Ellipsoidal and Interval Estimation of State Vectors for Families of Linear and Nonlinear Discrete-Time Dynamic Systems
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ELLIPSOIDAL AND INTERVAL ESTIMATION OF STATE VECTORS FOR FAMILIES OF LINEAR AND NONLINEAR DISCRETE-TIME DYNAMIC SYSTEMS1 V. M. Kuntsevicha† and V. V. Volosov a‡
UDC 621.391
Abstract. This paper proposes constructive algorithms for ellipsoidal estimation of state vectors measured under bounded disturbances in families of linear and a rather wide class of nonlinear dynamic systems. The algorithms are based on the use of support functions and approximation of generally non-convex sets of estimates by ellipsoids. Keywords: dynamic system, nonlinearity, uncertainty, measurement noise, state vector, robust method of ellipsoidal estimation. INTRODUCTION Determining estimates for the state vector of nonlinear dynamic objects that is measured under disturbances is a necessary stage of implementation of algorithms used to control such objects. At present, because of a number of causes for the estimation of state vectors measured under bounded disturbances in linear dynamic systems, ellipsoidal estimates are most widespread owing to works of A. B. Kurzhanskii [1, 2], F. C. Schweppe [3], F. L. Chernous’ko [4, 5], and others. In these works, different criteria for choosing approximating ellipsoids are used among which are the minimum-volume ellipsoid, trace of its matrix, etc. A more complicated problem is the determination of estimates for the state vector of nonlinear dynamic systems that is measured under bounded disturbances. At present, a few articles consider the solution of this problem for some special classes of nonlinear functions, in particular, quadratic ones (see, for example, [6]). This work presents an essential generalization and a further development of the general scheme of determining ellipsoidal estimates for the state vector of nonlinear discrete systems, which is described in [7] and proposes a method for determining ellipsoidal and interval minimum-volume estimates for families of linear and a rather wide class of families of nonlinear discrete systems. 1. ESTIMATES FOR THE STATE VECTOR OF FAMILIES OF LINEAR SYSTEMS The following equation of a family of discrete controllable systems is given: X n + 1 = AX n + BU n , n = 0, 1, 2,K ,
(1)
1
This work was financially supported by the NASU Target Complex Program on Scientific Space Research for 2012–2016. a
Space Research Institute of NASU and SSAU, Kyiv, Ukraine, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 73–84, January–February, 2015. Original article submitted September 2, 2014. 64
1060-0396/15/5101-0064
©
2015 Springer Science+Business Media New York
where X n Î R m , X n = ( x1,n , x 2 ,n , K , x m , n )T is a phase state vector; U n Î R k is a control vector; A is an m´ m matrix for row vectors A Ti of which their estimates A i are given, A iT Î A i , i = 1, m.
(2)
Here, A i is a given convex set and B is a given m´ k matrix. Row vectors (2) imply the following estimate for the matrix A: A Î A = A1 ´ A 2 ´ K ´ A m .
(3)
Assume that, for a vector X n , the following ellipsoidal estimate
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