Nonlinear discrete-time models: state-space vs. I/O representations

This paper compares state-space and input–output realizations for nonlinear discrete-time dynamic models. For linear models, these two realizations are essentially equivalent and their structures are closely related, but these statements do not hold for nonlinear models. We illustrate this point with simple, realistic examples for which only one of the two realizations exists or for which both exist but their structures are profoundly different. Overall, the main point of this paper is the importance of the choice of realization in the development of nonlinear dynamic models.     

State-space approach to discrete-time 2-optimal control with a causality constraint

Controller design with a causality constraint arises in periodic or multirate control systems. In this paper complete state-space solutions to the optimal and suboptimal ₂ control problems are developed with a causality constraint on controller feedthrough terms. Explicit formulas for the controllers are given in terms of solutions of two Riccati equations. The results are more implementable than existing frequency-domain solutions.     

State-space analysis and identification for a class of hysteretic systems

In this paper we present results on the twin subjects of system analysis and system identification for a class of state-space realizable dynamic systems under the influence of hysteresis. The class of systems in question consists of models in the form of a linear time-invariant dynamic system in series with a differential model of hysteresis. It will be demonstrated that under fairly light constraints on the differential model of hysteresis, it is possible to design a series of experiments leading towards the identification of the full state-space realization. The approach is tested successfully on a high-precision mechanical translation system affected by hysteresis.     

Subspace state space system identification for industrial processes

We give a general overview of the state-of-the-art in subspace system identification methods. We have restricted ourselves to the most important ideas and developments since the methods appeared in the late eighties. First, the basics of linear subspace identification are summarized. Different algorithms one finds in literature (such as N4SID, IV-4SID, MOESP, CVA) are discussed and put into a unifying framework. Further, a comparison between subspace identification and prediction error methods is made on the basis of computational complexity and precision of the methods by applying them on 10 industrial data sets.     

State-space truncation methods for parallel model reduction of large-scale systems

We discuss a parallel library of efficient algorithms for model reduction of large-scale systems with state-space dimension up to (10⁴). We survey the numerical algorithms underlying the implementation of the chosen model reduction methods. The approach considered here is based on state-space truncation of the system matrices and includes absolute and relative error methods for both stable and unstable systems. In contrast to serial implementations of these methods, we employ Newton-type iterative algorithms for the solution of the major computational tasks. Experimental results report the numerical accuracy and the parallel performance of our approach on a cluster of Intel Pentium II processors.     

Data driven local coordinates for multivariable linear systems and their application to system identification

In this paper we introduce a new parametrization for state-space systems: data driven local coordinates (DDLC). The parametrization is obtained by restricting the full state-space parametrization, where all matrix entries are considered to be free, to an affine plane containing a given nominal state-space realization. This affine plane is chosen to be perpendicular to the tangent space to the manifold of observationally equivalent state-space systems at the nominal realization. The application of the parametrization to prediction error identification is exemplified. Simulations indicate that the proposed parametrization has numerical advantages as compared to e.g. the more commonly used observable canonical form.     

An analysis of the parametrization by data driven local coordinates for multivariable linear systems

In this paper, we study a novel parametrization for state-space systems, namely data driven local coordinates (DDLC) which have recently been introduced and applied. Even though DDLC has meanwhile become the default parametrization used in the system identification toolbox of the software package MATLAB, an analysis of properties of DDLC, which are relevant to identification, has not been performed up to now. In this paper, we provide insights into the geometry and topology of the DDLC construction and show a number of results which are important for actual identification such as maximum likelihood-type estimation.     

Frequency domain subspace-based identification of discrete-time power spectra from nonuniformly spaced measurements

In this paper, we present a new subspace-based algorithm for the identification of multi-input/multi-output, square, discrete-time, linear-time invariant systems from nonuniformly spaced power spectrum measurements. The algorithm is strongly consistent and it is illustrated with one practical example that solves a stochastic road modeling problem.     

An analysis of separable least squares data driven local coordinates for maximum likelihood estimation of linear systems

In this paper we study a novel parametrization for state-space systems, namely separable least squares data driven local coordinates (slsDDLC). The parametrization by slsDDLC has recently been successfully applied to maximum likelihood estimation of linear dynamic systems. In a simulation study, the use of slsDDLC has led to numerical advantages in comparison to the use of more conventional parametrizations, including data driven local coordinates (DDLC). However, an analysis of properties of slsDDLC, which are relevant to identification, has not been performed up to now. In this paper, we provide insights into the geometry and topology of the slsDDLC construction and show a number of results which are important for actual identification, in particular for maximum likelihood estimation. We also prove that the separable least squares methodology is indeed guaranteed to be applicable to maximum likelihood estimation of linear dynamic systems in typical situations.     

Kernel methods for subspace identification of multivariable LPV and bilinear systems

Subspace identification methods for multivariable linear parameter-varying (LPV) and bilinear state-space systems perform computations with data matrices of which the number of rows grows exponentially with the order of the system. Even for relatively low-order systems with only a few inputs and outputs, the amount of memory required to store these data matrices exceeds the limits of what is currently available on the average desktop computer. This severely limits the applicability of the methods. In this paper, we present kernel methods for subspace identification performing computations with kernel matrices that have much smaller dimensions than the data matrices used in the original LPV and bilinear subspace identification methods. We also describe the integration of regularization in these kernel methods and show the relation with least-squares support vector machines. Regularization is an important tool to balance the bias and variance errors. We compare different regularization strategies in a simulation study.     

Robust likelihood inference for regression parameters in partially linear models

A robust likelihood approach is proposed for inference about regression parameters in partially-linear models. More specifically, normality is adopted as the working model and is properly corrected to accomplish the objective. Knowledge about the true underlying random mechanism is not required for the proposed method. Simulations and illustrative examples demonstrate the usefulness of the proposed robust likelihood method, even in irregular situations caused by the components of the nonparametric smooth function in partially-linear models.     

Fractional discrete-time consensus models for single- and double-summator dynamics

The leader–following consensus problem of fractional-order multi-agent discrete-time systems is considered. In the systems, interactions between opinions are defined like in Krause and Cucker–Smale models but the memory is included by taking the fractional-order discrete-time operator on the left-hand side of the nonlinear systems. In this paper, we investigate fractional-order models of opinions for the single- and double-summator dynamics of discrete-time by analytical methods as well as by computer simulations. The necessary and sufficient conditions for the leader–following consensus are formulated by proposing a consensus control law for tracking the virtual leader.     

Stability conditions for a class of time-varying discrete-time systems

This paper deals with the stability conditions for a class of dynamical discrete-time systems, called ‘P-invariants’ and ‘not P-invariants’. Stability conditions are given in terms of polyhedral convex cones and are obtained by using some extensions on M-matrices.     

Explicit solutions for multivariate, discrete-time control problems under uncertainty

We consider discrete time control problems, where only the support sets of the initial condition and of the disturbances are known. We study the applicability of the DP method and, as a counterpart of the LQ problem of the stochastic setting, we present a problem that admits an explicit analytic solution. In particular, we characterize a property of compatibility between the system dynamics and the norms of the spaces, that is crucial to obtain the analytic solution also in the general multidimensional case.     

离散广义系统的鲁棒严格正实控制

研究确定的以及具有范数有界不确定性的离散广义系统的严格正实分析和控制问题,首先给出了确定离散广义系统状态反馈扩展严格正实控制器的存在条件和设计方法;然后利用线性矩阵不等式分析了不确定离散广义系统广义二次稳定且扩展严格正实的条件,讨论了状态反馈使闭环系统广义二次稳定且扩展严格正实问题,给出了状态反馈鲁棒扩展严格正实控制器的综合方法;最后通过数值算例说明了所提出方法的有效性．     

Parameter constraints in generalized linear latent variable models

Parameter constraints in generalized linear latent variable models are discussed. Both linear equality and inequality constraints are considered. Maximum likelihood estimators for the parameters of the constrained model and corrected standard errors are derived. A significant reduction in the dimension of the optimization problem is achieved with the proposed methodology for fitting models subject to linear equality constraints.     

Simulation smoothing for state-space models: A computational efficiency analysis

Simulation smoothing involves drawing state variables (or innovations) in discrete time state-space models from their conditional distribution given parameters and observations. Gaussian simulation smoothing is of particular interest, not only for the direct analysis of Gaussian linear models, but also for the indirect analysis of more general models. Several methods for Gaussian simulation smoothing exist, most of which are based on the Kalman filter. Since states in Gaussian linear state-space models are Gaussian Markov random fields, it is also possible to apply the Cholesky Factor Algorithm (CFA) to draw states. This algorithm takes advantage of the band diagonal structure of the Hessian matrix of the log density to make efficient draws. We show how to exploit the special structure of state-space models to draw latent states even more efficiently. We analyse the computational efficiency of Kalman-filter-based methods, the CFA, and our new method using counts of operations and computational experiments. We show that for many important cases, our method is most efficient. Gains are particularly large for cases where the dimension of observed variables is large or where one makes repeated draws of states for the same parameter values. We apply our method to a multivariate Poisson model with time-varying intensities, which we use to analyse financial market transaction count data.