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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Strict-feedback form ： ウィキペディア英語版 Strict-feedback form In control theory, dynamical systems are in strict-feedback form when they can be expressed as :$\backslash begin\; \backslash dot)\; +\; g\_0(\backslash mathbf)\; z\_1\backslash \backslash$ \dot_1 = f_1(\mathbf,z_1) + g_1(\mathbf,z_1) z_2\\ \dot_2 = f_2(\mathbf,z_1,z_2) + g_2(\mathbf,z_1,z_2) z_3\\ \vdots\\ \dot_i = f_i(\mathbf,z_1, z_2, \ldots, z_, z_i) + g_i(\mathbf,z_1, z_2, \ldots, z_, z_i) z_ \quad \text 1 \leq i < k-1\\ \vdots\\ \dot_ = f_(\mathbf,z_1, z_2, \ldots, z_) + g_(\mathbf,z_1, z_2, \ldots, z_) z_k\\ \dot_k = f_k(\mathbf,z_1, z_2, \ldots, z_, z_k) + g_k(\mathbf,z_1, z_2, \dots, z_, z_k) u\end where * $\backslash mathbf\; \backslash in\; \backslash mathbb^n$ with $n\; \backslash geq\; 1$, * $z\_1,\; z\_2,\; \backslash ldots,\; z\_i,\; \backslash ldots,\; z\_,\; z\_k$ are scalars, * $u$ is a scalar input to the system, * $f\_0,\; f\_1,\; f\_2,\; \backslash ldots,\; f\_i,\; \backslash ldots,\; f\_,\; f\_k$ vanish at the origin (i.e., $f\_i(0,0,\backslash dots,0)\; =\; 0$), * $g\_1,\; g\_2,\; \backslash ldots,\; g\_i,\; \backslash ldots,\; g\_,\; g\_k$ are nonzero over the domain of interest (i.e., $g\_i(\backslash mathbf,z\_1,\backslash ldots,z\_k)\; \backslash neq\; 0$ for $1\; \backslash leq\; i\; \backslash leq\; k$). Here, ''strict feedback'' refers to the fact that the nonlinear functions $f\_i$ and $g\_i$ in the $\backslash dot\_i$ equation only depend on states $x,\; z\_1,\; \backslash ldots,\; z\_i$ that are ''fed back'' to that subsystem. That is, the system has a kind of lower triangular form. ==Stabilization== :(詳細はstabilized by recursive application of backstepping.〔 That is, # It is given that the system #::$\backslash dot)\; +\; g\_0(\backslash mathbf)\; u\_x(\backslash mathbf)$ #:is already stabilized to the origin by some control $u\_x(\backslash mathbf)$ where $u\_x(\backslash mathbf)\; =\; 0$. That is, choice of $u\_x$ to stabilize this system must occur using some other method. It is also assumed that a Lyapunov function $V\_x$ for this stable subsystem is known. # A control $u\_1(\backslash mathbf,z\_1)$ is designed so that the system #::$\backslash dot\_1\; =\; f\_1(\backslash mathbf,z\_1)\; +\; g\_1(\backslash mathbf,z\_1)\; u\_1(\backslash mathbf,z\_1)$ #:is stabilized so that $z\_1$ follows the desired $u\_x$ control. The control design is based on the augmented Lyapunov function candidate #::$V\_1(\backslash mathbf,z\_1)\; =\; V\_x(\backslash mathbf)\; +\; \backslash frac(\; z\_1\; -\; u\_x(\backslash mathbf)\; )^2$ #:The control $u\_1$ can be picked to bound $\backslash dot\_1$ away from zero. # A control $u\_2(\backslash mathbf,z\_1,z\_2)$ is designed so that the system #::$\backslash dot\_2\; =\; f\_2(\backslash mathbf,z\_1,z\_2)\; +\; g\_2(\backslash mathbf,z\_1,z\_2)\; u\_2(\backslash mathbf,z\_1,z\_2)$ #:is stabilized so that $z\_2$ follows the desired $u\_1$ control. The control design is based on the augmented Lyapunov function candidate #::$V\_2(\backslash mathbf,z\_1,z\_2)\; =\; V\_1(\backslash mathbf,z\_1)\; +\; \backslash frac(\; z\_2\; -\; u\_1(\backslash mathbf,z\_1)\; )^2$ #:The control $u\_2$ can be picked to bound $\backslash dot\_2$ away from zero. # This process continues until the actual $u$ is known, and # * The ''real'' control $u$ stabilizes $z\_k$ to ''fictitious'' control $u\_$. # * The ''fictitious'' control $u\_$ stabilizes $z\_$ to ''fictitious'' control $u\_$. # * The ''fictitious'' control $u\_$ stabilizes $z\_$ to ''fictitious'' control $u\_$. # * ... # * The ''fictitious'' control $u\_2$ stabilizes $z\_2$ to ''fictitious'' control $u\_1$. # * The ''fictitious'' control $u\_1$ stabilizes $z\_1$ to ''fictitious'' control $u\_x$. # * The ''fictitious'' control $u\_x$ stabilizes $\backslash mathbf$ to the origin. This process is known as backstepping because it starts with the requirements on some internal subsystem for stability and progressively ''steps back'' out of the system, maintaining stability at each step. Because * $f\_i$ vanish at the origin for $0\; \backslash leq\; i\; \backslash leq\; k$, * $g\_i$ are nonzero for $1\; \backslash leq\; i\; \backslash leq\; k$, * the given control $u\_x$ has $u\_x(\backslash mathbf)\; =\; 0$, then the resulting system has an equilibrium at the origin (i.e., where $\backslash mathbf=\backslash mathbf\backslash ,$, $z\_1=0$, $z\_2=0$, ... , $z\_=0$, and $z\_k=0$) that is globally asymptotically stable. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Strict-feedback form」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.