Publications about 'observability'
Books and proceedings
  1. E.D. Sontag. Polynomial Response Maps, volume 13 of Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 1979. [PDF] Keyword(s): realization theory, discrete-time, real algebraic geometry.
    (This is a monograph based upon Eduardo Sontag's Ph.D. thesis. The contents are basically the same as the thesis, except for a very few revisions and extensions.) This work deals the realization theory of discrete-time systems (with inputs and outputs, in the sense of control theory) defined by polynomial update equations. It is based upon the premise that the natural tools for the study of the structural-algebraic properties (in particular, realization theory) of polynomial input/output maps are provided by algebraic geometry and commutative algebra, perhaps as much as linear algebra provides the natural tools for studying linear systems. Basic ideas from algebraic geometry are used throughout in system-theoretic applications (Hilbert's basis theorem to finite-time observability, dimension theory to minimal realizations, Zariski's Main Theorem to uniqueness of canonical realizations, etc). In order to keep the level elementary (in particular, not utilizing sheaf-theoretic concepts), certain ideas like nonaffine varieties are used only implicitly (eg., quasi-affine as open sets in affine varieties) or in technical parts of a few proofs, and the terminology is similarly simplified (e.g., "polynomial map" instead of "scheme morphism restricted to k-points", or "k-space" instead of "k-points of an affine k-scheme").

Articles in journal or book chapters
  1. E.D. Sontag. Dynamic compensation, parameter identifiability, and equivariances. PLoS Computational Biology, 13:e1005447, 2017. Note: Preprint was in bioRxiv, 2016.[WWW] [PDF] Keyword(s): fcd, fold-change detection, scale invariance, dynamic compensation, identifiability, observability.
    A recent paper by Karin et al. introduced a mathematical notion called dynamical compensation (DC) of biological circuits. DC was shown to play an important role in glucose homeostasis as well as other key physiological regulatory mechanisms. Karin et al.\ went on to provide a sufficient condition to test whether a given system has the DC property. Here, we show how DC is a reformulation of a well-known concept in systems biology, statistics, and control theory -- that of parameter structural non-identifiability. Viewing DC as a parameter identification problem enables one to take advantage of powerful theoretical and computational tools to test a system for DC. We obtain as a special case the sufficient criterion discussed by Karin et al. We also draw connections to system equivalence and to the fold-change detection property.

  2. 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.
    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.

  3. 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.
    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.

  4. J.P. Hespanha, D. Liberzon, D. Angeli, and E.D. Sontag. Nonlinear norm-observability notions and stability of switched systems. IEEE Trans. Automat. Control, 50(2):154-168, 2005. [PDF] Keyword(s): observability, input to state stability, observability, invariance principle.
    This paper proposes several definitions of observability for nonlinear systems and explores relationships among them. These observability properties involve the existence of a bound on the norm of the state in terms of the norms of the output and the input on some time interval. A Lyapunov-like sufficient condition for observability is also obtained. As an application, we prove several variants of LaSalle's stability theorem for switched nonlinear systems. These results are demonstrated to be useful for control design in the presence of switching as well as for developing stability results of Popov type for switched feedback systems.

  5. M. Chaves and E.D. Sontag. State-Estimators for chemical reaction networks of Feinberg-Horn-Jackson zero deficiency type. European J. Control, 8:343-359, 2002. [PDF] Keyword(s): observability, zero-deficiency networks, systems biology, biochemical networks, observers, nonlinear stability, dynamical systems.
    This paper provides a necessary and sufficient condition for detectability, and an explicit construction of observers when this condition is satisfied, for chemical reaction networks of the Feinberg-Horn-Jackson zero deficiency type.

  6. E.D. Sontag. For differential equations with r parameters, 2r+1 experiments are enough for identification. J. Nonlinear Sci., 12(6):553-583, 2002. [PDF] Keyword(s): identifiability, observability, systems biology, biochemical networks, parameter identification.
    Given a set of differential equations whose description involves unknown parameters, such as reaction constants in chemical kinetics, and supposing that one may at any time measure the values of some of the variables and possibly apply external inputs to help excite the system, how many experiments are sufficient in order to obtain all the information that is potentially available about the parameters? This paper shows that the best possible answer (assuming exact measurements) is 2r+1 experiments, where r is the number of parameters.

  7. A. C. Antoulas, E. D. Sontag, and Y. Yamamoto. Controllability and Observability, pages 264-281. John Wiley & Sons, Inc., 2001. [WWW] [PDF] [doi:10.1002/047134608X.W1006] Keyword(s): reachability, controllability, observability, Lie algebra accessibility.

  8. D. Angeli and E.D. Sontag. Forward completeness, unboundedness observability, and their Lyapunov characterizations. Systems Control Lett., 38(4-5):209-217, 1999. [PDF] Keyword(s): observability, input to state stability, dynamical systems.
    A finite-dimensional continuous-time system is forward complete if solutions exist globally, for positive time. This paper shows that forward completeness can be characterized in a necessary and sufficient manner by means of smooth scalar growth inequalities. Moreover, a version of this fact is also proved for systems with inputs, and a generalization is also provided for systems with outputs and a notion (unboundedness observability) of relative completeness. We apply these results to obtain a bound on reachable states in terms of energy-like estimates of inputs.

  9. E.D. Sontag. Recurrent neural networks: Some systems-theoretic aspects. In M. Karny, K. Warwick, and V. Kurkova, editors, Dealing with Complexity: a Neural Network Approach, pages 1-12. Springer-Verlag, London, 1997. [PDF] Keyword(s): neural networks, recurrent neural networks.
    This paper provides an exposition of some recent results regarding system-theoretic aspects of continuous-time recurrent (dynamic) neural networks with sigmoidal activation functions. The class of systems is introduced and discussed, and a result is cited regarding their universal approximation properties. Known characterizations of controllability, observability, and parameter identifiability are reviewed, as well as a result on minimality. Facts regarding the computational power of recurrent nets are also mentioned.

  10. 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:] Keyword(s): identifiability, observability, realization theory.
    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.

  11. F. Albertini and E.D. Sontag. State observability in recurrent neural networks. Systems Control Lett., 22(4):235-244, 1994. [PDF] [doi:] Keyword(s): neural networks, recurrent neural networks, observability, identifiability.
    This paper concerns recurrent networks x'=s(Ax+Bu), y=Cx, where s is a sigmoid, in both discrete time and continuous time. Our main result is that observability can be characterized, if one assumes certain conditions on the nonlinearity and on the system, in a manner very analogous to that of the linear case. Recall that for the latter, observability is equivalent to the requirement that there not be any nontrivial A-invariant subspace included in the kernel of C. We show that the result generalizes in a natural manner, except that one now needs to restrict attention to certain special "coordinate" subspaces.

  12. R. Koplon, E.D. Sontag, and M. L. J. Hautus. Observability of linear systems with saturated outputs. Linear Algebra Appl., 205/206:909-936, 1994. [PDF] Keyword(s): observability, saturation.
    In this paper, we present necessary and sufficient conditions for observability of the class of output-saturated systems. These are linear systems whose output passes through a saturation function before it can be measured.

  13. F. Albertini and E.D. Sontag. For neural networks, function determines form. Neural Networks, 6(7):975-990, 1993. [PDF] Keyword(s): neural networks, identifiability, recurrent neural networks, realization theory, observability, neural networks.
    This paper shows that the weights of continuous-time feedback neural networks x'=s(Ax+Bu), y=Cx (where s is a sigmoid) are uniquely identifiable from input/output measurements. Under very weak genericity assumptions, the following is true: Assume given two nets, whose neurons all have the same nonlinear activation function s; if the two nets have equal behaviors as "black boxes" then necessarily they must have the same number of neurons and -except at most for sign reversals at each node- the same weights. Moreover, even if the activations are not a priori known to coincide, they are shown to be also essentially determined from the external measurements.

  14. R. Koplon and E.D. Sontag. Linear systems with sign-observations. SIAM J. Control Optim., 31(5):1245-1266, 1993. [PDF] [doi:] Keyword(s): observability.
    This paper deals with systems that are obtained from linear time-invariant continuous- or discrete-time devices followed by a function that just provides the sign of each output. Such systems appear naturally in the study of quantized observations as well as in signal processing and neural network theory. Results are given on observability, minimal realizations, and other system-theoretic concepts. Certain major differences exist with the linear case, and other results generalize in a surprisingly straightforward manner.

  15. Y. Wang and E.D. Sontag. Algebraic differential equations and rational control systems. SIAM J. Control Optim., 30(5):1126-1149, 1992. [PDF] Keyword(s): identifiability, observability, realization theory, input/output system representations.
    It is shown that realizability of an input/output operators by a finite-dimensional continuous-time rational control system is equivalent to the existence of a high-order algebraic differential equation satisfied by the corresponding input/output pairs ("behavior"). This generalizes, to nonlinear systems, the classical equivalence between autoregressive representations and finite dimensional linear realizability.

  16. Y. Wang and E.D. Sontag. Generating series and nonlinear systems: analytic aspects, local realizability, and i/o representations. Forum Math., 4(3):299-322, 1992. [PDF] Keyword(s): identifiability, observability, realization theory, input/output system representations.
    This paper studies fundamental analytic properties of generating series for nonlinear control systems, and of the operators they define. It then applies the results obtained to the extension of facts, which relate realizability and algebraic input/output equations, to local realizability and analytic equations.

  17. E.D. Sontag and Y. Wang. Input/output equations and realizability. In Realization and modelling in system theory (Amsterdam, 1989), volume 3 of Progr. Systems Control Theory, pages 125-132. Birkhäuser Boston, Boston, MA, 1990. [PDF] Keyword(s): identifiability, observability, realization theory.

  18. Y. Wang and E.D. Sontag. On two definitions of observation spaces. Systems Control Lett., 13(4):279-289, 1989. [PDF] [doi:] Keyword(s): observability, identifiability, observability, realization theory.
    This paper establishes the equality of the observation spaces defined by means of piecewise constant controls with those defined in terms of differentiable controls.

  19. 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.
    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.

  20. 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:] Keyword(s): identification, identifiability, observability, observation space.
    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.

  21. E.D. Sontag. Reachability, observability, and realization of a class of discrete-time nonlinear systems. In Encycl. of Systems and Control, pages 3288-3293. Pergamon Press, 1987. Keyword(s): observability.

  22. E.D. Sontag. A concept of local observability. Systems Control Lett., 5(1):41-47, 1984. [PDF] Keyword(s): observability.
    A notion of local observability, which is natural in the context of nonlinear input/output regulation. is introduced. A simple characterization is provided, a comparison is made with other local nonlinear observability definitions. and its behavior under constant-rate sampling is analyzed.

  23. M. L. J. Hautus and E.D. Sontag. An approach to detectability and observers. In Algebraic and geometric methods in linear systems theory (AMS-NASA-NATO Summer Sem., Harvard Univ., Cambridge, Mass., 1979), volume 18 of Lectures in Appl. Math., pages 99-135. Amer. Math. Soc., Providence, R.I., 1980. [PDF] Keyword(s): observability.
    This paper proposes an approach to the problem of establishing the existence of observers for deterministic dynamical systems. This approach differs from the standard one based on Luenberger observers in that the observation error is not required to be Markovian given the past input and output data. A general abstract result is given, which special- izes to new results for parametrized families of linear systems, delay systems and other classes of systems. Related problems of feedback control and regulation are also studied.

  24. E.D. Sontag. On the observability of polynomial systems. I. Finite-time problems. SIAM J. Control Optim., 17(1):139-151, 1979. [PDF] Keyword(s): observability, observability, polynomial systems.
    Different notions of observability are compared for systems defined by polynomial difference equations. The main result states that, for systems having the standard property of (multiple-experiment initial-state) observability, the response to a generic input sequence is sufficient for final-state determination. Some remarks are made on results for nonpolynomial and/or continuous-time systems. An identifiability result is derived from the above.

  25. E.D. Sontag. On split realizations of response maps over rings. Information and Control, 37(1):23-33, 1978. [PDF] Keyword(s): systems over rings.
    This paper deals with observability properties of realizations of linear response maps defined over commutative rings. A characterization is given for those maps which admit realizations which are simultaneously reachable and observable in a strong sense. Applications are given to delay-differential systems.

  26. E.D. Sontag. On finitely accessible and finitely observable rings. J. Pure Appl. Algebra, 8(1):97-104, 1976. [PDF] Keyword(s): systems over rings, observability, noncommutative rings.
    Two classes of rings which occur in linear system theory are introduced and compared. Characterizations of one of them are given in terms, of integral extensions (every finite extension of R is integral) and Cayley--Hamilton type matrix condition. A comparison is made in the case of no zero-divisors with Ore domains.

  27. E.D. Sontag and Y. Rouchaleau. On discrete-time polynomial systems. Nonlinear Anal., 1(1):55-64, 1976. [PDF] Keyword(s): identifiability, observability, polynomial systems, realization theory, discrete-time.
    Considered here are a type of discrete-time systems which have algebraic constraints on their state set and for which the state transitions are given by (arbitrary) polynomial functions of the inputs and state variables. The paper studies reachability in bounded time, the problem of deciding whether two systems have the same external behavior by applying finitely many inputs, the fact that finitely many inputs (which can be chosen quite arbitrarily) are sufficient to separate those states of a system which are distinguishable, and introduces the subject of realization theory for this class of systems.

Conference articles
  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. J.P. Hespanha, D. Liberzon, and E.D. Sontag. Nonlinear observability and an invariance principle for switched systems. In Proc. IEEE Conf. Decision and Control, Las Vegas, Dec. 2002, IEEE Publications, pages 4300-4305, 2002. [PDF] Keyword(s): observability.

  3. M. Chaves and E.D. Sontag. An alternative observer for zero deficiency chemical networks. In Proc. Nonlinear Control System Design Symposium, St. Petersburg, July 2001, pages 575-578, 2001. Keyword(s): observability, observers, zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems.

  4. M. Chaves and E.D. Sontag. Observers for certain chemical reaction networks. In Proc. 2001 European Control Conf., Sep. 2001, pages 3715-3720, 2001. Keyword(s): zero-deficiency networks, systems biology, biochemical networks, nonlinear stability, dynamical systems, observability, observers.

  5. 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.
    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.

  6. E.D. Sontag. Spaces of observables in nonlinear control. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Basel, pages 1532-1545, 1995. Birkhäuser. [PDF] Keyword(s): observability, dynamical systems.
    Invited talk at the 1994 ICM. Paper deals with the notion of observables for nonlinear systems, and their role in realization theory, minimality, and several control and path planning questions.

  7. E.D. Sontag and Y. Wang. Orders of I/O equations and uniformly universal inputs. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 1994, IEEE Publications, 1994, pages 1270-1275, 1994. Keyword(s): identifiability, observability, realization theory.

  8. F. Albertini and E.D. Sontag. State observability in recurrent neural networks. In Proc. IEEE Conf. Decision and Control, San Antonio, Dec. 1993, IEEE Publications, 1993, pages 3706-3707, 1993. Keyword(s): neural networks, observability, recurrent neural networks.

  9. E.D. Sontag and Y. Wang. I/O equations in discrete and continuous time. In Proc. IEEE Conf. Decision and Control, Tucson, Dec. 1992, IEEE Publications, 1992, pages 3661-3662, 1992. Keyword(s): identifiability, observability, realization theory.

  10. R. Schwarzschild and E.D. Sontag. Quantized systems, saturated measurements, and sign-linear systems. In Proc. Conf. Inform. Sci. and Systems, John Hopkins University, March 1991, pages 134-139, 1991. Keyword(s): observability.

  11. 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.
    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.

  12. Y. Wang and E.D. Sontag. Realization of families of generating series: differential algebraic and state space equations. In Proc. 11th IFAC World Congress, Tallinn, former USSR, 1990, pages 62-66, 1990. Keyword(s): identifiability, observability, realization theory.

  13. 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): identifiability, observability, realization theory.

  14. Y. Wang and E.D. Sontag. Realization and input/output relations: the analytic case. In Proceedings of the 28th IEEE Conference on Decision and Control, Vol. 1--3 (Tampa, FL, 1989), New York, pages 1975-1980, 1989. IEEE. Keyword(s): identifiability, observability, realization theory.



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