線性系統(tǒng)的分離原理——降維觀測(cè)器情形下的證明

2023-06-07 22:35 作者:斟好雨 0人讀過(guò) | 我要投稿

在現(xiàn)代控制理論中講解分離原理時(shí)一般都以全維觀測(cè)器為例進(jìn)行證明（劉豹《現(xiàn)代控制理論》p221-p222），事實(shí)上，這個(gè)定理對(duì)于降維觀測(cè)器也同樣成立，證明思路類似，但步驟較為繁瑣。下面給出證明。

????考慮如下形式的線性系統(tǒng)

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????這類系統(tǒng)無(wú)需進(jìn)行狀態(tài)變換就可進(jìn)行狀態(tài)觀測(cè)器設(shè)計(jì)，需要觀測(cè)的狀態(tài)向量為 $x_2$ ，則設(shè)計(jì)的狀態(tài)觀測(cè)器為

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

????引入 $x_1$ 的觀測(cè)值 $%5Chat%7Bx%7D_1$ ，且有 $%5Chat%7Bx%7D_1%3Dx_1%3Dy$ ，并將 $%5Chat%7Bx%7D_1$ 、 $%5Chat%7Bx%7D_2$ 合起來(lái)作為 $%5Chat%7Bx_2%7D$

????構(gòu)造狀態(tài)反饋控制律為

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

????由狀態(tài)反饋控制律的形式，可將狀態(tài)方程改寫為

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

????注意到

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

????則由式(3)(5)以及 $y%3Dx_1$ ,可將將(2)式改寫為

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

????則構(gòu)成2n維閉環(huán)系統(tǒng) $%5CSigma%20_%7BLK%7D$ 為

$%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09%5Cdot%7Bx%7D_1%5C%5C%20%09%5Cdot%7Bx%7D_2%5C%5C%20%09%5Cdot%7B%5Chat%7Bx%7D%7D_1%5C%5C%20%09%5Cdot%7B%5Chat%7Bx%7D%7D_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%3D%5Cleft%5B%20%5Cbegin%7Bmatrix%7D%20%09A_%7B11%7D%26%09%09A_%7B12%7D%26%09%09B_1k_1%26%09%09B_1k_2%5C%5C%20%09A_%7B21%7D%26%09%09A_%7B22%7D%26%09%09B_2k_1%26%09%09B_2k_2%5C%5C%20%09A_%7B11%7D%26%09%09A_%7B12%7D%26%09%09B_1k_1%26%09%09B_1k_2%5C%5C%20%09A_%7B21%7D%26%09%09LA_%7B12%7D%26%09%09B_2k_1%26%09%09%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%2BB_2k_2%20%5Cright)%5C%5C%20%5Cend%7Bmatrix%7D%20%5Cright%5D%20%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09x_1%5C%5C%20%09x_2%5C%5C%20%09%5Chat%7Bx%7D_1%5C%5C%20%09%5Chat%7Bx%7D_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20%2B%5Cleft%5B%20%5Cbegin%7Barray%7D%7Bc%7D%20%09B_1%5C%5C%20%09B_2%5C%5C%20%09B_1%5C%5C%20%09B_2%5C%5C%20%5Cend%7Barray%7D%20%5Cright%5D%20v%20%5Ctag%7B7%7D$ ????構(gòu)造變換矩陣為

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

????則狀態(tài)變換后的系統(tǒng)矩陣為

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

????則 $%5CSigma%20_%7BLK%7D$ 的特征多項(xiàng)式為

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

????其中 $%5Cdet%20%5Cleft(%20A%2BBK%20%5Cright)%20$ 為狀態(tài)反饋特征多項(xiàng)式， $%5Cdet%20%5Cleft(%20A_%7B22%7D-LA_%7B12%7D%20%5Cright)%20$ 為降維觀測(cè)器特征多項(xiàng)式。由此可知二者相互獨(dú)立，分離原理成立。

標(biāo)簽：

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