Abstract
This tour of four puzzles is to highlight fundamental and surprising open problems in linear systems and control theory, ranging from system identification to robust control. The first puzzle deals with complications caused by fragility of poles and zeros in standard parameterizations of systems due to unavoidable parameter inaccuracies. The second and the third puzzle are concerned with difficulties in generalizing robust H ∞ control theory to realistic persistent signal setups. Surprisingly it seems very difficult to generalize the theory to such setups and simultaneously keep the terminology robust H ∞ control intact. These puzzles have implications also to model validation. The fourth puzzle, due to Georgiou and Smith, deals with difficulties in doubly-infinite time axis formulations of input-output stabilization theory.
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References
N.I. Akhiezer and I.M. Glazman. Theory of Linear Operators in Hilbert Space, New York: Frederick Ungar Publ., 1961.
K.J. Aström and B. Wittenmark. Computer-Controlled Systems, Englewood Cliffs, NJ: Prentice-Hall, 1990.
P.E. Caines. Linear Stochastic Systems, New York: Wiley, 1988.
M.A. Dahleh and I.J. Diaz-Bobillo. Control of Uncertain Systems: A Linear Programming Approach, Englewood Cliffs, NJ: Prentice-Hall, 1995.
M.A. Dahleh and J.B. Pearson. l1 optimal feedback controllers for MIMO discrete-time systems. IEEE Trans. Automat. Control, vol. AC-32, pp. 314–322, 1987.
J.C. Doyle, B.A. Francis and A.R. Tannenbaum. Feedback Control Theory, New York: MacMillan, 1992.
P.L. Duren. Theory of H p Spaces, New York: Academic Press, 1970.
T.T. Georgiou and M.C. Smith. Intrinsic difficulties in using the doubly-infinite time axis for input-output control theory. IEEE Trans. Automat. Control, vol. 40, pp. 516–518, 1995.
M. Green and D.J.N. Limebeer. Linear Robust Control, Englewood Cliffs, NJ: Prentice-Hall, 1995.
L.H. Keel and S.P. Bhattacharyya. Robust, Fragile, or Optimal? IEEE Trans. Automat. Control, vol. 42, pp. 1098–1105, 1997.
E. Landau. Darstellung und Begründung einiger neuerer Ergebnisse der Funktionentheorie. Berlin: Springer-Verlag, 1929.
G. Li. On the structure of digital controllers with finite word length consideration. IEEE Trans. Automat. Control, vol. 43, pp. 689–693, 1998.
L. Ljung. System Identification, Englewood Cliffs, NJ: Prentice-Hall, 1987.
P.M. Mäkilä. Comments on „Robust, Fragile, or Optimal?“. IEEE Trans. Automat. Control, vol. 43, pp. 1265–1267, 1998.
P.M. Mäkilä and J.R. Partington. Robust stabilization — BIBO stability, distance notions and robustness optimization. Automatica, vol. 23, pp. 681–693, 1993.
P.M. Mäkilä and J.R. Partington. On bounded-error identification of feedback systems. Int. J. Adapt. Control Signal Proc., vol. 9, pp. 47–61, 1995.
P.M. Mäkilä and J.R. Partington. Lethargy results in LTI system modelling. Automatica, vol. 34, pp. 1061–1070, 1998.
P.M. Mäkilä, J.R. Partington and T.K. Gustafsson. Worst-case control-relevant identification. Automatica, vol. 31, pp. 1799–1819, 1995.
P.M. Mäkilä, J.R. Partington and T. Norlander. Bounded power signal spaces for robust control and modelling. SIAM J. Control & Optim., vol. 00, pp. 00–00, 1999. In print.
M. Milanese and A. Vicino. Information based complexity and nonparametric worst-case system identification. J. Complexity, vol. 2, pp. 78–94, 1993.
P. Moroney. Issues in the Implementation of Digital Compensators, Cambridge, U.S.A.: The MIT Press, 1983.
J.R. Partington. Interpolation, Identification and Sampling, Oxford: Oxford Univ. Press, 1997.
K. Poolla, P. Khargonekar, A. Tikku, J. Krause and K. Nagpal. A time-domain approach to model validation. IEEE Trans. Automat. Control, vol AC-39, 951–959, 1994.
W. Rudin. Real and Complex Analysis, New York: McGraw-Hill, 1987.
T. Söderström and P. Stoica. System Identification, Englewood Cliffs, NJ: Prentice-Hall, 1989.
M. Vidyasagar. Control System Synthesis, MA: MIT Press, 1985.
M. Vidyasagar. Optimal rejection of persistent bounded disturbances. IEEE Trans. Automat. Control, vol. 31, pp. 527–534, 1986.
N. Wiener. The Fourier Integral, Cambridge Univ. Press, 1933.
G. Zames. On the metric complexity of causal linear systems: ε-entropy and ε-dimension for continuous-time. IEEE Trans. Automat. Control, vol. 24, pp. 222–230, 1979.
G. Zames. Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Trans. Automat. Control, vol. AC-26, pp. 301–320, 1981.
K. Zhou, J.C. Doyle and K. Glover. Robust and Optimal Control, Englewood Cliffs, NJ: Prentice-Hall, 1996.
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Mäkilä, P.M. (1999). Puzzles in systems and control. In: Garulli, A., Tesi, A. (eds) Robustness in identification and control. Lecture Notes in Control and Information Sciences, vol 245. Springer, London. https://doi.org/10.1007/BFb0109872
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DOI: https://doi.org/10.1007/BFb0109872
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