Abstract
The large deviations of linear control system trajectories from the equilibrium position during the stabilization process, known as peak-effect, represent a serious obstacle to the design of cascade control systems and to guidance stabilization. Evaluation of such deviations is an important problem not only for control theory but also for other branches of mathematics where similar phenomenon appears as, for example, in numerical analysis, in the study of hydrodynamic stability and transition to turbulence, in the dynamics of vehicular platoons, and many others. The aim of this paper is to improve one of the main results concerning upper bounds for the peak-effect in linear control systems in the work by B.T. Polyak and myself in 2016. As the lower bound for overshoot proved by R.N. Izmailov in 1987 shows, the asymptotic estimate for the upper bound obtained here cannot be improved.
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REFERENCES
L. V. Balandin and M. M. Kogan, “Lyapunov function method for control law synthesis under one integral and several phase constraints,” Differ. Equations 45 (5), 670–679 (2009).
S. Boyd, L. El Ghaoui, E. Ferron, and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory (SIAM, Philadelphia, 1994).
A. Ya. Bulgakov, “An efficiently calculable parameter for the stability property of a system of linear differential equations with constant coefficients,” Sib. Math. J. 21 (3), 339–347 (1980).
Yu. M. Nechepurenko, “Bounds for the matrix exponential based on the Lyapunov equation and limits of the Hausdorff set,” Comput. Math. Math. Phys. 42 (2), 125–134 (2002).
A. A. Feldbaum, “On the distribution of roots of characteristic equations of control systems,” Avtom. Telemekh. 9 (2), 253–279 (1948).
M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming” (2014). http://www.stanford.edu/boyd/cvx/
D. Hinrichsen, E. Plischke, and F. Wurth, “State feedback stabilization with guaranteed transient bounds,” in Proceedings of the 15th International Symposium on Mathematical Theory of Networks & Systems (2002).
R. N. Izmailov, “The peak effect in stationary linear systems with scalar inputs and outputs,” Autom. Remote Control 48, 1018–1024 (1987).
A. M. Lyapunov, Problème Général de la Stabilité du Mouvement (Princeton Univ. Press, Princeton, 1947).
B. T. Polyak and G. V. Smirnov, “Large deviations for non-zero initial conditions in linear systems,” Automatica 74 (12), 297–307 (2016).
J. L. M. van Dorsselaer, J. F. B. M. Kraaijevanger, and M. N. Spijker, “Linear stability analysis in the numerical solution of initial value problems,” Acta Numer. 2 (1), 199–237 (1993).
H. Sussmann and P. Kokotovic, “The peaking phenomenon and the global stabilization of nonlinear systems,” IEEE Trans. Autom. Control 36 (4), 424–439 (1991).
J. F. Whidborne and J. McKernan, “On minimizing maximum transient energy growth,” IEEE Trans. Autom. Control 52 (9), 1762–1767 (2007).
J. F. Whidborne and N. Amar, “Computing the maximum transient energy growth,” BIT Numer. Math. 51 (2), 447–557 (2011).
Funding
This work was supported by the Portuguese Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/04650/2020.
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Dedicated to the blessed memory of Professor B.T. Polyak
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Smirnov, G.V. Asymptotic Upper Bound for the Peak-Effect in Linear Control Systems. Comput. Math. and Math. Phys. 64, 614–620 (2024). https://doi.org/10.1134/S0965542524700088
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DOI: https://doi.org/10.1134/S0965542524700088