Abstract
One of the most common methods of synthesis of the nonlinear control systems is the method of a feedback linearization (FL). The idea of this method consists in conversion of the original nonlinear system into a linear one by means of a state feedback and coordinate transformation. Then, the methods of control theory for the linear systems are used for the system design. If the original nonlinear system cannot be linearized exactly by the state feedback, the method of the approximate feedback linearization (AFL) is used. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system (not of the entire system). In this paper, the author proposes a method of an approximate feedback linearization control of the nonlinear singularly perturbed (SP) systems. The proposed method is based on a decomposition of the original SP system and construction of AFL control in the form of composite FL controls for the slow and fast subsystems. In general, a nonlinear SP system cannot be easily separated into slow and fast subsystems, because the conditions of Tikhonov theorem are not complied. In order to overcome this, the author proposes to perform the feedback linearization method at first for the system's part, which describes the fast state variables. Thus, a fast control is chosen, so that the conditions of Tikhonov theorem would be met. Then, using a standard singular perturbation technique, we obtain a slow subsystem. Further the problem of FL control for a slow subsystem is solved. The resulting AFL control is obtained in the form of a composite control. Application of the proposed approach is illustrated with two examples.
Highlights
Постановка задачиРассìатривается неëинейная синãуëярно возìущенная систеìа виäа x· (t) = F(x, z, u), x(0) = x0, y = φ(x),
Линеаризаöия с поìощüþ обратной связи явëяется эффективныì ìетоäоì синтеза неëинейных систеì управëения [1,2,3,4]
The author proposes a method of an approximate feedback linearization control
Summary
Рассìатривается неëинейная синãуëярно возìущенная систеìа виäа x· (t) = F(x, z, u), x(0) = x0, y = φ(x),. Ãäе x ∈ Rn1 , z ∈ Rn2 — переìенные состояния; y, u ∈ R1 — выхоä и управëение соответственно; F, f и g — равноìерно непрерывные и оãрани÷енные ãëаäкие векторные функöии с äостато÷ныì ÷исëоì произвоäных по всеì арãуìентаì; ε > 0 — ìаëый параìетр (синãуëярное возìущение). Требуется найти управëение u и преобразование T(x, z), не зависящие от параìетра ε и такие, ÷то при äостато÷но ìаëоì ε систеìа (1), (2) ìожет бытü прибëиженно (с то÷ностüþ äо O(ε)) преобразована к ëинейноìу виäу относитеëüно скаëярноãо выхоäа y = φ(x)
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