Unstable Dynamical Systems Research Articles

Semidiscretization for Time-Delayed Neural Balance Control

The observation that time-delayed feedback can stabilize an inverted pendulum motivates the formu- lation of models of human balance control in terms of delay differential equations (DDEs). Recently the intermittent, digital-like nature of the neural feedback control of balance has become evident. Here, semidiscretization methods for DDEs are used to investigate an unstable dynamic system sub- jected to a digital controller in the context of a switching model for postural control. In addition to limit cycle and chaotic (microchaos) oscillations, transiently stabilized balance states are possible even though both the open-loop and the closed-loop systems are globally unstable. The possibility that falls can be an intrinsic component of neural control of balance may provide new insights into how the risk of falling in the elderly can be minimized.

Extended phase space description of human-controlled systems dynamics

Humans are often incapable of precisely identifying and implementing the desired control strategy in controlling unstable dynamical systems. That is, the operator of a dynamical system treats the current control effort as acceptable even if it deviates slightly from the desired value, and starts correcting the actions only when the deviation has become evident. We argue that the standard Newtonian approach does not allow to model such behavior. Instead, the physical phase space of a controlled system should be extended with an independent phase variable characterizing the operator motivated actions. The proposed approach is illustrated via a simple non-Newtonian model capturing the operators fuzzy perception of their own actions. The properties of the model are investigated analytically and numerically; the results confirm that the extended phase space may aid in capturing the intricate dynamical properties of human-controlled systems.

Cascaded VFO Control for Non-Standard N-Trailer Robots

Articulated mobile robots consisting of a tractor and passively off-hooked trailers belong to a class of highly nonlinear, nonholonomic, structurally unstable, differentially non-flat, and underactuated dynamic systems. Due to the mentioned properties of their kinematics, motion control problems related to N-trailer robots (N-trailers) are non-trivial and challenging. Cascaded control strategy presented in this paper provides unified solution to the set-point and trajectory tracking control tasks for articulated robots equipped with arbitrary number of passively off-axle hitched trailers. Practically useful features of the proposed controller come from its high scalability and from application of the Vector-Field-Orientation (VFO) controller in the outer loop, which ensures fast errors convergence and simplicity of control implementation and tuning. Control input limitations of the robot are directly taken into account by utilization of a simple velocity scaling procedure which preserves an instantaneous motion curvature of a tractor. Description and theoretical substantiation of the concept are followed by the results of experimental validation tests conducted with usage of a 3-trailer semi-autonomous vehicle.

Application of chaos indicators in the study of dynamics of S-type extrasolar planets in stellar binaries

The orbits of two individual planets in two known binary star systems, \gamma Cephei and HD 196885 are numerically integrated using various numerical techniques to assess the chaotic or quasi-periodic nature of the dynamical system considered. The Hill stability (HS) function which measures the orbital perturbation of a planet around the primary star due to the secondary star is calculated for each system. The maximum Lyapunov exponent (MLE) time series are generated to measure the divergence/convergence rate of stable manifolds, which are used to differentiate between chaotic and non-chaotic orbits. Then, we calculate dynamical Mean Exponential Growth factor of Nearby Orbits (MEGNO) maps from solving the variational equations along with the equations of motion. These maps allow us to accurately differentiate between stable and unstable dynamical systems. The results obtained from the analysis of HS, MLE, and MEGNO maps are analysed for their dynamical variations and resemblance. The HS test for the planets shows stability and quasi-periodicity for at least ten million years. The MLE and the MEGNO maps have also indicated the local quasi-periodicity and global stability in a relatively short integration period. The orbital stability of the systems is tested using each indicator for various values of planet inclinations (i_{pl} \le 25^\circ) and binary eccentricities. The reliability of HS criterion is also discussed based on its stability results compared with the MLE and MEGNO maps.

Attractor local dimensionality, nonlinear energy transfers and finite-time instabilities in unstable dynamical systems with applications to two-dimensional fluid flows

We examine the geometry of the finite-dimensional attractor associated with fluid flows described by Navier–Stokes equations and relate its nonlinear dimensionality to energy exchanges between dynamical components (modes) of the flow. Specifically, we use a stochastic framework based on the dynamically orthogonal equations to perform efficient order-reduction and describe the stochastic attractor in the reduced-order phase space in terms of the associated probability measure. We introduce the notion of local fractal dimensionality to describe the geometry of the attractor and we establish a connection with the number of positive finite-time Lyapunov exponents. Subsequently, we illustrate in specific fluid flows that the low dimensionality of the stochastic attractor is caused by the synergistic activity of linearly unstable and stable modes as well as the action of the quadratic terms. In particular, we illustrate the connection of the low-dimensionality of the attractor with the circulation of energy: (i) from the mean flow to the unstable modes (due to their linearly unstable character), (ii) from the unstable modes to the stable ones (due to a nonlinear energy transfer mechanism) and (iii) from the stable modes back to the mean (due to the linearly stable character of these modes).

Realization of cascades on surfaces with finitely many moduli of topological conjugacy

A method for constructing cascades on surfaces is developed, which makes it possible to model structurally unstable discrete dynamical systems with finitely many orbits of heteroclinic tangency and preset moduli of topological conjugacy.

A dynamical system approach to realtime obstacle avoidance

This paper presents a novel approach to real-time obstacle avoidance based on Dynamical Systems (DS) that ensures impenetrability of multiple convex shaped objects. The proposed method can be applied to perform obstacle avoidance in Cartesian and Joint spaces and using both autonomous and non-autonomous DS-based controllers. Obstacle avoidance proceeds by modulating the original dynamics of the controller. The modulation is parameterizable and allows to determine a safety margin and to increase the robot's reactiveness in the face of uncertainty in the localization of the obstacle. The method is validated in simulation on different types of DS including locally and globally asymptotically stable DS, autonomous and non-autonomous DS, limit cycles, and unstable DS. Further, we verify it in several robot experiments on the 7 degrees of freedom Barrett WAM arm.

Receptance-Based Partial Pole Assignment for Asymmetric Systems Using State-Feedback

Many structures or machines interact with some internal nonconservative forces and present asymmetric systems in which the stiffness and damping matrices are asymmetric. Examples include friction-induced vibration and aeroelastic flutter. Asymmetric systems are prone to flutter instability as a result of the real parts of some poles becoming positive when certain system parameters vary. This paper presents a receptance-based inverse method for assigning a number of complex poles of second-order damped asymmetric systems while keeping other unassigned poles unchanged. It uses state-feedback (active damping and active stiff- ness) to shift the poles to desired locations where all poles have negative real parts. Receptances at only a small, limited number of degrees-of-freedom of the underlying symmetric system are required. Simulated numerical examples indicate that this is an effective method and is capable of assigning negative real parts to unstable poles to stabilise an otherwise unstable second-order dynamic system.

Receptance-Based Partial Pole Assignment for Asymmetric Systems Using State-Feedback

Many structures or machines interact with some internal nonconservative forces and present asymmetric systems in which the stiffness and damping matrices are asymmetric. Examples include friction-induced vibration and aeroelastic flutter. Asymmetric systems are prone to flutter instability as a result of the real parts of some poles becoming positive when certain system parameters vary.This paper presents a receptance-based inverse method for assigning a number of complex poles of second-order damped asymmetric systems while keeping other unassigned poles unchanged. It uses state-feedback (active damping and active stiffness) to shift the poles to desired locations where all poles have negative real parts. Receptances at only a small, limited number of degrees-of-freedom of the underlying symmetric system are required. Simulated numerical examples indicate that this is an effective method and is capable of assigning negative real parts to unstable poles to stabilise an otherwise unstable second-order dynamic system.

Lyapunov exponents of hybrid stochastic heat equations

In this paper, we investigate a class of hybrid stochastic heat equations. By explicit formulae of solutions, we not only reveal the sample Lyapunov exponents but also discuss the p th moment Lyapnov exponents. Moreover, several examples are established to demonstrate that unstable (deterministic or stochastic) dynamical systems can be stabilized by Markovian switching.

Maximal controllability of input constrained unstable systems by the addition of implicit constraints

The null controllable set of a system is the largest set of states that can be controlled to the origin. Control systems that have a region of attraction equal to the null controllable set are said to be maximally controllable closed-loop systems. In the case of open-loop unstable plants with amplitude constrained control it is well known that the null controllable set does not cover the entire state-space. Further the combination of input constraints and unstable system dynamics results in a set of state constraints which we call implicit constraints. It is shown that the simple inclusion of implicit constraints in a controller formulation results in a controller that achieves maximal controllability for a class of open-loop unstable systems.

Chaos and Gene expression: A Theoretical Approach

Intoduction: The aim of the study was to investigate whether chaotic phenomena (chaos theory) affects the process of Gene Expression. Methods: Modeling the genes X, Y and Z-which encode a certain protein P-a set of three first order differential equations has been developed and studied in phase-space (x,y,z). Results: The elementary equilibrium points in three dimensional phase portrait analysis, include attractors, saddles and repellors. Conclusions: Attractors indicate a stable equilibrium point which attenuates the production of the protein P, while the saddles and particularly the repellors correspond to an unstable dynamic system, which promotes either the production of a P-flaw protein or totally inhibits gene expression. Among other mechanisms-e.g. patents for gene treatment (US2007000515344) and modulation(US20040014083A1)-chaotic phenomena also seem to regulate in a particular way the DNA encoding , that calls for further theoretical and experimental research(e.g. cardiovascular disease, oncology). Keywords: DNA, chaos, phase-space, gene expression, cardiovascular disease

Mechatronic models for education -Robotic sea lion

AbstractThis paper deals with a mechatronic model called Robotic sea lion, which has been built for educational purposes at Department of Cybernetics in Pilsen. The model serves as a demonstration tool for evaluation of various control system design methods. At first a matematical model of the system is derived, next the control laws are designed. The important limitations for the control system performance due to the limited bandwidth of the sensors and actuators and problems coupled with unstable system dynamics are discussed. These limitations call for an advanced techniques of robust control, therefore a H∞ optimization was used to obtain suitable control algorithms.

Some remarks on stabilizability of linear stochastic hybrid systems

AbstractThe problem of the stabilizability of stochastic linear and bilinear hybrid systems is considered. Construction methods of stabilizing control signals for a class of linear and bilinear systems are derived. Also, the possibility of stabilizing of unstable dynamic systems with the hybrid control signal given in the form of a signal switching is discussed. The obtained results are illustrated by examples and simulations.

Stabilization and tracking control for a class of nonlinear systems

This paper discusses stabilization and tracking control using linear matrix inequalities for a class of systems with Lipschitz nonlinearities. A nonlinear state feedback stabilization control is proposed for systems containing a more general case of Lipschitz nonlinearity. The main objective of the present study is to provide, for multi-input multi-output nonlinear systems, a tracking control approach based on nonlinear state feedback, which guarantees global asymptotic output and state tracking with zero tracking error in the steady state. Further, the tracking control is formulated for optimal disturbance rejection, using L 2 gain reduction based performance criteria. The proposed methodologies are illustrated herein using two simulation examples of chaotic and unstable dynamical systems.

A numerical study of the hyperbolic manifolds in a priori unstable systems. A comparison with Melnikov approximations

Using numerical methods we study the hyperbolic manifolds in a model of a priori unstable dynamical system. We compare the numerically computed manifolds with their analytic expression obtained with the Melnikov approximation. We find that, at small values of the perturbing parameter, the topology of the numerically computed stable and unstable manifolds is the same as in their Melnikov approximation. Increasing the value of the perturbing parameter, we find that the stable and unstable manifolds have a peculiar topological transition. We find that this transition occurs near those values of the perturbing parameter for which the error terms of Melnikov approximations have a sharp increment. The transition value is also correlated with a change in the behaviour of dynamical quantities, such as the largest Lyapunov exponent and the diffusion coefficient.

Hybrid shooting – a new optimization method for unstable dynamical systems

Chemical processes are modeled by large-scale, highly-nonlinear process models often governed by unstable dynamics. Dynamic optimization is required to exploit economical performance of these processes in their unstable regions. On the one hand, direct single shooting is able to solve large-scale dynamic optimization problems, but lacks the ability to cope properly with unstable dynamical systems. On the other hand, the multiple shooting method is capable of dealing with instabilities but results in larger optimization problems. This work deals with a novel parameterization approach, termed hybrid shooting, which combines the advantages of single and multiple shooting into one approach. Similar to multiple shooting, stages are introduced and states with unstable properties are parameterized by free initial values at the beginning of each stage. This approach significantly enhances the behavior of the optimization problem by improving the condition of the underlying initial value problem (IVP).

Signalling problem in absolutely unstable systems

The dynamics of unstable systems crucially depends on the nature of the instability, either convective or absolute. The signalling problem, which is the study of the spatial response to a localized time-harmonic forcing, is generally believed to be relevant only for stable or convectively unstable systems and to be ill-posed for absolutely unstable systems, where the self-sustained perturbations grow faster than the forced harmonic response. The present investigation shows that the signalling problem may still be well posed for media displaying absolutely unstable regions. Considering weakly spatially inhomogenous systems, conditions are derived for the validity of the signalling problem. The complete spatial response to harmonic forcing is first analytically derived in terms of asymptotic approximations and then confirmed by direct numerical simulations.

Real-Time Optimal Control for Rotary Inverted Pendulum

Problem statement: The rotary inverted pendulum system was a highly nonlinear model, multivariable and absolutely unstable dynamic system. It was used for testing various design control techniques and in teaching modern control. The objectives of this study were to: (i) Develop a real rotary inverted pendulum which derived the mechanical model by using Euler-Lagrange and (ii) Design controller algorithm for self-erecting and balancing of a rotary inverted pendulum. Approach: Research shown a convenient way to implement a real-time control in self-erecting a pendulum from downward position and balancing the pendulum in vertical-upright position. An Energy based on PD controller was applied in self-erecting of the pendulum while LQR controller was applied to balance the pendulum. Results: Results of both control techniques from computer simulation and experiment were given to show the effectiveness of these controllers. Conclusion: Both simulations and experiments were confirmed the control efficiency of the method.

Cold atoms in double-well optical lattices

Cold atoms, loaded into an optical lattice with double-well sites, are considered. Pseudospin representation for an effective Hamiltonian is derived. The system in equilibrium displays two phases, ordered and disordered. The second-order phase transition between the phases can be driven either by temperature or by changing the system parameters. Collective pseudospin excitations have a gap disappearing at the phase-transition point. Dynamics of atoms is studied, when they are loaded into the lattice in an initially nonequilibrium state. It is shown that the temporal evolution of atoms, contrary to their equilibrium thermodynamics, cannot be described in the mean-field approximation, since it results in a structurally unstable dynamical system, but a more accurate description is necessary taking account of attenuation effects.