?p12?当?2??1时，特征矢量p2???
p?22??1由Ap2??2p2，得??41??p12???p12????? ??1??p22???p22??p12?p22??p12?1?即? ，可令p2???
??2??4p12?p22??p22?1?2???1??21?4?? 1??4??1??13t1?te?e4??22???13t?t???e?e4????14123t?t?e?4? 1?t??e?2?则T???1?21??1，T??2? eAt?1???21??e3t???2??0?10??2??t?e??1??2ee?13t
第二种方法，即拉氏反变换法：
?s?1sI?A????4?1?? s?1??sI?A??1??s?1??s?3??s?1??411?? s?1?1s?1??s?3s?1????? ??4???s?3??s?1???s?3??s?1??? ?s?1??s?3??s?1??111???????4?3?s?1?s? 111???????2?3?s?1?s??1?11?????2?s?3s?1? ???11??s?1?s?3eAt?13t1?t?2e?2e?1?1?L??sI?A???????e3t?e?t??1412ee3t3t?t?e?4? 1?t?e??2??1第三种方法，即凯莱—哈密顿定理 由第一种方法可知?1?3，?2??1
?1?e3t??4??t????e??1??43??13t3?t?e?e?3t????e444???t???? 1e13t1?t?????e?e???4???44???0??1??????1??13???1??1eAt?13t3?t??1??e?e??4?4??00??13t3?t??1???e?e??1??44??4?13t1?te?e1??22???1??3t?te?e??1412ee3t3t?t?e?4? 1?t??e?2??1
2-5 下列矩阵是否满足状态转移矩阵的条件，如果满足，试求与之对应的A阵。
?t?2t?2e?e（3）??t????t?2te?e??1?t3te?e?2t?t???22e?2e? （4）??t????2t?t?2e?e???e?t?e3t??0???I，所以该矩阵满足状态转移矩阵的条件 1?1?e?41e?2?t?e?e3t????
????t3t?解：（3）因为 ??0????1?0??t?A??t?0??2e?t?2e?2t???t?2t??e?2e?1?0?4e?t?2e??2t?t??4e?e??2tt?0?0???1?2?? ?3?（4）因为??0???0???I，所以该矩阵满足状态转移矩阵的条件 1???t?A??t?0?1?t33t??2e?2e???e?t?3e3t??14?e?t?12e?t??4?33t??e?2?e3t3?1???4t?01?? 1?
2-6 求下列状态空间表达式的解：
?0???x?01??0?x????u 0??1?y??1,0?x
?1?初始状态x?0????，输入u?t?时单位阶跃函数。
?1?解： A???0?01?? 0??1?? s??ssI?A???0?sI?A??1?1?s2?s?0?1?1??s???s??0??1?2s?? 1?s??t?? 1???t??eAt?L?1?1??sI?A??1?????0??0?因为 B??? ，u?t??I?t?
?1?x?t????t?x?0???1???0t??1?????1??1???t0??t???Bu???d?
?1??0t????0????d? 1??1?t0?t?1??????1??t0?t?????d? ?1??12??t?1??t??????2? ?1??t??12?t?t?1? ??2??t?1??y??10?x?12t?t?1
2
2-9 有系统如图2.2所示，试求离散化的状态空间表达式。设采样周期分别为T=0.1s和1s，而u1和u2为分段常数。
u1K/(s+1)x1X+u2-1/sx21+X+y2 图2.2 系统结构图 解：将此图化成模拟结构图
u1K-X∫x1u2-X∫x21+X+y2列出状态方程
?1?ku? x1x 1
?2?x1?u2 x
y?x2?2x1
??1x???1y??20??x?0??x1?1??? ?x2??k??00??u1???? ?1??u2?则离散时间状态空间表达式为 x?k?1??G?T?x?k??H?T?u?k? y?k??cx?k??Du?k?
由G?T??e??1A???1At?1At和H?T???T0edtB得：
0??2?T? C??? ?1??1?At0??k? B??0??0?1?1e?L??sI?A??T0T0??s?1??L??????10??dt1??k??00???e?T????s???1?e?T0??1?e?T????1??T?1?e?T0?? 1?0??k??T??0?T0??k?1?e?????1??k?T?1?e?T?H??edt?At??e?t??T?1?e?0?? ?T???e?1当T=1时 x?k?1????1?1?e0??x?k??1??k?1?e?1???1??ke0??u?k? ?1?? y?k?1???21??k ?x1?k?1?e?0.???k?e?0.1?0.9???e?0.10?当T=0.1时 x?k?1????x?k???0.11??1?e0??u?k? ?0.1?? y?k?1???2
1??k ?x第三章习题
3-1判断下列系统的状态能控性和能观测性。系统中a,b,c,d的取值对能控性和能观性是否有关，若有关，其取值条件如何？
（1）系统如图3.16所示：
u+-?a?x1y+-?x2+x3-+--?x4bcd图3.16 系统模拟结构图
解：由图可得：
?x1??ax1?u?x2??bx2x3??cx3?x2?x1?x1?x2?cx3
??x4?x3?dx4y?x3状态空间表达式为：
???x??1???a?x??0??2????x??13????0???x4??y??0??0?b1010?x00?c10??x1??1??????x00??2????u0??x3??0???????d??x4??0?
0由于x2、x3、x4与u无关，因而状态不能完全能控，为不能控系统。由于y只与x3有关，因而系统为不完全能观的，为不能观系统。 （3）系统如下式：
???x??1???1?x???0??2???x???03?????cy???0001?10d??x0?0??x1??2????0x?a??2???2????x3????b1??0u?0???
解：如状态方程与输出方程所示，A为约旦标准形。要使系统能控，控制矩阵b中相对于约旦块的最后一行元素不能为0，故有a?0,b?0。
要使系统能观，则C中对应于约旦块的第一列元素不全为0，故有c?0,d?0。 3-2时不变系统