Teljes Matlab script kiegészítő függvényekkel.
File: d2018_01_31_K_prelim_L_codesign_v6_rhofuggoBL_min.m Directory: projects/3_outsel/2017_11_13_lpv_passivity Author: Peter Polcz (ppolcz@gmail.com)
Created on 2018. January 31.
global VERBOSE SCOPE_DEPTH
VERBOSE = 1;
SCOPE_DEPTH = -1;
s = tf('s');
H = @(s) [
(s-1)/(s-2)/(s+1) 1/(s+3)/(s-0.1)
(s-7)/(s+1)/(s+5) (s-6)/(s^2+5*s+6)
];
sys = minreal( ss( H(s) ) );
[A0,B0,C,D] = deal(sys.a, sys.b, sys.c, sys.d);
tol = 1e-10;
prec = -log10(tol);
A0 = round(A0,prec);
B0 = round(B0,prec);
C = round(C,prec);
D = round(D,prec);
[POLES,ZEROS] = pzmap(sys)
2 states removed.
POLES =
-5.0000
-1.0000
2.0000
-3.0000
-2.0000
0.1000
ZEROS =
5.8894 + 0.0000i
-4.3164 + 0.0000i
0.7635 + 0.7978i
0.7635 - 0.7978i
rho_lim = [
-1 1
];
rho_rand = @(varargin) rand(varargin{:})*(rho_lim(2) - rho_lim(1)) + rho_lim(1);
A1 = A0;
A1(abs(A0) < 1) = 0;
A1 = A1 .* randn(size(A1))/10;
B1 = [ B0(:,2)/2 B0(:,2)*0 ];
% _fh: `funcion handle`
A_fh = @(rho) A0 + rho*A1;
B_fh = @(rho) B0 + rho*B1;
% Dimenziok
n_x = size(A0,1)
n_u = size(B0,2)
n_y = size(C,1)
n_r = n_u
n_yp = n_r
n_x =
6
n_u =
2
n_y =
2
n_r =
2
n_yp =
2
Kvadratikus stabilitas
Q = sdpvar(n_x);
N = sdpvar(n_u,n_x,'full');
Big_M = @(rho) A_fh(rho)'*Q + Q*A_fh(rho) - B_fh(rho)*N - N'*B_fh(rho)';
I = eye(n_x);
Constraints = [
Q - I >= 0
Big_M(rho_lim(1)) + 10*I <= 0
Big_M(rho_lim(2)) + 10*I <= 0
];
optimize(Constraints)
N = value(N);
Q = value(Q);
K = N/Q;
G = eye(n_u);
% for rho = rho_lim(1):0.25:rho_lim(2)
% eig(A_fh(rho) - B_fh(rho)*K)
% end
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 33
Cones : 0
Scalar variables : 0
Matrix variables : 3
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 : 33
Cones : 0
Scalar variables : 0
Matrix variables : 3
Integer variables : 0
Optimizer - threads : 4
Optimizer - solved problem : the primal
Optimizer - Constraints : 33
Optimizer - Cones : 0
Optimizer - Scalar variables : 0 conic : 0
Optimizer - Semi-definite variables: 3 scalarized : 63
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 : 561 after factor : 561
Factor - dense dim. : 0 flops : 5.11e+04
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.4e+00 6.0e+00 6.2e+01 0.00e+00 -2.520000000e+02 0.000000000e+00 1.0e+00 0.00
1 3.5e-01 8.8e-01 1.8e+00 -1.31e+00 -8.372130494e+02 0.000000000e+00 1.5e-01 0.01
2 5.4e-02 1.4e-01 3.3e-01 -1.35e+00 -2.163114617e+02 0.000000000e+00 2.3e-02 0.01
3 4.3e-04 1.1e-03 5.5e-02 8.66e-01 -1.369345252e+00 0.000000000e+00 1.8e-04 0.01
4 3.5e-13 8.9e-13 9.2e-12 9.96e-01 -1.630210353e-09 0.000000000e+00 1.5e-13 0.01
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -1.6302103533e-09 nrm: 4e-11 Viol. con: 2e-10 barvar: 9e-18
Dual. obj: 0.0000000000e+00 nrm: 1e+02 Viol. con: 0e+00 barvar: 5e-11
Optimizer summary
Optimizer - time: 0.02
Interior-point - iterations : 4 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
ans =
struct with fields:
yalmiptime: 0.2242
solvertime: 0.0250
info: 'Successfully solved (MOSEK)'
problem: 0
C1 = sdpvar([n_yp n_yp], [n_x n_x],'full');
D1 = sdpvar([n_yp n_yp], [n_y n_y],'full');
Q = sdpvar(n_x,n_x,'symmetric');
S = sdpvar(n_x,n_x,'symmetric');
N = sdpvar([n_x n_x],[n_y n_y],'full');
P = blkdiag(Q,S);
C1_fh = @(rho) C1{1} + rho*C1{2};
D1_fh = @(rho) D1{1} + rho*D1{2};
N_fh = @(rho) N{1} + rho*N{2};
% Open loop matrices
Bo_fh = @(rho) [
B_fh(rho)
zeros(n_x,n_u)
];
Co_fh = @(rho) [
D1_fh(rho)*C+C1_fh(rho) -C1_fh(rho)
];
% Closed loop matrices
Bc_fh = @(rho) Bo_fh(rho)*G;
AcP_PAc_fh = @(rho) [
Q*(A_fh(rho)-B_fh(rho)*K) + (A_fh(rho)-B_fh(rho)*K)'*Q , Q*B_fh(rho)*K
K'*B_fh(rho)'*Q , S*A_fh(rho) + A_fh(rho)'*S - N_fh(rho)*C - C'*N_fh(rho)'
];
W = eye(n_yp);
Lambda2 = @(rho) [
AcP_PAc_fh(rho) , P*Bc_fh(rho)-Co_fh(rho)' , Co_fh(rho)'
Bc_fh(rho)'*P-Co_fh(rho) , zeros(n_r,n_r) , zeros(n_r,n_yp)
Co_fh(rho) , zeros(n_yp,n_r) , -inv(W)
];
Constraints = [
Lambda2(rho_lim(1)) <= 0
Lambda2(rho_lim(2)) <= 0
P - 0.0001*eye(size(P)) >= 0
]
++++++++++++++++++++++++++++++++++ | ID| Constraint| ++++++++++++++++++++++++++++++++++ | #1| Matrix inequality 16x16| | #2| Matrix inequality 16x16| | #3| Matrix inequality 12x12| ++++++++++++++++++++++++++++++++++
sdpopts = sdpsettings('solver','sedumi');
optimize(Constraints,[],sdpopts)
check(Constraints)
The coefficient matrix is not full row rank, numerical problems may occur.
SeDuMi 1.3 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, theta = 0.250, beta = 0.500
eqs m = 98, order n = 45, dim = 657, blocks = 4
nnz(A) = 1962 + 0, nnz(ADA) = 9604, nnz(L) = 4851
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 3.12E+01 0.000
1 : 0.00E+00 8.21E+00 0.000 0.2632 0.9000 0.9000 2.00 1 1 8.9E+00
2 : 0.00E+00 2.82E+00 0.000 0.3430 0.9000 0.9000 1.26 1 1 3.2E+00
3 : 0.00E+00 1.21E+00 0.000 0.4306 0.9000 0.9000 1.08 1 1 1.7E+00
4 : 0.00E+00 5.93E-01 0.000 0.4893 0.9000 0.9000 1.04 1 1 1.1E+00
5 : 0.00E+00 2.05E-01 0.000 0.3448 0.9000 0.9000 1.01 1 1 7.6E-01
6 : 0.00E+00 7.53E-02 0.000 0.3681 0.9000 0.9000 1.00 1 1 6.5E-01
7 : 0.00E+00 2.89E-02 0.000 0.3838 0.9000 0.9000 0.98 1 1 6.2E-01
8 : 0.00E+00 9.92E-03 0.000 0.3432 0.9000 0.9000 0.97 1 1 5.3E-01
9 : 0.00E+00 3.57E-03 0.000 0.3601 0.9000 0.9000 0.96 1 1 2.0E-01
10 : 0.00E+00 1.21E-03 0.000 0.3379 0.9000 0.9000 0.95 1 1 7.2E-02
11 : 0.00E+00 4.44E-04 0.000 0.3679 0.9000 0.9000 0.92 1 1 2.9E-02
12 : 0.00E+00 1.49E-04 0.000 0.3346 0.9000 0.9000 0.88 1 1 1.1E-02
13 : 0.00E+00 5.52E-05 0.000 0.3718 0.9000 0.9000 0.79 1 1 4.9E-03
14 : 0.00E+00 1.89E-05 0.000 0.3427 0.9000 0.9000 0.67 1 1 2.2E-03
15 : 0.00E+00 7.23E-06 0.000 0.3818 0.9000 0.9000 0.49 1 1 1.2E-03
16 : 0.00E+00 2.49E-06 0.000 0.3452 0.9000 0.9000 0.35 1 1 5.3E-04
17 : 0.00E+00 9.46E-07 0.000 0.3793 0.9000 0.9000 0.19 1 1 4.1E-06
18 : 0.00E+00 3.22E-07 0.000 0.3407 0.9000 0.9000 0.14 1 1 2.2E-06
19 : 0.00E+00 1.19E-07 0.000 0.3687 0.9000 0.9000 0.05 1 1 1.4E-06
20 : 0.00E+00 4.04E-08 0.000 0.3397 0.9000 0.9000 0.06 1 1 7.4E-07
21 : 0.00E+00 1.48E-08 0.000 0.3665 0.9000 0.9000 0.00 1 1 4.7E-07
22 : 0.00E+00 5.06E-09 0.000 0.3420 0.9000 0.9000 0.04 1 1 2.6E-07
23 : 0.00E+00 1.86E-09 0.000 0.3680 0.9000 0.9000 -0.02 1 1 1.6E-07
24 : 0.00E+00 6.37E-10 0.000 0.3420 0.9000 0.9000 0.03 1 1 9.1E-08
25 : 0.00E+00 2.34E-10 0.000 0.3674 0.9000 0.9000 -0.02 1 1 5.8E-08
26 : 0.00E+00 7.98E-11 0.000 0.3409 0.9000 0.9000 0.02 1 1 3.2E-08
27 : 0.00E+00 2.92E-11 0.000 0.3663 0.9000 0.9000 -0.03 1 2 2.1E-08
28 : 0.00E+00 9.96E-12 0.000 0.3406 0.9000 0.9000 0.02 1 2 1.1E-08
29 : 0.00E+00 3.65E-12 0.000 0.3662 0.9000 0.9000 -0.03 2 2 7.3E-09
30 : 0.00E+00 1.24E-12 0.000 0.3409 0.9000 0.9000 0.02 2 2 4.0E-09
31 : 0.00E+00 4.56E-13 0.000 0.3666 0.9000 0.9000 -0.03 2 2 2.6E-09
32 : 0.00E+00 1.55E-13 0.000 0.3410 0.9000 0.9000 0.02 2 2 1.4E-09
33 : 0.00E+00 5.70E-14 0.000 0.3665 0.9000 0.9000 -0.03 2 2 9.1E-10
iter seconds digits c*x b*y
33 0.4 11.4 -4.3643988485e-07 0.0000000000e+00
|Ax-b| = 8.9e-10, [Ay-c]_+ = 3.1E-11, |x|= 1.1e+05, |y|= 3.1e+05
Detailed timing (sec)
Pre IPM Post
8.067E-03 1.522E-01 2.444E-03
Max-norms: ||b||=0, ||c|| = 1,
Cholesky |add|=0, |skip| = 2, ||L.L|| = 7.65576.
ans =
struct with fields:
yalmiptime: 0.1251
solvertime: 0.1635
info: 'Numerical problems (SeDuMi-1.3)'
problem: 4
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| ID| Constraint| Primal residual| Dual residual|
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| #1| Matrix inequality| -4.0565e-11| 8.9323e-12|
| #2| Matrix inequality| -3.4027e-11| 9.2439e-12|
| #3| Matrix inequality| 5.2031e-05| 1.5087e-12|
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Q = value(Q)
S = value(S)
P = blkdiag(Q,S);
N = cellfun(@value, N, 'UniformOutput', false);
C1 = cellfun(@value, C1, 'UniformOutput', false);
D1 = cellfun(@value, D1, 'UniformOutput', false);
L = cellfun(@(N) S\N, N, 'UniformOutput', false);
L_fh = @(rho) L{1} + rho*L{2};
C1_fh = @(rho) C1{1} + rho*C1{2};
D1_fh = @(rho) D1{1} + rho*D1{2};
Q =
1.0e-03 *
0.2975 0.0494 -0.0952 0.0089 0.0946 0.0837
0.0494 0.3665 -0.1271 -0.0580 0.1117 0.0901
-0.0952 -0.1271 0.2795 0.0651 -0.0970 -0.1050
0.0089 -0.0580 0.0651 0.5451 0.0279 -0.0237
0.0946 0.1117 -0.0970 0.0279 0.3054 0.0232
0.0837 0.0901 -0.1050 -0.0237 0.0232 0.3069
S =
1.0e+05 *
0.1098 0.0638 -0.0051 0.1726 0.0094 -0.1239
0.0638 0.2504 0.0325 0.2648 -0.0724 -0.1064
-0.0051 0.0325 0.1283 0.3389 -0.0283 0.0026
0.1726 0.2648 0.3389 2.3854 -0.1701 -0.5792
0.0094 -0.0724 -0.0283 -0.1701 0.0792 -0.0114
-0.1239 -0.1064 0.0026 -0.5792 -0.0114 0.3070
\begin{align} {\LARGE(5) \quad}
L(\rho) =
\left(\begin{array}{cc}
-4.72\rho -24.5 & 0.11\rho -7.53 \\
0.957\rho -46.4 & 1.25\rho -10.9 \\
2.36\rho +112.0 & 25.1-2.77\rho \\
-0.666\rho -45.9 & 1.2\rho -10.7 \\
1.67\rho -126.0 & 3.33\rho -30.8 \\
-2.83\rho -116.0 & 2.89\rho -28.4 \\
\end{array}\right)
\end{align}
\begin{align} {\LARGE(6) \quad}
C_1(\rho) =
\left(\begin{array}{cccccc}
1.7410^{-4}-6.0410^{-5}\rho & 2.3510^{-5}\rho -1.8610^{-4} & 1.8710^{-4}\rho +3.5910^{-4} & -1.1210^{-6}\rho -1.9110^{-4} & 1.7310^{-4}\rho -3.7510^{-4} & 8.2810^{-6}\rho -1.6710^{-4} \\
-3.910^{-10}\rho -1.2110^{-4} & 1.5310^{-8}\rho +4.710^{-5} & 7.5510^{-9}\rho +3.7310^{-4} & 1.8710^{-8}\rho -2.2510^{-6} & 3.4610^{-4}-1.0310^{-8}\rho & 5.6110^{-10}\rho +1.6610^{-5} \\
\end{array}\right)
\end{align}
\begin{align} {\LARGE(7) \quad}
D_1(\rho) =
\left(\begin{array}{cc}
9.3310^{-5}\rho +1.5110^{-4} & 6.2810^{-5}\rho +1.1410^{-4} \\
1.8710^{-4}-2.0210^{-8}\rho & 5.4110^{-9}\rho +1.2610^{-4} \\
\end{array}\right)
\end{align}
Ao_fh = @(rho) [
A_fh(rho) zeros(n_x,n_x)
zeros(n_x,n_x) A_fh(rho)-L_fh(rho)*C
];
Bo_fh = @(rho) [
B_fh(rho)
zeros(n_x,n_u)
];
Co_fh = @(rho) [
D1_fh(rho)*C+C1_fh(rho) -C1_fh(rho)
];
Do = zeros(n_yp,n_r);
% Closed loop matrices
Ac_fh = @(rho) [
A_fh(rho)-B_fh(rho)*K B_fh(rho)*K
zeros(n_x,n_x) A_fh(rho)-L_fh(rho)*C
];
Bc_fh = @(rho) Bo_fh(rho)*G;
for rho = rho_lim(1):0.25:rho_lim(2)
sys = ss(Ao_fh(rho),Bo_fh(rho),Co_fh(rho),Do);
[POLES,ZEROS] = pzmap(sys);
fprintf('\nzeros if rho = %g\n', rho)
disp(ZEROS)
pcz_info(all(real(ZEROS) < 0), 'Zeros are all negative');
end
zeros if rho = -1 -13.4718 + 0.0000i -4.0252 + 3.4690i -4.0252 - 3.4690i -3.4947 + 0.0000i -2.9024 + 0.0000i -0.8885 + 0.0000i -0.9692 + 1.1803i -0.9692 - 1.1803i -0.3061 + 0.4313i -0.3061 - 0.4313i [ OK ] Zeros are all negative zeros if rho = -0.75 -13.4125 + 0.0000i -3.9254 + 3.1196i -3.9254 - 3.1196i -3.3557 + 0.4947i -3.3557 - 0.4947i -0.8829 + 0.0000i -1.0098 + 1.1401i -1.0098 - 1.1401i -0.3013 + 0.4341i -0.3013 - 0.4341i [ OK ] Zeros are all negative zeros if rho = -0.5 -13.3005 + 0.0000i -3.6045 + 2.7449i -3.6045 - 2.7449i -3.7652 + 0.5596i -3.7652 - 0.5596i -0.8675 + 0.0000i -1.0502 + 1.0963i -1.0502 - 1.0963i -0.2967 + 0.4365i -0.2967 - 0.4365i [ OK ] Zeros are all negative zeros if rho = -0.25 -13.1260 + 0.0000i -5.1928 + 0.0000i -3.1104 + 2.6552i -3.1104 - 2.6552i -3.5702 + 0.0000i -0.8475 + 0.0000i -1.0904 + 1.0485i -1.0904 - 1.0485i -0.2922 + 0.4386i -0.2922 - 0.4386i [ OK ] Zeros are all negative zeros if rho = 0 -12.8703 + 0.0000i -6.4228 + 0.0000i -2.7690 + 2.7445i -2.7690 - 2.7445i -3.3492 + 0.0000i -0.8266 + 0.0000i -1.1305 + 0.9962i -1.1305 - 0.9962i -0.2879 + 0.4404i -0.2879 - 0.4404i [ OK ] Zeros are all negative zeros if rho = 0.25 -12.4941 + 0.0000i -7.4830 + 0.0000i -2.5334 + 2.8517i -2.5334 - 2.8517i -3.2056 + 0.0000i -0.8074 + 0.0000i -1.1703 + 0.9384i -1.1703 - 0.9384i -0.2838 + 0.4419i -0.2838 - 0.4419i [ OK ] Zeros are all negative zeros if rho = 0.5 -11.8821 + 0.0000i -8.6347 + 0.0000i -2.3559 + 2.9533i -2.3559 - 2.9533i -3.0869 + 0.0000i -0.7911 + 0.0000i -1.2100 + 0.8741i -1.2100 - 0.8741i -0.2799 + 0.4432i -0.2799 - 0.4432i [ OK ] Zeros are all negative zeros if rho = 0.75 -10.4831 + 0.8955i -10.4831 - 0.8955i -2.2157 + 3.0458i -2.2157 - 3.0458i -2.9804 + 0.0000i -0.7783 + 0.0000i -1.2494 + 0.8016i -1.2494 - 0.8016i -0.2762 + 0.4444i -0.2762 - 0.4444i [ OK ] Zeros are all negative zeros if rho = 1 -10.6759 + 2.0415i -10.6759 - 2.0415i -2.1022 + 3.1291i -2.1022 - 3.1291i -2.8808 + 0.0000i -0.7692 + 0.0000i -1.2887 + 0.7184i -1.2887 - 0.7184i -0.2727 + 0.4455i -0.2727 - 0.4455i [ OK ] Zeros are all negative
concat = @(fh) { fh(0) , fh(1)-fh(0) };
A = concat(Ao_fh);
B = concat(Bo_fh);
C = concat(Co_fh);
D = {Do, Do*0};
AA = [A{:}];
BB = [B{:}];
Im_B = IMA(BB);
L = ORTCO(Im_B);
tol = 1e-5;
Ker_C = INTS(KER(C{1}), KER(C{2}));
Ker_C = pcz_vecalg_ints(pcz_vecalg_null(C{1},tol), pcz_vecalg_null(C{2},tol),tol);
[R,V] = CSA(AA, Im_B, Ker_C);
dim_V = rank(V);
dim_Ker_V = size(V,1) - dim_V;
dim_L = rank(L);
assert(rank(R) == 0 && rank([V Ker_C]) == rank(Ker_C), ...
'Strong invertibility condition is not satisfied!');
assert(rank(INTS(V, Im_B)) == 0)
assert(dim_L >= dim_V);
T = [ ORTCO(V) L(:,1:dim_V) ]';
N = numel(A);
tA = cell(1,N);
tB = cell(1,N);
tC = cell(1,N);
tD = cell(1,N);
for i = 1:numel(A)
tA{i} = T*A{i}/T;
tB{i} = T*B{i};
tC{i} = C{i}/T;
tD{i} = D{i};
end
tA_fh = @(rho) tA{1} + tA{2}*rho;
tB_fh = @(rho) tB{1} + tB{2}*rho;
tC_fh = @(rho) tC{1} + tC{2}*rho;
tD_fh = @(rho) tD{1} + tD{2}*rho;
(Ennek most nincs jelentosege)
tP = T' \ P / T;
tP22 = tP(dim_Ker_V+1:end,dim_Ker_V+1:end);
for rho = rho_lim
tA_rho = tA_fh(rho);
tA22 = tA_rho(dim_Ker_V+1:end,dim_Ker_V+1:end);
eig(tP22 * tA22 + tA22' * tP22)
end
ans = 1.0e+04 * -3.6239 -3.3649 -0.7888 -1.3679 -2.1188 -2.1188 -0.0000 -0.0000 -0.0000 -0.0000 ans = 1.0e+04 * -3.4864 -3.9769 -0.9946 -2.1395 -2.1188 -2.1190 -0.0000 -0.0000 -0.0000 -0.0000
Using the output zeroing input
syms rho drho
A_sym = Ao_fh(rho);
B_sym = Bo_fh(rho);
C_sym = Co_fh(rho);
D_sym = Do;
dC_sym = Co_fh(drho);
A_zd = A_sym - B_sym*((C_sym*B_sym)\(C_sym*A_sym + dC_sym));
A_zd_fh = matlabFunction(A_zd, 'vars', [rho,drho]);
IS_OK = 1;
for rho = rho_lim(1):0.25:rho_lim(2)
for drho = -10:5:10
if pcz_info(all(real(eig(A_zd_fh(rho,drho))) < 0), ...
'Zero dynamics is stable if rho = %g, drho = %g.', rho, drho)
A_zd_eigvals = eig(A_zd_fh(rho,drho))
IS_OK = 0;
end
end
end
pcz_info(IS_OK, 'The zero dynamics is stable in everry sampling point')
[ OK ] Zero dynamics is stable if rho = -1, drho = -10. [ OK ] Zero dynamics is stable if rho = -1, drho = -5. [ OK ] Zero dynamics is stable if rho = -1, drho = 0. [ OK ] Zero dynamics is stable if rho = -1, drho = 5. [ OK ] Zero dynamics is stable if rho = -1, drho = 10. [ OK ] Zero dynamics is stable if rho = -0.75, drho = -10. [ OK ] Zero dynamics is stable if rho = -0.75, drho = -5. [ OK ] Zero dynamics is stable if rho = -0.75, drho = 0. [ OK ] Zero dynamics is stable if rho = -0.75, drho = 5. [ OK ] Zero dynamics is stable if rho = -0.75, drho = 10. [ OK ] Zero dynamics is stable if rho = -0.5, drho = -10. [ OK ] Zero dynamics is stable if rho = -0.5, drho = -5. [ OK ] Zero dynamics is stable if rho = -0.5, drho = 0. [ OK ] Zero dynamics is stable if rho = -0.5, drho = 5. [ OK ] Zero dynamics is stable if rho = -0.5, drho = 10. [ OK ] Zero dynamics is stable if rho = -0.25, drho = -10. [ OK ] Zero dynamics is stable if rho = -0.25, drho = -5. [ OK ] Zero dynamics is stable if rho = -0.25, drho = 0. [ OK ] Zero dynamics is stable if rho = -0.25, drho = 5. [ OK ] Zero dynamics is stable if rho = -0.25, drho = 10. [ OK ] Zero dynamics is stable if rho = 0, drho = -10. [ OK ] Zero dynamics is stable if rho = 0, drho = -5. [ OK ] Zero dynamics is stable if rho = 0, drho = 0. [ OK ] Zero dynamics is stable if rho = 0, drho = 5. [ OK ] Zero dynamics is stable if rho = 0, drho = 10. [ OK ] Zero dynamics is stable if rho = 0.25, drho = -10. [ OK ] Zero dynamics is stable if rho = 0.25, drho = -5. [ OK ] Zero dynamics is stable if rho = 0.25, drho = 0. [ OK ] Zero dynamics is stable if rho = 0.25, drho = 5. [ OK ] Zero dynamics is stable if rho = 0.25, drho = 10. [ OK ] Zero dynamics is stable if rho = 0.5, drho = -10. [ OK ] Zero dynamics is stable if rho = 0.5, drho = -5. [ OK ] Zero dynamics is stable if rho = 0.5, drho = 0. [ OK ] Zero dynamics is stable if rho = 0.5, drho = 5. [ OK ] Zero dynamics is stable if rho = 0.5, drho = 10. [ OK ] Zero dynamics is stable if rho = 0.75, drho = -10. [ OK ] Zero dynamics is stable if rho = 0.75, drho = -5. [ OK ] Zero dynamics is stable if rho = 0.75, drho = 0. [ OK ] Zero dynamics is stable if rho = 0.75, drho = 5. [ OK ] Zero dynamics is stable if rho = 0.75, drho = 10. [ OK ] Zero dynamics is stable if rho = 1, drho = -10. [ OK ] Zero dynamics is stable if rho = 1, drho = -5. [ OK ] Zero dynamics is stable if rho = 1, drho = 0. [ OK ] Zero dynamics is stable if rho = 1, drho = 5. [ OK ] Zero dynamics is stable if rho = 1, drho = 10. [ OK ] The zero dynamics is stable in everry sampling point
tol = 1e-10;
prec = -log10(tol);
is_negdef = @(A) all( round(eig(A), prec) <= 0 )
is_negdef( Ao_fh(rho_lim(1))'*P + P*Ao_fh(rho_lim(1)) )
is_negdef( Ao_fh(rho_lim(2))'*P + P*Ao_fh(rho_lim(2)) )
is_negdef =
function_handle with value:
@(A)all(round(eig(A),prec)<=0)
ans =
logical
0
ans =
logical
0