取Q?I,则矩阵由下式确定:
ATP?PA??I 则1?1??1??00?0?30?P11P12P13?P11P12P13??3030?????100??1???111???P21P22P23??0????0?10??0??P21P22P23??????303030???30??11??1???P31P32P33????P31P32P33????00?1???0?00???3012?12?????
1515??15?27??1545345?? 解出 P??249??2?15345432??4949??7? 用塞尔维斯判据检验P的正定性: ?1?P11?15?0 ?2?P11P12P21P22152?0 ?15452215151527P131545345P23??0 2249P33153454327494915P11P12 ?3?P21P22P31P32可知P?0,正定,系统在原点处的平衡状态是渐近稳定的.
7.采用状态反馈将系统的闭环极点配置到合适的值,使得闭环系统阶跃响应的上升时间比开环系统阶跃响应的上升时间缩短3倍左右
>>step(G)
开环阶跃响应如图一,上升时间约为200s 图一:
>>M=[-0.15 -0.15 -0.4]; >>K=acker(A,B,M) >>eig(A-B*K)
>>G1=ss(A-B*K,B,C,D); >>G2=tf(G1) >>step(G2)
期望反馈闭环极点为-0.15,-0.15,-0.4
? 状态反馈矩阵为K??17.500011.59092.1429验证配置结果ans=-0.1500,-0.1500,-0.4000
a =
x1 x2 x3 x1 -0.03333 0.03333 0 x2 0 -0.03333 0.03333 x3 -4.492 -2.975 -0.6333 b =
u1 x1 0 x2 0 x3 0.2567 c =
x1 x2 x3 y1 1 0 0 d =
u1 y1 0
Continuous-time model.
经过状态反馈后,系统的传递函数阵为 Transfer function: 0.0002852
-------------------------------- s^3 + 0.7 s^2 + 0.1425 s + 0.009
0.0002582G(s)?C(sI?(A?BK))?1B?3
s?0.7s2?0.1425?0.009并且经过状态反馈后,系统的阶跃响应为图二,上升时间约为50s 图二:
8.为系统设计全维状态观测器，将观测器的极点配置到合适的值
(1)配置状态观测器的极点
>>N=[-3 -3 -8]; >>L=acker(A',C',N)' >>eig(A-L*C)
配置观测器期望极点为-3,-3,-8
?14??1661全维状态观测器的反馈阵为L???????60612?
验证配置结果,ans=-8.0000,-3.0000,-3.0000
(2)生成系统状态估计器
>>est=estim(G,L)
a =
x1 x2 x3 x1 -13.88 0.03333 0 x2 -1661 -0.03333 0.03333 x3 -6.061e+004 0 -0.08333 b =
u1 x1 13.85 x2 1661 x3 6.061e+004 c =
x1 x2 x3 y1 1 0 0 y2 1 0 0 y3 0 1 0 y4 0 0 1 d =
u1 y1 0 y2 0 y3 0 y4 0
Input groups: Name Channels Measurement 1 Output groups: Name Channels OutputEstimate 1 StateEstimate 2,3,4 Continuous-time model.
(3)状态观测器的阶跃响应
>>step(
(4)输出和观测状态的传递函数
>>tf(est)
Transfer function from input to output...
13.85 s^2 + 56.99 s + 72 #1: ------------------------ s^3 + 14 s^2 + 57 s + 72
13.85 s^2 + 56.99 s + 72 #2: ------------------------ s^3 + 14 s^2 + 57 s + 72
1661 s^2 + 2214 s + 71.96 #3: ------------------------- s^3 + 14 s^2 + 57 s + 72
6.061e004 s^2 + 4041 s + 67.35 #4: ------------------------------ s^3 + 14 s^2 + 57 s + 72 Input groups: Name Channels Measurement 1 Output groups: Name Channels OutputEstimate 1 StateEstimate 2,3,4
9.画出带状态观测器的状态反馈控制系统的状态变量图
>>T=1:300;plot(T/50,y(1:300,:));ylabel('y');legend('系统','观测'); >>T=1:300;plot(T/50,x1(1:300,:));ylabel('x1');legend('系统','观测'); >>T=1:300;plot(T/50,x2(1:300,:));ylabel('x2');legend('系统','观测');