《现代控制理论》第3版(刘豹_唐万生)课后习题答案

《现代控制理论参考答案》

第一章答案

1-1 试求图1-27系统的模拟结构图，并建立其状态空间表达式。

U(s)+-K1Kps?K1+-Kps?K1s+-1J1sKbJ2s2?(s)Kns图1-27系统方块结构图

解：系统的模拟结构图如下：

Kp+U(s)-K1Kp+-K1Kp?x6+-K1x5++?+-1J1?x4x3KbJ2?x2??(s)x1Kn?图1-30双输入--双输出系统模拟结构图

系统的状态方程如下：

?x1?x2?x2??KbJ2Kx3px3???J1x3?KnJ1x4?1J1x5?KpJ1x6

x4?x3?x5??K1x3?K1X6?x6??K1Kpx1?K1Kpx6?K1Kpu令?(s)?y，则y?x1

所以，系统的状态空间表达式及输出方程表达式为

?0????x??1??0?x????2???x??03??????x4??0????0?x5???????K1?x6?????Kp100000?0KbJ2K00p001J000J11?KnJ1000?K10?x1???x?2??x3?0????x4??x?5????x6?????0?0??x1???0???x??Kp??2??0???xJ1??3???0?u?x??4?0???0???xK1?5?K1?????K1??x6???K?p?? ???Kp??0y??10000

1-2有电路如图1-28所示。以电压u(t)为输入量，求以电感中的电流和电容上的电压作为状态变量的状态方程，和以电阻R2上的电压作为输出量的输出方程。

R1L1L2i1CUi2---------Uc---------R2图1-28 电路图

解：由图，令i1?x1,i2?x2,uc?x3，输出量y?R2x2

??x1???R1L1R2L21Cx1?x2?1L11x3?x31L1uR1x1?L1x1?x3?u?有电路原理可知：L2x2?R2x2?x3? 既得

x2???L2x2

x1?x2?Cx3x3??x1?1Cy?R2x2写成矢量矩阵形式为：

?R1????L1x1?。???x???02?。???x??13???????C。0??R2L21C?1??1?L1??x????1L1????1?x2?0uL2??????0????x3???? 0?????y??0R2?x1???0?x2????x3??

1-4 两输入u1，u2，两输出y1，y2的系统，其模拟结构图如图1-30所示，试求其状态空间表达式和传递函数阵。

u1b1+---??y1a1a5a6?++a21u2b2+---?a3

y2

a4图1-30双输入--双输出系统模拟结构图解：系统的状态空间表达式如下所示：

?1??0?x?????a2x?2????x?3??1????4??0?x1?a10?a5?x1???x2??0??x3????x4?000?a40??x1??0?????a6x2b?????11??x3??0?????a3??x4??00??0?u0??b2?

y??101?s?a2?(sI?A)???1??0?1s?a10a500sa40??a6? ?1??a3??1s?a10a500sa40??a6??1??a3??1Wux?s?a2?1(s)?(sI?A)B????1??0?0?b?1?0??00??0? 0??b2?Wuy(s)?C(sI?A)?1B??101?s?a20????1??0?1s?a10a500sa40??a6??1??a3??1?0?b?1?0??00??0? 0??b2?

1-5系统的动态特性由下列微分方程描述

?......(2)y?5y?7y?3y?u?3u?2u

列写其相应的状态空间表达式，并画出相应的模拟结构图。 解：令x1?y，x2?y，x3?y，则有

?。?x1?。??0?x???02?。???x????33????y??2310?7?x1???1?x2????x3??0??x1??0??????1x?0u??2????5????x3????1??...

相应的模拟结构图如下：

13u+---?5x3?x2?x12++y73

1-6 （2）已知系统传递函数W(s)?6(s?1)s(s?2)(s?3)102,试求出系统的约旦标准型的实现，并画出相应的模拟结构图

解：W(s)?6(s?1)s(s?2)(s?3)1?30000?202??4(s?3)2??13?3?3

s?3s?2s?1???3?x????x0?2????3??0?x????4??0?x0??x1??0??????0x21??????u0??x3??1??????0??x4??1??y??4???1033?x1???1??x2?3???x3????x4?

1-7 给定下列状态空间表达式

?1??0?x????x??2?2???3???x????1y??001?31?x1???1?x2????x3??0??x1??0??????0x?1u??2????3????x3????2??‘

（1） 画出其模拟结构图

（2） 求系统的传递函数 解：

?s?（2）W(s)?(sI?A)?2???12?1s?3?10??0 ?s?3??sI?A?s(s?3)?2(s?3)?(s?3)(s?2)(s?1)

(sI?A)?1??s?3?2?1???2(s?3)(s?3)(s?2)(s?1)??s?5?s?3s(s?3)s?1??0? (s?1)(s?2)??0s?3s(s?3)s?1??0????0??1?(s?1)(s?2)??2????0Wux??s?3?2?1?1(s)?(sI?A)B???2(s?3)(s?3)(s?2)(s?1)??s?5?(s?3)??1???s(s?3)?(s?3)(s?2)(s?1)???(2s?1)(s?3)??

Wuy(s)?C(sI?A)?1B??00(s?3)??1??1?s(s?3)??(s?3)(s?2)(s?1)? ?(2s?1)(s?3)???(2s?1)(s?2)(s?1)

1-8 求下列矩阵的特征矢量 ?0?3（3）A?????1210?70??2 ??6?????解：A的特征方程 ?I?A??3???12?1?7??32?2???6??11??6?0

???6??0解之得：?1??1,?2??2,?3??3

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