Minimum phase robustness for uncertain state-space systems

The minimum phase robustness of an uncertain state-space system with affine parametric uncertainties in the state-space matrices is studied. A tolerable margin in terms of the structured singular value is given for uncertain parameters to guarantee the minimum phase property of the system. Based on the linear fractional transformation methodology, the matrix sizes involved in the computation of structured singular value are reduced significantly to improve computational burden. The approach can be applied to the proper or strictly proper linear uncertain systems.     

On simultaneous on-line state and parameter estimation in non-linear state-space models

On-line estimation plays an important role in process control and monitoring. Obtaining a theoretical solution to the simultaneous state-parameter estimation problem for non-linear stochastic systems involves solving complex multi-dimensional integrals that are not amenable to analytical solution. While basic sequential Monte-Carlo (SMC) or particle filtering (PF) algorithms for simultaneous estimation exist, it is well recognized that there is a need for making these on-line algorithms non-degenerate, fast and applicable to processes with missing measurements. To overcome the deficiencies in traditional algorithms, this work proposes a Bayesian approach to on-line state and parameter estimation. Its extension to handle missing data in real-time is also provided. The simultaneous estimation is performed by filtering an extended vector of states and parameters using an adaptive sequential-importance-resampling (SIR) filter with a kernel density estimation method. The approach uses an on-line optimization algorithm based on Kullback–Leibler (KL) divergence to allow adaptation of the SIR filter for combined state-parameter estimation. An optimal tuning rule to control the width of the kernel and the variance of the artificial noise added to the parameters is also proposed. The approach is illustrated through numerical examples.     

A framework for ranking uncertain distributed database

Distribution and uncertainty are considered as the most important design issues in database applications nowadays. A lot of ranking or top-k query processing techniques are introduced to solve the problems of communication cost and centralized processing. On the other hand, many techniques are also developed for modeling and managing uncertain databases. Although these techniques were efficient, they didn't deal with distributed data uncertainty. This paper proposes a framework that deals with both data distribution and uncertainty based on ranking queries. Within the proposed framework, communication and computation-efficient algorithms are investigated for retrieving the top-k tuples from distributed sites. The main objective of these algorithms is to reduce the communication rounds utilized and amount of data transmitted while achieving efficient ranking. Experimental results show that both proposed techniques have a great impact in reducing communication cost. Both techniques are efficient but in different situations. The first one is efficient in the case of low number of sites while the other achieves better performance at higher number of sites.     

Online variational inference for state-space models with point-process observations

We present a variational Bayesian (VB) approach for the state and parameter inference of a state-space model with point-process observations, a physiologically plausible model for signal processing of spike data. We also give the derivation of a variational smoother, as well as an efficient online filtering algorithm, which can also be used to track changes in physiological parameters. The methods are assessed on simulated data, and results are compared to expectation-maximization, as well as Monte Carlo estimation techniques, in order to evaluate the accuracy of the proposed approach. The VB filter is further assessed on a data set of taste-response neural cells, showing that the proposed approach can effectively capture dynamical changes in neural responses in real time.     

Variational learning for switching state-space models

We introduce a new statistical model for time series that iteratively segments data into regimes with approximately linear dynamics and learnsthe parameters of each of these linear regimes. This model combines and generalizes two of the most widely used stochastic time-series models -- hidden Markov models and linear dynamical systems -- and is closely related to models that are widely used in the control and econometrics literatures. It can also be derived by extending the mixture of experts neural network (Jacobs, Jordan, Nowlan, & Hinton, 1991) to its fully dynamical version, in which both expert and gating networks are recurrent. Inferring the posterior probabilities of the hidden states of this model is computationally intractable, and therefore the exact expectation maximization (EM) algorithm cannot be applied. However, we present a variational approximation that maximizes a lower bound on the log-likelihood and makes use of both the forward and backward recursions for hidden Markov models and the Kalman filter recursions for linear dynamical systems. We tested the algorithm on artificial data sets and a natural data set of respiration force from a patient with sleep apnea. The results suggest that variational approximations are a viable method for inference and learning in switching state-space models.     

Multidimensional state-space models: A comparative overview

This paper provides a comparative survey of the existing multidimensional state-space models. The 2-Dimensional state-space models are first presented and compared to each other from both the mathematical and the applications point of view. Then a canonical 3-Dimensional state space realization is considered and its applications are briefly discussed. Finally the special cases of 2-D and 3-D first-order transfer functions are investigated and necessary and sufficient conditions for existence of respective minimal state-space realizations are introduced. The models considered in the paper have been used for treating various design problems of multidimensional systems such as model mathing, transfer function separation, etc., as can be seen in the corresponding references.     

A robust estimation approach for uncertain systems with perturbed measurements

This paper deals with state estimation problem for uncertain continuous‐time systems. A numerical treatment is proposed for designing interval observers that ensures guaranteed upper and lower bounds on the estimated states. In order to take into account possible perturbations on the system and its outputs, a new type of interval observers is introduced. Such interval observers consist of two coupled general Luenberger‐type observers that involve dilatation functions. In addition, we provide an optimality criterion in order to find optimal interval observers that lead to tight interval error estimation. The proposed existence and optimality conditions are expressed in terms of linear programming. Also, some illustrative examples are given. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

A note on the estimation of variances in state-space models using the maximum a posteriori procedure

This note examines an estimation procedure for the unknown parameters in a state-space model proposed by Tsang, Glover, and Bach. The method is based on the maximum a posteriori (MAP) principle. Contrary to the claims of Tsang et al. it is shown that the algorithm performs very poorly compared to maximum likelihood. Some insight into the shortcomings of the MAP procedure is obtained by comparing it to the computation of maximum likelihood estimators by the EM algorithm.     

Robust stability of state-space models with structureduncertainties

A method for robust eigenvalue location analysis of linear state-space models affected by structured real parametric perturbations is proposed. The approach, based on algebraic matrix properties, deals with state-space models in which system matrix entries are perturbed by polynomial functions of a set of uncertain physical parameters. A method converting the robust stability problem into nonsingularity analysis of a suitable matrix is proposed. The method requires a check of the positivity of a multinomial form over a hyperrectangular domain in parameter space. This problem, which can be reduced to finding the real solutions of a system of polynomial equations, simplifies considerably when cases with one or two uncertain parameters are considered. For these cases, necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems     

Stability robustness bounds for linear state-space models with structured uncertainty

In this note, we consider the robust stability analysis problem in linear state-space models. We consider systems with structured uncertainty. Some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived. These bounds are shown to be less conservative than the existing ones.     

On-line structure selection for multivariable state-space models

An algorithm is described for the selection of model structure for identifying state-space models of ‘black box’ character. The algorithm receives as ‘input’ a given system in a given parametrization. It is then tested whether this parametrization is suitable (well conditioned) for identification purposes. If not, a better one is selected and the transformation of the system to the new representation is performed. This algorithm can be used as a block both in an iterative, off-line identification procedure, and for recursive, on-line identification. It can be called whenever there is some indication that the model structure is ill-conditioned. It is discussed how the model structure selection algorithm can be interfaced with an off-line identification procedure. A complete procedure is described and tested on real and simulated data.     

Interval estimation for uncertain systems with time-varying delays

The estimation problem for uncertain time-delay systems is addressed. A design method of reduced-order interval observers is proposed. The observer estimates the set of admissible values (the interval) for the state at each instant of time. The cases of known fixed delays and uncertain time-varying delays are analysed. The proposed approach can be applied to linear delay systems and nonlinear time-delay systems in the output canonical form. It involves the properties of quasi-monotone/Metzler/cooperative systems. In this framework, it is shown that if under a suitable coordinate transformation the delay-free subsystem is cooperative, then the delayed estimation error dynamics inherits this property. The conditions to find the observer gains are formulated in the form of LMI. The framework efficiency is demonstrated on examples of nonlinear systems.     

A robust design criterion for interpretable fuzzy models with uncertain data

We believe that nonlinear fuzzy filtering techniques may be turned out to give better robustness performance than the existing linear methods of estimation (H/sup 2/ and H/sup /spl infin// filtering techniques), because of the fact that not only linear parameters (consequents), but also the nonlinear parameters (membership functions) attempt to identify the uncertain behavior of the unknown system. However, the fuzzy identification methods must be robust to data uncertainties and modeling errors to ensure that the fuzzy approximation of unknown system's behavior is optimal in some sense. This study presents a deterministic approach to the robust design of fuzzy models in the presence of unknown but finite uncertainties in the identification data. We consider online identification of an interpretable fuzzy model, based on the robust solution of a regularized least-squares fuzzy parameters estimation problem. The aim is to resolve the difficulties associated with the robust fuzzy identification method due to lack of a priori knowledge about upper bounds on the data uncertainties. The study derives an optimal level of regularization that should be provided to ensure the robustness of fuzzy identification strategy by achieving an upper bound on the value of energy gain from data uncertainties and modeling errors to the estimation errors. A time-domain feedback analysis of the proposed identification approach is carried out with emphasis on stability, robustness, and steady-state issues. The simulation studies are provided to show the superiority of the proposed fuzzy estimation over the classical estimation methods.     

Unknown input observers for 2D state-space models

In this paper, dead-beat unknown input observers (UIOs) for two-dimensional (2D) state-space models are investigated. Dead-beat UIOs are observers which produce an exact estimate of the original system state trajectory, after a finite number of evolution steps, independently of the system and observer initial conditions and of the unknown disturbances that affect the system functioning. Necessary and sufficient conditions for the existence of dead-beat UIOs are provided, as well as a parameterization of all dead-beat UIO transfer matrices. Comparisons with Luenberger-type UIOs are also carried out, and the extension of the paper results to the case of asymptotic UIOs with a given rate of convergence and to Roesser models are finally discussed.     

A study of uncertain state estimation

In this paper, we present results of uncertain state estimation of systems that are monitored with limited accuracy. For these systems, the representation of state uncertainty as confidence intervals offers significant advantages over the more traditional approaches with probabilistic representation of noise. While the filtered-white-Gaussian noise model can be defined on grounds of mathematical convenience, its use is necessarily coupled with a hope that an estimator with good properties in idealised noise will still perform well in real noise. In this study we propose a more realistic approach of matching the noise representation to the extent of prior knowledge. Both interval and ellipsoidal representation of noise illustrate the principle of keeping the noise model simple while allowing for iterative refinement of the noise as we proceed. We evaluate one nonlinear and three linear state estimation technique both in terms of computational efficiency and the cardinality of the state uncertainty sets. The techniques are illustrated on a synthetic and a real-life system.     

A scattering framework for decentralized estimation problems

In this paper we develop a comprehensive framework for the study of decentralized estimation problems. This approach imbeds a decentralized estimation problem into an equivalent scattering problem, and makes use of the super-position principle to relate local and centralized estimates. Some decentralized filtering and smoothing algorithms are obtained for a simple estimation structure consisting of a central processor and of two local processors. The case when the local processors exchange some information is considered, as well as the case when the local state-space models differ from the central model.     

A unified SVM framework for signal estimation

This paper presents a review in the form of a unified framework for tackling estimation problems in Digital Signal Processing (DSP) using Support Vector Machines (SVMs). The paper formalizes our developments in the area of DSP with SVM principles. The use of SVMs for DSP is already mature, and has gained popularity in recent years due to its advantages over other methods: SVMs are flexible non-linear methods that are intrinsically regularized and work well in low-sample-sized and high-dimensional problems. SVMs can be designed to take into account different noise sources in the formulation and to fuse heterogeneous information sources. Nevertheless, the use of SVMs in estimation problems has been traditionally limited to its mere use as a black-box model. Noting such limitations in the literature, we take advantage of several properties of Mercerʼs kernels and functional analysis to develop a family of SVM methods for estimation in DSP. Three types of signal model equations are analyzed. First, when a specific time-signal structure is assumed to model the underlying system that generated the data, the linear signal model (so-called Primal Signal Model formulation) is first stated and analyzed. Then, non-linear versions of the signal structure can be readily developed by following two different approaches. On the one hand, the signal model equation is written in Reproducing Kernel Hilbert Spaces (RKHS) using the well-known RKHS Signal Model formulation, and Mercerʼs kernels are readily used in SVM non-linear algorithms. On the other hand, in the alternative and not so common Dual Signal Model formulation, a signal expansion is made by using an auxiliary signal model equation given by a non-linear regression of each time instant in the observed time series. These building blocks can be used to generate different novel SVM-based methods for problems of signal estimation, and we deal with several of the most important ones in DSP. We illustrate the usefulness of this methodology by defining SVM algorithms for linear and non-linear system identification, spectral analysis, non-uniform interpolation, sparse deconvolution, and array processing. The performance of the developed SVM methods is compared to standard approaches in all these settings. The experimental results illustrate the generality, simplicity, and capabilities of the proposed SVM framework for DSP.     

Sufficient sequences and state-space models for random processes

Let (U_{1}, U_{2}, ...) be a sequence of observed random variables whose probability distributions are described by a parameterized family of density functions{p_{k}(u_{1}, ..., u_{k}; theta)}. If there exists a sequence of sufficient statistics fortheta(T_{1}(U_{1}),T_{2}(U_{1}, U_{2}), ...), and if a realizability assumption holds, then there is a finite-dimensional state-space model whose output process agrees with (U_{1}, U_{2}, ...) in distribution.     

A model predictive control framework for constrained uncertain positive systems

In this paper, a new model predictive control framework is proposed for positive systems subject to input/state constraints and interval/polytopic uncertainty. Instead of traditional quadratic performance index, simple linear performance index, linear Lyapunov function, cone invariant set with linear form and linear computation tool are first adopted. Then, a control law that can handle the constraints and robustly stabilise the systems is proposed. The advantages of the new framework lie in the following facts: (1) an equivalent linear problem is formulated that can be easily solved than other problems including the quadratic ones, (2) simple linear index and linear tool can be used based on the essential property of positive systems to achieve the desired control performance and (3) a general model predictive control law without sign restriction is designed. Finally, an attempt of application on mitigating viral escape is provided to verify the effectiveness of the proposed approach.