smith标准型_线性系统理论（八）多项式矩阵Smith-McMillan标准型计算方法

1 参考^[1]

Chenglin Li：线性系统理论（七）finite- and infinite-zeroszhuanlan.zhihu.com

多项式矩阵Smith-McMillan标准型确定方法分析

2 单模矩阵法

Chenglin Li：线性系统理论（七）finite- and infinite-zeros

3 最大公因子法

3.1 原理

对于多项式矩阵G(s)，其Smith型

$equation?tex=%5CLambda%28s%29$ 的对角元素为

$equation?tex=%5Clambda_1%2C%5Clambda_2%2C%5Clambda_3%2C...$ 即不变多项式。

令Di(s)=gcd{G(s)的所有

$equation?tex=i%5Ctimes+i$ 子式，i=1,2,3...}，

$equation?tex=D_0%28s%29%3D1.+$

那么可得：

$equation?tex=%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+%5Clambda_1%3D%26%5Cfrac%7BD_1%28s%29%7D%7BD_0%28s%29%7D%2C%5C%5C+%5Clambda_2%3D%26%5Cfrac%7BD_2%28s%29%7D%7BD_1%28s%29%7D%2C%5C%5C+%5Clambda_3%3D%26%5Cfrac%7BD_3%28s%29%7D%7BD_2%28s%29%7D%2C+...++%5Cend%7Baligned%7D+%5Cright.%5C%5C$

3.2 举例

$equation?tex=H%28%5Clambda%29%3D%5Cleft%28+%5Cbegin%7Bmatrix%7D+-%5Cfrac%7B1%7D%7B%5Clambda-1%7D%260%260%5C%5C+0%26-%5Cfrac%7B%5Clambda%7D%7B%5Clambda-1%7D%260%5C%5C+0+%260%26%5Cfrac%7B%28%5Clambda-1%29%5E2%7D%7B%5Clambda%5E2%7D++%5Cend%7Bmatrix%7D++%5Cright%29.%5C%5C$

$equation?tex=H%28%5Clambda%29$ 所有元素分母的最小公倍数为

$equation?tex=d%28%5Clambda%29%3D%5Clambda%5E2%28%5Clambda-1%29%2C+N%28%5Clambda%29%3D+%5Cleft%28+%5Cbegin%7Bmatrix%7D+-%5Clambda%5E2+%26+0+%26+0%5C%5C++0+%26+-%5Clambda%5E3+%26+0%5C%5C++0+%26+0+%26+%28%5Clambda-1%29%5E3++%5Cend%7Bmatrix%7D+++%5Cright%29.%5C%5C$

$equation?tex=N%28%5Clambda%29$ 各阶子式的最大公因子为：

$equation?tex=%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+D_0%3D%261%2C%5C%5C+D_1%3D%26+gca+%5Cleft%5C%7B-%5Clambda%5E2%2C-%5Clambda%5E3%2C%28%5Clambda-1%29%5E3+%5Cright%5C%7D%3D1%2C%5C%5C+D_2%3D%26+gca+%5Cleft%5C%7B-%5Clambda%5E5%2C-%5Clambda%5E2%28%5Clambda-1%29%5E3%2C%5Clambda%5E3%28%5Clambda-1%29%5E3+%5Cright%5C%7D%3D%5Clambda%5E2%2C%5C%5C+D_1%3D%26+gca+%5Cleft%5C%7B%5Clambda%5E5%28%5Clambda-1%29%5E3+%5Cright%5C%7D%3D%5Clambda%5E5%28%5Clambda-1%29%5E3.%5C%5C+++%5Cend%7Baligned%7D+%5Cright.%5C%5C$

Smith不变多项式为（需检查是否满足首一性）：

$equation?tex=%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+%5Clambda_1%3D%261%2Fd%28%5Clambda%29%2C%5C%5C+%5Clambda_2%3D%26%5Clambda%5E2%2Fd%28%5Clambda%29%2C%5C%5C+%5Clambda_3%3D%26+%5Clambda%5E3%28%5Clambda-1%29%5E3%2Fd%28%5Clambda%29.++%5Cend%7Baligned%7D+%5Cright.%5C%5C$

Smith-McMillan形式为：

$equation?tex=M%28%5Clambda%29%3D%5Cleft%28+%5Cbegin%7Bmatrix%7D+%5Cfrac%7B1%7D%7B%5Clambda%5E2%28%5Clambda-1%29%7D%260%260%5C%5C+0%26%5Cfrac%7B1%7D%7B%5Clambda-1%7D%260%5C%5C+0+%260%26%5Clambda%28%5Clambda-1%29%5E2++%5Cend%7Bmatrix%7D++%5Cright%29.%5C%5C$

4 结构指数法

4.1 评价值

$equation?tex=%7CG%7C%5E%7Bi%7D$ 为G(s)的所有

$equation?tex=i%5Ctimes+i$ 子式，i=1,2,3...}，则G(s)在

$equation?tex=s%3D%5Cxi_k$ 上的第

$equation?tex=i$ 阶评价值为

$equation?tex=v_%7B%5Cxi_k%7D%5E%7B%28i%29%7D%28G%29%3Dmin%5Cleft%5C%7B+v_%7B%5Cxi_k%7D+%5Cleft%28+%7CG%7C%5Ei++%5Cright%29+%5Cright%5C%7D.%5C%5C$

4.2 举例分析

$equation?tex=G%28s%29%3D%5Cleft%28+%5Cbegin%7Bmatrix%7D+%5Cfrac%7Bs%7D%7B%28s%2B1%29%5E2%28s%2B2%29%5E2%7D+%26%5Cfrac%7Bs%7D%7B%28s%2B2%29%5E2%7D%5C%5C+-%5Cfrac%7Bs%7D%7B%28s%2B2%29%5E2%7D+%26-%5Cfrac%7Bs%7D%7B%28s%2B2%29%5E2%7D%5C%5C++%5Cend%7Bmatrix%7D++%5Cright%29%5C%5C$

G(s)1阶子式：

$equation?tex=%7CG%7C%5E1%3D%5Cfrac%7Bs%7D%7B%28s%2B1%29%5E2%28s%2B2%29%5E2%7D%2C%5Cfrac%7B-s%7D%7B%28s%2B2%29%5E2%7D%2C%5Cfrac%7Bs%7D%7B%28s%2B1%29%5E2%7D%2C%5Cfrac%7B-s%7D%7B%28s%2B2%29%5E2%7D.%5C%5C$

G(s)2阶子式：

$equation?tex=%7CG%7C%5E2%3D%5Cfrac%7Bs%5E3%7D%7B%28s%2B1%29%5E2%28s%2B2%29%5E3%7D.%5C%5C$

G(s)在s=0处的评价值：

$equation?tex=%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+v_0%5E%7B%281%29%7D%28G%29%3D%26min%5Cleft%5C%7B+1%2C1%2C1%2C1+%5Cright%5C%7D%3D1%2C%5C%5C+v_0%5E%7B%282%29%7D%28G%29%3D%26min%5Cleft%5C%7B+3+%5Cright%5C%7D%3D3.%5C%5C++%5Cend%7Baligned%7D++%5Cright.%5C%5C$

G(s)在s=-1处的评价值：

$equation?tex=%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+v_%7B-1%7D%5E%7B%281%29%7D%28G%29%3D%26min%5Cleft%5C%7B+-2%2C0%2C0%2C0+%5Cright%5C%7D%3D-2%2C%5C%5C+v_%7B-1%7D%5E%7B%282%29%7D%28G%29%3D%26min%5Cleft%5C%7B+-2+%5Cright%5C%7D%3D-2.%5C%5C++%5Cend%7Baligned%7D++%5Cright.%5C%5C$

G(s)在s=-2处的评价值：

$equation?tex=%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+v_%7B-2%7D%5E%7B%281%29%7D%28G%29%3D%26min%5Cleft%5C%7B+-2%2C-2%2C-2%2C-2+%5Cright%5C%7D%3D-2%2C%5C%5C+v_%7B-2%7D%5E%7B%282%29%7D%28G%29%3D%26min%5Cleft%5C%7B+-3+%5Cright%5C%7D%3D-3.%5C%5C++%5Cend%7Baligned%7D++%5Cright.%5C%5C$

4.3 结构指数组

$equation?tex=%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+%5Csigma_1%28%5Cxi%29%3D%26v_%7B%5Cxi%7D%5E%7B%281%29%7D%28G%29%2C%5C%5C+%5Csigma_2%28%5Cxi%29%3D%26v_%7B%5Cxi%7D%5E%7B%282%29%7D%28G%29-v_%7B%5Cxi%7D%5E%7B%281%29%7D%28G%29%2C%5C%5C+%5Csigma_3%28%5Cxi%29%3D%26v_%7B%5Cxi%7D%5E%7B%283%29%7D%28G%29-v_%7B%5Cxi%7D%5E%7B%282%29%7D%28G%29%2C%5C%5C+...+++%5Cend%7Baligned%7D++%5Cright.%5C%5C$

对于4.2的例子

$equation?tex=%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+%5Csigma_1%280%29%3D%261%2C%5C%5C+%5Csigma_2%280%29%3D%263-1%3D2.%5C%5C+%5Cend%7Baligned%7D++%5Cright.+%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+%5Csigma_1%28-1%29%3D%26-2%2C%5C%5C+%5Csigma_2%28-1%29%3D%26-2-%28-2%29%3D0.%5C%5C+%5Cend%7Baligned%7D++%5Cright.+%5Cleft%5C%7B+%5Cbegin%7Baligned%7D+%5Csigma_1%28-2%29%3D%26-2%2C%5C%5C+%5Csigma_2%28-2%29%3D%26-3-%28-2%29%3D-1.%5C%5C+%5Cend%7Baligned%7D++%5Cright.%5C%5C$

4.4 构造Smith-McMillan型

$equation?tex=M%28s%29%3D+%5Cleft%28+%5Cbegin%7Bmatrix%7D+%5Cprod_%7Bk%3D1%7D%5E%7Br%7DM_%7B%5Cxi_k%7D%28s%29+%26+0%5C%5C+0+%26+0%5C%5C++%5Cend%7Bmatrix%7D+%5Cright%29.%5C%5C$

$equation?tex=M_%7B%5Cxi_k%7D%28s%29%3D+%5Cleft%28+%5Cbegin%7Bmatrix%7D+%28s-%5Cxi_k%29%5E%7B%5Csigma_1%28%5Cxi_k%29%7D+%26+0+%26...%5C%5C+0+%26+%28s-%5Cxi_k%29%5E%7B%5Csigma_2%28%5Cxi_k%29%7D+%26...+%5C%5C+0%26...%26...%5C%5C+0+%26...%26%28s-%5Cxi_k%29%5E%7B%5Csigma_r%28%5Cxi_k%29%7D+++%5Cend%7Bmatrix%7D+%5Cright%29.%5C%5C$

对于4.2的例子，G(s)的Smith-McMillan型如下：

$equation?tex=M%28s%29%3D+%5Cleft%28+%5Cbegin%7Bmatrix%7D+s%28s%2B1%29%5E%7B-2%7D%28s%2B2%29%5E%7B-2%7D+%26+0%5C%5C+0+%26+s%5E2%28s%2B2%29%5E%7B-1%7D+%5C%5C++%5Cend%7Bmatrix%7D+%5Cright%29.%5C%5C$