Tracking the Trajectory of a Fractional Dynamical System When Measuring Part of State Vector Coordinates
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L THEORY
Tracking the Trajectory of a Fractional Dynamical System When Measuring Part of State Vector Coordinates P. G. Surkov1∗ 1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, Yekaterinburg, 620108 Russia e-mail: ∗ [email protected]
Received January 16, 2020; revised June 26, 2020; accepted June 26, 2020
Abstract—We consider the trajectory tracking problem for a dynamical system described by nonlinear fractional differential equations with an unknown input disturbance. Based on regularization methods and constructions from guaranteed positional control theory, we propose an information noise- and computational error-robust algorithm solving the problem for the case in which only part of the system parameters can be measured. DOI: 10.1134/S0012266120110075
1. INTRODUCTION The trajectory tracking problem for a dynamical system is one classical problem that can be studied, say, by methods of positional control theory [1–4]. This class of problems attracts great interest and, at the same time, is hard to solve, because it involves constructing a feedback control simultaneously with the system operation, i.e., online. Such problems often arise when modeling processes in information technology (tracking and shadowing objects), medicine (viral diseases and their treatment), and other fields. For example, tracking problems are applied to differential games in [5, 6] and to robotics in [7]. Here we design a control algorithm ensuring that the trajectory of a given system shadows the trajectory of another system on a finite time interval. The present paper continues and generalizes the results in [8], the difference being that only part of the system state vector coordinates are assumed to be measured. If the systems in question have the form of ordinary differential equations, the controls are subjected to instantaneous constraints in the form of a compact set, and all the coordinates are measured, then the problem can be solved with the use of Krasovskii’s extremal aiming principle. Its modification for the case of constraint-free controls can be found in [9]. For systems with fractional derivatives, the solution of the problem was obtained in [10] for the case in which all state vector components are measured and the controls are subjected to instantaneous constraints. For delay systems and distributed-parameter systems, the tracking problem was studied in [11–15] with the use of the same approach as in the present paper, all state coordinates being measured in [11, 13, 15] and the case of measuring part of the coordinates being discussed in [12, 14]. 2. STATEMENT OF THE PROBLEM Before we state the problem considered in the present paper, let us recall some notions of fractional analysis. Definition 1 [16, p. 42]. The fractional integral of order γ of an arbitrary function f ∈ L1 (T, Rn ) with initial point σ is defined by the formula 1 [I f ](t) = Γ(γ) γ
Zt
(t − s)γ−1 f (s) ds,
γ ∈ (0, 1),
σ
where Γ(·) is the Euler gamma function [16, p. 29]. 1463
t ∈ T = [σ, θ],
θ
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