Stability analysis techniques for competitive neural networks with different time-scales

Attitude control systems naturally evolve on nonlinear configurations, such as S2 and SO(3). The nontrivial topological properties of these configurations result in interesting and complicated nonlinear dynamics when studying the corresponding closed loop attitude control systems. In this paper, we review some global analysis and simulation techniques that allow us to describe the global nonlinear stable manifolds of the hyperbolic equilibria of these closed loop systems. A deeper understanding of these invariant manifold structures are critical to understanding the global stabilization properties of closed loop attitude control systems, and these global analysis techniques are applicable to a broad range of problems on nonlinear configuration manifolds.

We present a new method of analyzing the dynamics of a biological relevant system with different time scales based on the theory of flow invariance. We are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the K-monotone theory, singular perturbation theory, or those based on supervised synaptic learning. We prove the existence and the uniqueness of the equilibrium. A strict Lyapunov function for the flow of a competitive neural system with different time scales is given and based on it we are able to prove the global exponential stability of the equilibrium point.

The theory of singular perturbations and time scales has played an important role in aircraft performance analysis and in the development of feedback guidance laws for aerospace vehicles. This paper presents a survey of the applications of the theory of singular perturbations and time scales in guidance, navigation, and control of air and space transportation systems. In particular, emphasis will be placed on problem formulation and solution approaches that are useful in applying the theory for various types of problems arising in aerospace systems. Fhrther, the survey e m braces the related fields such as fluid dynamics, space structures and space robotics.

In this paper, the regulation problem of a class of nonlinear singularly perturbed discrete-time systems is investigated. Using the theory of singular perturbations and time scales, the nonlinear system is decoupled into reduced-order slow and fast (boundary layer) subsystems. Then, a composite controller consisting of two sub-controllers for the slow and fast subsystems is developed using the discrete-time state-dependent Riccati equation (D-SDRE). It is proved that the equilibrium point of the original closed-loop system with a composite controller is locally asymptotically stable. Moreover, the region of attraction of the closed-loop system is estimated by using linear matrix inequality. One example is given to illustrate the effectiveness of the results obtained.

Indirect adaptive control of multi-input-multi-output nonlinear singularly perturbed systems with model uncertainties

Nonlinear analyses of structures under dynamic excitation are becoming increasingly important in structural design and performance evaluation, but large computational effort is the main factor that limits their application. Because nonlinearity is commonly confined within small regions, many studies have been devoted to improving the efficiency of solving such local nonlinear problems by maintaining the structural stiffness elasticity and simulating the effects of local nonlinearity through fictitious nonlinear forces or local modification of the elastic structural response. For dynamic analysis, the nature of the stiffness matrix determines the formulation of the widely used Rayleigh damping; as a result, these local nonlinear analysis methods often use elastic stiffness-based Rayleigh damping models. However, such a damping model can generate unexpected artificial damping forces for inelastic systems and consequently produce inaccurate or even invalid results. Although a tangent stiffness–based Rayleigh damping model has been proven to be a reasonable basis for performing highly accurate dynamic analyses, this type of damping model has difficulty achieving direct compatibility with local nonlinear analysis methods. The present research focuses on the implementation of a tangent stiffness–based Rayleigh damping model for use in efficient local nonlinear analysis methods developed based on the Woodbury formula, which can calculate the structural inelastic behavior by updating the elastic solution rather than by updating the stiffness. By representing tangent stiffness–based Rayleigh damping as a low-rank perturbation to the elastic stiffness-based damping matrix, this study derives a modified dynamic Woodbury formula in which an additional influence coefficient is introduced to reflect the effects of local nonlinearity on the structural damping properties. Moreover, to overcome the potential solution difficulty caused by abrupt changes in the damping forces in certain steps and further improve the efficiency of the proposed method, a variable time step solution scheme that can meet the computational requirements of the Woodbury formula is presented. Verifications demonstrate the validity and high efficiency of the proposed method.

We present a new method of analyzing the dynamics of self-organizing neural networks with different time scales based on the theory of flow invariance. We are able to show the conditions under which the solutions of such a system are bounded being less restrictive than with the K-monotone theory, singular perturbation theory, or those based on supervised synaptic learning. We prove the existence and the uniqueness of the equilibrium. A strict Lyapunov function for the flow of a competitive neural system with different time scales is given, and based on it we are able to prove the global exponential stability of the equilibrium point.

Singular Perturbations and Singular Layers in Variational Inequalities

Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality.

This paper discusses the in near–real time processing of Global Navigation Satellite An alternative approach based on using Vertical Electrical Sounding (VES) measurements and Artificial Neural Network (ANN) technique is newly proposed for computing the hydraulic conductivity K and the transmissivity T of an aquifer. VES measurements in the locations, where available water samples exist are required in such an approach, in order to train a neural network with fitting capability to evaluate both the hydraulic conductivity and transmissivity. The hydraulic conductivity and transmissivity are thereafter extrapolated by the use of trained neural network, even in the VES points where no water samples exist. This approach is practiced and tested in the Khanasser valley, Northern Syria, where the hydraulic conductivity and the transmissivity of the Quaternary aquifer is computed. We find an acceptable agreement between the hydraulic conductivity values obtained by the new approach and those obtained by the pumping test, which range between 0.864 and 8.64 m/day.

A new approach based on sliding control is presented for modeling and simulation of thermo-fluid systems described by differential-algebraic equations (DAEs). The dynamics of thermo-fluid systems are often complicated by momentum interactions that occur on a time scale that is orders of magnitude faster than the time scale of interest. To address this problem the momentum equation is often modeled using algebraic steady state approximations. This will, in general, result in a model described by nonlinear DAEs for which few control methods are currently applicable. In this paper, the modeling problem is addressed using an approach that systematically constructs an explicit state space approximation of the DAEs. The state space model can in turn be used with existing control methods. This procedure, known as realization, is achieved by solving an associated nonlinear control problem by combining boundary layer sliding control with the singular perturbation method. The necessary criteria for key properties such as convergence are established. Further, the new approach is illustrated using a vapor compression cycle example. This demonstrates significant advantages over directly modeling momentum interactions.

This study presents an (ε, μ)−uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank–Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's 1/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature.

For a singular perturbation $$A = A_0 + \sum {_{i,j = 1}^n } t_{ij} \left\langle {\psi _j , \cdot } \right\rangle \psi _i ,n \leqslant \infty$$ , n ≤ ∞, of a positive self-adjoint operator A 0 with Lebesgue spectrum, the spectral analysis of the corresponding self-adjoint operator realizations A T is carried out and the scattering matrix $$\mathfrak{S}(A_T ,A_0 )(\delta )$$ is calculated in terms of parameters t ij under some additional restrictions on singular elements ψ j . The results obtained enable one to apply the Lax-Phillips approach in scattering theory.

A sliding manifold based control strategy is proposed for the attitude control problem. By using the theory of singular perturbation and the globally nonsingular unit quaternion representation for the orientation error a nonlinear feedback control is designed. The use of quaternions allows us to avoid the drawbacks of Euler angle approach, namely singularities which can occur in large maneuvers. The feedback control system exhibits robustness properties with respect to disturbances and plant parametric uncertainties. The resulting control signal remains, after a fast transient, in a neighborhood of the well-defined equivalent control avoiding the peaking phenomenon of high gain systems. The proposed control technique is then applied to a complete satellite rotation about a fixed axis.

In this paper, we consider a multimedia data transmission system over a wireless channel, where packets are queued at the transmitter. Multimedia data transmission over wireless networks often suffers from delay, jitter and packet loss. The main problem to implement a wireless network is the high and variable bit error rate in the radio link (fading, shadowing etc). Among the most important performance measures for real time applications are the packet loss probability. In order to improve the radio link, one often retransmits packets that have not been well received (using the Automatic Retransmission reQuest). This however may lead to queuing phenomena and increased delay due to retransmissions, and to losses of packets due to buffer overflow. In this paper, we allow that the number of retransmission is finite. We model the system state by a three-dimensional Markov chain which represents the evolution the radio link state, the number of packets in the buffer and the number of transmission of a packet in the service. We use an advanced approach based on the theory of singular perturbation in order to compute some performance of interest.