线性系统的分离原理——降维观测器情形下的证明

2023-06-07 22:35 作者:斟好雨 0人读过 | 我要投稿

在现代控制理论中讲解分离原理时一般都以全维观测器为例进行证明（刘豹《现代控制理论》p221-p222），事实上，这个定理对于降维观测器也同样成立，证明思路类似，但步骤较为繁琐。下面给出证明。

考虑如下形式的线性系统

$%5Cleft%5C%7B%20%5Cbegin%7Barray%7D%7Bl%7D%0A%09%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09%5Cdot%7Bx%7D_1%5C%5C%0A%09%5Cdot%7Bx%7D_2%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%0A%09A_%7B11%7D%26%09%09A_%7B12%7D%5C%5C%0A%09A_%7B21%7D%26%09%09A_%7B22%7D%5C%5C%0A%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09x_1%5C%5C%0A%09x_2%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%2B%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09B_1%5C%5C%0A%09B_2%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20u%5C%5C%0A%09y%3D%5Cleft%5B%20I%5C%20%5C%200%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09x_1%5C%5C%0A%09x_2%5C%5C%0A%5Cend%7Barray%7D%20%5Cright%5D%20%3Dx_1%5C%5C%0A%5Cend%7Barray%7D%20%5Cright.%20%5Ctag%7B1%7D$

这类系统无需进行状态变换就可进行状态观测器设计，需要观测的状态向量为 $x_2$ ，则设计的状态观测器为

$%5Cdot%7B%5Chat%7Bx%7D%7D_2%3D%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%20%5Cright)%20%5Chat%7Bx%7D_2%2BL%5Cleft(%20%5Cdot%7By%7D-A_%7B11%7Dy-B_1u%20%5Cright)%20%2BA_%7B21%7Dy%2BB_2u%5Ctag%7B2%7D$

引入 $x_1$ 的观测值 $%5Chat%7Bx%7D_1$ ，且有 $%5Chat%7Bx%7D_1%3Dx_1%3Dy$ ，并将 $%5Chat%7Bx%7D_1$ 、 $%5Chat%7Bx%7D_2$ 合起来作为 $%5Chat%7Bx_2%7D$

构造状态反馈控制律为

$u%3DK%5Chat%7Bx%7D%2Bv%3D%5Cleft%5B%20k_1%5C%20k_2%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09%5Chat%7Bx%7D_1%5C%5C%20%09%5Chat%7Bx%7D_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%2Bv%3Dk_1%5Chat%7Bx%7D_1%2Bk_2%5Chat%7Bx%7D_2%2Bv%5Ctag%7B3%7D$

由状态反馈控制律的形式，可将状态方程改写为

$%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09%5Cdot%7Bx%7D_1%5C%5C%20%09%5Cdot%7Bx%7D_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09A_%7B11%7D%26%09%09A_%7B12%7D%5C%5C%20%09A_%7B21%7D%26%09%09A_%7B22%7D%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09x_1%5C%5C%20%09x_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%2B%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09B_1%5C%5C%20%09B_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%5Cleft%5B%20k_1%5C%20k_2%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09%5Chat%7Bx%7D_1%5C%5C%20%09%5Chat%7Bx%7D_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%2B%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09B_1%5C%5C%20%09B_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20v%20%5C%5C%20%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09A_%7B11%7D%26%09%09A_%7B12%7D%5C%5C%20%09A_%7B21%7D%26%09%09A_%7B22%7D%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09x_1%5C%5C%20%09x_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%2B%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09B_1k_1%26%09%09B_1k_2%5C%5C%20%09B_2k_1%26%09%09B_2k_2%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09%5Chat%7Bx%7D_1%5C%5C%20%09%5Chat%7Bx%7D_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%2B%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09B_1%5C%5C%20%09B_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20v%5Ctag%7B4%7D$

注意到

$%5Cdot%7By%7D%3D%5Cdot%7Bx%7D_1%3DA_%7B11%7Dx_1%2BA_%7B12%7Dx_2%2BB_1k_1%5Chat%7Bx%7D_1%2BB_1k_2%5Chat%7Bx%7D_2%2BB_1v%5Ctag%7B5%7D$

则由式(3)(5)以及 $y%3Dx_1$ ,可将将(2)式改写为

$%5Cdot%7B%5Chat%7Bx%7D%7D_2%3D%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%20%5Cright)%20%5Chat%7Bx%7D_2%2BL%5Cleft(%20%5Cdot%7By%7D-A_%7B11%7Dy-B_1u%20%5Cright)%20%2BA_%7B21%7Dy%2BB_2u%20%5C%5C%20%3D%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%20%5Cright)%20%5Chat%7Bx%7D_2%2BLA_%7B11%7Dx_1%2BLA_%7B12%7Dx_2%2BLB_1k_1%5Chat%7Bx%7D_1%2BLB_1k_2%5Chat%7Bx%7D_2%2BLB_1v%20%0A%5C%5C-LA_%7B11%7Dx_1%2BA_%7B21%7Dx_1-LB_1%5Cleft(%20k_1%5Chat%7Bx%7D_1%2Bk_2%5Chat%7Bx%7D_2%2Bv%20%5Cright)%20%2BB_2%5Cleft(%20k_1%5Chat%7Bx%7D_1%2Bk_2%5Chat%7Bx%7D_2%2Bv%20%5Cright)%20%20%5C%5C%20%3DA_%7B21%7Dx_1%2BLA_%7B12%7Dx_2%2BB_2k_1%5Chat%7Bx%7D_1%2B%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%2BB_2k_2%20%5Cright)%20%5Chat%7Bx%7D_2%2BB_2v%20%5Ctag%7B6%7D$

则构成2n维闭环系统 $%5CSigma%20_%7BLK%7D$ 为

$%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09%5Cdot%7Bx%7D_1%5C%5C%20%09%5Cdot%7Bx%7D_2%5C%5C%20%09%5Cdot%7B%5Chat%7Bx%7D%7D_1%5C%5C%20%09%5Cdot%7B%5Chat%7Bx%7D%7D_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09A_%7B11%7D%26%09%09A_%7B12%7D%26%09%09B_1k_1%26%09%09B_1k_2%5C%5C%20%09A_%7B21%7D%26%09%09A_%7B22%7D%26%09%09B_2k_1%26%09%09B_2k_2%5C%5C%20%09A_%7B11%7D%26%09%09A_%7B12%7D%26%09%09B_1k_1%26%09%09B_1k_2%5C%5C%20%09A_%7B21%7D%26%09%09LA_%7B12%7D%26%09%09B_2k_1%26%09%09%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%2BB_2k_2%20%5Cright)%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09x_1%5C%5C%20%09x_2%5C%5C%20%09%5Chat%7Bx%7D_1%5C%5C%20%09%5Chat%7Bx%7D_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%2B%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09B_1%5C%5C%20%09B_2%5C%5C%20%09B_1%5C%5C%20%09B_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20v%20%5Ctag%7B7%7D$ 构造变换矩阵为

$T%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09I%26%09%09O%5C%5C%20%09I%26%09%09-I%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Ctag%7B8%7D$

则状态变换后的系统矩阵为

$%5Cbar%7BA%7D_c%3DT%5E%7B-1%7DA_cT%20%5C%5C%20%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09I_n%26%09%09O%5C%5C%20%09I_n%26%09%09-I_n%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09A_%7B11%7D%26%09%09A_%7B12%7D%26%09%09B_1k_1%26%09%09B_1k_2%5C%5C%20%09A_%7B21%7D%26%09%09A_%7B22%7D%26%09%09B_2k_1%26%09%09B_2k_2%5C%5C%20%09A_%7B11%7D%26%09%09A_%7B12%7D%26%09%09B_1k_1%26%09%09B_1k_2%5C%5C%20%09A_%7B21%7D%26%09%09LA_%7B12%7D%26%09%09B_2k_1%26%09%09%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%2BB_2k_2%20%5Cright)%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09I_n%26%09%09O%5C%5C%20%09I_n%26%09%09-I_n%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%20%5C%5C%20%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09A_%7B11%7D%26%09%09A_%7B12%7D%26%09%09B_1k_1%26%09%09B_1k_2%5C%5C%20%09A_%7B21%7D%26%09%09A_%7B22%7D%26%09%09B_2k_1%26%09%09B_2k_2%5C%5C%20%090%26%09%090%26%09%090%26%09%090%5C%5C%20%090%26%09%09A_%7B22%7D-LA_%7B12%7D%26%09%090%26%09%09-%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%20%5Cright)%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09I_n%26%09%09O%5C%5C%20%09I_n%26%09%09-I_n%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%20%5C%5C%20%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09A_%7B11%7D%2BB_1k_1%26%09%09A_%7B12%7D%2BB_1k_2%26%09%09-B_1k_1%26%09%09-B_1k_2%5C%5C%20%09A_%7B21%7D%2BB_2k_1%26%09%09A_%7B22%7D%2BB_2k_2%26%09%09-B_2k_1%26%09%09-B_2k_2%5C%5C%20%090%26%09%090%26%09%090%26%09%090%5C%5C%20%090%26%09%090%26%09%090%26%09%09A_%7B22%7D-LA_%7B12%7D%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%20%5C%5C%20%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09A%2BBK%26%09%09-BK%5C%5C%20%09O%26%09%09%5Cbegin%7Bmatrix%7D%20%090%26%09%090%5C%5C%20%090%26%09%09A_%7B22%7D-LA_%7B12%7D%5C%5C%20%5Cend%7Bmatrix%7D%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%20%5Ctag%7B9%7D$

则 $%5CSigma%20_%7BLK%7D$ 的特征多项式为

$%5Cdet%20%5Cleft(%20%5Clambda%20I-A_c%20%5Cright)%20%3D%5Cdet%20%5Cleft(%20%5Clambda%20I-%5Cbar%7BA%7D_c%20%5Cright)%20%20%5C%5C%20%3D%5Cdet%20%5Cleft(%20A%2BBK%20%5Cright)%20%5Ccdot%20%5Cdet%20%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%20%5Cright)%20%20%5Ctag%7B10%7D$

其中 $%5Cdet%20%5Cleft(%20A%2BBK%20%5Cright)%20$ 为状态反馈特征多项式， $%5Cdet%20%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%20%5Cright)%20$ 为降维观测器特征多项式。由此可知二者相互独立，分离原理成立。

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线性系统的分离原理——降维观测器情形下的证明

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