Publications by Eduardo D. Sontag in year 1999 |
Books and proceedings |
Articles in journal or book chapters |
In this expository paper, we deal with several questions related to stability and stabilization of nonlinear finite-dimensional continuous-time systems. We review the basic problem of feedback stabilization, placing an emphasis upon relatively new areas of research which concern stability with respect to "noise" (such as errors introduced by actuators or sensors). The table of contents is as follows: Review of Stability and Asymptotic Controllability, The Problem of Stabilization, Obstructions to Continuous Stabilization, Control-Lyapunov Functions and Artstein's Theorem, Discontinuous Feedback, Nonsmooth CLF's, Insensitivity to Small Measurement and Actuator Errors, Effect of Large Disturbances: Input-to-State Stability, Comments on Notions Related to ISS. |
This paper shows that, for time varying systems, global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control-Lyapunov function with respect to the set. |
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. |
This paper shows 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 respectively for input-to-state stability, input-to-state exponential stability, and the property of finite square-norm gain ("nonlinear H-infty"). The results are shown for systems of any dimension not equal to 4 or 5. |
One of the fundamental facts in control theory (Artstein's theorem) is the equivalence, for systems affine in controls, between continuous feedback stabilizability to an equilibrium and the existence of smooth control Lyapunov functions. This equivalence breaks down for general nonlinear systems, not affine in controls. One of the main results in this paper establishes that the existence of smooth Lyapunov functions implies the existence of (in general, discontinuous) feedback stabilizers which are insensitive to small errors in state measurements. Conversely, it is shown that the existence of such stabilizers in turn implies the existence of smooth control Lyapunov functions. Moreover, it is established that, for general nonlinear control systems under persistently acting disturbances, the existence of smooth Lyapunov functions is equivalent to the existence of (possibly) discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances. |
We consider recurrent analog neural nets where the output of each gate is subject to Gaussian noise, or any other common noise distribution that is nonzero on a large set. We show that many regular languages cannot be recognized by networks of this type, and we give a precise characterization of those languages which can be recognized. This result implies severe constraints on possibilities for constructing recurrent analog neural nets that are robust against realistic types of analog noise. On the other hand we present a method for constructing feedforward analog neural nets that are robust with regard to analog noise of this type. |
We provide an explicit KL stability or input-to-state stability (ISS) estimate for a sampled-data nonlinear system in terms of the KL estimate for the corresponding discrete-time system and a K function describing inter-sample growth. It is quite obvious that a uniform inter-sample growth condition, plus an ISS property for the exact discrete-time model of a closed-loop system, implies uniform ISS of the sampled-data nonlinear system; our results serve to quantify these facts by means of comparison functions. Our results can be used as an alternative to prove and extend results of Aeyels et al and extend some results by Chen et al to a class of nonlinear systems. Finally, the formulas we establish can be used as a tool for some other problems which we indicate. |
This paper provides a precise result which shows that insensitivity to small measurement errors in closed-loop stabilization can be attained provided that the feedback controller ignores observations during small time intervals. |
This paper studies controllability properties of recurrent neural networks. The new contributions are: (1) an extension of the result in "Complete controllability of continuous-time recurrent neural networks" 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 deals with several related notions of output stability with respect to inputs (which may be thought of as disturbances). The main such notion is called input to output stability (IOS), and it reduces to input to state stability (ISS) when the output equals the complete state. For systems with no inputs, IOS provides a generalization of the classical concept of partial stability. Several variants, which formalize in different manners the transient behavior, are introduced. The main results provide a comparison among these notions |
Conference articles |
This paper studies the input-to-state stability (ISS) property for discrete-time nonlinear systems. We show that many standard ISS results may be extended to the discrete-time case. More precisely, we provide a Lyapunov-like sufficient condition for ISS, and we show the equivalence between the ISS property and various other properties, as well as provide a small gain theorem. |
This paper continues the investigation of the recently introduced integral version of input-to-state stability (iISS). We study the problem of designing control laws that achieve iISS disturbance attenuation. The main contribution is an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The results are compared and contrasted with the ones available for the ISS case. |
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