Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed

The paper deals with Cauchy problems for first-order systems of linear ordinary differential equations with unknown data. It is assumed that the right-hand sides of equations belong to certain bounded sets in the space of square-integrable vector-functions, and the information about the initial conditions is absent. From indirect noisy observations of solutions to the Cauchy problems on a finite system of points and intervals, the guaranteed mean square estimates of linear functionals on unknown solutions of the problems under consideration are obtained. Under an assumption that the statistical characteristics of noise in observations are not known exactly, it is proved that such estimates can be expressed in terms of solutions to well-defined boundary value problems for linear systems of impulsive ordinary differential equations.

On some estimation problems for nonlinear dynamic systems

Two problems of nonlinear guaranteed estimation for states of dynamical systems are considered. It is supposed that unknown measurable in $t$ disturbances are linearly included in the equation of motion and are additive in the measurement equations. These disturbances are constrained by nonlinear integral functionals, one of which is analog of functional of the generalized work. The studied problem consists in creation of the information sets according to measurement data containing the true position of the trajectory. The dynamic programming approach is used. If the first functional requires solving a nonlinear equation in partial derivatives of the first order which is not always possible, then for functional of the generalized work it is enough to find a solution of the linear Lyapunov equation of the first order that significantly simplifies the problem. Nevertheless, even in this case it is necessary to impose additional conditions on the system parameters in order for the system trajectory of the observed signal to exist. If the motion equation is linear in state variable, then many assumptions are carried out automatically. For this case the issue of mutual approximation of information sets via inclusion for different functionals is discussed. In conclusion, the most transparent linear quadratic case is considered. The statement is illustrated by examples.

ADAPTIVE GUARANTEED ESTIMATION OF A CONSTANT SIGNAL UNDER UNCERTAINTY OF MEASUREMENT ERRORS

In the guaranteed estimation problems under uncertainty relative to disturbances and meas-urement errors, the admissible sets of their possible values are determined. The solution is chosen due to conditions of guaranteed bounded estimates optimization corresponding to the worst realiza-tion of disturbances and measurement errors. The result of the guaranteed estimation is an unim-provable bounded estimate (information set), which turns to be overly pessimistic (reinsurance) if a prior admissible set of measurement errors is too large compared to their realized values. The admis-sible sets of disturbances and measurement errors can turn to be only rough upper estimates on a short observation interval. The goal of research is the accuracy enhancement problem of guaran-teed estimation when measurement errors are not realized in the worst way, i.e. the environment in which the object operates does not behave as aggressively as it is built in a priori data on the permis-sible set of error values. Research design. The problem of adaptive guaranteed estimation of a con-stant signal from noisy measurements is considered. The adaptive filtering problem is, according to the results of measurement processing, from the whole set of possible realizations of errors, to choose the one that would generate the measurement sequence. Results. An adaptive guaranteed estimation algorithm is presented. The adaptive algorithm construction is based on a multi-alternative method based on the Kalman filter bank. The method uses a set of filters, each of which is tuned to a specific hypoth-esis about the measurement error model. Filter residuals are used to compute estimates of realized measurement errors. The choice of the realization of possible errors is performed using a function that has the meaning of the residual variance over a short time interval. Conclusion. The computa-tional scheme of the adaptive algorithm, the numerical example, and comparative analysis of ob-tained estimates are presented.

Robust Estimation of Vehicle Motion States Utilizing an Extended Set-Membership Filter

Reliable vehicle motion states are critical for the precise control performed by vehicle active safety systems. This paper investigates a robust estimation strategy for vehicle motion states by feat of the application of the extended set-membership filter (ESMF). In this strategy, a system noise source is only limited as unknown but bounded, rather than the Gaussian white noise claimed in the stochastic filtering algorithms, such as the unscented Kalman filter (UKF). Moreover, as one part of this strategy, a calculation scheme with simple structure is proposed to acquire the longitudinal and lateral tire forces with acceptable accuracy. Numerical tests are carried out to verify the performance of the proposed strategy. The results indicate that as compared with the UKF-based one, it not only has higher accuracy, but also can provide a 100% hard boundary which contains the real values of the vehicle states, including the vehicle’s longitudinal velocity, lateral velocity, and sideslip angle. Therefore, the ESMF-based strategy can proffer a more guaranteed estimation with robustness for practical vehicle active safety control.