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Authors
Mark A. Petersen; Arjan J. van der Schaft

Digital Object Identifier (DOI)
10.3182/20020721-6-ES-1901.00293

Page Numbers:
291-291

Index Terms
nonlinear inner systems,spectral factorization,process,geometric,and H-infinity control

Abstract
This paper extends some results involving linear inner systems to the nonlinear case. In this regard, the arithmetic of nonlinear inner systems is developed further and some new connections with nonlinear spectral and all-pass factorization and control theory are discussed. In particular, explicit formulas for the (state space) realizations of nonlinear inner systems in terms of the components of extremal spectral factors is provided. Relationships between inner systems and process control, geometric control and H∞-control are also discussed.

References

[1] Ball, J. A. and J. W. Helton (1992). Inner-outer
    factorization of nonlinear operators. Journal of
    Functional Analysis, 104, 363-413.
[2] Ball, J. A. and M. A. Petersen (2002). Nonlinear
    Minimal Square Spectral Factorization. International
     Journal of Control, to appear.
[3] Ball, J. A., M. A. Petersen and A. J. van der Schaft
    (2001). Inner-Outer Factorization and Chemical
    Process Control for Noninvertible Nonlinear Systems,
     IEEE Transactions on Automatic Control,
    submitted.
[4] Ball, J. A. and A. J. van der Schaft (1996), J-
    inner-outer factorization, J-spectral factorization
    and robust control for nonlinear systems. IEEE
    Transactions on Automatic Control, AC-41, 379-
    392.
[5] Crouch, P. E. and A. J. van der Schaft (1987).
    Variational and Hamiltonian Control Systems.
    LNCIS, 101, Springer, Berlin.
[6] Ferrante, A., G. Michaletzky and M. Pavon
    (1993). Parametrization of all minimal square
    spectral factors. Systems and Control Letters, 21,
    249-254.
[7] Finesso, L. and G. Picci (1982). A characterization
     of minimal spectral factors. IEEE Transactions
     on Automatic Control, AC-27, 122-127.
[8] Fuhrmann, P. A. (1981). Duality on Polynomial
    Models with Some Applications to Geometric
    Control Theory. IEEE Transactions on Automatic
    Control, AC-26, 284-295.
[9] Fuhrmann, P. A. (1995). On the Characterization
    and Parametrization of minimal spectral factors.
    Journal of Mathematical Systems, Estimation and
    Control, 5, 383-444.
[10] Fuhrmann, P. A. and A. Gombani (1998). On a
     Hardy space approach to the analysis of spectral
     factors. International Journal of Control, 71, 277-
     357.
[11] Fuhrmann, P. A. and A. Gombani (2000). On the
     Lyapunov equation, coinvariant subspaces and
     some problems related to spectral factorizations.
     International Journal of Control, 73, 1129-1159.
[12] Helton, J. W. and M. R. James (1999). Extending
      H∞-control to Nonlinear Systems: Control of
     Nonlinear Systems to Achieve Performance Objectives
     , SIAM Frontiers in Applied Mathematics.
[13] Hill, D. J. and P. J. Moylan, Dissipative Dynamical
     Systems: basic input and state properties, Journal
     of the Franklin Institute, 309, 322-357.
[14] Isidori, A. (2001). The Differential Geometric Approach
      to Detection of Faults in Nonlinear Systems,
      Plenary session at NOLCOS 2001.
[15] Isidori, A. and A. Astolfi. (1992a). Disturbance attenuation
      and H∞-control via measurement feed-back
      in nonlinear systems. IEEE Transactions on
     Automatic Control, AC-37, 1283-1293.
[16] Isidori, A. and A. Astolfi. (1992b). Nonlinear
     H∞-control via measurement feedback. Journal of
     Mathematical Systems, Estimation and Control,
     2, 31-35.
[17] James, M. R. (1993). A partial differential inequality
      for dissipative nonlinear systems, Systems and
     Control Letters, 21, 315-320.
[18] James, M. R. (2001). L∞-Bounded Robustness:
     State Feedback Analysis and Synthesis. NOLCOS
     2001, 784-788.
[19] Petersen, M. A. (2001). On Nonlinear (j, J)-Inner
     Systems. Proceedings of the IEEE's 3rd International
      Conference on Control Theory and Applications
     , 231-235.
[20] Petersen, M. A. and A. C. M. Ran (2001a). Minimal
      Square Spectral Factors via Triples. SIAM
     Journal on Matrix Analysis and Applications, 22,
     1222-1244.
[21] Petersen, M. A. and A. C. M. Ran (2001b). Minimal
     Nonsquare Spectral Factors. Special Issue on Linear
      Systems and Control Theory of Linear Algebra
     and its Applications, to appear.
[22] Petersen, M. A. and A. C. M. Ran (2001c). Nonsquare
      Spectral Factors via Factorizations of Unitary
      Matrices. Special Issue on Linear Systems
     and Control Theory of Linear Algebra and its
     Applications, to appear.
[23] Petersen, M. A. and A. C. M. Ran (2001d). Minimal
     Nonsquare J-Spectral Factorization, Generalized
     Bezoutians and Common Zeros of Rational Matrix
      Functions. Integral Equations and Operator
     Theory, submitted.
[24] Petersen, M. A. and A. J. van der Schaft (2001).
     On a Connection between Nonlinear Nonsquare
     Spectral Factors and Hamilton-Jacobi Equations.
     NOLCOS 2001, 1578-1583.
[25] Petersen, M. A. and A. J. van der Schaft (2002).
     Nonlinear Nonsquare Spectral Factorization, IEEE
     Transactions on Automatic Control, submitted.
[26] Schumacher, J. M. (1979). (C, A)-invariant sub-spaces:
      Some facts and uses, Wiskunde Seminarium,
      Vrije Universiteit, Amsterdam, Report 110.
[27] van der Schaft, A. J. (1992). L2-gain analysis of
     nonlinear systems and nonlinear state feedback
     H∞-control. IEEE Transactions on Automatic
     Control, AC-37, 770-784.
[28] van der Schaft, A. J. (1996). L2-gain and passivity
      techniques in nonlinear control, LNCIS, 2,
     Springer-Verlag, Berlin.
[29] Willems, J. C. (1972). Dissipative Dynamical Systems,
      Part I: General Theory, Archive for Rational
     Mechanics and Analysis, 45, 321-351.
[30] Willems, J. C. and C. Commault. (1991). Disturbance
      decoupling by measurement feedback with
     stability or pole placement. SIAM Journal on
     Control Optimization 19, 490-504.
[31] Wonham, W. M. (1974). Linear Multivariable Control
     , Spring.