Bilinear State-space Model Research Articles

State Estimation Moving Window Gradient Iterative Algorithm for Bilinear Systems Using the Continuous Mixed p-norm Technique

This paper studies the parameter estimation problems of the nonlinear systems described by the bilinear state space models in the presence of disturbances. A bilinear state observer is designed for deriving identification algorithms to estimate the state variables using the input-output data. Based on the bilinear state observer, a novel gradient iterative algorithm is derived for estimating the parameters of the bilinear systems by means of the continuous mixed p-norm cost function. The gain at each iterative step adapts to the data quality so that the algorithm has good robustness to the noise disturbance. Furthermore, to improve the performance of the proposed algorithm, a dynamic moving window is designed which can update the dynamical data by removing the oldest data and adding the newest measurement data. A numerical example of identification of bilinear systems is presented to validate the theoretical analysis.

Hierarchical multi-innovation stochastic gradient identification algorithm for estimating a bilinear state-space model with moving average noise

This paper considers the combined parameter and state estimation problem of a bilinear state space system with moving average noise. There are product terms of state variables and control variables in bilinear systems, which brings difficulties to parameter and state estimation. By designing a bilinear state estimator based on Kalman filter and using input–output data to estimate the state, a hierarchical multi-innovation stochastic gradient (i.e., H-MISG) algorithm based on the state estimator is proposed to jointly estimate unknown states and parameters. In addition, compared with the hierarchical stochastic gradient algorithm, H-MISG algorithm introduces the innovation length parameter, makes full use of the system input and output data information, and improves the accuracy of parameter estimation. Numerical simulation examples verify the effectiveness of the proposed algorithm.

Synchronous Optimization Schemes for Dynamic Systems Through the Kernel-Based Nonlinear Observer Canonical Form

Although some dynamic behaviors are commonly modeled as the linear or bilinear systems, recent studies have consistently noted the existence of significant nonlinearities in these behaviors. Here, this work attempts to devise effective models and optimization approaches to represent and analyze the nonlinear phenomena ground on the online measured data. A kernel-based nonlinear observer canonical form is put forward by exploiting the fitting superiorities of kernel functions, which comprises the classical linear and bilinear state space models as the special cases. The utilization of polynomial properties reduces the difficulty of identification while preserving the capacity of state space models to fit nonlinear characteristics. Aiming to fulfilling simultaneous state and parameter estimation, a nonlinear state observer-based separable hierarchical forgetting gradient (NSO-SHFG) algorithm is devised in according with the extended Kalman filtering and the decomposition technique. The convergence analysis is established to verify the theoretical results, and the feasibility of the NSO-SHFG algorithm is validated by simulations.

Discrete Bilinear Biquadratic Regulator

Bilinear state-space models, a type of simple nonlinear model, have different applications. Optimal control of such models is often used for performance improvement. A solution for a new class of optimization problem, which is defined as inhomogeneous discrete-time bilinear plant with control trajectory constraints and a biquadratic performance index, is presented. Krotov's method is used for computing a candidate solution for this problem. A suitable sequence of improving functions, which are essential for the utilization of this method is formulated. The obtained algorithm is convergent and monotonic by means of the defined performance index. The method's effectiveness is demonstrated by a numerical example.

Variational Bayesian identification for bilinear state space models with Markov‐switching time delays

SummaryThis article studies the parameter identification problem for bilinear state space models with time‐varying time delays. Considering the correlation of time delays, the Markov chain switching mechanism is adopted to model the delay sequence. Based on the observer canonical form, the bilinear state space model is transformed into a regressive form. A bilinear state observer is designed to estimate the state variables. Under the variational Bayesian scheme, the system parameters, discrete delays, and the Markov transition probabilities are identified by using the measurement data. A numerical example and a continuous stirred tank reactor simulation are employed to validate the effectiveness of the proposed algorithm.

Extension of the Constrained Bilinear Quadratic Regulator to the Excited Multi-input Case

This paper introduces a new method for the computation of an optimal semi-active feedback of a plant, which is controlled by multiple semi-active control dampers and subjected to external, a priori known deterministic disturbance input. The control force is written in bilinear form in equivalent damping gains and a linear combination of the states. This form leads to a bilinear state-space model with corresponding damping gain constraints. An optimal control problem, denoted as constrained bilinear quadratic regulator, is formulated with a performance index, which is quadratic in the states and the equivalent damping gains. The methodology, which is used for solving this problem, is Krotov’s method. In this study, the sequence of improving functions, which enables the use of Krotov’s method in this case, is formulated. Its incorporation in Krotov’s algorithm leads to the suggested novel algorithm for solution of the constrained bilinear quadratic regulator problem for excited optimal semi-active control design. The results are demonstrated numerically by constrained bilinear quadratic regulator control design of controlled structure with seismic disturbance input.

Bilinear reduced order approximate model of parabolic distributed solar collectors

This paper proposes a novel, low dimensional and accurate approximate model for the distributed parabolic solar collector, based on a modified Gaussian interpolation along the spatial domain. The proposed reduced model, written in a form of a low dimensional bilinear state representation, enables the reproduction of the heat transfer dynamics along the collector tube for system analysis. Moreover, presented as a reduced order bilinear state space model, the well established control theory for this class of systems can be applied. The approximation efficiency has been proven by several simulation tests, which have been performed considering parameters of the Acurex field with real external working conditions. Model accuracy has been evaluated by comparison to the analytical solution of the hyperbolic distributed model and its semi discretized approximation highlighting the benefits of using the proposed numerical scheme. Furthermore, model sensitivity to the different parameters of the Gaussian interpolation has been studied.

Optimal bilinear observers for bilinear state-space models by interaction matrices

This paper formulates optimal bilinear observers for bilinear state-space models. Observers in bilinear form, as opposed to other nonlinear forms, are required to develop an extension of observer/Kalman filter identification for simultaneous identification of a bilinear state-space model and an associated bilinear observer from noisy input–output measurements. The paper establishes the relationship between the bilinear observer gains and the interaction matrices which are used to convert the original bilinear state-space model to a form that simplifies the identification of such a model. Techniques to find the interaction matrices are developed. In the absence of noises, these matrices produce the gains of the fastest converging observer. In the presence of noises, they minimise the state estimation error in the same manner as a standard steady-state Kalman filter. Numerical examples illustrate both the theoretical and computational aspects of the proposed algorithms.

A Superspace Method for Discrete-Time Bilinear Model Identification by Interaction Matrices

This paper presents a method for discrete-time bilinear state-space model identification from a single set of input–output data. The initial state can be unknown. By extending the interaction matrix formulation from linear to bilinear state-space models, the bilinear system state is expressed in terms of input–output measurements. This relationship is used to convert the bilinear model to an Equivalent Linear Model (ELM) which can be identified by any linear identification method such as a superspace method presented here. The bilinear state-space model is then extracted from the identified ELM.

Development of bilinear power system representations for small signal stability analysis

In this paper, a new analysis technique for predicting and characterizing nonlinear behavior of stressed power networks is proposed. Making use of an analytical approximation of the system nonlinear model via the use of a truncated Carleman linearization technique, a bilinear state-space model of the power system is developed in which the second and higher order nonlinear terms are explicitly incorporated in the series expansion representation of the system model. The proposed framework enables the study of the dynamic behavior of nonlinear systems, both analytically and numerically, and can be used to represent a wide class of non-linear systems and oscillatory processes. Analytical criteria are developed based on the structural properties of the bilinear state-space model matrices, which predict the existence and stability character of modal interaction in terms of the eigenstructure of the linear system representation. The properties and behavior of the bilinear model are then investigated, and a number of useful results are derived. The present method is quite general and extends readily to higher-dimensional systems. A simplified 2-area, 4-machine system is used to illustrate the proposed procedure. Detailed nonlinear time-domain simulations are conducted to identify the strength of nonlinear behavior arising from interaction of the fundamental modes of oscillation, as well as to check the validity of the analysis.

Model reduction of bilinear systems described by input–output difference equation

A class of single input single output bilinear systems described by their input–output difference equation is considered. A simple expression for the Volterra kernels of the system is derived in terms of the coefficients of difference equation. An algorithm, based on the singular value decomposition of a generalized Hankel matrix, is also developed. The algorithm is then used to find a reduced-order bilinear state-space model. The Hankel approach will be extensively studied under different data length cases and different orders of the state-space models. A numerical example is presented to illustrate the effectiveness of the proposed algorithm.

Optimization algorithms for bilinear model–based predictive control problems

AbstractModel‐based predictive control (MPC) for discrete‐time bilinear state–space models is considered. The optimization problem of the bilinear MPC algorithm is nonlinear in general. It is demonstrated that the structural properties of the bilinear state–space model provide a way to formulate the nonlinear optimization problem as a sequence of quadratic programming problems that exactly represent the original objective function. The proposed optimization algorithm is compared to one that is based on a linearization about an input trajectory. To benefit from the advantages of both algorithms, a hybrid algorithm is proposed, which outperforms the other two in most cases. The applicability of the proposed bilinear MPC algorithm is demonstrated on a polymerization process. 2004 American Institute of Chemical Engineers AIChE J, 50:1453–1461, 2004

Bilinear state space realization for polynomial stochastic systems

The Wiener approach to the non-linear stochastic systems permits the representation of single-valued systems with memory for which a small perturbation of the input produces a small perturbation of the output. The Wiener functional series representation contains infinitely many transfer functions to describe entirely the input output connections. One of the most important class of stochastic systems, especially in the statistical point of view, is the case when all the transfer functions are determined by finitely many parameters. The question of realizability, i.e., to construct a state space model for a system given in terms of a set of transfer functions, is a central problem in many cases for example in engineering. There are several results in this direction for systems with deterministic inputs see [1,2]. The realization of linear systems with stochastic Gaussian input was treated by Akaike [3] and studied intensively afterwards, see [4,5]. The bilinear system seems to be the next step leading from the linear towards non-linear ones. The realization problem of bilinear systems appeared in the papers [6–8], mainly by the time series side. In this work, we follow the early Akaike work and the results for the deterministic systems by Rugh [2] and also Brillinger [9], who investigated the identification of finite degree functional series by spectral analysis. First the bilinear realization problem is considered together with a simplest non-linear system, i.e., the second Hermite degree bilinear process. In the second section, the transfer function system with its main properties is given by a recursive formula for the bilinear states. Next, we show that these properties are necessary and sufficient for the transfer functions to be bilinear realizable. In section four, the stationarity of the bilinear state space model is investigated. We are constructing the abstract bilinear minimal realization for a time invariant stationary degree- N non-linear system driven by Gaussian white noise. Concerning to the identification of the bilinear model with Hermite degree-2 explicit formula of the bispectrum is given for this family of models.

3-D motion estimation using a sequence of noisy stereo images: models, estimation, and uniqueness results

A kinematic model-based approach for the estimation of 3-D motion parameters from a sequence of noisy stereo images is discussed. The approach is based on representing the constant acceleration translational motion and constant precession rotational motion in the form of a bilinear state-space model using standard rectilinear states for translation and quaternions for rotation. Closed-form solutions of the state transition equations are obtained to propagate the quaternions. The measurements are noisy perturbations of 3-D feature points represented in an inertial coordinate system. It is assumed that the 3-D feature points are extracted from the stereo images and matched over the frames. Owing to the nonlinearity in the state model, nonlinear filters are designed for the estimation of motion parameters. Simulation results are included. The Cramer-Rao performance bounds for motion parameter estimates are computed. A constructive proof for the uniqueness of motion parameters is given. It is shown that with uniform sampling in time, three noncollinear feature points in five consecutive binocular image pairs contain all the spatial and temporal information. Both nondegenerate and degenerate motions are analyzed. A deterministic algorithm to recover motion parameters from a stereo image sequence is summarized from the constructive proof.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>