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24
Existence and Learning of Oscillations in Recurrent Neural Networks
, 1999
"... In this paper we study a particular class of nnode recurrent neural networks (RNNs). In the 3node case we use monotone dynamical systems theory to show, for a welldefined set of parameters, that, generically, every orbit of the RNN is asymptotic to a periodic orbit. Then, within the usual `learni ..."
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Cited by 8 (0 self)
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In this paper we study a particular class of nnode recurrent neural networks (RNNs). In the 3node case we use monotone dynamical systems theory to show, for a welldefined set of parameters, that, generically, every orbit of the RNN is asymptotic to a periodic orbit. Then, within the usual `learning' context of Neural Networks, we investigate whether RNNs of this class can adapt their internal parameters so as to `learn' and then replicate autonomously certain external periodic signals. Our learning algorithm is similar to identification algorithms in adaptive control theory. The main feature of the adaptation algorithm is that global exponential convergence of parameters is guaranteed. We also obtain partial convergence results in the nnode case.
ON CONDITIONS FOR ASYMPTOTIC STABILITY OF DISSIPATIVE INFINITEDIMENSIONAL SYSTEMS WITH INTERMITTENT DAMPING
, 2012
"... Abstract. We study the asymptotic stability of a dissipative evolution in a Hilbert space subject to intermittent damping. We observe that, even if the intermittence satisfies a persistent excitation condition, if the Hilbert space is infinitedimensional then the system needs not being asymptotical ..."
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Cited by 5 (2 self)
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Abstract. We study the asymptotic stability of a dissipative evolution in a Hilbert space subject to intermittent damping. We observe that, even if the intermittence satisfies a persistent excitation condition, if the Hilbert space is infinitedimensional then the system needs not being asymptotically stable (not even in the weak sense). Exponential stability is recovered under a generalized observability inequality, allowing for timedomains that are not intervals. Weak asymptotic stability is obtained under a similarly generalized unique continuation principle. Finally, strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition, evaluating their total contribution to the decay of the trajectories of the damped system. Our results are discussed using the example of the wave equation, Schrödinger’s equation and, for strong stability, also the special case of finitedimensional systems.
On the stabilization of persistently excited linear systems
 SIAM J. Control Optim
"... We consider control systems of the type ˙x = Ax+α(t)bu, where u ∈ R, (A, b) is a controllable pair and α is an unknown timevarying signal with values in [0, 1] satisfying a persistent excitation condition i.e., ∫ t+T t α(s)ds ≥ µ for every t ≥ 0, with 0 < µ ≤ T independent on t. We prove that such ..."
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Cited by 5 (5 self)
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We consider control systems of the type ˙x = Ax+α(t)bu, where u ∈ R, (A, b) is a controllable pair and α is an unknown timevarying signal with values in [0, 1] satisfying a persistent excitation condition i.e., ∫ t+T t α(s)ds ≥ µ for every t ≥ 0, with 0 < µ ≤ T independent on t. We prove that such a system is stabilizable with a linear feedback depending only on the pair (T, µ) if the eigenvalues of A have nonpositive real part. We also show that stabilizability does not hold for arbitrary matrices A. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter µ/T. 1
Model Reference Adaptive Control of Distributed Parameter Systems
 SIAM J. Control Optim
, 1995
"... A model reference adaptive control law is defined for nonlinear distributed parameter systems. The reference model is assumed to be governed by a strongly coercive linear operator defined with respect to a Gelfand triple of reflexive Banach and Hilbert spaces. The resulting nonlinear closed loop sys ..."
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Cited by 3 (0 self)
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A model reference adaptive control law is defined for nonlinear distributed parameter systems. The reference model is assumed to be governed by a strongly coercive linear operator defined with respect to a Gelfand triple of reflexive Banach and Hilbert spaces. The resulting nonlinear closed loop system is shown to be well posed. The tracking error is shown to converge to zero, and regularity results for the control input and the output are established. With an additional richness, or persistence of excitation assumption, the parameter error is shown to converge to zero as well. A finite dimensional approximation theory is developed. Examples involving both first (parabolic) and second (hyperbolic) order systems and linear and nonlinear systems are discussed, and numerical simulation results are presented. Supported in part by DFG. y Supported in part by the Air Force Office of Scientific Research under grant AFOSR F496209310198, and in part by NASA under grant NAG11600. z S...
On the Persistence of Excitation in the Adaptive Estimation of Distributed Parameter Systems
 IEEE Trans. Automat. Control
, 1996
"... Persistence of excitation is a sufficient condition for parameter convergence in adaptive identification schemes for dynamical systems. For abstract parabolic and hyperbolic distributed parameter systems, this condition requires that a family of bounded linear functionals be norm bounded away from z ..."
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Cited by 3 (2 self)
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Persistence of excitation is a sufficient condition for parameter convergence in adaptive identification schemes for dynamical systems. For abstract parabolic and hyperbolic distributed parameter systems, this condition requires that a family of bounded linear functionals be norm bounded away from zero. The level of persistence of excitation of the plant and its implications are considered for a simple parabolic and hyperbolic system. Its effect on the qualitative and quantitative behavior of the estimators is investigated. This research was supported in part by the Air Force Office of Scientific Research under grant AFOSR 910076. 1 Introduction In this short note we consider the persistence of excitation condition in the adaptive, or online, identification of abstract distributed parameter systems. For a certain class of schemes which we have studied in depth in [5] (see also [4]), the persistence of excitation of the plant is a sufficient condition for parameter convergence. T...
Adaptive Identification of Second Order Distributed Parameter Systems
 Inverse Problems
, 1997
"... The adaptive (online) estimation of parameters for a class of second order distributed parameter systems is considered. This class of systems, which includes abstract wave and beam equations with a variety of forms of damping, is frequently used to model the vibration of large flexible structures. ..."
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Cited by 3 (3 self)
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The adaptive (online) estimation of parameters for a class of second order distributed parameter systems is considered. This class of systems, which includes abstract wave and beam equations with a variety of forms of damping, is frequently used to model the vibration of large flexible structures. A combined state and parameter estimator is constructed as an initial value problem for an infinite dimensional evolution equation in weak or variational form. State convergence is established via a Lyapunovlike estimate. The finite dimensional notion of persistence of excitation is extended to the infinite dimensional case and used to establish parameter convergence. A finite dimensional approximation theory is presented and a convergence result is proven. An example involving the identification of a damped onedimensional wave equation is discussed and results of a numerical study are presented. This research was supported in part by the Air Force Office of Scientific Research under gran...
Summationtype conditions for uniform asymptotic convergence in discretetime systems: applications in identification
, 2005
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Towards uniform linear timeinvariant stabilization of systems with persistency of excitation
, 2007
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Range Identification for Perspective Dynamic Systems Using Linear Approximation
"... Abstract — This paper presents linear approximation ideas to range identification problem for a perspective dynamic system (PDS). Using a recently introduced linear approximation technique, the perspective dynamic system, which is a special class of nonlinear systems, can be approximated by a sequen ..."
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Abstract — This paper presents linear approximation ideas to range identification problem for a perspective dynamic system (PDS). Using a recently introduced linear approximation technique, the perspective dynamic system, which is a special class of nonlinear systems, can be approximated by a sequence of linear, timevarying (LTV) subsystems. Observer design problem of the original nonlinear PDS reduces to the observer design of this sequence of LTV subsystems. For each LTV subsystem, existing standard observer methods can be applied. I.