Identification of multivariable errors in variable models with dynamics

This note considers the task of identifying a causal, linear, dynamic, multivariable system excited by stationary, zero-mean noise of unknown spectrum, and given measurements of the system inputs and outputs contaminated by independent, additive noise also of unknown spectra. Although the solution is in general not unique, finite-dimensional parameterizations of the solution set are given, even though the various spectra may not be rational.     

A parallel filtering algorithm for linear systems with unknown time varying noise statistics

The problem of estimating the state of a linear dynamic system driven by additive Gaussian noise with unknown time varying statistics is considered. Estimates of the state of the system are obtained which are based on all past observations of the system. These observations are linear functions of the state contaminated by additive white Gaussian noise. A previously developed algorithm designed for use in the case of stationary noise is modified to allow estimation of an unknown Kalman gain and thus the system state in the presence of unknown time varying noise statistics. The algorithm is inherently parallel in nature and if implemented in a computer with parallel processing capability should only be slightly slower than the stationary Kalman filtering algorithm with known noise statistics.     

On-line identification of linear dynamic systems with applications to Kalman filtering

Kalman gave a set of recursive equations for estimating the state of a linear dynamic system. However, the Kalman filter requires a knowledge of all the system and noise parameters. Here it is assumed that all these parameters are unknown and therefore must be identified before use in the Kalman filter. A correlation technique which identifies a system in its canonical form is presented. The estimates are shown to be asymptotically normal, unbiased, and consistent. The scheme is capable of being implemented on-line and can be used in conjunction with the Kalman filter. A technique for more efficient estimation by using higher order correlations is also given. A recursive technique is given to determine the order of the system when the dimension of the system is unknown. The results are first derived for stationary processes and are then extended to nonstationary processes which are stationary in theqth increment. An application of the results to a practical problem is presented.     

Wiener theory of digital linear-quadratic control

The linear quadratic regulator problem is considered for discrete time systems with time delay. The theory forms a close parallel to that for stationary prediction and filtering. For coloured noise disturbance at the plant input it is shown that the optimal control decomposes into the solution for white noise disturbance plus a feedforward controller. For unit time delay the solution becomes equivalent to that given by the stationary Kalman theory.     

Identification of optimum filter steady-state gain for systems with unknown noise covariances

A discrete linear stationary system is considered for which the input noise covarianceQand the output noise covarianceRare unknown. A stable filter with a suboptimal gain is assumed. An identification scheme is presented which uses the autocorrelation functions of the innovations sequence of the suboptimal filter to determine the optimum filter steady state gainGammadirectly without the intermediate determination of the unknown covariancesQandR. The approach used is to identify an output equivalent representation of the original system which does not involve the unknown covariances directly.     

Estimating performance of the robust control system under unknown upper disturbance boundaries and measurement noise

Consideration was given to estimation of the robust performance of the closed-loop control system under unknown upper disturbance boundaries. The linear stationary discrete plant with scalar output and control served as a model of the nominal plant. The disturbances included external bounded disturbance, measurement noise, and operator disturbances in output and control. Consideration was given to calculation of the asymptotic upper boundaries of the measurable and nonmeasurable tracking errors coordinated with the measurement data.     

Symmetric Laplace transform and its application to parametric identification of linear systems

Properties of bilateral symmetric integral transform, which possesses useful features for identification of linear systems, are investigated. Relationships for covariance functions of stochastic stationary processes of measurement errors and disturbances are determined. The optimal solution is proposed to the identification problem for linear multivariable system with measured state coordinates in the transformation parameter domain, with account for restricted bandwidth of dynamic processes in the system at hand. The problem is solved for unknown initial state of the system and complex measurement errors including correlated noise, constant errors and linear trends.     

On the identification of variances and adaptive Kalman filtering

A Kalman filter requires an exact knowledge of the process noise covariance matrixQand the measurement noise covariance matrixR. Here we consider the case in which the true values ofQandRare unknown. The system is assumed to be constant, and the random inputs are stationary. First, a correlation test is given which checks whether a particular Kalman filter is working optimally or not. If the filter is suboptimal, a technique is given to obtain asymptotically normal, unbiased, and consistent estimates ofQandR. This technique works only for the case in which the form ofQis known and the number of unknown elements inQis less thann times rwherenis the dimension of the state vector andris the dimension of the measurement vector. For other cases, the optimal steady-state gain Kopis obtained directly by an iterative procedure without identifyingQ. As a corollary, it is shown that the steady-state optimal Kalman filter gain Kopdepends only onn times rlinear functionals ofQ. The results are first derived for discrete systems. They are then extended to continuous systems. A numerical example is given to show the usefulness of the approach.     

Extended optimality properties of the linear quadratic regulatorand stationary Kalman filter

It is shown that the constant gain state-feedback solution to the infinite-time linear quadratic regulator problem is optimal not only for arbitrarily initial conditions or white noise disturbances, but also for worst-case L¹ disturbances. Using a similar technique, it is shown that in the stationary Kalman filter, the white disturbance and measurement noise can be replaced by unknown bounded energy signals, and that optimality still holds if the performance criterion is a time domain L_∞ normal of the state estimation errors in the presence of worst-case energy signals     

A comparison between non-gaussian closure and statistical linearization techniques for random vibration of a nonlinear oscillator

The non-Gaussian closure technique is applied to a nonlinear single degree of freedom oscillator subjected to a stationary white noise excitation. The nonlinear restoring force in this system is a softening spring with a hyperbolic tangent behavior. This type of nonlinear system has practical applications in package cushioning. The governing differential equation of motion is used to generate relations between stationary response statistics. These relations are then employed to evaluate a corresponding number of unknown coefficients in a non-Gaussian probability distribution function. A truncated Gram-Charlier expansion is chosen for the probability density function. Up to the sixth order moments of the response process are obtained. The probability density functions predicted by this technique are then compared with density functions constructed by exact solution via the Fokker-Plank-Kolmogorov equation. A comparison is also made with the density function evaluated by statistical linearization for this system. Limitations of this non-Gaussian closure method are discussed.     

Minimax Filtering of Sequences with Periodically Stationary Increments

The authors consider the problem of optimal filtering of functionals that depend on unknown values of the stochastic sequence with periodically stationary increments based on observations of the sequence with a stationary noise. For sequences with known spectral densities, formulas are obtained for the root-mean-square errors and spectral characteristics of the optimal estimates of the functionals. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of functionals are proposed in the case where spectral densities of the sequence are not known exactly while some sets of feasible spectral densities are given.

     

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.     

An algorithm of measurement processing for piecewise continuous noise

The problem of processing of measurements containing dynamic observation noise is solved. The obtained method is invariant to piecewise continuous noise of the deterministic structure with unknown parameters. The method does not require an extension of the state space and provides higher efficiency of solution of the estimation problem. An illustrative example proving the efficiency of the method is given.     

On the existence of stationary points for the Steiglitz-McBridealgorithm

Most convergence results for adaptive identification algorithms have been developed in sufficient order settings, involving an unknown system with known degree. Reduced-order settings, in which the degree of the unknown system is underestimated, are more common, but more difficult to analyze. Deducing stationary points in these cases typically involves solving nonlinear equations, hence the sparseness of results for reduced-order cases. If we allow ourselves the tractable case in which the input to an identification experiment is white noise, we shall show that the Steiglitz-McBride method (1965) indeed admits a stationary point in reduced-order settings for which the resulting model is stable. Our interest in this study stems from a previous result, showing an attractive a priori bound on the mismodeling error at any such stationary point     

On closed-loop system identification using polyspectral analysis given noisy input–output time-domain data

The problem of closed-loop system identification given noisy input–output measurements is considered. It is assumed that the closed-loop system operates under an external non-Gaussian input which is not measured. If the external input has non-vanishing integrated bispectrum (IB) and data IB is used for identification, then the various disturbances/noise processes affecting the system are assumed to be zero-mean stationary with vanishing IB. If the external input has non-vanishing integrated trispectrum (IT) and data IT is used for identification, then the various disturbances/noise processes affecting the system are assumed to be zero-mean stationary Gaussian. Noisy measurements of the (direct) input and output of the plant are assumed to be available. The closed-loop system must be stable but it is allowed to be unstable in open loop. Parametric modeling of the various noise sequences affecting the system is not needed. First the open-loop transfer function is estimated using the integrated polyspectrum and cross-polyspectrum of the time-domain input–output measurements. Then two existing techniques for parametric system identification given consistent estimates of the underlying transfer function, are exploited. The parameter estimators are strongly consistent. Asymptotic performance analysis is also carried out. A computer simulation example using an unstable open-loop system is presented to illustrate the proposed approach.     

Robust H∞ filtering of stationary continuous-timelinear systems with stochastic uncertainties

The problem of applying H_∞ filters on stationary, continuous-time, linear systems with stochastic uncertainties in the state-space signal model is addressed. These uncertainties are modeled via white noise processes. The relevant cost function is the expected value of the standard H_∞ performance index with respect to the uncertain parameters. The solution is obtained via a stochastic bounded real lemma that results in a modified Riccati inequality. This inequality is expressed in the form of a linear matrix inequality whose solution provides the filter parameters. The method proposed is also applied to the case where, in addition to the stochastic uncertainty, other deterministic parameters of the system are not perfectly known and are assumed to lie in a given polytope. The problem of mixed H₂ /H_∞ filtering for the above system is also treated. The theory developed is demonstrated by a practical example     

Sliding-mode observers for nonlinear systems with unknown inputs and measurement noise

This paper deals with the design of observers for Lipschitz nonlinear systems with not only unknown inputs but also measurement noise when the observer matching condition is not satisfied. First, an augmented vector is introduced to construct an augmented system, and an auxiliary output vector is constructed such that the observer matching condition is satisfied and then a high-gain sliding mode observer is considered to get the exact estimates of both the auxiliary outputs and their derivatives in a finite time. Second, for nonlinear system with both unknown inputs and measurement noise, an adaptive robust sliding mode observer is developed to asymptotically estimate the system’s states, and then an unknown input and measurement noise reconstruction method is proposed. Finally, a numerical simulation example is given to illustrate the effectiveness of the proposed methods.     

单输出系统最优自校正滤波新方法及其在跟踪系统中的应用

对于带未知噪声统计且含未知模型参数的单输出系统,本文用现代时间序列分析方法提出了一种新的自校正滤波方法,给出了具有渐近最优性的自校正滤波器,新方法的特点是基于ARMA新息模型通过计算自校正输出预报器和自校正观测噪声滤波器就可得到自校正状态滤波器,文中给出了在跟踪系统中的应用例子,仿真结果说明了新方法的有效性。     

Robust filtering of process in the stationary difference stochastic system

A technique to construct the robust Kalman filter for process estimation in the difference linear stationary stochastic system with an unknown covariance observation error matrix was developed. Consideration was given to the algorithm of constructing the set of permissible covariance matrices from a priori statistical data. A numerical method for solution of the general minimax optimization problem was proposed; and on its basis an iterative algorithm to calculate the robust filter parameters was developed, and its convergence was proved. Results of the numerical experiment were presented.