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247
Interconnected Automata and Linear Systems: A Theoretical Framework in DiscreteTime
 in Hybrid Systems III: Verification and Control
, 1996
"... . This paper summarizes the definitions and several of the main results of an approach to hybrid systems, which combines finite automata and linear systems, developed by the author in the early 1980s. Some related more recent results are briefly mentioned as well. 1 Introduction  The Need for Hybri ..."
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Cited by 51 (0 self)
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. This paper summarizes the definitions and several of the main results of an approach to hybrid systems, which combines finite automata and linear systems, developed by the author in the early 1980s. Some related more recent results are briefly mentioned as well. 1 Introduction  The Need for Hybrid Systems Linear control theory is welldeveloped and highly sophisticated, and is widely applied in areas ranging from aerospace to automotive control. Linear systems provide highly accurate models of many physical systems; furthermore, the use of linear systems as "robust" controllers often allows the tolerance of even severe model nonlinearities and uncertainties. However, it remains a fact that many continuous physical processes cannot be satisfactorily modeled linearly, nor can be adequately regulated by means of linear controllers alone. It has long been recognized that the control of more complex systems than those handled by the linear theory will require switching mechanisms (disco...
Exponential Stabilization of Driftless Nonlinear Control Systems
, 1995
"... This dissertation lays the foundation for practical exponential stabilization of driftless control systems. Driftless systems have the form, x = X 1 (x)u 1 + \Delta \Delta \Delta + Xm (x)um ; x 2 R n : Such systems arise when modeling mechanical systems with nonholonomic constraints. In engineer ..."
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Cited by 50 (3 self)
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This dissertation lays the foundation for practical exponential stabilization of driftless control systems. Driftless systems have the form, x = X 1 (x)u 1 + \Delta \Delta \Delta + Xm (x)um ; x 2 R n : Such systems arise when modeling mechanical systems with nonholonomic constraints. In engineering applications it is often required to maintain the mechanical system around a desired configuration. This task is treated as a stabilization problem where the desired configuration is made an asymptotically stable equilibrium point. The control design is carried out on an approximate system. The approximation process yields a nilpotent set of input vector fields which, in a special coordinate system, are homogeneous with respect to a nonstandard dilation. Even though the approximation can be given a coordinatefree interpretation, the homogeneous structure is useful to exploit: the feedbacks are required to be homogeneous functions and thus preserve the homogeneous structure in the close...
Structure and stability of certain chemical networks and applications to the kinetic proofreading model of Tcell receptor signal transduction
 IEEE Trans. Autom. Control
, 2001
"... Abstract This paper deals with the theory of structure, stability, robustness, and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks. The results given here extend, but are also heavily based upon, certain previous work by Feinberg, Horn, and ..."
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Cited by 48 (8 self)
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Abstract This paper deals with the theory of structure, stability, robustness, and stabilization for an appealing class of nonlinear systems which arises in the analysis of chemical networks. The results given here extend, but are also heavily based upon, certain previous work by Feinberg, Horn, and Jackson, of which a selfcontained and streamlined exposition is included. The theoretical conclusions are illustrated through an application to the kinetic proofreading model proposed by McKeithan for Tcell receptor signal transduction. Keywords:. chemical networks, stability, feedback, system structure, kinetic proofreading, immunology
Stability and stabilization: discontinuities and the effect of disturbances, in “Nonlinear analysis, differential equations and control
, 1998
"... ..."
Input to state stability: Basic concepts and results
 Nonlinear and Optimal Control Theory
, 2006
"... The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classes of systems for which systematic decompositi ..."
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Cited by 46 (6 self)
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The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classes of systems for which systematic decomposition approaches can result in effective
InputOutputtoState Stability
 SIAM J. Control Optim
, 1999
"... This work explores Lyapunov characterizations of the inputoutputtostate stability (oss) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zerodetectability property used in the linear case. The main contribution of this work is to establish a compl ..."
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Cited by 44 (18 self)
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This work explores Lyapunov characterizations of the inputoutputtostate stability (oss) property for nonlinear systems. The notion of IOSS is a natural generalization of the standard zerodetectability property used in the linear case. The main contribution of this work is to establish a complete equivalence between the inputoutputtostate stability property and the existence of a certain type of smooth Lyapunov function. As corollaries, one shows the existence of "normestimators", and obtains characterizations of nonlinear detectability in terms of relative stability and of finiteenergy estimates.
Finiteness Results for Sigmoidal "Neural" Networks
 In Proceedings of 25th Annual ACM Symposium on the Theory of Computing
, 1993
"... ) Angus Macintyre Mathematical Inst., University of Oxford Oxford OX1 3LB, England, UK Email: ajm@maths.ox.ac.uk Eduardo D. Sontag 3 Dept. of Mathematics, Rutgers University New Brunswick, NJ 08903 Email: sontag@hilbert.rutgers.edu Abstract Proc. 25th Annual Symp. Theory Computing , San Diego, ..."
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Cited by 44 (12 self)
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) Angus Macintyre Mathematical Inst., University of Oxford Oxford OX1 3LB, England, UK Email: ajm@maths.ox.ac.uk Eduardo D. Sontag 3 Dept. of Mathematics, Rutgers University New Brunswick, NJ 08903 Email: sontag@hilbert.rutgers.edu Abstract Proc. 25th Annual Symp. Theory Computing , San Diego, May 1993 This paper deals with analog circuits. It establishes the finiteness of VC dimension, teaching dimension, and several other measures of sample complexity which arise in learning theory. It also shows that the equivalence of behaviors, and the loading problem, are effectively decidable, modulo a widely believed conjecture in number theory. The results, the first ones that are independent of weight size, apply when the gate function is the "standard sigmoid" commonly used in neural networks research. The proofs rely on very recent developments in the elementary theory of real numbers with exponentiation. (Some weaker conclusions are also given for more general analytic gate functions...
The Theory of Timed I/O Automata
, 2003
"... This paper presents the Timed Input/Output Automaton (TIOA) modeling framework, a basic mathematical framework to support description and analysis of timed systems. An important feature of this model is its support for decomposing timed system descriptions. In particular, the framework includes a no ..."
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Cited by 44 (24 self)
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This paper presents the Timed Input/Output Automaton (TIOA) modeling framework, a basic mathematical framework to support description and analysis of timed systems. An important feature of this model is its support for decomposing timed system descriptions. In particular, the framework includes a notion of external behavior for a timed I/O automaton, which captures its discrete interactions with its environment. The framework also denes what it means for one TIOA to implement another, based on an inclusion relationship between their external behavior sets, and de nes notions of simulations, which provide sucient conditions for demonstrating implementation relationships. The framework includes a composition operation for TIOAs, which respects external behavior, and a notion of receptiveness, which implies that a TIOA does not block the passage of time. The TIOA framework supports the statement and verication of safety and liveness properties for timed systems. It denes what it means for a property to be a safety or a liveness property, includes basic results about safetyliveness classication, and
Feedback Stabilization Using TwoHiddenLayer Nets
 IEEE Trans. Neural Networks
, 1992
"... This paper compares the representational capabilities of one hidden layer and two hidden layer nets consisting of feedforward interconnections of linear threshold units. It is remarked that for certain problems two hidden layers are required, contrary to what might be in principle expected from the ..."
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Cited by 40 (6 self)
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This paper compares the representational capabilities of one hidden layer and two hidden layer nets consisting of feedforward interconnections of linear threshold units. It is remarked that for certain problems two hidden layers are required, contrary to what might be in principle expected from the known approximation theorems. The differences are not based on numerical accuracy or number of units needed, nor on capabilities for feature extraction, but rather on a much more basic classification into “direct ” and “inverse ” problems. The former correspond to the approximation of continuous functions, while the latter are concerned with approximating onesided inverses of continuous functions —and are often encountered in the context of inverse kinematics determination or in control questions. A general result is given showing that nonlinear control systems can be stabilized using two hidden layers, but not in general using just one. Key words: Neural nets, nonlinear control systems, feedback 1
On the InputtoState Stability Property
 Systems & Control Letters
, 1995
"... The "input to state stability" (iss) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunovtheoretic, and dissipation terms. We sketch s ..."
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Cited by 35 (5 self)
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The "input to state stability" (iss) property provides a natural framework in which to formulate notions of stability with respect to input perturbations. In this expository paper, we review various equivalent definitions expressed in stability, Lyapunovtheoretic, and dissipation terms. We sketch some applications to the stabilization of cascades of systems and of linear systems subject to control saturation. 1 Introduction There are two very conceptually different ways of formulating the notion of stability of control systems. One of them, which we may call the input/output approach, relies on operatortheoretic techniques. Among the main contributions to this area, one may cite the foundational work by Zames, Sandberg, Desoer, Safanov, Vidyasagar, and others. In this approach, a "system" is a causal operator F between spaces of signals, and "stability" is taken to mean that F maps bounded inputs into bounded outputs, or finiteenergy inputs into finiteenergy outputs. More stringe...