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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 Derivation of the Routh array ： ウィキペディア英語版 Derivation of the Routh array The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial. Central to the field of control systems design, the Routh–Hurwitz theorem and Routh array emerge by using the Euclidean algorithm and Sturm's theorem in evaluating Cauchy indices. == The Cauchy index == Given the system: : $\backslash begin$ f(x) & \quad (1) \\ & Assuming no roots of $f(x)\; =\; 0\backslash ,$ lie on the imaginary axis, and letting : $N\backslash ,$ = The number of roots of $f(x)\; =\; 0\backslash ,$ with negative real parts, and : $P\backslash ,$ = The number of roots of $f(x)\; =\; 0\backslash ,$ with positive real parts then we have : $N+P=n\; \backslash quad\; (3)\; \backslash ,$ Expressing $f(x)\backslash ,$ in polar form, we have : $f(x)\; =\; \backslash rho(x)e^\; \backslash quad\; (4)\; \backslash ,$ where : $\backslash rho(x)\; =\; \backslash sqrt^2()\}\; \backslash quad\; (5)$ and : $\backslash theta(x)\; =\; \backslash tan^\backslash big(\backslash mathfrak()/\backslash mathfrak()\backslash big)\; \backslash quad\; (6)$ from (2) note that : $\backslash theta(x)\; =\; \backslash theta\_(x)+\backslash theta\_(x)+\backslash cdots+\backslash theta\_(x)\; \backslash quad\; (7)\backslash ,$ where : $\backslash theta\_(x)\; =\; \backslash angle(x-r\_i)\; \backslash quad\; (8)\backslash ,$ Now if the ith root of $f(x)\; =\; 0\backslash ,$ has a positive real part, then (using the notation y=(RE(),IM())) : $\backslash begin$ \theta_(x)\big|_ & = \angle(x-r_i)\big|_ \\ & = \angle(0-\mathfrak(),\infty-\mathfrak()) \\ & = \angle(-\mathfrak(),\infty) \\ & = \lim_\tan^\phi=-\frac \quad (9)\\ \end and : $\backslash theta\_(x)\backslash big|\_\; =\; \backslash angle(-\backslash mathfrak(),-\backslash infty)\; =\; \backslash lim\_\backslash tan^\backslash phi=\backslash frac\; \backslash quad\; (10)\backslash ,$ Similarly, if the ith root of $f(x)=0\backslash ,$ has a negative real part, : $\backslash theta\_(x)\backslash big|\_\; =\; \backslash angle(-\backslash mathfrak(),\backslash infty)\; =\; \backslash lim\_\backslash tan^\backslash phi=\backslash frac\backslash ,\; \backslash quad\; (11)$ and : $\backslash theta\_(x)\backslash big|\_\; =\; \backslash angle(-\backslash mathfrak(),-\backslash infty)\; =\; \backslash lim\_\backslash tan^\backslash phi=-\backslash frac\backslash ,\; \backslash quad\; (12)$ Therefore, $\backslash theta\_(x)\backslash Big|\_^\; =\; -\backslash pi\backslash ,$ when the ith root of $f(x)\backslash ,$ has a positive real part, and $\backslash theta\_(x)\backslash Big|\_^\; =\; \backslash pi\backslash ,$ when the ith root of $f(x)\backslash ,$ has a negative real part. Alternatively, : $\backslash theta(x)\backslash big|\_\; =\; \backslash angle(x-r\_1)\backslash big|\_+\backslash angle(x-r\_2)\backslash big|\_+\backslash cdots+\backslash angle(x-r\_n)\backslash big|\_\; =\; \backslash fracN-\backslash fracP\; \backslash quad\; (13)\backslash ,$ and : $\backslash theta(x)\backslash big|\_\; =\; \backslash angle(x-r\_1)\backslash big|\_+\backslash angle(x-r\_2)\backslash big|\_+\backslash cdots+\backslash angle(x-r\_n)\backslash big|\_\; =\; -\backslash fracN+\backslash fracP\; \backslash quad\; (14)\backslash ,$ So, if we define : $\backslash Delta=\backslash frac\backslash theta(x)\backslash Big|\_^\; \backslash quad\; (15)\backslash ,$ then we have the relationship :$N\; -\; P\; =\; \backslash Delta\; \backslash quad\; (16)\backslash ,$ and combining (3) and (16) gives us : $N\; =\; \backslash frac\backslash ,$ and $P\; =\; \backslash frac\; \backslash quad\; (17)\backslash ,$ Therefore, given an equation of $f(x)\backslash ,$ of degree $n\backslash ,$ we need only evaluate this function $\backslash Delta\backslash ,$ to determine $N\backslash ,$, the number of roots with negative real parts and $P\backslash ,$, the number of roots with positive real parts. Equations (13) and (14) show that at $x=\backslash pm\backslash infty\backslash ,$, $\backslash theta=\backslash theta(x)\backslash ,$ is an integer multiple of $\backslash pi/2\backslash ,$. Note now, in accordance with (6) and Figure 1, the graph of $\backslash tan(\backslash theta)\backslash ,$ vs $\backslash theta\backslash ,$, that varying $x\backslash ,$ over an interval (a,b) where $\backslash theta\_a=\backslash theta(x)|\_\backslash ,$ and $\backslash theta\_b=\backslash theta(x)|\_\backslash ,$ are integer multiples of $\backslash pi\backslash ,$, this variation causing the function $\backslash theta(x)\backslash ,$ to have increased by $\backslash pi\backslash ,$, indicates that in the course of travelling from point a to point b, $\backslash theta\backslash ,$ has "jumped" from $+\backslash infty\backslash ,$ to $-\backslash infty\backslash ,$ one more time than it has jumped from $-\backslash infty\backslash ,$ to $+\backslash infty\backslash ,$. Similarly, if we vary $x\backslash ,$ over an interval (a,b) this variation causing $\backslash theta(x)\backslash ,$ to have decreased by $\backslash pi\backslash ,$, where again $\backslash theta\backslash ,$ is a multiple of $\backslash pi\backslash ,$ at both $x\; =\; ja\backslash ,$ and $x\; =\; jb\backslash ,$, implies that $\backslash tan\; \backslash theta\; (x)\; =\; \backslash mathfrak()/\backslash mathfrak()\backslash ,$ has jumped from $-\backslash infty\backslash ,$ to $+\backslash infty\backslash ,$ one more time than it has jumped from $+\backslash infty\backslash ,$ to $-\backslash infty\backslash ,$ as $x\backslash ,$ was varied over the said interval. Thus, $\backslash theta(x)\backslash Big|\_^\backslash ,$ is $\backslash pi\backslash ,$ times the difference between the number of points at which $\backslash mathfrak()/\backslash mathfrak()\backslash ,$ jumps from $-\backslash infty\backslash ,$ to $+\backslash infty\backslash ,$ and the number of points at which $\backslash mathfrak()/\backslash mathfrak()\backslash ,$ jumps from $+\backslash infty\backslash ,$ to $-\backslash infty\backslash ,$ as $x\backslash ,$ ranges over the interval $(-j\backslash infty,+j\backslash infty\backslash ,)$ provided that at $x=\backslash pm\; j\backslash infty$, $\backslash tan()\backslash ,$ is defined. In the case where the starting point is on an incongruity (i.e. $\backslash theta\_a=\backslash pi/2\; \backslash pm\; i\backslash pi\backslash ,$, ''i'' = 0, 1, 2, ...) the ending point will be on an incongruity as well, by equation (16) (since $N\backslash ,$ is an integer and $P\backslash ,$ is an integer, $\backslash Delta\backslash ,$ will be an integer). In this case, we can achieve this same index (difference in positive and negative jumps) by shifting the axes of the tangent function by $\backslash pi/2\backslash ,$, through adding $\backslash pi/2\backslash ,$ to $\backslash theta\backslash ,$. Thus, our index is now fully defined for any combination of coefficients in $f(x)\backslash ,$ by evaluating $\backslash tan()=\backslash mathfrak()/\backslash mathfrak()\backslash ,$ over the interval (a,b) = $(+j\backslash infty,\; -j\backslash infty)\backslash ,$ when our starting (and thus ending) point is not an incongruity, and by evaluating : $\backslash tan()=\backslash tan(+\; \backslash pi/2)\; =\; -\backslash cot()\; =\; -\backslash mathfrak()/\backslash mathfrak()\; \backslash quad\; (18)\backslash ,$ over said interval when our starting point is at an incongruity. This difference, $\backslash Delta\backslash ,$, of negative and positive jumping incongruities encountered while traversing $x\backslash ,$ from $-j\backslash infty\backslash ,$ to $+j\backslash infty\backslash ,$ is called the Cauchy Index of the tangent of the phase angle, the phase angle being $\backslash theta(x)\backslash ,$ or $\backslash theta\text{'}(x)\backslash ,$, depending as $\backslash theta\_a\backslash ,$ is an integer multiple of $\backslash pi\backslash ,$ or not. 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「Derivation of the Routh array」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.