Deterministic and stochastic linear periodic systems

  • Sergio Bittanti
Conference paper
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 86)


Riccati Equation Periodic System Monodromy Matrix Lyapunov Equation Canonical Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bailey, J.E. (1973): Periodic Operation of Chemical Reactors: A Review. Chem. Eng. Commun. 1, 111–124.Google Scholar
  2. Bekir, E. and R.S. Bucy (1976): Periodic Equilibria for Matrix Riccati Equations. Stochastics 2, 1–104.Google Scholar
  3. Belevitch, V. (1968): Classical Network Theory. Holden Day, San Francisco.Google Scholar
  4. Bernstein, D.S and E.G. Gilbert (1980): Optimal Periodic Control: The п Test Revisited. IEEE Trans. Automatic Control AC-25, 673–684.Google Scholar
  5. Bittanti, S. and P. Bolzern (1984,a): Can the Kalman Canonical Decomposition be performed for a Discrete-time Linear Periodic System? 1st Latin American Conference on Automatica, Campina Grande, Brazil, 449–453.Google Scholar
  6. Bittanti, S. and P. Bolzern (1984,b): Canonical Decomposition and Discrete-time Linear Systems. 23rd Conference of Decision and Control, Las Vegas, U.S.A., 1737, 1738.Google Scholar
  7. Bittanti, S. and P. Bolzern (1984,c): Four Equivalent Notions of Stabilizability of Periodic Linear Systems. 3rd American Control Conference, San Diego, U.S.A., 1321–1323.Google Scholar
  8. Bittanti, S. and P. Bolzern (1985,a): Reachability and Controllability of Discrete-time Linear Systems. IEEE Trans. Automatic Control 30, 399–491.Google Scholar
  9. Bittanti, S. and P. Bolzern (1985,b): Discrete-time Linear Periodic Systems: Grammian and Modal Criteria for Reachability and Controllability. International J. Control 41, 899–928.Google Scholar
  10. Bittanti, S. and P. Bolzern (1985,c): Stabilizability and Detectability of Linear Periodic Systems. Systems and Control Letters 6, 141–145. Plus Addendum, to appear in Systems and Control Letters (1986), 7, 73.Google Scholar
  11. Bittanti, S. and P. Bolzern (1986): On the Structure Theory of Discrete-time Linear Systems. International J. Systems Science, 17, 33–47.Google Scholar
  12. Bittanti, S., P. Bolzern and P. Colaneri (1984): Stability Analysis of Linear Periodic Systems via the Lyapunov Equation. 9th IFAC World Congress, Budapest, 8, 169–172.Google Scholar
  13. Bittanti, S., P. Bolzern and P. Colaneri (1985): The Extended Periodic Lyapunov Lemma. Automatica 5, 603–605.Google Scholar
  14. Bittanti, S., P. Bolzern, P. Colaneri and G. Guardabassi (1983): H and K-Controllability of Linear Periodic Systems. 22nd Conference on Decision and Control, S. Antonio, U.S.A., 1376–1379.Google Scholar
  15. Bittanti, S. and P. Colaneri (1986): Lyapunov and Riccati Equations: Periodic Inertia Theorems. IEEE Trans. Automatic Control (to appear).Google Scholar
  16. Bittanti, S., P. Colaneri and G. De Nicolao (1986): Discrete-time Periodic Systems: a note on the Reachability and Controllability interval length. Centro Teoria Sistemi, Politecnico di Milano, Int. Rep. 86-003.Google Scholar
  17. Bittanti, S., P. Colaneri and G. Guardabassi (1984): H-Controllability and Observability of Linear Periodic Systems. SIAM J. Control and Optimization 22, 889–893.Google Scholar
  18. Bittanti, S., G. Fronza and G. Guardabassi (1973): Periodic Control: A Frequency Domain Approach. IEEE Trans. Automatic Control 18, 33–38.Google Scholar
  19. Bittanti, S., G. Guardabassi, C. Maffezzoni and L. Silverman (1978): Periodic Systems: Controllability and the Matrix Riccati Equation. SIAM J. Control and Optimization 16, 37–40.Google Scholar
  20. Bittanti, S. and D.B. Hernandez (1986): The Simple Pendulum as an Illustrative Example of the Periodic Control Problem. Centro Teoria dei Sistemi, Politecnico di Milano, Int. Rep. 86-010.Google Scholar
  21. Bolzern, P. (1986): Criteria for Reachability, Controllability and Stabilizability of Discrete-time Linear Periodic Systems. V Polish-English Seminar on Real-Time Process Control, Warsaw, Poland.Google Scholar
  22. Bolzern, P. and P. Colaneri (1986): Existence and Uniqueness Conditions for the Periodic Solutions of the Discrete-time Periodic Lyapunov Equation. Centro Teoria dei Sistemi, Politecnico di Milano, Int. Rep. 86-011.Google Scholar
  23. Brockett, R.W. (1970): Finite Dimensional Linear Systems. J. Wiley and Sons.Google Scholar
  24. Brunovsky, P. (1969): Controllability and Linear Closed loop Controls in Linear Periodic Systems. J. Differential Equations 6, 296–313.Google Scholar
  25. Chen, C.T. (1970): Introduction to Linear System Theory. Holt, Rinehart and Winston.Google Scholar
  26. Colonius, F. (1985,a): Optimality for Periodic Control of Functional Differential Systems. J. Mathematical Analysis and Applications (to appear).Google Scholar
  27. Colonius, F. (1985,b): The High Frequency Pi-Criterion for Retarded Systems. IEEE Trans. Automatic Control 11, 1045–1048.Google Scholar
  28. DaPrato, G. (1984): Periodic Solutions of Infinite Dimensional Riccati Equations. Rendiconti Accademia Nazionale dei Lincei, (to appear).Google Scholar
  29. Dorato, P. and A.H. Levis (1971): Optimal Linear Regulators: the Discrete-time Case. IEEE Trans. Automatic Control 6, 613–620.Google Scholar
  30. Dorato, P. and H.K. Knudsen (1979): Periodic Optimization with Applications to Solar Energy Control. Automatica 15, 673–679.Google Scholar
  31. Gardner, W.A. and D.E. Franks (1975): Characterization of Cyclo-stationary Random Processes. IEEE Trans. Information Theory 21, 1–24.Google Scholar
  32. Gilbert, E.G. (1977): Optimal Periodic Control: A General Theory of Necessary Conditions. SIAM J. Control and Optimization 15, 717–746.Google Scholar
  33. Gilbert, E.G. and D.T. Lyons (1981): The Improvement of Aircraft Specific Range by Periodic Control. AIAA Guidance and Control Conference, Albuquerque.Google Scholar
  34. Graham, A. (1981): Kronecker Products and Matric Calculus with Applications. Ellis Horwood Limited, Chichester.Google Scholar
  35. Grasselli, O.M. (1984): A Canonical Decomposition of Linear Periodic Discrete-time Systems. International J. Control 40, 201–214.Google Scholar
  36. Guardabassi, G. (1971): Optimal Steady State Versus Periodic Control. Ricerche di Automatica 2, 240–252.Google Scholar
  37. Guardabassi, G. (1976): The Optimal Periodic Control Problem. Journal A 17, 75–83.Google Scholar
  38. Halanay, A. (1966): Differential Equations. Academic Press, New York.Google Scholar
  39. Hautus, M.L.J. (1969): Controllability and Observability Conditions of Linear Autonomous Systems. Indag. Math. 72 443–448.Google Scholar
  40. Hernandez, V. and L. Jodar (1985): Boundary Problems and Periodic Riccati Equations. IEEE Trans. Automatic Control 11, 1131–1133.Google Scholar
  41. Hewer, G.A. (1975): Periodicity, Detectability and the Matrix Riccati Equation. SIAM J. Control 13, 1235–1251.Google Scholar
  42. Horn, F.J.M. and R.C. Lin (1967): Periodic Processes: A Variational Approach. Ind. Eng. Chem. Process Des. Dev. 6, I, 21–30.Google Scholar
  43. Horn, F.J.M. and J.E. Bailey (1968): An Application of the Theorem of Relaxed Control to the Problem of Increasing Catalyst Selectivity. J. Optimization Theory and Applications 2, 441–449.Google Scholar
  44. Houlihan, S.C., E.M. Cliff and H.J. Kelley (1982): Study of Chattering Cruise, Journal Aircraft 19, 119–124.Google Scholar
  45. Johnson, C.D. (1966): Invariant Hyperplanes for Linear Dynamical Systems. IEEE Trans. Automatic Control 11, 113–116.Google Scholar
  46. Kabamba, P.T. (1985): Monodromy Eigenvalue Assignment in Linear Periodic Systems. 24th Conference on Decision and Control, Ft. Lauderdale, U.S.A., 177, 178.Google Scholar
  47. Kalman, R.E. (1969): Theory of Regulators for Linear Plants. In: Kalman R.E., P.L. Falb and M.A. Arbib: Topics in Mathematical System Theory. McGraw-Hill Co., New York.Google Scholar
  48. Kano, H. and T. Nishimura (1979): Periodic Solutions of Matrix Riccati Equations with Detectability and Stabilizability. International J. Control 29, 471–487.Google Scholar
  49. Kern, G. (1980): Linear Closed-loop Control in Linear Periodic Systems with Application to Spin-stabilized Bodies. International J. Control 31, 905–916.Google Scholar
  50. Khandelwal, D.N., J. Sharma and L.M. Ray (1979): Optimal Periodic Maintenance of a Machine. IEEE Trans. Automatic Control 24, 513.Google Scholar
  51. Khargonekar, P.P., K. Poolla and A. Tannenbaum (1985): Robust Control of Linear Time-invariant Plants Using Periodic Compensation. IEEE Trans. Automatic Control 11, 1088–1098.Google Scholar
  52. Kono, M. (1980): Eigenvalue Assignment in Linear Periodic Discrete-time Systems. International J. Control 1, 149–158.Google Scholar
  53. Maffezzoni, C. (1974): Hamilton-Jacobi Theory for Periodic Control Problems. J. Optimization Theory and Applications 14, 21–29.Google Scholar
  54. Markus, L. (1973): Optimal Control of Limit Cycles or what Control Theory can do to Cure a Heart Attack or to Cause One. Symposium on Ordinary Differential Equations, Minneapolis, Minnesota (1972). W.A. Harris, Y. Sibuya, eds., Springer-Verlag, Berlin.Google Scholar
  55. Matsubara, M., N. Nishimura, N. Watanabe and K. Onogi (1981): Periodic Control Theory and Applications. Research Reports of Automatic Control Laboratory Vol. 28, Faculty of Engineering, Nagoya University.Google Scholar
  56. Matsubara, M. and K. Onogi (1978): Stabilized Suboptimal Periodic Control of a Chemical Reactor. IEEE Trans. Automatic Control 23, 1005–1008.Google Scholar
  57. Meyer, R.A. and C.S. Burrus (1976): Design and Implementation of Multirate Digital Filters. IEEE Trans. Acoustics, Speech and Signal Processing 1, 53–58.Google Scholar
  58. Nistri, P. (1983): Periodic Control Problems for a Class of Nonlinear Periodic Differential Systems. Nonlinear Analysis, Theory, Methods and Applications 7, 79–90.Google Scholar
  59. Noldus, E. (1975): A Survey of Optimal Periodic Control of Continuous Systems. Journal A 16, 11–16.Google Scholar
  60. Onogi, K. and M. Matsubara (1980): Structure Analysis of Periodically Controlled Chemical Processes. Chem. Eng. Sci. 34, 1009–1019.Google Scholar
  61. Popov, V.M. (1973): Hyperstability of Control Systems. Springer, Berlin.Google Scholar
  62. Schädlich, K., U. Hoffmann and H. Hofmann (1983): Periodical Operation of Chemical Processes and Evaluation of Conversion Improvements. Chemical Engineering Science 38, 1375–1384.Google Scholar
  63. Shayman, M.A. (1984): Inertia Theorems for the Periodic Lyapunov Equation and Periodic Riccati Equation. Systems and Control Letters 4, 27–32.Google Scholar
  64. Shayman, M.A. (1985): On the Phase Portrait of the Matrix Riccati Equation Arising from the Periodic Control Problem. SIAM. J. Control and Optimization 23, 717–751.Google Scholar
  65. Sincic, D. and J.E. Bailey (1978): Optimal Periodic Control of Variable Time-delay Systems. International J. Control 27, 547–555.Google Scholar
  66. Speyer, J.L. (1973): On the Fuel Optimality of Cruise, J. Aircraft 10, 763–764.Google Scholar
  67. Speyer, J.L. (1976): Non-optimality of Steady-state Cruise for Aircraft. AIAA Journal 14, 1604–1610.Google Scholar
  68. Speyer, J.L. and R.T. Evans (1984): A Second Variational Theory of Optimal Periodic Processes. IEEE Trans. Automatic Control 29, 138–148.Google Scholar
  69. Valko, P. and G.A. Almasy (1982): Periodic Optimization of Hammerstein-type Systems. Automatica 18, 245–148.Google Scholar
  70. Watanabe, N., Y. Nishimura and M. Matsubara (1976): Singular Control Test for Optimal Periodic Control Problems. IEEE Trans. Automatic Control 21, 609–610.Google Scholar
  71. Watanabe, N., K. Onogi and M. Matsubara (1981): Periodic Control of Continuous Stirred Tank Reactors — I, Chem. Eng. Sci. 36, 809–818, II ibid. 37, 745–752.Google Scholar
  72. Watanabe, N., H. Kurimoto and M. Matsubara (1984): Periodic Control of Continuous Stirred Tank Reactors — III, Case of multistage reactors. Chem. Eng. Sci. 39, 31–36.Google Scholar
  73. Wimmer, H.K. and A.D. Ziebur (1975): Remarks on Inertia Theorems for Matrices. Czechoslovak Mathematical Journal 25, 556–561.Google Scholar
  74. Wong, E. and B. Hajek (1985): Stochastic Processes in Engineering. Springer-Verlag, Berlin.Google Scholar
  75. Wonham, W.M. (1968): On a Matrix Riccati Equation for Stochastic Control. SIAM Journal Control 6, 681–698.Google Scholar
  76. Yakubovich, V.A. and V.M. Starzhinskii (1975): Linear Differential Equations with Periodic Coefficients. J. Wiley, New York.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Sergio Bittanti

There are no affiliations available

Personalised recommendations