assignment#05
DESCRIPTION
LCSTRANSCRIPT
LINEAR CONTROL SYSTEMS
ASSIGNMENT #05
Submitted By:
NS Umair Zulfiqar
NS Waseem Tariq
NS M. Naeem
Submitted To:
Madam Muwahida
College of Electrical and Mechanical Engineering
NUST
Question # 01
Take a 4x4 system where q=2. Decompose the system into controllable part.
Matlab Code:
A=[ 3 3 2 4; 4 2 2 4; 1 2 3 4; 2 1 1 3] B=[1;1;-1;1] W = ctrb(A,B) rank_of_W=rank(W)rank_of_first_two_column=rank(W(:,1:2)) p3=[3;1;1;2]p4=[7;2;2;1] P=[W(:,1:2) p3 p4] rank_of_P=rank(P) PI=inv(P) A_hat = PI*A*P B_hat = PI*B
A11_hat=A_hat(1:2,1:2)A22_hat=A_hat(3:4,3:4)B11_hat=B_hat(1:3,:)
Output:
A =
3 3 2 4
4 2 2 4
1 2 3 4
2 1 1 3
B =
1
1
-1
1
W =
1 8 76 740
1 8 76 740
-1 4 56 568
1 5 43 413
rank_of_W =
2
rank_of_first_two_column =
2
p3 =
3
1
1
2
p4 =
7
2
2
1
P =
1 8 3 7
1 8 1 2
-1 4 1 2
1 5 2 1
rank_of_P =
4
PI =
0.1333 0.1667 -0.6556 0.0444
-0.0667 0.1667 0.0778 -0.0222
0 -0.5000 0.1667 0.6667
0.2000 0 -0.0667 -0.2667
A_hat =
0.0000 -12.0000 -2.9333 -1.5000
1.0000 11.0000 3.4667 5.5000
-0.0000 -0.0000 -0.0000 -2.5000
0.0000 0.0000 -0.4000 0.0000
B_hat =
1.0000
-0.0000
0
0
A11_hat =
0.0000 -12.0000
1.0000 11.0000
A22_hat =
-0.0000 -2.5000
-0.4000 0.0000
B11_hat =
1.0000
-0.0000
0
Question#02
Take a 6 order system with three outputs and represent it in observer form.
Matlab Code:
A=[2 4 4 4 3 2;2 4 7 1 1 3; 3 6 2 4 5 1; 1 3 1 6 3 4; 5 1 6 1 2 4; 2 5 1 3 3 4]C=[4 3 2 6 7 1; 4 2 7 4 4 5; 6 7 5 1 2 1]rank_of_C=rank(C)O = obsv(A,C)rank_of_O=rank(O)eeta_C1=2eeta_C2=2eeta_C3=2Oa=[C(1,:); C(1,:)*A; C(2,:); C(2,:)*A; C(3,:); C(3,:)*A;]Rank_of_Oa=rank(Oa)J=inv(Oa)j=J(:,2)j1=J(:,4)j2=J(:,6)PI=[j A*j j1 A*j1 j2 A*j2 ]P=inv(PI)AH=P*A*PICH=C*PI
Output:
A =
2 4 4 4 3 2 2 4 7 1 1 3 3 6 2 4 5 1 1 3 1 6 3 4 5 1 6 1 2 4 2 5 1 3 3 4
C =
4 3 2 6 7 1 4 2 7 4 4 5 6 7 5 1 2 1
rank_of_C =
3
O =
4 3 2 6 7 1 4 2 7 4 4 5 6 7 5 1 2 1 63 70 90 73 60 75 67 107 77 89 84 73 54 92 97 62 60 54 1059 1726 1430 1405 1273 1258 1234 1874 1837 1520 1347 1516 1053 1682 1530 1290 1207 1185 20146 31498 29479 25159 22588 24470 23014 36941 32846 29173 26563 27459 19755 31122 28763 24516 22330 23410 378764 603865 545185 476950 433394 453133 435354 688273 632345 544359 493235 522477 371029 589014 536306 464850 422640 442663 7150999 11331535 10362928 8954154 8123119 8568216 8199777 13027275 11850263 10285889 9343038 9808156 6992380 11088513 10123179 8758083 7951450 8366018
rank_of_O =
6
eeta_C1 =
2
eeta_C2 =
2
eeta_C3 =
2
Oa =
4 3 2 6 7 1 63 70 90 73 60 75 4 2 7 4 4 5 67 107 77 89 84 73 6 7 5 1 2 1 54 92 97 62 60 54
Rank_of_Oa =
6
J =
0.0768 0.0295 -0.0555 -0.0101 0.2015 -0.0273 -0.0763 -0.0107 -0.0949 0.0130 0.0168 0.0071 0.0373 0.0051 -0.0050 -0.0248 -0.0408 0.0269 0.2601 0.1089 -0.8547 -0.0663 -0.1452 0.0154 -0.0681 -0.1026 0.7536 0.0578 0.0147 -0.0044 -0.2367 -0.0315 0.3704 0.0444 -0.0063 -0.0275
j =
0.0295 -0.0107 0.0051 0.1089 -0.1026 -0.0315
j1 =
-0.0101 0.0130 -0.0248 -0.0663 0.0578 0.0444
j2 =
-0.0273 0.0071 0.0269 0.0154 -0.0044 -0.0275
PI =
0.0295 0.1015 -0.0101 -0.0704 -0.0273 0.0750 -0.0107 -0.0362 0.0130 -0.0170 0.0071 0.0910 0.0051 -0.0742 -0.0248 0.0662 0.0269 0.0268 0.1089 0.2220 -0.0663 -0.0426 0.0154 -0.0099 -0.1026 -0.0549 0.0578 0.0406 -0.0044 -0.0712 -0.0315 -0.0965 0.0444 0.1721 -0.0275 -0.0691
P =
-18.6716 12.9687 -20.3111 2.2973 -16.8920 6.0062 4.0000 3.0000 2.0000 6.0000 7.0000 1.0000 -26.7443 35.1129 -33.4058 9.2647 -5.9721 9.0817 4.0000 2.0000 7.0000 4.0000 4.0000 5.0000 -25.8433 26.2309 -6.4298 13.7884 5.1702 -3.3069 6.0000 7.0000 5.0000 1.0000 2.0000 1.0000
AH =
-0.0000 -16.0970 -0.0000 -5.5357 0.0000 -9.3264 1.0000 2.8540 -0.0000 12.5609 -0.0000 3.3354 -0.0000 -34.9789 0.0000 21.6738 -0.0000 -5.4487 -0.0000 5.3107 1.0000 10.7363 0.0000 4.9261 0 -17.6143 0.0000 19.9929 -0.0000 0.8328 0.0000 0.2085 0 10.1377 1.0000 6.4097
CH =
0.0000 1.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 1.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 1.0000