Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems
Abstract
:1. Notations, Motivation and Introduction
1.1. Notations
1.2. Motivation and Introduction
2. Results
2.1. Auxiliary Lemma
2.2. Main Result
- (A1)
- in some left neighborhood of where is the largest pointwise eigenvalue of
- (A2)
- for all is where function is continuous on and satisfies
- (A3)
- (i)
- When A is negative definite, then Assumption A1 of Theorem 1 is automatically satisfied because is also negative definite ([34] Corollary 14.2.7), and
- (ii)
- in connection with Assumption A3, it is worth noting that Assumption A3 reduces to as ensuring the vanishing at infinity of all solutions of perturbed system cf. Example 2 below, where non-constant allows convergence to zero of all solutions of perturbed system for a wider class of perturbations, where even unbounded perturbations are admissible.
3. Simulation Experiments in MATLAB®
4. Conclusions
Funding
Acknowledgments
Conflicts of Interest
References
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f = @(t,x) [(-2)*x(1)+(exp(t))*x(2)+(atan(x(1)+x(2))/(t+1)); (-exp(t))*x(1)+(-2)*x(2)+(exp(-t)/(x(1)^2+1))] [t,xa] = ode45(f,[0 4],[50 -20]); hold~on pbaspect([2 1 1]) plot(t,xa(:,1), ’k’, ’LineWidth’,1.5) % 1 or 2 grid on xlabel(’t’) ylabel(’x_1’) % 1 or 2 print(’example_first_x_1’,’-deps’) % 1 or 2 |
syms b b=5; f = @(t,x) [(-t^2+sin(t))*x(1)+(b)*x(2)+t^(1.5); (0)*x(1)+(1-t^2+sin(t))*x(2)+3*cos(t*x(1)-x(2))] [t,xa] = ode45(f,[0 4],[10 -5]); hold~on pbaspect([2 1 1]) plot(t,xa(:,1), ’k’, ’LineWidth’,1.5) % 1 or 2 grid on xlabel(’t’) ylabel(’x_1’) % 1 or 2 print(’example_second_x_1’,’-deps’) % 1 or 2 |
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Vrabel, R. Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems. Symmetry 2019, 11, 569. https://doi.org/10.3390/sym11040569
Vrabel R. Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems. Symmetry. 2019; 11(4):569. https://doi.org/10.3390/sym11040569
Chicago/Turabian StyleVrabel, Robert. 2019. "Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems" Symmetry 11, no. 4: 569. https://doi.org/10.3390/sym11040569