arXiv subjects
Publications and source records attributed to Eduardo D. Sontag.
In many applications, such as those arising from the field of cellular networks, it is often desired to determine the interaction (graph) structure of a set of differential equations, using as data measured sensitivities. This note proposes an approach to this problem.
This note provides a simple result showing, under suitable technical assumptions, that if a system S adapts to a class of external signals U, then S must necessarily contain a subsystem which is capable of generating all the signals in U. It is not assumed that regulation is robust, nor is there a prior requirement for the system to be partitioned into separate plant and controller components.
In previous work the notion of input to state stability (ISS) has been generalized to systems with outputs, yielding a number of useful concepts. When considering a system whose output is to be kept small (i.e. an error output), the notion of input to output stability (IOS) arises. Alternatively, when considering a system whose output is meant to provide information about the state (i.e. a measurement output), one arrives at the detectability notion of output to state stability (OSS). Combining these concepts, one may consider a system with two outputs, an error and a measurement. This leads naturally to a notion of partial detectability we call measurement to error stability (MES). This property characterizes systems in which the error signal is detectable through the measurement signal. This paper provides a partial Lyapunov characterization of the MES property. A closely related property of stability in three measures (SIT) is introduced, which characterizes systems for which the error decays whenever it dominates the measurement. The SIT property is shown to imply MES, and the two are shown to be equivalent under an additional boundedness assumption. A nonsmooth Lyapunov characterization of the SIT property is provided, which yields the partial characterization of MES. The analysis is carried out on systems described by differential inclusions -- implicitly incorporating a disturbance input with compact value-set.
The notions of asymptotic amplitude for signals, and Cauchy gain for input/output systems, and an associated small-gain principle, are introduced. These concepts allow the consideration of systems with multiple, and possibly feedback-dependent, steady states. A Lyapunov-like characterization allows the computation of gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades and feedback loops. An application in biology (MAPK signaling) is worked out in detail.
A notion of "asymptotic Cauchy gain" for input/output systems, and an associated small-gain principle, are introduced. A Lyapunov-like characterization allows the computation of these gains for state-space systems, and the formulation of sufficient conditions insuring the lack of oscillations and chaotic behaviors in a wide variety of cascades and feedback loops.
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.
A general ISS-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.
The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wazewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is presented, which provides approximations on infinite intervals, but does not guarantee that the approximation and the reference trajectory satisfy the same initial condition.
This paper introduces and studies the notion of output-input stability, which represents a variant of the minimum-phase property for general smooth nonlinear control systems. The definition of output-input stability does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. In the spirit of the ``input-to-state stability'' philosophy, it requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of output-input stable systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. As an application, we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control.
This paper takes a computational learning theory approach to a problem of linear systems identification. It is assumed that input signals have only a finite number k of frequency components, and systems to be identified have dimension no greater than n. The main result establishes that the sample complexity needed for identification scales polynomially with n and logarithmically with k.
One of the most interesting questions in control theory is that of constructing observers. Observers compute estimates of the internal states of a dynamical system, using data provided by measurement probes or partial state information. For linear systems, Luenberger observers (also known as "deterministic Kalman filters" since they amount to Kalman filters designed without regard to the statistics of measurement noise) provide a general solution, but, for nonlinear systems, establishing generally applicable conditions for existence and convergence of observers is an open and active area of research. 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.
This note presents a self-contained and streamlined exposition of chemical-network results due to Feinberg, Horn, and Jackson. As an application, it shows the global asymptotic stability to equilibria of McKeithan's kinetic proofreading model for T-cell receptor signal transduction.
This paper deals with the regularity of solutions of the Hamilton-Jacobi Inequality which arises in H-infinity control. It shows by explicit counterexamples that there are gaps between existence of continuous and locally Lipschitz (positive definite and proper) solutions, and between Lipschitz and continuously differentiable ones. On the other hand, it is shown that it is always possible to smooth-out solutions, provided that an infinitesimal increase in gain is allowed.
This work explores Lyapunov characterizations of the input-output-to-state stability (IOSS) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zero-detectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the input-output-to-state stability property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of ``norm-estimators'', and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.
This paper deals with several related notions of output stability with respect to inputs. The inputs may be thought of as disturbances; when there are no inputs, one obtains generalizations of the classical concepts of partial stability. The main notion studied is called input to output stability (IOS), and it reduces to input to state stability (ISS) when the output equals the complete state. Several variants, which formalize in different manners the transient behavior, are introduced. The main results provide a comparison among these notions. A companion paper establishes necessary and sufficient Lyapunov-theoretic characterizations.
In this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables. The same is shown for input-to-state stability and input-to-state exponential stability, and for input-to-state exponential stability and a nonlinear $H_\infty$ estimate.
This paper studies controllability properties of recurrent neural networks. The new contributions are: (1) an extension of the result in the previous paper "Complete controllability of continuous-time recurrent neural networks" (Sontag and Sussmann) to a slightly different model, where inputs appear in an affine form, (2) a formulation and proof of a necessary and sufficient condition, in terms of local-local controllability, and (3) a complete analysis of the 2-dimensional case for which the hypotheses made in previous work do not apply
This paper presents necessary and sufficient characterizations of several notions of input to output stability. Similar Lyapunov characterizations have been found to play a key role in the analysis of the input to state stability property, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated. The work is related to partial stability of differential equations (the particular case which arises when there are no external inputs).