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Original Articles

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

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Pages 1396-1405 | Received 05 Oct 2015, Accepted 10 Dec 2015, Published online: 22 Jan 2016

References

  • Abou-Kandil, H., Freiling, G., Jank, G., & Ionescu, V. (2003). Matrix Riccati Equation in Control and Systems Theory Basel, Boston, Berlin: Birkhäser Verlag.
  • Bekir, E., & Bucy, R.S. (1976). Periodic equilibria for matrix Riccati equations. Stochastics, 2, 1–104.
  • Bernstein, S.N. (1926). Leçons sur les Proprietes Extremales et la Meilleure Approximation des Fonctions Analytiques d'une Variable Reele [Lectures on extremal properties and best approximation of analytic functions of a real variable]. Paris: Gauthier-Villars.
  • Nicolao, G. (1989). A note on the maximal solution of the periodic Riccati equation. IEEE Transactions on Automatic Control, 34(12), 1316–1319.
  • Bittanti, S., Colaneri, P., & De Nicolao, G. (1991). The periodic Riccati equation. In S. Bittanti, J.A. Laub, & J-C. Willems (Eds.), The Riccati equation (pp. 127–162). Berlin, Heidelberg: Springer Verlag.
  • Boyd, S., El Chaoui, L., Feron, E., & Balakrishnan, V. (1994). Linear matrix inequalities in system and control theory Philadelphia, PA: SIAM.
  • Byrnes, C.I., Gusev, S.V., & Lindquist, A. (2001). From finite covariance windows to modeling filters: A convex optimization approach. SIAM Reviews, 43(4), 645–675.
  • Chen, Y.Z., Liu, J.Q., & Chen, S.B. (1989). Comparison and uniqueness theorems for periodic Riccati differential equations. International Journal of Control, 69(3), 467–473.
  • Coppel, W.A. (1971). Disconjugacy. Lecture Notes in mathematics, vol. 220. Berlin, Heidelberg, New York: Springer-Verlag.
  • Feng, Y., Varga, A., Anderson, B.D.O., & Lovera, M. (2011). A new iterative algorithm to solve periodic Riccati differential equations with sign indefinite quadratic terms. IEEE Transactions on Automatic Control, 56(4), 929–934.
  • Freidovich, L.B., Robertsson, A., Shiriaev, A.S., & Johansson, R. (2008). Periodic motions of the Pendubot via virtual holonomic constraints: Theory and experiments. Automatica, 44(3), 785–791.
  • Gahinet, P., Nemirovski, A., Laub, A.J., & Chilali, M. (1995). LMI control toolbox for use with Matlab. User guide. Natick, MA: MathWorks.
  • Gusev, S., Johansson, S., Kågström, B., Shiriaev, A., & Varga, A. (2010). A numerical evaluation of solvers for the periodic Riccati differential equation. BIT Numerical Mathematics, 50, 301–329.
  • Gusev, S.V. (2009). Kalman-Yakubovich-Popov lemma for matrix frequency domain inequality. System and Control Letters, 58, 469–473.
  • Gusev, S.V., Shiriaev, A.S., & Freidovich, L.B. (2007). LMI approach for solving periodic matrix Riccati equation. In G. Leonov & A. Fradkov (Eds.), 3rd IFAC Workshop on Periodic Control Systems. (Vol. 3, Part 1, pp. 254–256) St. Petersburg. Russia: IFAC.
  • Kalman, R.E. (1960). Contributions to the theory of optimal control. Boletín de la Sociedad Matemática Mexicana, 5, 102–119.
  • Kilgore, T. (1993). An elementary simulteneous approximation theorem. Proceedings of the American Mathematical Society, 118(2), 529–536.
  • Kleinman, D.L. (1968). On an iterative technique for Riccati equation computations. IEEE Transactions on Automatic Control, 13(1), 114–115.
  • La Hera, P.X., Freidovich, L.B., Shiriaev, A.S., & Mettin, U. (2009). New approach for swinging up the Furuta pendulum: Theory and experiments. Mechatronics, 19(8), 1240–1250.
  • Larin, V.B. (2002). Algorithm for solving the discrete periodic Riccati equation. Journal of Automation and Information Sciences, 34(1), 1–7. http://dx.doi.org/10.1615/JAutomatInfScien.v34.i1.50.
  • Laub, A.L. (1979). A Schur method for solving algebraic Riccati equations. IEEE Transactions on Automatic Control, 24, 913–921.
  • Löfberg, J (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. Proceedings of IEEE International Symposium on Computer Aided Control Systems Design. Taipei.
  • Parrilo, P.A. (1999). On the numerical solution of LMIs derived from the KYP Lemma. In C.G. Cassandras (Ed.), Proceedings of the 38th IEEE Conference on Decision and Control (pp. 2334–2338). Phoenix, AZ: IEEE.
  • Peng, H.-J., Wu, Z.-G., & Zhong, W.-X. (2011). Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations. Journal of Computational and Applied Mathematics, 235(12), 3571–3588.
  • Shayman, M.A. (1985). On the phase portrait of the matrix Riccati equation arising from the periodic control problem. SIAM Journal on Control and Optimization, 23(5), 717–751.
  • Shiriaev, A.S., Freidovich, L.B., & Gusev, S.V. (2010). Transverse linearization for controlled mechanical systems with several passive degrees of freedom. IEEE Transactions on Automatic Control, 55(4), 893–906.
  • Shiriaev, A.S., Freidovich, L.B., Robertsson, A., Johansson, R., & Sandberg, A. (2007). Virtual-holonomic-constraints-based design of stable oscillations of Furuta pendulum: Theory and experiments, IEEE Transactions on Robotics, 23(8), 827–832.
  • Shiriaev, A.S., Perram, J.W., & Canudas-de-Wit, C. (2005). Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach. IEEE Transactions on Automatic Control, 50(8), 1164–1176.
  • Tutuncu, R.H., Toh, K.C., & Todd, M.J. (2003). Solving semidefinite-quadratic-linear programs using SDPT3. Mathematical Programming, Series B, 95, 189–217.
  • Varga, A. (2008). On solving periodic Riccati equation. Numerical Linear Algebra with Applications, 15(9), 809–835.
  • Varga, A. (2013). Computational issues for linear periodic systems: Paradigms, algorithms, open problems. International Journal of Control, 86(7), 1227–1239.
  • Willems, J. (1971). Least squares stationary optimal control and the algebraic Riccati equation. IEEE Transactions on Automatic Control, 16, 621–634.
  • Yakubovich, V. (1986). Linear-quadratic optimization problem and frequency theorem for periodic systems. Siberian Mathematical Journal, 27(4), 181–200.

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