State-space Dynamics Research Articles

Projected single-spin-flip dynamics in the Ising model

We study transition matrices for projected dynamics in the energy-magnetization, magnetization, and energy spaces. Several single-spin-flip dynamics are considered, such as the Glauber and Metropolis canonical ensemble dynamics, and the Metropolis dynamics for three multicanonical ensembles: the flat energy-magnetization, the flat energy, and the flat magnetization histograms. From the numerical diagonalization of the matrices for the projected dynamics we obtain the subdominant eigenvalues and the largest relaxation times for systems of varying size. Although the projected dynamics is an approximation to the full state space dynamics, comparison with some available results, obtained by other authors, shows that projection in the magnetization space is a reasonably accurate method to study the scaling of relaxation times with system size. For each system size, the transition matrices for arbitrary single-spin-flip dynamics are obtained from a single Monte Carlo estimate of the infinite-temperature transition matrix. This makes the method an efficient tool for evaluating the relative performance of any arbitrary local spin-flip dynamics. We also present results for appropriately defined average tunneling times of magnetization and compare their finite-size scaling exponents with results of energy tunneling exponents available for the flat energy histogram multicanonical ensemble.

Nonlinear feedback control of low-thrust orbital transfer in a central gravitational field

This paper investigates the problem of continuous-thrust orbital transfer using orbital elements feedback from a nonlinear control standpoint, utilizing concepts of controllability, feedback stabilizability and their interaction. Gauss's variational equations (GVEs) are used to model the state-space dynamics of motion under a central gravitational field. First, the notion of accessibility is reviewed. It is then shown that the GVEs are globally accessible. Based on the accessibility result, a nonlinear feedback controller is derived which asymptotically steers a spacecraft form an initial elliptic orbit to any given elliptic orbit. The performance of the new controller is illustrated by simulating an orbital transfer between two geosynchronous Earth orbits. It is shown that the low-thrust controller requires less fuel than an impulsive maneuver for the same transfer time. Closed-form, analytic expressions for the new orbital transfer controller are given. Finally, it is proven, based on a topological nonlinear stabilizability test, that there does not exist a continuous closed-loop controller that can transfer a spacecraft onto a parabolic escape trajectory.

Nonlinear Behaviors of Bandpass Sigma–Delta Modulators With Stable System Matrices

It has been established that a class of bandpass sigma-delta modulators may exhibit state space dynamics which are represented by elliptical or fractal patterns confined within trapezoidal regions when the system matrices are marginally stable. In this brief, it is found that fractal or irregular chaotic patterns may also be exhibited in the phase plane when the system matrices are strictly stable

The Crystallizing Substochastic Sequential Machine Extractor: CrySSMEx

This letter presents an algorithm, CrySSMEx, for extracting minimal finite state machine descriptions of dynamic systems such as recurrent neural networks. Unlike previous algorithms, CrySSMEx is parameter free and deterministic, and it efficiently generates a series of increasingly refined models. A novel finite stochastic model of dynamic systems and a novel vector quantization function have been developed to take into account the state-space dynamics of the system. The experiments show that (1) extraction from systems that can be described as regular grammars is trivial, (2) extraction from high-dimensional systems is feasible, and (3) extraction of approximative models from chaotic systems is possible. The results are promising, and an analysis of shortcomings suggests some possible further improvements. Some largely overlooked connections, of the field of rule extraction from recurrent neural networks, to other fields are also identified.

Organization of the state space of a simple recurrent network before and after training on recursive linguistic structures

Recurrent neural networks are often employed in the cognitive science community to process symbol sequences that represent various natural language structures. The aim is to study possible neural mechanisms of language processing and aid in development of artificial language processing systems. We used data sets containing recursive linguistic structures and trained the Elman simple recurrent network (SRN) for the next-symbol prediction task. Concentrating on neuron activation clusters in the recurrent layer of SRN we investigate the network state space organization before and after training. Given a SRN and a training stream, we construct predictive models, called neural prediction machines, that directly employ the state space dynamics of the network. We demonstrate two important properties of representations of recursive symbol series in the SRN. First, the clusters of recurrent activations emerging before training are meaningful and correspond to Markov prediction contexts. We show that prediction states that naturally arise in the SRN initialized with small random weights approximately correspond to states of Variable Memory Length Markov Models (VLMM) based on individual symbols (i.e. words). Second, we demonstrate that during training, the SRN reorganizes its state space according to word categories and their grammatical subcategories, and the next-symbol prediction is again based on the VLMM strategy. However, after training, the prediction is based on word categories and their grammatical subcategories rather than individual words. Our conclusion holds for small depths of recursions that are comparable to human performances. The methods of SRN training and analysis of its state space introduced in this paper are of a general nature and can be used for investigation of processing of any other symbol time series by means of SRN.

Evolutionary Dynamics and Long-Run Selection

We study the emergence and nature of long-run behavior of continuous state space dynamics that are subjected to random shocks. It is shown that the fine details of the underlying deterministic dynamics may be crucial in determining the evolution of the system. In particular, a risk dominated strategy can emerge in an evolutionary game subject to symmetric decaying perturbations.

Stationkeeping on unstable orbits: Generalization to the elliptic restricted three-body problem

We develop new methods for generating periodic orbits about the collinear libration points and for stabilizing motion on libration orbits using the general formalism of the elliptic restricted three-body problem (ER3BP). Calculation of periodic orbits is accomplished by formulating the ER3BP as a control problem. This approach yields halo-like orbits that do not exist without applying active control, having arbitrarily small amplitudes. Linearization about the libration points in pulsating coordinates yields an unstable linear parameter-varying (LPV) system with periodic coefficients. We introduce a continuous acceleration control term into the state-space dynamics and use an LPV-generalized version of the pole-assignment technique to find linear periodic reference trajectories. The nonlinear terms of the equations of motion are then treated as periodic disturbances. A disturbance-accommodating control is used to track the libration-point reference orbit in the presence of nonlinear periodic disturbances. Simulation experiments show that stationkeeping is robust to propulsive dispersions.

Robust stabilization of a class of polytopic linear time-varying continuous systems under point delays and saturating controls

This paper investigates the stabilization dependent on the delays of, in general, time-varying linear systems with multiple constant point time-delays under saturating state-feedback controls. The matrices describing the state-space dynamics and control belong to polytopes. Also, the controller gain matrix is characterized as belonging to another polytope whose vertices are computed from the knowledge of a closed bounded ball containing a set of values of the state norm and also the component-wise saturating gains and saturation parameterizations. This knowledge defines the polytope vertices through scaling diagonal matrices being associated with the various operation modes in the linear and saturated zones of each input component. In these conditions, the resulting closed-loop system is of polytopic nature whose whole number of vertices is (at most) equal to the product of both numbers of vertices of the above two polytopes characterizing the plant parameterization and the saturation. The closed-loop sufficiency-type stability conditions are obtained from Lyapunov’s stability theory by constructing candidates for each vertex of the polytopic closed-loop system each satisfying, in the most general case, a Riccati matrix differential inequality. Some conditions guaranteeing the stability conditions are obtained from a general Kalman–Yakubovitch–Popov (KYP) Lemma and some weaker stability conditions are also obtained for the time-invariant case from a set of linear matrix inequalities associated with the set of vertices.

Voltage-calcium state-space dynamics during initiation of reentry

Voltage-calcium state-space dynamics during initiation of reentry

Study of nonlinear phenomena in a multi-bit bandpass sigma delta modulator

Bandpass sigma delta modulators (SDMs) have applications in areas such as digital radio demodulation. It is well known that bandpass SDMs with single bit quantizers can exhibit nonlinear and complex state space dynamics like elliptical fractal patterns. These fractal patterns are usually confined in trapezoidal regions. In this paper, we consider bandpass SDMs with multi-bit quantizers. Their nonlinear dynamics is studied.

Parameter dependent Lyapunov functions for robust stability of time-varying linear systems under point delays

This paper investigates the stabilization dependent on the delays of, in general, time-varying linear systems with multiple constant point time-delays. The matrices describing the state-space dynamics are parameterized by, in general, time-varying function matrices governed by coefficient functions in compact sets. The closed-loop sufficiency-type stability conditions are obtained from Lyapunov’s stability theory by constructing parameter-dependent candidates which satisfy, in the most general case, a Riccati matrix differential inequality.

A geometric description of the intermediate behaviour for spatially homogeneous models

A new approach is suggested for the study of geometric symmetries in general relativity, leading to an invariant characterization of the evolutionary behaviour for a class of spatially homogeneous (SH) vacuum and orthogonal γ-law perfect fluid models. Exploiting the 1 + 3 orthonormal frame formalism, we express the kinematical quantities of a generic symmetry using expansion-normalized variables. In this way, a specific symmetry assumption leads to geometric constraints that are combined with the associated integrability conditions, coming from the existence of the symmetry and the induced expansion-normalized form of Einstein's field equations (EFE), to give a close set of compatibility equations. By specializing to the case of a kinematic conformal symmetry (KCS), which is regarded as the direct generalization of the concept of self-similarity, we give the complete set of consistency equations for the whole SH dynamical state space. An interesting aspect of the analysis of the consistency equations is that, at least for class A models which are locally rotationally symmetric or lying within the invariant subset satisfying Nαα = 0, a proper KCS always exists and reduces to a self-similarity of the first or second kind at the asymptotic regimes, providing a way for the ‘geometrization’ of the intermediate epoch of SH models.

Method for Determining a Dynamical State–Space Model for Control of Thermal MEMS Devices

A method is presented for determining lumped dynamical models of thermal microelectromechanical systems (MEMS) devices for purposes of feedback control. As a case study, an electrothermal actuator is used. The physical properties and a set of assumptions are used to determine the basic structure of the dynamical model, which requires the development of the electrical, thermal, and mechanical dynamics. The importance of temperature-dependent parameters is emphasized for dynamical modeling for purposes of feedback control. To confront temperature dependence in a practical yet effective manner, an average temperature is introduced to preserve the energy balance inside the structure. This allows the development of a practical method that combines structure of the model, through the average body temperature, with finite element analysis (FEA) in novel way to perform system identification and identify the unknown parameters. The result is a lumped dynamical model of a MEMS device that can be used for the design of feedback control systems. We compare computer simulated results using the dynamical model with experimental behavior of the actual device to show that our procedure indeed generates an accurate model. This dynamical model is intended for further synthesis of driving signal and control system but also gives a qualitative insight into the relationship between device's geometry and its behavior. The method enables fast development of the model by conducting relatively few static FEA and is verifiable with dynamic experimental results even when temperature measurements are not available. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$hfillhbox[1406]$</tex>

OCCURRENCE OF ELLIPTICAL FRACTAL PATTERNS IN MULTI-BIT BANDPASS SIGMA DELTA MODULATORS

It has been established that the class of bandpass sigma delta modulators (SDMs) with single bit quantizers could exhibit state space dynamics represented by elliptic or fractal patterns confined within trapezoidal regions. In this letter, we find that elliptical fractal patterns may also occur in bandpass SDMs with multibit quantizers, even for the case when the saturation regions of the multibit quantizers are not activated and a large number of bits are used for the implementation of the quantizers. Moreover, the fractal pattern may occur for low bit quantizers, and the visual appearance of the phase portraits between the infinite state machine and the finite state machine with high bit quantizers is different. These phenomena are different from those previously reported for the digital filter with two's complement arithmetic. Furthermore, some interesting phenomena are found. A bit change of the quantizer can result in a dramatic change in the fractal patterns. When the trajectories of the corresponding linear systems converge to a fixed point, the regions of the elliptical fractal patterns diminish in size as the number of bits of the quantizers increases.

Canonical Modelling of Relative Spacecraft Motion Via Epicyclic Orbital Elements

This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an example, closed-form solutions for J2-invariant orbits are obtained.

Equilibrium points of the tilted perfect fluid Bianchi VIh state space

We present the full set of evolution equations for the spatially homogeneous cosmologies of type VI$_h$ filled with a tilted perfect fluid and we provide the corresponding equilibrium points of the resulting dynamical state space. It is found that only when the group parameter satisfies $h>-1$ a self-similar solution exists. In particular we show that for $h>-{\frac 19}$ there exists a self-similar equilibrium point provided that $\gamma \in ({\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}},{\frac 32}) $ whereas for $h<-{\frac 19}$ the state parameter belongs to the interval $\gamma \in (1,{\frac{2(3+\sqrt{-h})}{5+3\sqrt{-h}}}) $. This family of new exact self-similar solutions belongs to the subclass $n_\alpha ^\alpha =0$ having non-zero vorticity. In both cases the equilibrium points have a five dimensional stable manifold and may act as future attractors at least for the models satisfying $n_\alpha ^\alpha =0$. Also we give the exact form of the self-similar metrics in terms of the state and group parameters. As an illustrative example we provide the explicit form of the corresponding self-similar radiation model ($\gamma={\frac 43}$), parametrised by the group parameter $h$. Finally we show that there are no tilted self-similar models of type III and irrotational models of type VI$_h$.

Enhanced Nonlinear Dynamics for Accurate Identification of Stiffness Loss in a Thermo-Shielding Panel

The dynamics of a panel forced by transverse loads and undergoing limit cycle oscillations and chaos is investigated. The nonlinear von Karman plate theory is used to obtain a model for healthy and damaged panels. Damage is modeled by a loss of stiffness in a portion of the plate. The presence of low levels of damage is identified by using an external nonlinear excitation and analyzing the attractor of the resulting dynamics in state space. Most of the current studies of such problems are based on linear theories and linear structures. In contrast, the results presented are obtained by using and enhancing nonlinear and chaotic dynamics, and have the advantage of an increased accuracy in detecting damage and monitoring structural health.

Self-similar Bianchi models: II. Class B models

In a companion article (referred to hereafter as paper I) a detailed study of the simply transitive spatially homogeneous (SH) models of class A concerning the existence of a simply transitive similarity group has been given. The present work (paper II) continues and completes the above study by considering the remaining set of class B models. Following the procedure of paper I we find all SH models of class B subjected only to the minimal geometric assumption to admit a proper homothetic vector field (HVF). The physical implications of the obtained geometric results are studied by specializing our considerations to the case of vacuum and γ-law perfect fluid models. As a result, we regain all the known exact solutions regarding vacuum and non-tilted perfect fluid models. In the case of tilted fluids, we find the general self-similar solution for the exceptional type VI−1/9 model and we identify it as an equilibrium point in the corresponding dynamical state space. It is found that this new exact solution belongs to the subclass of models nαα = 0, is defined for , and although it has a five-dimensional stable manifold there always exist two unstable modes in the restricted state space. Furthermore, the analysis of the remaining types guarantees that tilted perfect fluid models of types III, IV, V and VIIh cannot admit a proper HVF, strongly suggesting that these models either may not be asymptotically self-similar (type V) or may be extreme tilted at late times. Finally, for each Bianchi type, we give the extreme tilted equilibrium points of their state space.

Nonlinear Optimal Control Using Dynamic Programming in Cell Space — Application to Nonlinear CSTR

Optimal control problems for nonlinear systems are generally difficult to study. Closed-form solutions are often limited to linear systems and quadratic performance indices. This paper proposes a numerical approach to solve nonlinear optimal control problems using Bellman's principle of optimality in discrete time and space. The approach is based on Simple Cell Mapping (SCM), a numerical procedure to approximately describe nonlinear state-space dynamics. The paper discusses the construction of a general control database and performance index database along with a backward search algorithm to formulate optimal control policy. The cell space dynamic programming methodology is investigated on a nonlinear CSTR model. Numerical studies dealing with the influence of space and time discretization on computational feasibility are presented

Accessibility, stabilizability, and feedback control of continuous orbital transfer.

This paper investigates the problem of low-thrust orbital transfer using orbital element feedback from a control-theoretic standpoint, concepts of controllability, feedback stabilizability, and their interaction. The Gauss variational equations (GVEs) are used to model the state-space dynamics. First, the notion of accessibility, a weaker form of controllability, is presented. It is then shown that the GVEs are globally accessible. Based on the accessibility result, a nonlinear feedback controller is derived that asymptotically steers a vehicle from an initial elliptic Keplerian orbit to any given elliptic Keplerian orbit. The performance of the new controller is illustrated by simulating an orbital transfer between two geosynchronous Earth orbits. It is shown that the low-thrust controller requires less fuel than an impulsive maneuver for the same transfer time. Closed-form, analytic expressions for the new orbital transfer controller are given. Finally, it is proved, based on a topological nonlinear stabilizability test, that there does not exist a continuous closed-loop controller that can transfer a spacecraft to a parabolic escape trajectory.