1. Y. Wang and E.D. Sontag. Orders of input/output differential equations and state-space dimensions. SIAM J. Control Optim., 33(4):1102-1126, 1995. [PDF] [doi:http://dx.doi.org/10.1137/S0363012993246828] Keyword(s): identifiability, observability, realization theory, real-analytic functions. Abstract:

   This paper deals with the orders of input/output equations satisfied by nonlinear systems. Such equations represent differential (or difference, in the discrete-time case) relations between high-order derivatives (or shifts, respectively) of input and output signals. It is shown that, under analyticity assumptions, there cannot exist equations of order less than the minimal dimension of any observable realization; this generalizes the known situation in the classical linear case. The results depend on new facts, themselves of considerable interest in control theory, regarding universal inputs for observability in the discrete case, and observation spaces in both the discrete and continuous cases. Included in the paper is also a new and simple self-contained proof of Sussmann's universal input theorem for continuous-time analytic systems.




2. Y. Wang and E.D. Sontag. On two definitions of observation spaces. Systems Control Lett., 13(4):279-289, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90116-3] Keyword(s): observability, identifiability, observables, observation space, realization theory. Abstract:

   This paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls.




3. E.D. Sontag. Bilinear realizability is equivalent to existence of a singular affine differential I/O equation. Systems Control Lett., 11(3):181-187, 1988. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(88)90057-6] Keyword(s): identification, identifiability, observability, observation space, real-analytic functions. Abstract:

   For continuous time analytic input/output maps, the existence of a singular differential equation relating derivatives of controls and outputs is shown to be equivalent to bilinear realizability. A similar result holds for the problem of immersion into bilinear systems. The proof is very analogous to that of the corresponding, and previously known, result for discrete time.

1. E.D. Sontag and Y. Wang. I/O equations for nonlinear systems and observation spaces. In Proc. IEEE Conf. Decision and Control, Brighton, UK, Dec. 1991, IEEE Publications, 1991, pages 720-725, 1991. [PDF] Keyword(s): identifiability, observability, realization theory, real-analytic functions. Abstract:

   This paper studies various types of input/output representations for nonlinear continuous time systems. The algebraic and analytic i/o equations studied in previous papers by the authors are generalized to integral and integro-differential equations, and an abstract notion is also considered. New results are given on generic observability, and these results are then applied to give conditions under which that the minimal order of an equation equals the minimal possible dimension of a realization, just as with linear systems but in contrast to the discrete time nonlinear theory.




2. Y. Wang and E.D. Sontag. A new result on the relation between differential-algebraic realizability and state space realizations. In Proc. Conf. Info. Sciences and Systems, Johns Hopkins University Press, 1989, pages 143-147, 1989. Keyword(s): observables, observation space, identifiability, observability, realization theory.

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