Linear Time-varying State-space Model Research Articles

Real-time estimation of fuel injection rate and injection volume in high-pressure common rail systems

The injection rate and injection volume are essential parameters for the combustion process, which greatly affect the engine efficiency and emission performance. However, at the current time the injection information cannot be obtained in real time during the actual operation of engine. This paper presents a novel real-time estimation method for the injection rate and injection volume based on the measurable rail pressure. A comprehensive dynamic model is derived by considering both the effect of fuel injection and supply process on the pressure fluctuation. Then a linear time-varying state space model is constructed by choosing three appropriate state variables. On this basis, considering the measurement noise and model uncertainty, a Kalman filter-based estimation method of the injection information is investigated. And the noise covariance matrix Q is optimally designed to adapt to a wide range of operating conditions. The proposed estimation method is validated on the different conditions, and the results show the injection rate estimation errors are within 4.5 %, the injection volume estimation errors are within 4 %. And on the situation with overlap of injection and supply process, the injection rate and injection volume estimation errors are all within 4.5 %.

Noise Covariances Identification by MDM: Weighting, Recursion, and Implementation

The problem of noise covariance matrix identification of stochastic linear time-varying state-space models is addressed. The measurement difference method (MDM) is generalized to time-varying dimensions of the measurement and control. Three MDM identification techniques that differ in weighting used in the underlying least squares method are proposed. The techniques differ in estimate quality and computational complexity. In addition, recursive forms are designed for two techniques. The performance of the proposed techniques is analyzed using two numerical examples. The implementation of techniques is enclosed with the paper.

Measurement Difference Method: A Universal Tool for Noise Identification

This article deals with noise identification of a system described by the linear time-varying state-space model using correlation methods. In particular, the stress is laid on the measurement difference method (MDM) as a universal tool allowing estimation of moments and parameters of the state and measurement noises. The recent results are summarized in a common framework and the full (and weighted) MDM implementation is developed. This implementation provides unbiased and weakly consistent estimate of an arbitrary raw or central moment of the state and measurement noises. The performance of the method is shown in a numerical study.

Design of measurement difference autocovariance method for estimation of process and measurement noise covariances

The paper deals with the estimation of the process and measurement noise covariance matrices of a system described by the linear time-varying state–space model. In particular, the stress is laid on the correlation methods and a novel method, the measurement difference autocovariance method, is designed. The proposed method is based on the statistical analysis of an augmented measurement prediction error leading to a system of linear matrix equations for the elements of the noise covariance matrices. Compared to other correlation methods, the proposed method provides unbiased estimates even for a finite number of measurements. The theoretical results are discussed and illustrated in a numerical example.

Noise Moment and Parameter Estimation of State-Space Model

The paper deals with the estimation of the moments and parameters of the state and measurement noises of a system described by a linear time-varying state-space model. In particular, a novel correlation measurement difference method is designed. The method is based on a statistical analysis of an augmented measurement prediction error leading to a system of linear matrix equations enabling higher-order noise moments estimation. The estimated moments, shown to converge to the true values, are then used for noise parameters computation. The theoretical results are discussed and illustrated in a numerical example.

Estimation of Heat Source Term and Thermal Diffusion in Tokamak Plasmas Using a Kalman Filtering Method in the Early Lumping Approach

In this paper, early lumping estimation of space-time varying diffusion coefficient and source term for a nonhomogeneous linear parabolic partial differential equation (PDE) describing tokamak plasma heat transport is considered. The analysis of this PDE is achieved in a finite-dimensional framework using the cubic b-splines finite element method with the Galerkin formulation. This leads to a finite-dimensional linear time-varying state-space model with unknown parameters and inputs. The extended Kalman filter with unknown inputs without direct feedthrough (EKF-UI-WDF) is applied to simultaneously estimate the unknown parameters and inputs and an adaptive fading memory coefficient is introduced in the EKF-UI-WDF, to deal with time varying parameters. Conditions under which the direct problem is well posed and the reduced order model converges to the initial one are established. In silico and real data simulations are provided to evaluate the performances of the proposed technique.

State-space least mean square

In this paper, we present a generalized form of the well-known least mean square (LMS) filter. The proposed filter incorporates linear time-varying state-space model of the underlying environment and hence is termed as state-space LMS (SSLMS). This attribute results in marked improvement in its tracking performance over the standard LMS. Furthermore, the use of SSLMS in state estimation in control systems is straightforward. Overall performance of SSLMS, however, depends on factors like model uncertainty and time-varying nature of the problem. SSLMS with adaptive memory, having time-varying step-size parameter, provides solutions to such cases. The step-size parameter is iteratively tuned by stochastic gradient method so as to minimize the mean square value of the prediction error. Different computer simulations demonstrate the ability of the algorithms suggested in this paper. A detailed study of computational complexities of the proposed algorithms is carried out at the end.

A New Iterative Learning Control Scheme for Linear Time-varying Discrete Systems

In this paper we use a repetitive process setting to develop a new iterative learning control algorithm for plants modelled by a discrete linear time-varying state space model. As the next step in evaluating its performance, the results of a simulation based study are given where the plant model used is that obtained from frequency responses tests on a gantry robot.

Methods for stability evaluation for linear time varying, discrete-time systems on finite time horizon

This paper develops a method for determining stability of discrete-time (DT), linear time-varying (LTV) systems defined on a finite time horizon (FTH). In our considerations we use a collection of stability definitions for linear time invariant (LTI) and LTV systems which are defined on an infinite time horizon (ITH). Based on the analysis carried out and the introduced operator-matrix notation we define four stability functions. These functions allow examination of the stability of a system described by an LTV state space model. Moreover, using these functions we introduce the time stability margin and making use of the operator-matrix notation we propose a method of determining the stability on a finite time horizon. The theoretical considerations are numerically verified on two examples of LTV systems with a variable degree of non-stationarity depending on a parameter ε. Several examples illustrate the application of the introduced concepts and definitions to examination of the system stability also depending on parameter ε.

Data-Driven Modeling of Batch Processes

A one dimensional grid of interdependent linear models obtained from operation data is proposed for modeling repeated finite horizon, non linear and nonstationary process operations. Such finite horizon process operations include startups, grade transitions. shut-downs, and of course batch, semi-batch and periodic processes. The model grid is identified from data using a novel interpretation of generalized ridge regression that penalizes weighted discrepancies betweell one linear model and the models in its neighborhood. It is furthermore outlined how different representations of such a model grid may be used off-line as well as Oil-line. for prediction. monitoring, control, and optimization. Among these representations is a linear time-varying state space model which may be used for design in established linear monitoring and control methodologies.

Multivariable adaptive control using an observer based on a recurrent neural network

A real-time learning control technique for a general non-linear multivariable process is presented and applied to a laboratory plant. The proposed technique is a hybrid approach, which combines the ability of a recurrent neural network for modelling purposes and a linear pole placement control law to design the controller, providing a bridge between the field of neural networks and the well-known linear adaptive control methods. An Elman-type recurrent neural network strategy is introduced to model the behaviour of the non-linear plant, using available input–output data (an unmeasurable state problem is assumed). Following a linearization technique a linear time-varying state-space model is obtained, which allows simultaneous estimation of parameters and states. Once the neural model is linearized, some well-established standard linear control strategies can be applied. With simultaneous online training of the neural network and controller synthesis, the resulting structure is an indirect adaptive self-tuning strategy. The identification and control performances of the proposed approach are investigated on a non-linear multivariable three-tank laboratory system. Experimental results show the effectiveness of the proposed hybrid structure. Copyright © 1999 John Wiley & Sons, Ltd.

New approach for the stability analysis of linear time-varying state-space models

A new approach is proposed to establish stability bounds on the parameters of linear time-varying systems. In particular, stability of the Mathieu equation is investigated and stability boundaries are established. The proposed method is computationally simple to apply and can be calculated easily by computer. Examples are given to illustrate the improvement of the proposed method over that reported by Yedavalli and Kolla (1988).