Combining particle MCMC with Rao-Blackwellized Monte Carlo data association for parameter estimation in multiple target tracking

We consider state and parameter estimation in multiple target tracking problems with data association uncertainties and unknown number of targets. We show how the problem can be recast into a conditionally linear Gaussian state-space model with unknown parameters and present an algorithm for computationally efficient inference on the resulting model. The proposed algorithm is based on combining the Rao-Blackwellized Monte Carlo data association algorithm with particle Markov chain Monte Carlo algorithms to jointly estimate both parameters and data associations. Both particle marginal Metropolis–Hastings and particle Gibbs variants of particle MCMC are considered. We demonstrate the performance of the method both using simulated data and in a real-data case study of using multiple target tracking to estimate the brown bear population in Finland.     

Global adaptive control of nonlinearly parametrized systems

In this paper, we consider global adaptive control of nonlinearly parametrized systems in parametric-strict-feedback form. Unlike previous results, we do not require a priori bounds on the unknown parameters, which is as in the linear parametrization case. We also allow unknown parameters to be time-varying provided they are bounded. Our proposed adaptive controller is a switching type controller, in which the controller parameter is tuned in a switching manner via a switching logic. Global stability results of the closed-loop system have been proved.     

Data decomposition using independent component analysis with prior constraints

In many data analysis problems, it is useful to consider the data as generated from a set of unknown (latent) generators or sources. The observations we make of a system are then taken to be related to these sources through some unknown function. Furthermore, the (unknown) number of underlying latent sources may be less than the number of observations. Recent developments in independent component analysis (ICA) have shown that, in the case where the unknown function linking sources to observations is linear, such data decomposition may be achieved in a mathematically elegant manner. In this paper, we extend the general ICA paradigm to include a very flexible source model, prior constraints and conditioning on sets of intermediate variables so that ICA forms one part of a hierarchical system. We show that such an approach allows for efficient discovery of hidden representation in data and for unsupervised data partitioning.     

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.     

On sequential confidence estimation of parameters of stochastic dynamical systems with conditionally Gaussian noises

We consider the problem of non-asymptotical confidence estimation of linear parameters in multidimensional dynamical systems defined by general regression models with discrete time and conditionally Gaussian noises under the assumption that the number of unknown parameters does not exceed the dimension of the observed process. We develop a non-asymptotical sequential procedure for constructing a confidence region for the vector of unknown parameters with a given diameter and given confidence coefficient that uses a special rule for stopping the observations. A key role in the procedure is played by a novel property established for sequential least squares point estimates earlier proposed by the authors. With a numerical modeling example of a two-dimensional first order autoregression process with random parameters, we illustrate the possibilities for applying confidence estimates to construct adaptive predictions.     

Robust linear filtering for discrete-time hybrid Markov linear systems

In this paper we consider the robust linear filtering of hybrid discrete-time Markovian jump linear systems. We assume that only an output of the system is available, and therefore the values of the jump parameter are not known. It is desired to design a dynamic linear filter such that the closed loop system is mean square stable and minimizes the stationary expected value of the square error. We consider uncertainties on the parameters of the possible modes of operation of the system. A linear matrix inequalities (LMI) formulation is proposed to solve the problem. For the case in which there are no uncertainties on the modes of operation of the system, we show that the LMI formulation provides a filter with the same stationary mean square error as the one obtained from the Riccati equation approach.     

Identifiability of the stochastic semi-blind deconvolution problem for a class of time-invariant linear systems

Semi-blind deconvolution is the process of estimating the unknown input of a linear system, starting from output data, when the kernel of the system contains unknown parameters. In this paper, identifiability issues related to such a problem are investigated. In particular, we consider time-invariant linear models whose impulse response is given by a sum of exponentials and assume that smoothness is the sole available a priori information on the unknown signal. We state the semi-blind deconvolution problem in a Bayesian setting where prior knowledge on the smoothness of the unknown function is mathematically formalized by describing the system input as a Brownian motion. This leads to a Tychonov-type estimator containing unknown smoothness and system parameters which we estimate by maximizing their marginal likelihood/posterior. The mathematical structure of this estimator is studied in the ideal situation of output data noiseless with their number tending to infinity. Simulated case studies are used to illustrate the practical implications of the theoretical findings in system modeling. Finally, we show how semi-blind deconvolution can be improved by proposing a new prior for signals that are initially highly nonstationary but then become, as time progresses, more regular.     

Sparse estimation from noisy observations of an overdetermined linear system

This note studies a method for the estimation of a finite number of unknown parameters from linear equations, which are perturbed by Gaussian noise. In the case the unknown parameters have only few nonzero entries, the proposed estimator performs more efficiently than a traditional approach. The method consists of three steps: (1) a classical Least Squares Estimate (LSE); (2) the support is recovered through a Linear Programming (LP) optimization problem which can be computed using a soft-thresholding step; (3) a de-biasing step using a LSE on the estimated support set. The main contribution of this note is a formal derivation of an associated ORACLE property of the final estimate. That is, with probability 1, the estimate equals the LSE based on the support of the true parameters when the number of observations goes to infinity.     

Bayes and empirical Bayes semi-blind deconvolution using eigenfunctions of a prior covariance

We consider the semi-blind deconvolution problem; i.e., estimating an unknown input function to a linear dynamical system using a finite set of linearly related measurements where the dynamical system is known up to some system parameters. Without further assumptions, this problem is often ill-posed and ill-conditioned. We overcome this difficulty by modeling the unknown input as a realization of a stochastic process with a covariance that is known up to some finite set of covariance parameters. We first present an empirical Bayes method where the unknown parameters are estimated by maximizing the marginal likelihood/posterior and subsequently the input is reconstructed via a Tikhonov estimator (with the parameters set to their point estimates). Next, we introduce a Bayesian method that recovers the posterior probability distribution, and hence the minimum variance estimates, for both the unknown parameters and the unknown input function. Both of these methods use the eigenfunctions of the random process covariance to obtain an efficient representation of the unknown input function and its probability distributions. Simulated case studies are used to test the two methods and compare their relative performance.     

Adaptive control of a class of nonlinear time-varying systems with multiple models

The adaptive control of nonlinear systems that are linear in the unknown but time-varying parameters are treated in this paper. Since satisfactory transient performance is an important factor, multiple models are required as these parameters change abruptly in the parameter space. In this paper we consider both the multiple models with switching and tuning methodology as well as multiple models with second level adaptation for this class of systems. We demonstrate that the latter approach is better than the former.     

Parametrizations of linear dynamical systems: Canonical forms and identifiability

We consider the problem of what parametrizations of linear dynamical systems are appropriate for identification (i.e., so that the identification problem has a unique solution, and all systems of a particular class can be represented). Canonical forms for controllable linear systems under similarity transformation are considered and it is shown that their use in identification may cause numerical difficulties, and an alternate approach is proposed which avoids these difficulties. Then it is assumed that the system matrices are parametrized by some unknown parameters from a priori system knowledge. The identiability of such an arbitrary parametrization is then considered in several situations. Assuming that the system transfer function can be identified asymptotically, conditions are derived for local and global identifiability. Finally, conditions for identifiability from the output spectral density are given for a system driven by unobserved white noise.     

Stochastic system identification with noisy input-outputmeasurements using polyspectra

Two new classes of parametric, frequency domain approaches are proposed for estimation of the parameters of scalar, linear “errors-in-variables” models, i.e., linear systems where measurements of both input and output of the system are noise contaminated. The first approach consists of linear estimators where using the bispectrum or the integrated polyspectrum of the input and the cross-bispectrum or the integrated cross-polyspectrum, the system transfer function is first estimated at a number of frequencies exceeding one-half the number of unknown parameters. The estimated transfer function is then used to estimate the unknown parameters using an overdetermined linear system of equations. In the second class of approaches, quadratic transfer function matching criteria are optimized by using the results of the linear estimators as initial guesses. Both classes of the parameter estimators are shown to be consistent in any measurement noise that has symmetric probability density function when the bispectral approaches are used. The proposed parameter estimators are shown to be consistent in Gaussian measurement noise when trispectral approaches are used     

Control for a system of linear agents based on a high order adaptation algorithm

We solve the problem of synchronizing a network of linear agents with unknown parameters and unknown network topology given that the Laplacian that defines it has no complex eigenvalues. To solve this problem, we use a modified high order adaptation algorithm. We obtain conditions for reaching consensus with the proposed algorithm. We show modeling results that demonstrate the efficiency of the proposed approach.     

Probabilistic guarantees for high-level robot behavior in the presence of sensor error

This paper presents an approach for reasoning about the effects of sensor error on high-level robot behavior. We consider robot controllers that are synthesized from high-level, temporal logic task specifications, such that the resulting robot behavior is guaranteed to satisfy these specifications when assuming perfect sensors and actuators. We relax the assumption of perfect sensing, and calculate the probability with which the controller satisfies a set of temporal logic specifications. We consider parametric representations, where the satisfaction probability is found as a function of the model parameters, and numerical representations, allowing for the analysis of large examples. We also consider models in which some parts of the environment and sensor have unknown transition probabilities, in which case we can determine upper and lower bounds for the probability. We illustrate our approach with two examples that provide insight into unintuitive effects of sensor error that can inform the specification design process.     

Robust distributed maximum likelihood estimation with dependent quantized data

In this paper, we consider the distributed maximum likelihood estimation (MLE) with dependent quantized data under the assumption that the structure of the joint probability density function (pdf) is known, but it contains unknown deterministic parameters. The parameters may include different vector parameters corresponding to marginal pdfs and parameters that describe the dependence of observations across sensors. Since MLE with a single quantizer is sensitive to the choice of thresholds due to the uncertainty of pdf, we concentrate on MLE with multiple groups of quantizers (which can be determined by the use of prior information or some heuristic approaches) to fend off against the risk of a poor/outlier quantizer. The asymptotic efficiency of the MLE scheme with multiple quantizers is proved under some regularity conditions and the asymptotic variance is derived to be the inverse of a weighted linear combination of Fisher information matrices based on multiple different quantizers which can be used to show the robustness of our approach. As an illustrative example, we consider an estimation problem with a bivariate non-Gaussian pdf that has applications in distributed constant false alarm rate (CFAR) detection systems. Simulations show the robustness of the proposed MLE scheme especially when the number of quantized measurements is small.     

A self‐tuning iterative learning controller for time variant systems

We consider the iterative learning control problem from an adaptive control viewpoint. The self‐tuning iterative learning control systems (STILCS) problem is formulated in a general case, where the underlying linear system is time‐variant and its parameters are all unknown and where its initial conditions are not constant and not determinable in various iterations. A procedure for solving this problem will be presented. The Lyapunov technique is employed to ensure the convergence of the presented STILCS. Computer simulation results are included to illustrate the effectiveness of the proposed STILCS. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Stabilization of Markovian jump linear system over networks with random communication delay

This paper is concerned with the stabilization problem for a networked control system with Markovian characterization. We consider the case that the random communication delays exist both in the system state and in the mode signal which are modeled as a Markov chain. The resulting closed-loop system is modeled as a Markovian jump linear system with two jumping parameters, and a necessary and sufficient condition on the existence of stabilizing controllers is established. An iterative linear matrix inequality (LMI) approach is employed to calculate a mode-dependent solution. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.     

Scenario-based verification of uncertain parametric MDPs

We consider parametric Markov decision processes (pMDPs) that are augmented with unknown probability distributions over parameter values. The problem is to compute the probability to satisfy a temporal logic specification with any concrete MDP that corresponds to a sample from these distributions. As solving this problem precisely is infeasible, we resort to sampling techniques that exploit the so-called scenario approach. Based on a finite number of samples of the parameters, the proposed method yields high-confidence bounds on the probability of satisfying the specification. The number of samples required to obtain a high confidence on these bounds is independent of the number of states and the number of random parameters. Experiments on a large set of benchmarks show that several thousand samples suffice to obtain tight and high-confidence lower and upper bounds on the satisfaction probability.

     

Sample Complexity for Computational Classification Problems

In a statistical setting of the classification (pattern recognition) problem the number of examples required to approximate an unknown labelling function is linear in the VC dimension of the target learning class. In this work we consider the question of whether such bounds exist if we restrict our attention to computable classification methods, assuming that the unknown labelling function is also computable. We find that in this case the number of examples required for a computable method to approximate the labelling function not only is not linear, but grows faster (in the VC dimension of the class) than any computable function. No time or space constraints are put on the predictors or target functions; the only resource we consider is the training examples. The task of classification is considered in conjunction with another learning problem - data compression. An impossibility result for the task of data compression allows us to estimate the sample complexity for pattern recognition.     

Multiple model-based event-triggered adaptive control of a class of discrete-time nonlinear systems

In this study, the problem of event-triggered-based adaptive control (ETAC) for a class of discrete-time nonlinear systems with unknown parameters and nonlinear uncertainties is considered. Both neural network (NN) based and linear identifiers are used to approximate the unknown system dynamics. The feedback output signals are transmitted, and the parameters and the NN weights of the identifiers are tuned in an aperiodic manner at the event sample instants. A switching mechanism is provided to evaluate the approximate performance of each identifier and decide which estimated output is utilised for the event-triggered controller design, during any two events. The linear identifier with an auxiliary output and an improved adaptive law is introduced so that the nonlinear uncertainties are no longer assumed to be Lipschitz. The number of transmission times are significantly reduced by incorporating multiple model schemes into ETAC. The boundedness of both the parameters of identifiers and the system outputs is demonstrated though the Lyapunov approach. Simulation results demonstrate the effectiveness of the proposed method.