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Teljes Matlab script kiegészítő függvényekkel.
Check controllability and observability of an LTI system using LMI conditions
File: LTI_ctrb_obsv.m Directory: 2_demonstrations/workspace/ccs/ccs_2018 Author: Peter Polcz (ppolcz@gmail.com)
Created on 2018. November 06.
a = 4; % Nr. of contr. and obs.
b = 2; % Nr. of contr. and unobs.
c = 1; % Nr. of uncontr. and obs.
d = 2; % Nr. of uncontr. and unobs.
n = a+b+c+d; % Nr. of states
m = 2; % Nr. of inputs
p = 3; % Nr. of outputs
is_stabilisable = 1;
[A,B,C,D] = LTI_generate_sys(a,b,c,d,m,p,is_stabilisable);
pcz_display(A,B,C,D)
A [9x9] =
Columns 1 through 7
-0.7133 -0.8096 0.1205 -1.8870 -1.1994 1.3781 -0.3345
-0.3155 -0.9061 0.1360 -0.5072 -0.1441 0.7988 -0.0119
-1.0551 -0.4048 0.6322 -0.2716 -0.0652 0.5923 -0.9039
-0.4383 1.1600 -0.5307 -0.2200 -0.7026 -0.0309 0.4819
0.7237 -1.0182 -0.4602 -0.6953 -0.3437 0.5771 0.6972
-0.6133 -0.3200 -0.0853 0.3476 0.1530 0.4782 0.9597
-0.6471 1.0909 0.1917 0.6902 -0.1175 -0.2533 -1.2189
0.3614 -0.6029 0.0237 -1.4579 -0.7253 0.6867 -0.5887
-0.2655 -0.6340 0.2827 -0.7155 -0.4540 1.0812 -0.3331
Columns 8 through 9
-1.9839 0.3631
-1.8753 -0.3619
-0.6597 -0.4457
-0.6175 0.6516
-0.7353 0.0525
-0.9051 -0.0375
-0.1694 -0.7943
-1.8883 0.0211
-0.6463 -0.6177
B [9x2] =
-0.1950 0.1173
-0.6804 -0.1330
0.2484 0.8463
-0.2721 -1.0088
0.0042 -0.2183
-0.3025 2.0736
-0.6414 0.9666
-1.6839 0.6606
0.2164 -0.0616
C [3x9] =
Columns 1 through 7
0.1422 -0.3378 0.1971 -0.5098 0.0441 -0.4651 -0.3756
1.5335 -0.8053 1.3047 -0.9906 0.6754 0.1004 -1.1887
0.3162 0.6132 0.4366 0.7592 0.6316 -0.0773 -0.0353
Columns 8 through 9
-0.2195 0.0869
0.4146 0.8364
0.2100 0.3036
D [3x2] =
0 0
0 0
0 0
Check wether the system is stabilisable.
Cn = ctrb(A,B);
[U,Sigma,V] = svd(Cn);
T = U';
A_Ctrb_Staircase = round(T*A*T',10)
B_Ctrb_Staircase = round(T*B,10)
k = c+d-1;
A_Unctrb = A_Ctrb_Staircase(end-k:end,end-k:end)
Uncontrollable_Eigenvalues = eig(A_Unctrb)
Check wether the system is detectable.
On = obsv(A,C);
[U,Sigma,V] = svd(On);
T = V';
A_Obsv_Staircase = round(T*A*T',10)
C_Obsv_Staircase = round(C*T',10)
k = b+d-1;
A_Unobsv = A_Obsv_Staircase(end-k:end,end-k:end)
Unobservable_Eigenvalues = eig(A_Unobsv)
A_Ctrb_Staircase =
Columns 1 through 7
-2.0455 -0.4262 2.6091 -2.5460 -1.6275 1.9781 -0.3326
0.0024 -0.3403 1.1716 0.0599 0.1856 -0.2883 -1.0109
-0.0008 0.6713 0.1646 1.2267 0.4327 -1.3781 0.8142
0.0002 -0.0091 0.0141 -0.4884 -0.0999 -1.3248 -0.6121
-0.0007 0.1879 0.0065 -0.1890 -0.6261 0.6130 0.5874
0.0001 -0.0500 0.0117 0.0075 0.0585 0.1515 0.0711
0 0 0 0 0 0 -0.2659
0 0 0 0 0 0 0.1487
0 0 0 0 0 0 -0.3722
Columns 8 through 9
-0.6745 -1.0431
-1.0971 0.6241
1.0481 0.2289
0.0626 -0.9662
-1.4221 0.6277
0.3063 0.1266
-0.8613 0.5724
0.0038 0.1983
0.2720 -1.3514
B_Ctrb_Staircase =
-0.9896 1.1768
0.0256 -0.5689
0.1923 1.3809
-1.4677 0.3477
-0.9060 0.0691
-0.1731 1.9355
0 0
0 0
0 0
A_Unctrb =
-0.2659 -0.8613 0.5724
0.1487 0.0038 0.1983
-0.3722 0.2720 -1.3514
Uncontrollable_Eigenvalues =
-1.0807
-0.3462
-0.1866
A_Obsv_Staircase =
Columns 1 through 7
-2.0457 -0.0012 0.0004 0.0001 0.0000 0 0
1.5594 -0.3092 0.2883 0.0646 0.0157 0 0
-2.8119 2.4336 0.2847 0.0585 0.0029 0 0
0.9918 -1.3934 0.8955 -0.1435 -0.1645 0 0
-2.2451 1.0432 -1.7931 -0.3435 0.0100 0 0
0.4903 -0.0436 -0.1690 -0.2983 -0.1739 -0.4915 0.1358
0.0500 1.0218 0.1962 -0.0520 0.1103 -0.9700 -1.1370
-0.3120 0.6560 -0.3933 -0.6671 -0.6981 1.1783 -0.0090
0.0710 0.1490 -0.1659 -0.2785 -0.2498 1.7491 0.4581
Columns 8 through 9
0 0
0 0
0 0
0 0
0 0
-0.0309 0.3837
-0.0795 -0.7095
-0.2954 -0.8172
0.1230 -0.6700
C_Obsv_Staircase =
Columns 1 through 7
-0.3458 0.2315 0.4981 -0.1520 -0.6355 0 0
-0.1624 1.1215 2.3569 -1.2628 0.0287 0 0
0.7369 -0.5872 0.1117 -0.8838 0.3227 0 0
Columns 8 through 9
0 0
0 0
0 0
A_Unobsv =
-0.4915 0.1358 -0.0309 0.3837
-0.9700 -1.1370 -0.0795 -0.7095
1.1783 -0.0090 -0.2954 -0.8172
1.7491 0.4581 0.1230 -0.6700
Unobservable_Eigenvalues =
-1.0807
-0.9322
-0.1866
-0.3944
Q = sdpvar(n,n,'symmetric');
N = sdpvar(m,n,'full');
CONS = [
Q - 0.01*eye(n) >= 0
Q*A' + A*Q - B*N - N'*B' + 0.01*eye(n) <= 0
];
sol = optimize(CONS)
Q = double(Q);
N = double(N);
P = inv(Q);
K = N/Q;
Check solution
Eigenvalues_of_the_closed_loop = eig(A - B*K)
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 63
Cones : 0
Scalar variables : 0
Matrix variables : 2
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator - tries : 0 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 63
Cones : 0
Scalar variables : 0
Matrix variables : 2
Integer variables : 0
Optimizer - threads : 4
Optimizer - solved problem : the primal
Optimizer - Constraints : 63
Optimizer - Cones : 0
Optimizer - Scalar variables : 0 conic : 0
Optimizer - Semi-definite variables: 2 scalarized : 90
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 2016 after factor : 2016
Factor - dense dim. : 0 flops : 2.16e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.4e+00 1.0e+00 8.2e-01 0.00e+00 -1.800000000e-01 0.000000000e+00 1.0e+00 0.00
1 2.4e-01 1.7e-01 2.9e-01 8.99e-01 -7.434373035e-02 0.000000000e+00 1.7e-01 0.01
2 4.8e-02 3.5e-02 1.0e-01 7.24e-01 -3.471435471e-02 0.000000000e+00 3.4e-02 0.01
3 4.9e-03 3.5e-03 3.6e-02 8.91e-01 -2.143662557e-03 0.000000000e+00 3.5e-03 0.01
4 3.3e-07 2.4e-07 2.8e-04 9.89e-01 -2.078577969e-07 0.000000000e+00 2.3e-07 0.01
5 6.1e-13 4.3e-13 3.4e-13 1.00e+00 -7.033482389e-13 0.000000000e+00 4.9e-13 0.01
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -7.0334823890e-13 nrm: 2e-11 Viol. con: 3e-12 barvar: 0e+00
Dual. obj: 0.0000000000e+00 nrm: 3e+00 Viol. con: 0e+00 barvar: 6e-13
Optimizer summary
Optimizer - time: 0.02
Interior-point - iterations : 5 time: 0.01
Basis identification - time: 0.00
Primal - iterations : 0 time: 0.00
Dual - iterations : 0 time: 0.00
Clean primal - iterations : 0 time: 0.00
Clean dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
sol =
struct with fields:
yalmiptime: 0.2342
solvertime: 0.0328
info: 'Successfully solved (MOSEK)'
problem: 0
Eigenvalues_of_the_closed_loop =
-0.8361 + 5.5836i
-0.8361 - 5.5836i
-0.5299 + 0.8968i
-0.5299 - 0.8968i
-0.1866 + 0.0000i
-0.3462 + 0.0000i
-0.4264 + 0.0000i
-1.0807 + 0.0000i
-0.9181 + 0.0000i