1,722,169 research outputs found
Regularity conditions for the stability margin problem with linear dependent perturbations
No full textIn this paper, the problem of continuity of the stability margin of a control system on problem input data is addressed. The case in which perturbations are linearly correlated is considered. It is shown that the existence of special points (called critical points) in the stability boundary manifold in parameter space plays a key role in the analysis of the problem. Several conditions, either sufficient or both necessary and sufficient, are given, ensuring continuity of the stability margin on problem data. The obtained conditions turn out to be easily checkable for practical applications. Numerical examples are presented to illustrate the proposed techniques
New conditions for global stability of neural networks with application to linear and quadratic programming problems
No full textIn this paper, we present new conditions ensuring existence, uniqueness, and Global Asymptotic Stability (GAS) of the equilibrium point for a large class of neural networks. The results are applicable to both symmetric and nonsymmetric interconnection matrices and allow for the consideration of all continuous nondecreasing neuron activation functions. Such functions may be unbounded (but not necessarily surjective), may have infinite intervals with zero slope as in a piece-wise-linear model, or both. The conditions on GAS rely on the concept of Lyapunov Diagonally Stable (or Lyapunov Diagonally Semi-Stable) matrices and are proved by employing a class of Lyapunov functions of the generalized Lur'e-Postnikov type. Several classes of interconnection matrices of applicative interest are shown to satisfy our conditions for GAS. In particular, the results are applied to analyze GAS for the class of neural circuits introduced for solving linear and quadratic programming problems. In this application, the principal result here obtained is that these networks are GAS also when the constraint amplifiers are dynamical, as it happens in any practical implementation
On the problem of local minima in backpropagation
No full textSupervised learning in multilayered neural networks (MLN's) has been recently proposed through the well-known backpropagation (BP) algorithm. This is a gradient method that can get stuck in local minima, as simple examples can show. In this paper, some conditions on the network architecture and the learning environment, which ensure the convergence of the BP algorithm, are proposed. It is proven in particular that the convergence holds if the classes are linearly separable. In this case, the experience gained in several experiments shows that MLN's exceed perceptrons in generalization to new examples
The Lojasiewicz exponent at an equilibrium point of a standard CNN is 1/2
No full textIn the sixties, Łojasiewicz proved a fundamental inequality for vector fields defined by the gradient of an analytic function, which gives a lower bound on the norm of the gradient in a neighborhood of a (possibly) non-isolated critical point. The inequality involves a number belonging to (0,1), which depends on the critical point, and is known as the Łojasiewicz exponent. In this paper, a class of vector fields which are defined on a hypercube of Rn, is considered. Each vector field is the gradient of a quadratic function in the interior of the hypercube, however it is discontinuous on the boundary of the hypercube. An extended Łojasiewicz inequality for this class of vector fields is proved, and it is also shown that the Łojasiewicz exponent at each point where a vector field vanishes is equal to 1/2. The considered fields include a class of vector fields which describe the dynamics of the output trajectories of a standard Cellular Neural Network (CNN) with a symmetric neuron interconnection matrix. By applying the extended Łojasiewicz inequality, it is shown that each output trajectory of a symmetric CNN has finite length, and as a consequence it converges to an equilibrium point. Furthermore, since the Łojasiewicz exponent at each equilibrium point of a symmetric CNN is equal to 1/2, it follows that each (state) trajectory, and each output trajectory, is exponentially convergent toward an equilibrium point, and this is true even in the most general case where the CNN possesses infinitely many nonisolated equilibrium points. In essence, the obtained results mean that standard symmetric CNNs enjoy the property of absolute stability of exponential convergence. © World Scientific Publishing Company
Absolute stability of analytic neural networks: An approach based on finite trajectory length
No full textA neural network is said to be convergent (or completely stable) when each trajectory tends to an equilibrium point (a stationary state). A stronger property is that of absolute stability, which means that convergence holds for any choice of the neural network parameters, and any choice of the nonlinear functions, within specified and well characterized sets. In particular, the property of absolute stability requires that the neural network be convergent also when, for some parameter values, it possesses nonisolated equilibrium points (e.g., a manifold of equilibria). Such a property, which is really well suited for solving several classes of signal processing tasks in real time, cannot be in general established via the classical LaSalle approach, due to its inherent limitations to study convergence in situations where the neural network has nonisolated equilibrium points. In this paper, a new method to address absolute stability is developed, based on proving that the total length of the neural network trajectories is finite. A fundamental result on absolute stability is given, under the hypothesis that the neural network possesses a Lyapunov function, and the nonlinearities involved (neuron activations, inhibitions, etc.) re modeled by analytic functions. At the core of the proof of finiteness of trajectory length is the use of some basic inequalities for analytic functions due to Lojasiewicz. The result is applicable to a large class of neural networks, which includes the networks proposed by Vidyasagar, the Hopfield neural networks, and the standard cellular neural networks introduced by Chua and Yang
A new method to analyze complete stability of PWL cellular neural networks
No full textIn recent years the standard Cellular Neural Networks (CNN's) introduced by Chua and Yang [1988] have been one of the most investigated paradigms for neural information processing. In a wide range of applications, the CNN's are required to be completely stable, i.e. each trajectory should converge toward some stationary state. However, a rigorous proof of complete stability, even in the simplest original setting of piecewise-linear (PWL) neuron activations and symmetric interconnections [Chua & Yang, 1988], is still lacking. This paper aims primarily at filling this gap in order to give a sound analytical foundation to the CNN paradigm. To this end a novel approach for studying complete stability is proposed. This is based on a fundamental limit theorem for the length of the CNN trajectories. The method differs substantially from the classic approach using LaSalle invariance principle, and permits to overcome difficulties encountered when using LaSalle approach to analyze complete stability of PWL CNN's. The main result obtained, is that a symmetric PWL CNN is completely stable for any choice of the network parameters, i.e. it possesses the Absolute Stability property of global pattern formation. This result is really general and shows that complete stability holds under hypotheses weaker than those considered in [Chua & Yang, 1988]. The result does not require, for example, that the CNN has binary stable equilibrium points only. It is valid even in degenerate situations where the CNN has infinite nonisolated equilibrium points. These features significantly extend the potential application fields of the standard CNN's
Frequency response of interval plant-controller families of transfer functions
No full textIn this paper a general frequency domain result for families of interval plant with a fixed linear controller is given. It is shown that the locus of the polar diagrams of frequency responses of the transfer functions of an interval plant-controller family is bounded by the polar plots relative to the 32 segments of transfer functions of the interval plant family. Easy proofs of several important results, such as the generalization of the Theorem of Kharitonov for feedback systems with interval plants or the robust version of the small gain theorem for the same class of systems, are constructed by using the general result. More importantly, an immediate consequence of the main theorem is that extremal phase/gain margins or sensitivity/complementary sensitivity peaks for systems of the family can be deduced from those of the 32 segments of the interval plant family
Robust stability of state space models with structured uncertainties
No full textIn this paper, a method for robust eigenvalue location analysis of linear state-space models affected by structured real parametric perturbations is proposed. The approach, based on algebraic matrix properties, deals with state-space models where system matrix entries are perturbed by polynomial functions of a set of physical uncertain parameters. A method converting the robust stability problem in the nonsingularity analysis of a suitable matrix is proposed. The method leads to positivity check of a multinomial form over a hyperrectangular domain in parameter space. This problem, which can be reduced to finding the real solutions of a system of polynomial equations, simplifies considerably when considering cases with one or two uncertain parameters. For these cases, necessary and sufficient conditions for stability are given in terms of the solution of suitable real eigenvalue problems. © 1990 IEE
A new fast algorithm for robust stability analysis of linear control systems with linearly correlated parametric uncertainties
No full textIn this paper a fast algorithm is proposed for the computation of stability margins in parameter space for linear control systems subject to structured uncertainties. The case in which plant transfer function coefficients are affine in a set of physical uncertain parameters is considered. The paper shows also how the proposed algorithm can be used to solve another interesting problem in robust control analysis, i.e. the determination of the region of pole location of a closed loop control system including an entire family of uncertain plants
On the margin of complete stability for a class of cellular neural networks
No full textIn this paper, the dynamical behavior of a class of third-order competitive cellular neural networks (CNNs) depending on two parameters, is studied. The class contains a one-parameter family of symmetric CNNs, which are known to be completely stable. The main result is that it is a generic property within the family of symmetric CNNs that complete stability is robust with respect to (small) nonsymmetric perturbations of the neuron interconnections. The paper also gives an exact evaluation of the complete stability margin of each symmetric CNN via the characterization of the whole region in the two-dimensional parameter space where the CNNs turn out to be completely stable. The results are established by means of a new technique to investigate trajectory convergence of the considered class of CNNs in the nonsymmetric case
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