Routh-Wurwitz 稳定判据_routh hurwizte-优快云博客

本文详细介绍了Routh-Hurwitz稳定判据及其应用，该判据用于确定闭环系统的稳定性，通过分析系统特征方程的根的位置来判断系统是否稳定。文章还列举了几种特殊情况下的计算方法。

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Routh-Hurwitz 稳定判据

系统稳定性概念

A stable system should exhibit a bounded output if the corresponding input is bounded.

The closed-loop system transfer function is written as:

T (s) = p ( s ) q ( s ) = \prod M i = 1 ( s + z i ) s N \prod Q k = 1 ( s + σ k ) \prod R m = 1 [ s 2 + 2 α m s + ( α 2 m + ω 2 m ) ]

$T(s)=\frac{p(s)}{q(s)}=\frac{\prod_{i=1}^M(s+z_i)}{s^N\prod_{k=1}^Q(s+\sigma_k)\prod_{m=1}^R[s^2+2\alpha_ms+(\alpha_m^2+\omega_m^2)]}$

q(s)=0 $q(s)=0$ is the characteristic equation whose roots are the poles of the closed-loop system. The output response for an impulse function input(When N = 0) is then:

y (t) = \sum k = 1 Q A k e - α k t + \sum m = 1 R B m (1 ω m e - α m t s i n (ω m t + θ m))

$y(t)=\sum_{k=1}^QA_ke^{-\alpha_kt}+\sum_{m=1}^RB_m(\frac{1}{\omega_m}e^{-\alpha_mt}sin(\omega_mt+\theta_m))$
So, to obtain a bounded response, the poles of the closed-loop system must be in the left-hand portion of the s-plane

A necessary and sufficient condition for a feedback system to be stable is that all the poles of the system transfer function have negative real parts.

判断系统稳定性的三种方法：

the s-plane approach;
the frequency plane ( $j\omega$ ) approach;
the time domain approach;

Routh-Hurwitz 稳定判据

特征方程：

Δ (s) = q (s) = a n s n + a n - 1 s n - 1 + \cdot s + a 1 s + a 0 = 0

$\Delta(s)=q(s)=a_ns^n+a_{n-1}s^{n-1}+\cdot s +a_1s+a_0=0$
上式可以写成：

a n (s - r 1) (s - r 2) \dots (s - r n) = 0

$a_n(s-r_1)(s-r_2)\cdots(s-r_n)=0$
其中：

r1,r2,⋯rn<0 $r_1,r_2,\cdots r_n<0$

又可以写成：

(s) = a n s n - a n (r 1 + r 2 + \dots + r n) s n - 1 + a n (r 1 r 2 + r 1 r 3 + r 2 r 3 + \dots) s n - 2 - a n (r 1 r 2 r 3 + r 1 r 2 r 4 + \dots) s n - 3 + \dots + a n (- 1) n r 1 r 2 r 4 \dots r n = 0

$\begin{array}q(s)=&a_ns^n-a_n(r_1+r_2+\cdots+r_n)s^{n-1}\\ &+a_n(r_1r_2+r_1r_3+r_2r_3+\cdots)s^{n-2}\\ &-a_n(r_1r_2r_3+r_1r_2r_4+\cdots)s^{n-3}+\cdots\\ &+a_n(-1)^nr_1r_2r_4\cdots r_n=0 \end{array}$
由上式可得系统稳定的必要条件是特征方程多项式的系数同号且不为0，有系数0时可能为临界稳定状态。

s n s n - 1 s n - 2 s n - 3 ⋮ s 0 | | | | | | a n a n - 1 b n - 1 c n - 1 ⋮ h n - 1 a n - 2 a n - 3 b n - 3 c n - 3 ⋮ a n - 4 a n - 5 b n - 5 c n - 5 ⋮

$\begin{array}&s^n&|&a_n &a_{n-2} &a_{n-4}\\ s^{n-1}&| &a_{n-1}&a_{n-3}&a_{n-5}\\ s^{n-2}&|&b_{n-1}&b_{n-3}&b_{n-5}\\ s^{n-3}&|&c_{n-1}&c_{n-3}&c_{n-5}\\ \vdots&| &\vdots&\vdots&\vdots\\ s^0&|&h_{n-1} \end{array}$
其中：

b n - 1 = ( a n - 1 ) ( a n - 2 ) - a n ( a n - 3 ) a n - 1 b n - 3 = ( a n - 1 ) ( a n - 4 ) - a n ( a n - 5 ) a n - 1 c n - 1 = ( b n - 1 ) ( a n - 3 ) - ( a n - 1 ) ( b n - 3 ) b n - 1

$\begin{array}&b_{n-1}=\frac{(a_{n-1})(a_{n-2})-a_n(a_{n-3})}{a_{n-1}}\\ b_{n-3}=\frac{(a_{n-1})(a_{n-4})-a_n(a_{n-5})}{a_{n-1}}\\ c_{n-1}=\frac{(b_{n-1})(a_{n-3})-(a_{n-1})(b_{n-3})}{b_{n-1}} \end{array}$