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Numerical error bound of optimal control for homogeneous linear systems

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this article we focus on the balanced truncation linear quadratic regulator (LQR) with constrained states and inputs. For closed-loop, we want to use the LQR to find an optimal control that minimizes the objective function which called "the quadratic cost function” with respect to the constraints on the states and the control input. In order to do that we have used formal asymptotes for the Pontryagin maximum principle (PMP) and we introduce an approach using the so called The Hamiltonian Function and the underlying algebraic Riccati equation. The theoretical results are validated numerically to show that the model order reduction based on open-loop balancing can also give good closed-loop performance.
Rocznik
Strony
323--337
Opis fizyczny
Bibliogr. 25 poz., rys., tab., wzory
Twórcy
  • Department of Mathematics, Faculty of Science, An-Najah National University, Nablus, Palestine
  • Department of Mathematics, Faculty of Science, An-Najah National University, Nablus, Palestine
Bibliografia
  • [1] A. Daraghmeh and N. Qatanani: Error bound for non-zero initial condition using the singular perturbation approximation method. Mathematics, 6 (2018), 1–16.
  • [2] O. Alvarez and M. Bardi: Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM Journal on Control and Optimization, 40 (2002), 1159–1188.
  • [3] A. C. Antoulas: Approximation of large-scale dynamical systems. Advances in design and control, Society for Industrial and Applied Mathematics, 2005.
  • [4] A. Bensoussan and G. Blankenship: Singular perturbations in stochastic control. In: Singular Perturbations and Asymptotic Analysis in Control Systems, Springer, 1987, 171–260.
  • [5] Y. Chahlaoui and P. Van Dooren: Benchmark examples for model reduction of linear time invariant dynamical systems. In: Dimension Reduction of Large-Scale Systems, P. Benner, D.C. Sorensen, and V. Mehrmann, eds., vol. 45 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 2005, 379–392.
  • [6] R. F. Curtain and K. Glover: Balanced realisations for infinite dimensional systems. In: Operator theory and system, Proc. Workshop Amesterdam (1986), 87–104.
  • [7] An Introduction to Infinite-Dimensional Systems. Springer-Verlag. New York, 1995.
  • [8] A. Daraghmeh: Model Order Reduction of Linear Control Systems: Comparison of Balanced Truncation and Singular Perturbation Approximation with Application to Optimal Control. PhD Thesis, Berlin Freie Universitat,Germany, 2016.
  • [9] A. Daraghmeh, C. Hartmann, and N. Qatanan: Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation. Applied Mathematics and Computation, 353 (2019), 295–307.
  • [10] L. C. Evans: The perturbed test function method for viscosity solutions of nonlinear pde. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 111 (1989), 359–375.
  • [11] S. A. Ghoreishi, M. A. Nekoui, and S. O. Basiri: Optimal design of 1qr weighting matrices based on intelligent optimization methods. International Journal of Intelligent Information Processing, 2 (2011).
  • [12] K. Glover: All optimal Hankel-norm approximations of linear multivariable systems and their l∞-error bounds. International Journal of Control, 39 (1984), 1115–1193.
  • [13] K. Glover, R. F. Curtain, and J.R. Partington: Realisation and approximation of linear Infinite dimensional systems with error bounds. SIAM Journal on Control and Optimization, 26 (1988), 863–898.
  • [14] G. E. Knowles: An Introduction to Applied Optimal Control. Mathematics in Science and Engineering 159, Academic Press, 1981.
  • [15] P. V. Kokotovic, R. E. O’malley, and P. Sannuti: Singular perturbations and order reduction in control theory: an overview. Automatica, 12 (1976), 123–132.
  • [16] P.-L. Lions and P. E. Souganidis: Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Communications on Pure and Applied Mathematics, 56 (2003), 1501–1524.
  • [17] Y. Liu and B. D. Anderson: Singular perturbation approximation of balanced systems. International Journal of Control, 50 (1989), 1379–1405.
  • [18] B. Moore: Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control, 26 (1981), 17–32.
  • [19] R. M. Murray: Optimization-based control. California Institute of Technology, CA, 2009.
  • [20] G. Muscato, G. Nunnari, and L. Fortuna: Singular perturbation approximation of bounded real balanced and stochastically balanced transfer matrices. International Journal of Control, 66 (1997), 253–270.
  • [21] R. O’Malley, Jr.: The singularly perturbed linear state regulator problem. SIAM Journal on Control, 10 (1972), 399–413.
  • [22] R.E. O’Malley Jr: On two methods of solution for a singularly perturbed linear state regulator problem. SIAM Review, 17 (1975), 16–37.
  • [23] A. Sasane: Hankel norm approximation for infinite-dimensional systems, vol. 277. Springer Science & Business Media, 2002.
  • [24] S. Skogestad and I. Postlethwaite: Multivariable feedback control: analysis and design, vol. 2. Wiley New York, 2007.
  • [25] K. Zhou, J. C. Doyle, and K. Glover: Robust and optimal control. Prentice Hall, 1998.
Uwagi
EN
1. The financial support of the Palestinian Ministry of Higher Education to undertake this work under grant number ANNU-MoHE-1819-Sco14 is highly acknowledged.
PL
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-3dc36e8d-7478-458d-b4a4-d3d05c3e4c22
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