%% ================================================================ % Figure Generation — Paper 1 (Controllability) FINAL VERSION % ---------------------------------------------------------------- % "Equivalence of Hewer and Kalman Controllability for % Generalised Bilinear Matrix Periodic Systems" % Authors: Sri Venkata Durga Sudarsan Madhyannapu & % Sravanam Pradheep Kumar % % Generates 4 publication-quality figures at 500 DPI (vector PDF): % Fig1_Ctrl_EigModuli.pdf % Fig2_Ctrl_GramianSpectrum.pdf % Fig3_Ctrl_ScalingStudy.pdf % Fig4_Ctrl_StepSizeSensitivity.pdf % % FONT SIZES UPDATED per Springer requirement: % Axis labels : 16pt (was 13pt) % Tick labels : 15pt (was 13pt) % Annotations : 14pt (was 10-11pt) % Legend : 13pt (was 10pt) % Title : 16pt (was 14pt) % % Run AFTER Paper1_Controllability_Verification.m (uses workspace) % OR run standalone — all values hardcoded from MATLAB v3 output. % ================================================================ clear; clc; close all; %% ---------------------------------------------------------------- % VERIFIED VALUES FROM MATLAB v3 RUN % ---------------------------------------------------------------- % Monodromy eigenvalues (9 eigenvalues of M) eig_real = [0.7877, 0.7877, -1.5883, -0.6296, ... 0.9597, 0.9597, 1.0000, 1.0000, 1.0000]; eig_imag = [0.6160, -0.6160, 0.0000, 0.0000, ... 0.2812, -0.2812, 0.0000, 0.0000, 0.0000]; eig_moduli = abs(eig_real + 1i*eig_imag); [eig_moduli_sorted, ~] = sort(eig_moduli, 'descend'); % Gramian statistics (Table 1) — MATLAB v3 verified periods = [1, 2, 3 ]; lam_min_W = [0.5211, 0.5681, 0.5705]; lam_max_W = [4.084e5, 8.758e6, 1.516e8]; kappa_W = [7.837e5, 1.542e7, 2.657e8]; % Scaling study (Table 2) — MATLAB v3 verified n_vals = [2, 3 ]; kappa_mean = [2.128e10, 4.953e11]; kappa_std = [6.648e10, 1.357e12]; % Step-size sensitivity — MATLAB v3 verified tol_vals = [1e-3, 1e-4 ]; delta_vals = [9.6591e-2, 8.3716e-3 ]; %% ---------------------------------------------------------------- % GLOBAL STYLE — SPRINGER-COMPLIANT FONT SIZES % ---------------------------------------------------------------- FS_label = 16; % axis labels FS_tick = 15; % tick labels FS_title = 16; % figure titles FS_annot = 14; % value annotations FS_legend = 13; % legend text set(0, 'DefaultAxesFontSize', FS_tick); set(0, 'DefaultAxesFontName', 'Times New Roman'); set(0, 'DefaultTextFontName', 'Times New Roman'); set(0, 'DefaultTextFontSize', FS_annot); set(0, 'DefaultAxesLineWidth', 1.2); set(0, 'DefaultLineLineWidth', 2.2); clr_blue = [0.122, 0.471, 0.706]; clr_orange = [1.000, 0.498, 0.055]; clr_red = [0.839, 0.153, 0.157]; clr_gray = [0.50, 0.50, 0.50 ]; %% ================================================================ % FIGURE 1 — Eigenvalue Moduli of Monodromy M %% ================================================================ fig1 = figure('Units','centimeters','Position',[2,2,16,11]); b = bar(1:9, eig_moduli_sorted, 0.65); b.FaceColor = 'flat'; for k = 1:9 if eig_moduli_sorted(k) > 1 + 1e-6 b.CData(k,:) = clr_red; elseif abs(eig_moduli_sorted(k) - 1.0) <= 1e-3 b.CData(k,:) = clr_orange; else b.CData(k,:) = clr_blue; end end hold on; yline(1.0, '--k', 'LineWidth', 1.8); text(8.0, 1.06, '$|\lambda|=1$', ... 'Interpreter','latex','FontSize',FS_annot,'Color','k'); [~, idx_max] = max(eig_moduli_sorted); text(idx_max, eig_moduli_sorted(idx_max)+0.07, ... sprintf('$%.4f$', eig_moduli_sorted(idx_max)), ... 'Interpreter','latex','FontSize',FS_annot, ... 'HorizontalAlignment','center','Color',clr_red,'FontWeight','bold'); xlabel('Eigenvalue index $j$', 'Interpreter','latex', 'FontSize',FS_label); ylabel('$|\lambda_j(\mathcal{M})|$', 'Interpreter','latex', 'FontSize',FS_label); title({'Moduli of Monodromy Eigenvalues'; ... '$\mathcal{M} = \Phi_{\mathcal{A}}(2\pi,0),\quad n=3$'}, ... 'Interpreter','latex', 'FontSize',FS_title); hb1 = bar(NaN, NaN, 'FaceColor', clr_red); hb2 = bar(NaN, NaN, 'FaceColor', clr_orange); hb3 = bar(NaN, NaN, 'FaceColor', clr_blue); hl = plot(NaN, NaN, '--k', 'LineWidth',1.8); legend([hb1,hb2,hb3,hl], ... {'$|\lambda|>1$ (unstable)', ... '$|\lambda|=1$ (neutral)', ... '$|\lambda|<1$ (stable)', ... 'Unit circle'}, ... 'Interpreter','latex','Location','northeast', ... 'FontSize',FS_legend,'Box','on'); xlim([0.5,9.5]); ylim([0,1.9]); xticks(1:9); ax = gca; ax.FontSize = FS_tick; grid on; grid minor; box on; set(fig1,'PaperUnits','centimeters','PaperSize',[16,11], ... 'PaperPosition',[0,0,16,11]); print(fig1, 'Fig1_Ctrl_EigModuli', '-dpdf', '-r500'); fprintf('✅ Fig1_Ctrl_EigModuli.pdf saved.\n'); %% ================================================================ % FIGURE 2 — Gramian Spectrum: λ_min and λ_max across periods %% ================================================================ fig2 = figure('Units','centimeters','Position',[2,2,16,11]); yyaxis left h1 = semilogy(periods, lam_min_W, 'o-', ... 'Color',clr_blue,'MarkerFaceColor',clr_blue,'MarkerSize',10); ylabel('$\lambda_{\min}(\widetilde{W}_i)$','Interpreter','latex', ... 'FontSize',FS_label); ylim([0.40, 1.0]); ax = gca; ax.YAxis(1).Color = clr_blue; hold on; yyaxis right h2 = semilogy(periods, lam_max_W, 's--', ... 'Color',clr_red,'MarkerFaceColor',clr_red,'MarkerSize',10); ylabel('$\lambda_{\max}(\widetilde{W}_i)$','Interpreter','latex', ... 'FontSize',FS_label); ax.YAxis(2).Color = clr_red; % Annotations for i = 1:3 yyaxis left text(periods(i)+0.06, lam_min_W(i)*1.04, ... sprintf('%.4f', lam_min_W(i)), ... 'FontSize',FS_annot,'Color',clr_blue,'FontWeight','bold'); yyaxis right text(periods(i)+0.06, lam_max_W(i)*1.3, ... sprintf('$%.3e$', lam_max_W(i)), ... 'FontSize',FS_annot-1,'Color',clr_red, ... 'Interpreter','latex','FontWeight','bold'); end xlabel('Period index $i$','Interpreter','latex','FontSize',FS_label); title({'Controllability Gramian Spectrum vs.\ Period'; ... '$n=3$,\quad $T=2\pi$,\quad \texttt{ode45}'}, ... 'Interpreter','latex','FontSize',FS_title); legend([h1,h2], ... {'$\lambda_{\min}(\widetilde{W}_i)$ (stable $\approx 0.52$--$0.57$)', ... '$\lambda_{\max}(\widetilde{W}_i)$ (growing geometrically)'}, ... 'Interpreter','latex','Location','west', ... 'FontSize',FS_legend,'Box','on'); xticks([1,2,3]); xlim([0.7,3.5]); ax.FontSize = FS_tick; grid on; grid minor; box on; set(fig2,'PaperUnits','centimeters','PaperSize',[16,11], ... 'PaperPosition',[0,0,16,11]); print(fig2, 'Fig2_Ctrl_GramianSpectrum', '-dpdf', '-r500'); fprintf('✅ Fig2_Ctrl_GramianSpectrum.pdf saved.\n'); %% ================================================================ % FIGURE 3 — Scaling Study: Condition Number κ vs dimension n %% ================================================================ fig3 = figure('Units','centimeters','Position',[2,2,14,10]); kappa_mean = abs(kappa_mean); kappa_std = abs(kappa_std); lower = max(kappa_mean - kappa_std, 1e-12); upper = kappa_mean + kappa_std; h1 = errorbar(n_vals, kappa_mean, ... kappa_mean - lower, upper - kappa_mean, 'o-', ... 'Color',clr_blue,'MarkerFaceColor',clr_blue, ... 'MarkerSize',10,'LineWidth',2.2,'CapSize',12); hold on; p = polyfit(log10(n_vals), log10(kappa_mean), 1); n_f = linspace(1.8, 3.2, 100); h2 = plot(n_f, 10.^polyval(p,log10(n_f)), '--', ... 'Color',clr_gray,'LineWidth',1.8); set(gca,'YScale','log'); xlabel('State dimension $n$','Interpreter','latex','FontSize',FS_label); ylabel('$\kappa(\widetilde{W}_1)$ (mean $\pm$ std, 10 trials)', ... 'Interpreter','latex','FontSize',FS_label); title({'Gramian Condition Number Scaling'; ... '$\widetilde{W}_1$, random trials, \texttt{ode45}'}, ... 'Interpreter','latex','FontSize',FS_title); xticks([2,3]); xticklabels({'$n=2$\;($n^2=4$)','$n=3$\;($n^2=9$)'}); ax = gca; ax.XAxis.TickLabelInterpreter = 'latex'; ax.FontSize = FS_tick; xlim([1.7,3.4]); for i = 1:2 text(n_vals(i)+0.06, kappa_mean(i)*2.0, ... sprintf('$%.3e$',kappa_mean(i)), ... 'Interpreter','latex','FontSize',FS_annot, ... 'Color',clr_blue,'FontWeight','bold'); end text(3.05, kappa_mean(2)*0.12, ... {'$n=4$ excluded:';'$256{\times}256$ ODE';'$>5\,\mathrm{GB}$ memory'}, ... 'Interpreter','latex','FontSize',FS_annot-2,'Color',clr_red); legend([h1,h2], ... {'Mean $\pm$ std (10 trials)','Log-linear trend fit'}, ... 'Interpreter','latex','Location','northwest', ... 'FontSize',FS_legend,'Box','on'); grid on; grid minor; box on; set(fig3,'Toolbar','none','PaperUnits','centimeters', ... 'PaperSize',[14,10],'PaperPosition',[0,0,14,10]); print(fig3, 'Fig3_Ctrl_ScalingStudy', '-dpdf', '-r500'); fprintf('✅ Fig3_Ctrl_ScalingStudy.pdf saved.\n'); %% ================================================================ % FIGURE 4 — Step-Size Sensitivity (||ΔM||_F convergence) %% ================================================================ fig4 = figure('Units','centimeters','Position',[2,2,14,10]); h1 = loglog(tol_vals, delta_vals, 'o-', ... 'Color',clr_blue,'MarkerFaceColor',clr_blue, ... 'MarkerSize',10,'LineWidth',2.2); hold on; t_ref = logspace(-4.5, -2.5, 100); ref = delta_vals(1) * (t_ref / tol_vals(1)).^1; h2 = loglog(t_ref, ref, '--', 'Color',clr_gray,'LineWidth',1.8); % Annotations text(tol_vals(1)*1.08, delta_vals(1)*1.35, ... '$9.66\times10^{-2}$','Interpreter','latex', ... 'FontSize',FS_annot,'Color',clr_blue,'FontWeight','bold'); text(tol_vals(2)*1.08, delta_vals(2)*1.35, ... '$8.37\times10^{-3}$','Interpreter','latex', ... 'FontSize',FS_annot,'Color',clr_blue,'FontWeight','bold'); ratio = delta_vals(1)/delta_vals(2); text(2.5e-4, 3.5e-2, ... sprintf('Ratio $\\approx %.1f$ (1st-order)',ratio), ... 'Interpreter','latex','FontSize',FS_annot,'Color',clr_gray); xlabel('RelTol','Interpreter','latex','FontSize',FS_label); ylabel('$\|\Delta\mathcal{M}\|_F$','Interpreter','latex','FontSize',FS_label); title({'Step-Size Sensitivity of Monodromy Computation'; ... 'AbsTol $=10^{-10}$,\quad \texttt{ode45}'}, ... 'Interpreter','latex','FontSize',FS_title); legend([h1,h2], ... {'$\|\mathcal{M}_{\mathrm{coarse}}-\mathcal{M}_{\mathrm{fine}}\|_F$', ... 'First-order reference'}, ... 'Interpreter','latex','Location','northwest', ... 'FontSize',FS_legend,'Box','on'); set(gca,'XScale','log','YScale','log','FontSize',FS_tick); xlim([5e-5,5e-3]); ylim([1e-3,5e-1]); grid on; grid minor; box on; set(fig4,'Toolbar','none','PaperUnits','centimeters', ... 'PaperSize',[14,10],'PaperPosition',[0,0,14,10]); print(fig4, 'Fig4_Ctrl_StepSizeSensitivity', '-dpdf', '-r500'); fprintf('✅ Fig4_Ctrl_StepSizeSensitivity.pdf saved.\n'); fprintf('\n✅ All 4 figures saved. Vector PDF, 500 DPI, Springer font sizes.\n');