1. E.D. Sontag, Y. Wang, and A. Megretski. Input classes for identification of bilinear systems. IEEE Transactions Autom. Control, 54:195-207, 2009. Note: Also arXiv math.OC/0610633, 20 Oct 2006, and short version in ACC'07.[PDF] Keyword(s): realization theory, observability, identifiability, bilinear systems. Abstract:

   This paper asks what classes of input signals are sufficient in order to completely identify the input/output behavior of generic bilinear systems. The main results are that step inputs are not sufficient, nor are single pulses, but the family of all pulses (of a fixed amplitude but varying widths) do suffice for identification.




2. E.D. Sontag and Y. Wang. Uniformly Universal Inputs. In Alessandro Astolfi, editor, Analysis and Design of Nonlinear Control Systems, volume 224, pages 9-24. Springer-Verlag, London, 2007. [PDF] Keyword(s): observability, identification. Abstract:

   A result is presented showing the existence of inputs universal for observability, uniformly with respect to the class of all continuous-time analytic systems. This represents an ultimate generalization of a 1977 theorem, for bilinear systems, due to Alberto Isidori and Osvaldo Grasselli.




3. E.D. Sontag. Comments on integral variants of ISS. Systems Control Lett., 34(1-2):93-100, 1998. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(98)00003-6] Keyword(s): input-to-state stability. Abstract:

   This note discusses two integral variants of the input-to-state stability (ISS) property, which represent nonlinear generalizations of L2 stability, in much the same way that ISS generalizes L-infinity stability. Both variants are equivalent to ISS for linear systems. For general nonlinear systems, it is shown that one of the new properties is strictly weaker than ISS, while the other one is equivalent to it. For bilinear systems, a complete characterization is provided of the weaker property. An interesting fact about functions of type KL is proved as well.




4. E.D. Sontag. A Chow property for sampled bilinear systems. In C.I. Byrnes, C.F. Martin, and R. Saeks, editors, Analysis and Control of Nonlinear Systems, pages 205-211. North Holland, Amsterdam, 1988. [PDF] Keyword(s): discrete-time, bilinear systems. Abstract:

   This paper studies accessibility (weak controllability) of bilinear systems under constant sampling rates. It is shown that the property is preserved provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case, and which, for observability, results in the classical Nyquist theorem.




5. 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] 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.




6. E.D. Sontag. Controllability is harder to decide than accessibility. SIAM J. Control Optim., 26(5):1106-1118, 1988. [PDF] [doi:http://dx.doi.org/10.1137/0326061] Keyword(s): computational complexity, controllability, computational complexity. Abstract:

   The present article compares the difficulties of deciding controllability and accessibility. These are standard properties of control systems, but complete algebraic characterizations of controllability have proved elusive. We show in particular that for subsystems of bilinear systems, accessibility can be decided in polynomial time, but controllability is NP-hard.




7. E.D. Sontag. A remark on bilinear systems and moduli spaces of instantons. Systems Control Lett., 9(5):361-367, 1987. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(87)90064-8] Abstract:

   Explicit equations are given for the moduli space of framed instantons as a quasi-affine variety, based on the representation theory of noncommutative power series, or equivalently, the minimal realization theory of bilinear systems.




8. E.D. Sontag. An eigenvalue condition for sample weak controllability of bilinear systems. Systems Control Lett., 7(4):313-315, 1986. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(86)90045-9] Keyword(s): discrete-time. Abstract:

   Weak controllability of bilinear systems is preserved under sampling provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case.

1. E.D. Sontag, Y. Wang, and A. Megretski. Remarks on Input Classes for Identification of Bilinear Systems. In Proceedings American Control Conf., New York, July 2007, pages 4345-4350, 2007. Keyword(s): realization theory, observability, identifiability, bilinear systems.



2. E.D. Sontag. From linear to nonlinear: some complexity comparisons. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2916-2920, 1995. [PDF] Keyword(s): theory of computing and complexity, computational complexity, controllability, observability. Abstract:

   This paper deals with the computational complexity, and in some cases undecidability, of several problems in nonlinear control. The objective is to compare the theoretical difficulty of solving such problems to the corresponding problems for linear systems. In particular, the problem of null-controllability for systems with saturations (of a "neural network" type) is mentioned, as well as problems regarding piecewise linear (hybrid) systems. A comparison of accessibility, which can be checked fairly simply by Lie-algebraic methods, and controllability, which is at least NP-hard for bilinear systems, is carried out. Finally, some remarks are given on analog computation in this context.

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