State estimation in continuous dynamic systems by the ellipsoid method

Abstract

519.6 Ellipsoidal approximation of information sets uses the intersection of some initial ellipsoid with a half-space or a layer bounded by parallel hyperplanes. The ideas of ellipsoidal approximation have found many uses in minmax estimation and control problems [1][2][3][4][5][6], and also in mathematical programming [7, 8]. In the first category, the ellipsoidal information sets contain the unknown states or the parameters of the controlled system; in the second category, they contain the sought minimum points of the functionals. A comprehensive list of studies using ellipsoidal approximation for minmax estimation is given in [3, 4].The present article is a direct continuation and extension of [9]. Among previous publications on similar topics, it is closest to [10] by problem formulation. It differs, however, by its new estimation algorithms that are easier to use and by a new proof of their convergence based on the direct Lyapunov method. STATEMENT OF THE PROBLEMConsider the following equations of motion for the system and its observer: y(t) =hr(t) x(t) +~(t).(2)Here t is continuous time, t E [0, oo); x(t) is the state vector x E R n, R n is the n-dimensional real Euclidean space; u(t) is the control vector u E Rm; y(t) is the observed signal, which is the result of observing the system output h r (t)x(t) mixed with additive noise ~(t), y(t) E R1; A(t), B(t), and h(t) are appropriately dimensioned matrices and vector with bounded continuous elements. We assume that the noise is continuous and its absolute value is bounded by a given constant I ~(t) I < c v t >__ 0. The superscript T denotes the transpose of vectors and matrices.Let K(t) be the fundamental matrix of the homogeneous system 2 = A(t)x, corresponding to system (1), and K(t, to) its transition matrix (the Cauchy matrix, the matricant) [11][12][13]. Here K(t, to) = K(t) K-1(to), where K(t o) = K(t)[t=to and the matrices K(0 and K(t, to) satisfy the matrix equationsat -A (O K (O' at -A (O g (t, to).Following [14], we say that system (1), (2) is everywhere observable with observability index T, 0 < T < oo, if the matrix t Q (t) = f K T(0, t) h (0) h T(0) K (O, t)dO t-T

Structural and Parametrical Synthesis of the Laws of Critical Control

“…Let us consider the problem of optimizing the parameters of the control law (regulator) (17) in the proposal of the minimum-phase object (5) and show that the optimal controller can be written as…”

“…For the synthesis of a critical control system, preliminary construction of both models of the control object itself and the environment is required. And if the object model can be described in terms of «Input -output», then the influence of the environment can be taken into account using a special description of the signals acting on the object [2][3][4][5].…”

Partial stability and control: The state-of-the-art and development prospects

“…Research on partial stability (stabilization) and control also covers -problems in celestial mechanics [37,38,138,227], -convergence of certain optimization algorithms [94], -abstract nonlinear Cauchy problems [154], -stability of a proper vortex n-gon [69], -coordinate synchronization of dynamic systems [327,199,14,15,30,200] related with safe communication (in synchronization of chaotic systems), -estimation of states and domain of partial stability of dynamic systems [16,269], -stability of radar systems [150], -computer network traffic balancing [252], and -stability of biotechnical processes [271]. Let us consider examples on the partial stability of the zero equilibrium position of a holonomic Lagrangian system, partial stabilization of the permanent rotation of an asymmetric solid with a flywheel, and partial control of the angular motion of a solid.…”