求当u=1(t)时,系统的输出y。
2解:det(sI?A)?s?5s?6,s1??2,s2??3 (1分)
?11??1?31? P???,P???2?1? (2分)?2?3?????e?2te?P??0At0??1?e?2tP???3t??2te???2ee?3t??31??3e?2t?2e?3t????3t???3e???2?1???6e?2t?6e?3te?2t?e?3t???2e?2t?3e?3t?(2分)
x(t)?ex0??e0AttA(t??)1?1??e?2t?e?3t??(1?e?2t)?(1?e?3t)?Bu(?)d????2 (3分) 3?2t?3t????2e?3e?2t?3t????1?e?1?e??11?2t2?3t??e?e?112??62,y(t)?x1(t)??e?2t?e?3t (2分) 3??623?2t?3t?e?2e????Ax的状态转移矩阵?(t)如下,求其逆矩阵?(t)和系统12.(5分)已知齐次状态方程x矩阵A。
?1?3e?t?2e?2t?(t)???t?2t??3e?3e解:
2e?t?2e?2t?。 ?t?2t??2e?3e?2et?2e2t? (2分) t2t??2e?3e??3et?2e2t2.(5分) ?(t)??(?t)??t2t??3e?3e?1?12?? A??(t)|t?0??? (3分) -3-4??13.(6分) 已知系统的状态方程为
?01??1?&x???x??2?u?6?5???? y??10?x初始条件为x(0)??11?试求系统在单位阶跃输入作用下的状态响应。 (6分) 解法1:
2?3??s?3?1?1s?2??(t)?L?1?(sI?A)?L?????6?6??s?2s?311???2t?3ts?2s?3??3e?2e????23???6e?2t?6e?3t?s?2s?3??e?2t?e?3t???2e?2t?3e?3t?T (3分)
6
?4e?2t?3e?3t?t?5e?2??4e?3??x(t)??(t)x(0)???(?)Bu(t??)d??????d??2t?3t??2??3??00??8e?9e???10e?12e?47??38?5??4e?2t?3e?3t???e?2t?e?3t???e?2t?e?3t?7????236?23?2t?3t??????8e?9e?2t?3t?2t?3t??5e?4e?1?3e?5e?1???? (3分)
t解法2:
?3?2t8?3t?e?e?7?x(t)?L?1[(sI?A)?1]x(0)?L?1[(sI?A)?1BU(s)]??23???2t?3t?3e?5e?1?? (6分)
14.(3分) 已知线性定常系统的矩阵指数函数e为:
At?3t6t2e?e??3t6t??6e?6e 求其系统矩阵A。
11??e3t?e6t?333t6t??e?2e?
deAt解:A?dtt?0?01???? (3分) 189??15. (6分) 已知系统的状态空间描述及初始条件如下:
?-11??1?&x???x??1?u,y??00?2????1?x x1?0??1,x2?0??0
求当u(t)??(t)时,系统的输出y。
2解:(1分)det(sI?A)?s?3s?2,s1??1,s2??2
?11??1?21?(1分) P???,P???1?1?
?1?2????(2分)
?e?te?P??At??1?e?tP???2t?e??0Ate?t?e?2t?? ?2te?(1分) x(t)?ex0??e0tA(t??)?e?t??2e?t?2e?2t)??3e?t?2e?2t?Bu(?)d?????????? ?2t?2te?0????e??2t(1分) y(t)?x1(t)?e
方法2:
7
?s?1?1? (1分) sI?A???s?2??01?1??s?21??s?1(s?1)(s?2)?1?1? (2分) (sI?A)?????0s?1(s?1)(s?2)?1????0??s?2???3e?t?e?2t?x(t)?L[(sI?A)x(0)?(sI?A)BU(s)]??? (2分) ?2t?e??1?1?1y(t)?cx(t)?e?2t (1分)
16.已知线性定常系统如下所示,其状态转移矩阵?(t)是( A )
?0?2??1?& x???x??2?u
1?3?????2e?t?e?2t A ??t?2t?e?e?2et?e?3t C ?t?3t?e?e?2et?e?2t?2e?t?2e?2t? B ?t?2t?t?2t??e?2e??e?e?2et?2e?3t?? D
?et?2e?3t??2e?t?e?2t??t?2t?e?2e?2et?2e?2t? t?2t??e?2e??2e?t?e?2t??
?e?t?2e?2t?17.(8分) 已知系统的状态空间描述为:
&?? x?01??0?x?u,y??1?????2?3??1?1?x
若系统的初始状态为x(0)?0,求在单位阶跃信号作用下的系统输出y(t)。 解:状态转移矩阵为:
?2e?t?e?2t e???t?2t??2e?2eAte?t?e?2t? ?t?2t??e?2e? (2分)
状态响应为:
t x(t)?ex(0)?e0At?A(t??)?0.5?e?t?0.5e?2t?Bu(?)d???? (4分) ?t?2te?e???2t 系统输出y(t)?Cx(t)?0.5?0.5e
(2分)
8
数学物理方法第三章答案完整版
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