considered for the first time in this paper. A novel approach to design a minimum-order observer is proposed to guarantee that the observer error is ϵ-convergent, which means that the estimate converges robustly within an ϵ-bound of the true state. Conditions for the existence of this observer are first derived. Then, by utilising an extended Lyapunov-Krasovskii functional and the free-weighting matrix technique, a sufficient condition for ϵ-convergence of the observer error system is given. This condition is presented in terms of linear matrix inequalities with two parameters needed to be tuned, so that it can be efficiently solved by incorporating a two-dimensional search method into convex optimisation algorithms to obtain the smallest possible value for ϵ. Three numerical examples, including the well-known single-link flexible joint robotic system, are given to illustrate the feasibility and effectiveness of our results.

Design of reduced-order positive linear functional observers for positive time-delay systems

This brief considers a new problem of designing reduced-order positive linear functional observers for positive time-delay systems. The order of the designed functional observers is equal to the dimension of the functional state vector to be estimated. The designed functional observers always nonnegative at any time and they converge asymptotically to the true functional state vector. Moreover, conditions for the existence of such positive linear functional observers are formulated in terms of linear programming (LP). Numerical examples and simulation results are given to illustrate the effectiveness of the proposed design method.

PySSM : a Python module for Bayesian inference of linear Gaussian state space models

PySSM is a Python package that has been developed for the analysis of time series using linear Gaussian state space models (SSM). PySSM is easy to use; models can be set up quickly and efficiently and a variety of different settings are available to the user. It also takes advantage of scientific libraries Numpy and Scipy and other high level features of the Python language. PySSM is also used as a platform for interfacing between optimised and parallelised Fortran routines. These Fortran routines heavily utilise Basic Linear Algebra (BLAS) and Linear Algebra Package (LAPACK) functions for maximum performance. PySSM contains classes for filtering, classical smoothing as well as simulation smoothing.

On a linear steady-state system of equations in microcontinuum fluid mechanics

Galerkin representations and integral representations are obtained for the linearized system of coupled differential equations governing steady incompressible flow of a micropolar fluid. The special case of 2-dimensional Stokes flows is then examined and further representation formulae as well as asymptotic expressions, are generated for both the microrotation and velocity vectors. With the aid of these formulae, the Stokes Paradox for micropolar fluids is established.

Rao-Blackwellized particle smoothers for mixed linear/nonlinear state-space models

We consider the smoothing problem for a class of conditionally linear Gaussian state-space (CLGSS) models, referred to as mixed linear/nonlinear models. In contrast to the better studied hierarchical CLGSS models, these allow for an intricate cross dependence between the linear and the nonlinear parts of the state vector. We derive a Rao-Blackwellized particle smoother (RBPS) for this model class by exploiting its tractable substructure. The smoother is of the forward filtering/backward simulation type. A key feature of the proposed method is that, unlike existing RBPS for this model class, the linear part of the state vector is marginalized out in both the forward direction and in the backward direction. © 2013 IEEE.

Numerical solution of a PDE system with non-linear steady state conditions that translates the air stripping pollutants removal

This work deals with the numerical simulation of air stripping process for the pre-treatment of groundwater used in human consumption. The model established in steady state presents an exponential solution that is used, together with the Tau Method, to get a spectral approach of the solution of the system of partial differential equations associated to the model in transient state.