Identification and model approximation for continuous-time systems on finite parameter sets

Almost-sure convergence of the maximum likelihood and the maximum a posteriori probability estimates of unknown parameters of continuous-time stochastic dynamical linear time-invariant systems is investigated. The unknown parameter set is assumed to be finite. The situation where the ture parameter does not belong to the unknown parameter set is considered, as well as the situation where the true model is included in the unknown parameter set.     

Estimation of the autoregressive parameters of a mixed autoregressive moving-average time series

The problem of estimating the autoregressive parameters of a mixed autoregressive moving-average (ARMA) time series (of known order) using the output data alone is treated. This problem is equivalent to the estimation of the denominator terms of the scalar transfer function of a stationary, linear discrete time system excited by an unobserved unenrrelated sequence input by employing only the observations of the scalar output. The solution of this problem solves the problem of the identification of the dynamics of a white-noise excited continuous-time linear stationary system using sampled data. The latter problem was suggested by Bartlett in 1946. The problem treated here has appeared before in the engineering literature. The earlier treatment yielded biased parameter estimates. An asymptotically unbiased estimator of the autoregressive parameters is obtained as the solution of a modified set of Yule-Walker equations. The asymptotic estimator covariance matrix behaves like a least-squares parameter estimate of an observation set with unknown error covariances. The estimators are also shown to be unbiased in the presence of additive independent observation noise of arbitrary finite correlation time. An example illustrates the performance of the estimating procedures.     

Input estimation in nonlinear dynamical systems using differential algebra techniques

The paper considers the problem of estimating the unknown input of a nonlinear dynamical system, described by polynomial or rational differential equations, from a finite set of noisy output samples. Without additional information this problem is ill-posed since the unknown function may belong to an arbitrary infinite-dimensional space. We tackle this difficulty by designing a novel class of fast regularization algorithms that relies upon differential algebra techniques. Monte Carlo studies are used to demonstrate the effectiveness of the new approach.     

Continuous time systems identification with unknown noise covariance

In identifying parameters of a continuous-time dynamical system, a difficulty arises when the observation noise covariance is unknown. The present paper solves this problem in the case of a linear time-invariant system with white noise affecting additively both the state and the observation. The problem is that the likelihood functional cannot be obtained when the observation noise covariance is unknown. A related procedure is suggested, however, and the estimates are obtained by finding roots of an appropriate functional. It is shown that the estimates obtained are consistent.     

Robust Kalman filtering for uncertain discrete-time systems with probabilistic parameters bounded within a polytope

We consider the problem of finite horizon discrete-time Kalman filtering for systems with parametric uncertainties. Specifically, we consider unknown but deterministic uncertainties where the uncertain parameters are assumed to lie in a convex polyhedron with uniform probability density. The condition and a procedure for the construction of a suboptimal filter that minimizes an expected error covariance over-bound are derived.     

Design of input signals for parameter estimation

The optimum energy-constrained and time-constrained input signal is obtained for estimating the parameters of a system. The output is corrupted by nonstationary, nonwhite additive observation noise, and the observation time is finite. The reproducing kernel Hilbert space formulation is used to obtain the parameter estimates and the error covariance matrix in terms of the input. The performance index, assumed to be a function of the error covariance matrix, is minimized by a variational procedure. A necessary condition for optimality is that the input satisfy a nonlinear Fredholm equation. An example estimates the gain of a single time constant system where the observation noise has an exponential autocorrelation function. For broadband noise, the optimum input is a portion of a sinusoid. For a noise bandwidth narrower than the system bandwidth, the optimum input switches sign as rapidly as possible, but near-optimum performance can be obtained with a relatively high frequency sinusoidal input.     

Parameter identification in a class of nonlinear systems

Two approaches are proposed for on-line identification of parameters in a class of nonlinear discrete-time systems. The system is modeled by state equations in which state and input variables enter nonlinearly in general polynomial form, while unknown parameters and random disturbances enter linearly. All states and inputs must be observed with measurement errors represented by white Gaussian noise having known covariance. System disturbances are also white and Gaussian with finite, but unknown, covariance. One method of parameter estimation is based upon a least squares approach, the second is a related stochastic approximation algorithm (SAA). Under fairly mild conditions the estimate derived from the least squares algorithm (LSA) is shown to converge in probability to the correct parameter; the SAA yields an estimate which converges in mean square and with probability 1. Examples illustrate convergence of the LSA which even in recursive form requires inversion of a matrix at each step. The SAA requires no matrix inversions, but experience with the algorithm indicates that convergence is slow relative to that of the LSA.     

Parameter consistency and quadratically constrained errors-in-variables least-squares identification

In this article, we investigate the consistency of parameter estimates obtained from least-squares identification with a quadratic parameter constraint. For generality, we consider infinite impulse-response systems with coloured input and output noise. In the case of finite data, we show that there always exists a possibly indefinite quadratic constraint depending on the noise realisation that results in a constrained optimisation problem that yields the true parameters of the system when a persistency condition is satisfied. When the noise covariance matrix is known to within a scalar multiple, we prove that solutions of the quadratically constrained least-squares (QCLs) estimator with a semidefinite constraint matrix are both unbiased and consistent in the sense that the averaged problem and limiting problem produce, respectively, unbiased and true (with probability 1) estimators. In addition, we provide numerical results that illustrate these properties of the QCLS estimator.     

非方广义系统的未知输入有限时间观测器设计

研究了含未知输入的非方广义系统的有限时间输入解耦观测器设计问题,在一定条件下基于非方广义系统的结构特征,引入一个输入-状态对的非奇异转换,把含未知输入的非方广义系统等价地转化为输入已知的正常状态空间系统．用传统的设计正常状态空间系统观测器的方法去构造含未知输入的非方广义系统的未知输入观测器,并给出了观测器存在的充分条件,由此得出了有限时间观测器的设计步骤．     

Improved parameter estimates for non-linear dynamical models using a bootstrap method

It is known that the least-squares (LS) class of algorithms produce unbiased estimates providing certain assumptions are met. There are many practical problems, however, where the required assumptions are violated. Typical examples include non-linear dynamical system identification problems, where the input and output observations are affected by measurement uncertainty and possibly correlated noise. This will result in biased LS estimates and the identified model will exhibit poor generalisation properties. Model estimation for this type of error-in-variables problem is investigated in this study, and a new identification scheme based on a bootstrap algorithm is proposed to improve the model estimates for non-linear dynamical system identification.     

Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance

We address the class of stochastic output-feedback nonlinear systems driven by noise whose covariance is time varying and bounded but the bound is not known a priori. This problem is analogous to deterministic disturbance attenuation problems. We first design a controller which guarantees that the solutions converge (in probability) to a residual set proportional to the unknown bound on the covariance. Then, for the case of a vanishing noise vector field, we design an adaptive controller which, besides global stability in probability, guarantees regulation of the state of the plant to zero with probability one.     

Convergence of continuous-time partitioned adaptive state estimators

The asymptotic behaviour of Bayes optimal adaptive state estimation schemes (also called the partitioned adaptive estimation algorithms) for continuous-time linear dynamic Gauss-Markov systems with unknown parameters is investigated. The unknown system parameters are asssumed to belong to a finite set. The results are developed through, weak consistency of the maximum likelihood and the maximum a posteriori probability estimates of the unknown parameters.     

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.     

Input selection for parameter identification in discrete systems

In this paper the problem of selecting an optimal input for identifying an unknown parameter of a known discrete system by observing its output in the presence of Gaussian noise is considered. The system is assumed to be a generalized discrete system in which the inputs and possible parameter values are members of a finite set. The criterion for the optimal input is defined as that which maximizes the probability of correctly determining the true parameter value from a multiple hypothesis test. Although the above criterion totally orders the set of inputs, it is a difficult task to select the best inputs. Some theorems are presented which yield a partial ordering whose extension is the desired total ordering. In the special case of strong noise, it is shown that the ordering of inputs can be related to the perimeter in the output vector space. The results of the paper are applicable to the selection of preset input lengths or to adaptive identification.     

Optimal linear recursive estimation with uncertain system parameters

In an estimation problem the statistics of various random processes involved may not be known exactly. Using linear state space modeling techniques, this lack of information can often be represented by allowing certain system model parameters to assume any of a finite set of possible known values with corresponding a priori known probabilities. In this short paper a recursive minimum variance estimator, restricted to be a linear function of the observation data sequence, is obtained for an estimation problem which can be described by a linear discrete time system model with uncertain parameters; all initial information relative to these uncertain parameters is utilized by the estimator. The estimation error covariance matrix, in a recursive form, is also obtained. An example is given to illustrate the usefulness of this estimator.     

Discrete-time inverse linear quadratic optimal control over finite time-horizon under noisy output measurements

In this paper, the problem of inverse quadratic optimal control over finite time-horizon for discrete-time linear systems is considered. Our goal is to recover the corresponding quadratic objective function using noisy observations. First, the identifiability of the model structure for the inverse optimal control problem is analyzed under relative degree assumption and we show the model structure is strictly globally identifiable. Next, we study the inverse optimal control problem whose initial state distribution and the observation noise distribution are unknown, yet the exact observations on the initial states are available. We formulate the problem as a risk minimization problem and approximate the problem using empirical average. It is further shown that the solution to the approximated problem is statistically consistent under the assumption of relative degrees. We then study the case where the exact observations on the initial states are not available, yet the observation noises are known to be white Gaussian distributed and the distribution of the initial state is also Gaussian (with unknown mean and covariance). EM-algorithm is used to estimate the parameters in the objective function. The effectiveness of our results are demonstrated by numerical examples.     

To the optimal identification of multivariate systems under perturbations of unknown covariances

We consider the problem of parameter and covariance estimation for multivariate stochastic systems described by regression models with special structure perturbations of unknown covariance. Sufficient conditions of uniformly optimal estimations are obtained for system parameters and covariances. The observation vector distribution family is factorized, and the full sufficient statistics is found under those conditions. Equations for uniformely optimal unbiased estimates of covariance parameters are obtained.     

The optimal control of linear systems with unknown parameters

The optimal open-loop controller for a linear system with some unknown parameters is determined. It is assumed that the unknown parameters do not vary during the process, and that their probability distribution function is given. The criterion for optimality is a quadratic function of the state and the input whoso expectation with respect to the unknown parameters has to be minimized. An explicit solution for the optimal controller is obtained by using the methods of the calculus of variations. Further, an alternate approach is presented which, for the problem solved, leads to the same optimal controller.     

Mixture Density Estimation Based on Maximum Likelihood and Sequential Test Statistics

We address the problem of estimating an unknown probability density function from a sequence of input samples. We approximate the input density with a weighted mixture of a finite number of Gaussian kernels whose parameters and weights we estimate iteratively from the input samples using the Maximum Likelihood (ML) procedure. In order to decide on the correct total number of kernels we employ simple statistical tests involving the mean, variance, and the kurtosis, or fourth moment, of a particular kernel. We demonstrate the validity of our method in handling both pattern classification (stationary) and time series (nonstationary) problems.