Script proba_LTI_passivity
file: proba_LTI_passivity.m author: Peter Polcz <ppolcz@gmail.com>
Created on 2017.03.18. Saturday, 22:48:49
Contents
% Automatically generated stuff global SCOPE_DEPTH SCOPE_DEPTH = 0; TMP_QVgVGfoCXYiYXzPhvVPX = pcz_dispFunctionName; try c = evalin('caller','persist'); catch; c = []; end persist = pcz_persist(mfilename('fullpath'), c); clear c; persist.backup(); %clear persist
┌proba_LTI_passivity │ - Persistence for `proba_LTI_passivity` reused (inherited) [run ID: 3492, 2017.03.18. Saturday, 23:28:54] │ - Script `proba_LTI_passivity` backuped
LTI SISO Passivity
Ez a rendszer NEM passziv, mert relativ foka 2! Controller
formbol adodoan: 
.
A = [
-2 -2 -1
1 0 0
0 1 0
];
B = [ 1 ; 0 ; 0 ];
C = [ 0 1 1 ];
P = sdpvar(3);
Constraints = [
P >= 0 , ...
[
A'*P + P*A , P*B - C'
B'*P - C , 0
] <= 0
];
optimize(Constraints)
MOSEK Version 7.1.0.51 (Build date: 2016-4-5 06:54:35)
Copyright (c) 1998-2016 MOSEK ApS, Denmark. WWW: http://mosek.com
Platform: Linux/64-X86
Computer
Platform : Linux/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 6
Cones : 0
Scalar variables : 0
Matrix variables : 2
Integer variables : 0
Optimizer started.
Conic interior-point optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator - tries : 0 time : 0.00
Eliminator - elim's : 0
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Optimizer - threads : 4
Optimizer - solved problem : the primal
Optimizer - Constraints : 6
Optimizer - Cones : 0
Optimizer - Scalar variables : 0 conic : 0
Optimizer - Semi-definite variables: 2 scalarized : 16
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 : 21 after factor : 21
Factor - dense dim. : 0 flops : 8.65e+02
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 5.0e+00 1.0e+00 1.0e+00 0.00e+00 0.000000000e+00 0.000000000e+00 1.0e+00 0.00
1 1.1e+00 2.1e-01 2.1e-01 -2.08e-01 -1.029830788e+00 0.000000000e+00 2.1e-01 0.01
2 3.1e-16 1.3e-01 8.4e-02 -5.42e-01 -1.029830788e+00 0.000000000e+00 8.6e-02 0.01
Interior-point optimizer terminated. Time: 0.01.
MOSEK DUAL INFEASIBILITY REPORT.
Problem status: The problem is dual infeasible
Optimizer terminated. Time: 0.02
Interior-point solution summary
Problem status : DUAL_INFEASIBLE
Solution status : DUAL_INFEASIBLE_CER
Primal. obj: -4.8168824580e-01 Viol. con: 3e-16 barvar: 0e+00
Optimizer summary
Optimizer - time: 0.02
Interior-point - iterations : 2 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
Clean primal-dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Primal-dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
ans =
yalmiptime: 0.1597
solvertime: 0.0365
info: 'Infeasible problem (MOSEK)'
problem: 1
LTI SISO with observer
Azonban ha tervezek hozza egy allapotbecslot, akkor az egyuttes rendszer mar 1-es relativ foku lesz, vagyis passziv.
L = place(A',C',[-1 -0.5 -2])';
A_ = [
A , zeros(3,3)
L*C , A - L*C
];
B_ = [ B ; B ];
C_ = [ zeros(1,3) ones(1,3) ];
C_*B_
P_ = sdpvar(6);
Constraints = [
P_ >= 0 , ...
[
A_'*P_ + P_*A_ , P_*B_ - C_'
B_'*P_ - C_ , 0
] <= 0
];
optimize(Constraints)
P_ = double(P_);
x0 = [ rand(3,1) ; zeros(3,1) ];
sys = ss(A_,B_,C_,0);
[u_num,t_num] = gensig('square',20,100,0.1);
u = @(t) interp1(t_num, u_num, t);
[t_ode,x_ode] = ode45(@(t,x) A_*x + B_*u(t), [0,100], x0);
y_ode = x_ode * C_';
u_ode = u(t_ode);
V = diag(x_ode * P_ * x_ode');
dV = diag(x_ode * (A_'*P_ + P_*A_) * x_ode') + ...
diag(u_ode * (B_'*P_) * x_ode') + ...
diag(x_ode * (P_*B_) * u_ode');
s = 2 * y_ode .* u_ode;
figure('Position', [ 305 281 1553 633 ], 'Color', [1 1 1]), subplot(223)
plot(t_ode,[u_ode,y_ode]),
ptitle('Bemenet, kimenet az ido fuggvenyeben')
plabel 'x' 'time $t$'
plabel 'y' '$u(t), y(t)$'
subplot(224)
plot(t_ode,dV - s)
title 'Passivity test (SISO): $\dot V(t) \le 2\cdot u(t) \cdot y(t) $' interpreter latex
plabel 'x' 'time $t$'
plabel 'y' '$\dot V(t) - 2\cdot u(t) \cdot y(t)$'
subplot(211)
plot(t_ode,x_ode)
ptitle('Allapotvaltozok az ido fuggvenyeben')
plabel 'x' 'time $t$'
plabel 'y' '$x(t)$'
ans =
1
MOSEK Version 7.1.0.51 (Build date: 2016-4-5 06:54:35)
Copyright (c) 1998-2016 MOSEK ApS, Denmark. WWW: http://mosek.com
Platform: Linux/64-X86
Computer
Platform : Linux/64-X86
Problem
Name :
Objective sense : min
Type : CONIC (conic optimization problem)
Constraints : 21
Cones : 0
Scalar variables : 0
Matrix variables : 2
Integer variables : 0
Optimizer started.
Conic interior-point optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator - tries : 0 time : 0.00
Eliminator - elim's : 0
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.00
Optimizer - threads : 4
Optimizer - solved problem : the primal
Optimizer - Constraints : 21
Optimizer - Cones : 0
Optimizer - Scalar variables : 0 conic : 0
Optimizer - Semi-definite variables: 2 scalarized : 49
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 : 231 after factor : 231
Factor - dense dim. : 0 flops : 1.45e+04
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 5.0e+00 1.0e+00 1.0e+00 0.00e+00 0.000000000e+00 0.000000000e+00 1.0e+00 0.00
1 8.7e-01 1.7e-01 1.7e-01 -2.53e-01 -7.632831160e-01 0.000000000e+00 1.7e-01 0.01
2 2.3e-01 4.6e-02 4.6e-02 6.63e-01 -7.486100366e-02 0.000000000e+00 4.6e-02 0.01
3 2.8e-02 5.6e-03 5.6e-03 1.37e+00 -4.542576317e-03 0.000000000e+00 5.6e-03 0.01
4 2.8e-03 5.6e-04 5.6e-04 1.05e+00 -4.114345352e-04 0.000000000e+00 5.6e-04 0.01
5 1.7e-04 3.3e-05 3.3e-05 1.01e+00 -2.479762506e-05 0.000000000e+00 3.3e-05 0.01
6 8.9e-06 1.8e-06 1.8e-06 9.98e-01 -1.384659975e-06 0.000000000e+00 1.8e-06 0.01
7 5.0e-07 1.0e-07 1.0e-07 9.95e-01 -8.541464338e-08 0.000000000e+00 1.0e-07 0.01
8 3.2e-08 6.4e-09 6.4e-09 9.87e-01 -6.511520713e-09 0.000000000e+00 6.4e-09 0.01
9 2.6e-09 5.3e-10 5.3e-10 9.72e-01 -6.848746048e-10 0.000000000e+00 5.3e-10 0.01
Interior-point optimizer terminated. Time: 0.01.
Optimizer terminated. Time: 0.01
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: -6.8487460477e-10 Viol. con: 5e-09 barvar: 0e+00
Dual. obj: 0.0000000000e+00 Viol. con: 0e+00 barvar: 9e-10
Optimizer summary
Optimizer - time: 0.01
Interior-point - iterations : 9 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
Clean primal-dual - iterations : 0 time: 0.00
Simplex - time: 0.00
Primal simplex - iterations : 0 time: 0.00
Dual simplex - iterations : 0 time: 0.00
Primal-dual simplex - iterations : 0 time: 0.00
Mixed integer - relaxations: 0 time: 0.00
ans =
yalmiptime: 0.1554
solvertime: 0.0248
info: 'Successfully solved (MOSEK)'
problem: 0
End of the script.
pcz_dispFunctionEnd(TMP_QVgVGfoCXYiYXzPhvVPX);
clear TMP_QVgVGfoCXYiYXzPhvVPX
└ 1.308 [sec]