Estimates of reachable sets of control systems with bilinear–quadratic nonlinearities

Abstract

The problem of estimating reachable sets of nonlinear impulsive control systems with quadratic nonlinearity and with uncertainty in initial states and in the matrix of system is studied. The problem is studied under uncertainty conditions with set-membership description of uncertain variables, which are taken to be unknown but bounded with given bounds. We study the case when the system nonlinearity is generated by the combination of two types of functions in related differential equations, one of which is bilinear and the other one is quadratic. The problem may be reformulated as the problem of describing the motion of set-valued states in the state space under nonlinear dynamics with state velocities having bilinear-quadratic kind. Basing on the techniques of approximation of the generalized trajectory tubes by the solutions of control systems without measure terms and using the techniques of ellipsoidal calculus we present here a state estimation algorithms for the studied nonlinear impulsive control problem bilinear-quadratic type.

State estimation for bilinear impulsive control systems under uncertainties

“…Here we develop the set-membership approach based on ellipsoidal calculus for the considered system. Also we generalize earlier results [Filippova and Matviychuk, 2015;Filippova, 2016;Matviychuk, 2017a], in particular we consider more complicated model of the control system than in [Matviychuk, 2017b]. In this paper the control function of studied bilinear impulsive control system is a pair of a classical (measurable) control and an impulsive control function.…”

“…The proof of this theorem uses the procedure of external ellipsoidal estimating a sum of two ellipsoids [Chernousko, 1994;Kurzhanski and Valyi, 1997]. Applying the scheme from [Filippova and Matviychuk, 2011;Filippova and Matviychuk, 2015] and using results of the Theorem 6 we can find the upper estimatesfor reachable sets W (t 0 + σ) of the differential inclusion (20). Remark 2.…”

Viscosity solutions to evolution problems of star-shaped reachable sets

“…It is also worth noting that currently there is a great amount of research undertaken to approximate of the reachable sets with ellipsoids and offer some rapid and reliable computational methods [1,19,20]. Although those results usually impose some additional restriction on the system, such as bilinearity or some other specific form of the mapping F, they are promising and offer a serious constructive alternative for the approach discussed in this paper.…”

A guaranteed state estimation of linear retarded neutral type systems

“…A finite-dimensional approximation of the problem is given and an illustrating example is considered. The problem under investigation were considered, for example, for different systems in Bertsecas and Rhodes [12], Schweppe [13], Ananyev and Anikin [14], Ananiev [15], Anan'ev [8], Ananyev [16], Filippova and Matviychuk [17], Gusev and Zykov [18].…”