8_3
已知两个系统
???x??01??0?1????1?2??x1???1?u1
?y1??13?x?1?u1???x??01???x?0?2?2?????1?31?u2
?y?1?4?x??2?2求按串联、并联、单位负反馈、单位正反馈连接时的系统状态方程。 解：
a1=[0,1;-1,-2]; b1=[0;1]; c1=[1,3]; d1=[1];
a2=[0,1;-1,-3]; b2=[0;1]; c2=[1,4]; d2=[0];
sys1=ss(a1,b1,c1,d1); sys2=ss(a2,b2,c2,d2);
title='Series state space matrix' sys=series(sys2,sys1)
title='parallel state space matrix'
[a,b,c,d]=parallel(a1,b1,c1,d1,a2,b2,c2,d2) title='Negative Feedback state space matrix' [a,b,c,d]=feedback(a1,b1,c1,d1,a2,b2,c2,d2) title='positive Feedback state space matrix' [a,b,c,d]=feedback(a1,b1,c1,d1,a2,b2,c2,d2,+1)
title =
Series state space matrix a =
x1 x2 x3 x4 x1 0 1 0 0 x2 -1 -2 1 4 x3 0 0 0 1 x4 0 0 -1 -3 b =
u1 x1 0 x2 0 x3 0 x4 1 c =
x1 x2 x3 x4
例 y1 1 3 1 4 d =
u1 y1 0
Continuous-time model. title =
parallel state space matrix a =
0 1 0 0 -1 -2 0 0 0 0 0 1 0 0 -1 -3 b = 0 1 0 1 c =
1 3 1 4 d = 1 title =
Negative Feedback state space matrix a =
0 1 0 0 -1 -2 -1 -4 0 0 0 1 1 3 -2 -7 b = 0 1 0 1 c =
1 3 -1 -4 d = 1 title =
positive Feedback state space matrix a =
0 1 0 0 -1 -2 1 4 0 0 0 1 1 3 0 1 b = 0 1 0 1 c =
1 3 1 4 d = 1 回目录
例8_4
时不变系统
???31??11??x?x???1?3??11?u????? ??11??y???x?1?1???判别系统的能控性和能观性。
解：
a=[-3,1;1,-3]; b=[1,1;1,1]; c=[1,1;1,-1]; cam=ctrb(a,b); rcam=rank(cam) oam=obsv(a,c); roam=rank(oam) rcam = 1 roam =
2
回目录
例8_5
线性系统
H(s)?当?分别取－1，0，＋1 时，判别系统的可控性和可观性，并求出相应的状态方程。 解：
for alph=[-1:1]; alph
num=[1,alph];
den=[1 10 27 18];
[a b c d]=tf2ss(num,den); cam=ctrb(a,b); rcam=rank(cam) oam=obsv(a,c); coam=rank(oam) end;
alph = -1 rcam = 3 coam = 3 alph = 0 rcam = 3 coam = 3 alph = 1 rcam = 3 coam = 2
回目录
s??
s3?10s2?27s?18
例8_6
利用Lyapunov方程确定线性时不变系统
?1??x1?2x2?x ???x2?x1?4x2的稳定性。 解：a??1?2????
1?4??令q=I，求解Lyapunov方程
ap?paT??q
然后确定p的正定性来证实系统的稳定性。
a=[-1,-2;1,-4]; q=[1,0;0,1]; p=lyap(a,q); detp=det(p)
detp =
0.0567
回目录