Gitlab Runner Update from atc-aux-files/master:Merge branch 'ExerciseChapter12' into 'master'

ATC/exercise-files/LPV/Exercise1_1/exercise_Hidden_Coupling.m

+%% Hidden Coupling - Instability of the linearized closed loop
+% -------------------------------------------------------------------------
+% script   : exercise_Hidden_Coupling
+% -------------------------------------------------------------------------
+% Author   : Christian Hoffmann
+% Version  : May, 28th 2021
+% Copyright: CH, 2013
+% Update   : PG, 2021
+% -------------------------------------------------------------------------
+%
+% 1. Construct linearized closed loop
+% 2. Plot eigenvalues for different parameter values
+%
+% -------------------------------------------------------------------------
+
+%% 1.
+rho = -0.9:0.01:0.9;
+xKe = atanh(rho);
+
+urho = ureal('urho', 0       , 'Range', [-1, 1]);
+uxKe = ureal('uxKe', atanh(0), 'Range', [min(xKe), max(xKe)]);
+
+uAcl = [ -1 + (-1/3+2*urho*uxKe)/(1-urho^2)   ,    1/(1-urho^2)   ;...
+         -(1-urho^2)                  ,    0              ];
+     
+     
+for ii = 1:length(rho)
+    Aclsubs = usubs(uAcl    , 'uxKe', xKe(ii));
+    Aclsubs = usubs( Aclsubs, 'urho', rho(ii));
+    eigVals(:, ii) = eig(Aclsubs);
+end
+
+%% 2.
+
+ReEig = real(eigVals);
+
+plot(rho, ReEig(1, :)); hold on
+plot(rho, ReEig(2, :), 'r'); hold on

ATC/exercise-files/LPV/Exercise1_2/exercise_LTI_LTV_Stability.m

+%% Investigate the Fact that a LTV Stability Cannot be Inferred from LTI
+% Stability
+% -------------------------------------------------------------------------
+% script   : exercise_LTI_LTV_Stability
+% -------------------------------------------------------------------------
+% Author   : Christian Hoffmann
+% Version  : October, 27th 2015
+% Copyright: CH, 2013
+% Update   : PC, 2015
+% -------------------------------------------------------------------------
+%
+% 1. Define plant
+% 2. Plot phase diagrams for various scenarios
+%
+% -------------------------------------------------------------------------
+%% Initialization
+clc;clear all;close all;warning off;s  = tf('s');
+
+% Add subdirectories to path
+addpath(genpath('.'))
+
+thisdate = datestr(now, 'yyyy-mm-dd_HH-MM');
+
+%% 1. Plant Definition
+% Parameterized plant definition
+A  = @(k, b, m) [  0      1 ;...
+                  -k     -b ];
+
+x0 = [0; 1];
+
+k1 = 3;
+k2 = 1/3;
+
+b1 = 0.5;
+b2 = 0;
+
+[V1, D1] = eig(A(k1, b2, 1))
+[V2, D2] = eig(A(k2, b2, 1))
+              
+% Draw phase portraits
+x10_grid = -1:2:1;
+x20_grid = -1:2:1;
+x0_grid  = combvec(x10_grid, x20_grid);
+% x0_grid  = x0_grid(:, 8);
+time     = 0:0.01:100;
+colors   = {'k', 'b', 'r'};
+
+sys{1} = ss(A(k1, b2, 1), zeros(2, 1), eye(2), zeros(2, 1));
+sys{2} = ss(A(k2, b2, 1), zeros(2, 1), eye(2), zeros(2, 1));
+% sys{3} = ss(A(k1, b2, 1), zeros(2, 1), eye(2), zeros(2, 1));
+
+figure(1); hold on
+% phasePlotCompute(sys, x0_grid, time, colors)
+
+[~,~,x1] = initial(sys{1},x0,10);
+[~,~,x2] = initial(sys{2},x0,20);
+
+plot(x1(:,1),x1(:,2),'r',x2(:,1),x2(:,2),'k')
+xlabel('$y$','Interpreter','Latex')
+ylabel('$\dot{y}$','Interpreter','Latex')
+axis([-2 2 -2 2])
+grid on
+%%
+boolSwitchingLaw    = 0;
+boolSmoothKFunction = 0;
+
+sim('model_LTI_LTV_Stability')
+x_sys{1, 1} = x.signals.values;
+
+figure(1); hold on
+phasePlot(x_sys, {'k'}, 2)
\ No newline at end of file