Simple Dynamical System Research Articles

Resource and revenue potential of California residential load participation in ancillary services

Increasing penetrations of intermittent renewable energy resources will require additional power system services. California recently adopted an energy storage mandate to support its renewable portfolio standard, which requires 33% of delivered energy from renewables by 2020. The objective of this paper is to estimate the amount of energy storage that could be provided by residential thermostatically controlled loads, such as refrigerators and air conditioners, and the amount of revenue that could be earned by loads participating in ancillary services markets. We model load aggregations as virtual energy storage, and use simple dynamical system models and publicly available data to generate our resource and revenue estimates. We find that the resource potential is large: 10–40GW/8–12GWh, which is significantly more than that required by the mandate. We also find that regulation and spinning/non-spinning reserve revenues vary significantly depending upon type of load and, for heat pumps and air conditioners, climate zone. For example, mean regulation revenues for refrigerators are $11/year, for electric water heaters are $24/year, for air conditioners are $0-32/year, and for heat pumps are $22–56/year. Both consumer choices, such as appliance settings, and policy, such as the design of ancillary service compensation and appliance standards, could increase revenue potentials.

Leader and Predecessor Following Robust Controller Synthesis for String Stable Heterogeneous Vehicle Platoons

Conditions of string (in)stability for look-ahead interconnected vehicle systems are presented based on a compact characterization of the interconnected system. Continuous-time local models describe the temporal evolution of the state-variables of one vehicle. Discrete spatially varying systems describe the spatial evolution of local systems. The problem of string stability analysis of large scale interconnected systems is reduced to the stability analysis of two simple dynamic systems. The evaluation of worst-case spacing errors, as responses to bounded L2 leader acceleration, can be upper-bounded by computing the step-response to a simple discrete linear system. Based on the compact characterization, it is also shown that the string stability requirement can be directly converted to a standard control problem. The efficiency of the synthesis method is demonstrated on a numerical example.

One-year dynamic monitoring of a masonry tower

The paper presents some results of the continuous dynamic monitoring program carried out on the tallest historic tower in Mantua, Italy. This project follows an extensive diagnostic investigation aimed at assessing the structural condition of the tower after the Italian earthquakes of May 2012. A simple dynamic monitoring system was permanently installed in the upper part of the building and automatic modal identification was performed. The results allow to evaluate the effects of changing temperature on automatically identified natural frequencies, to verify the practical feasibility of damage detection methods based on natural frequencies shifts and provide clear evidence of the possible key role of continuous dynamic monitoring in the preventive conservation of historic towers.

Global Finite-Time Stabilization for Nonholonomic Mobile Robots Based on Visual Servoing

In this paper, the global finite-time stabilization problem is considered for nonholonomic mobile robots based on visual servoing with uncalibrated visual parameters, control direction and unmatched external disturbances. Firstly, the simple dynamic chained-form systems is obtained by using a state and input transformation of the kinematic robot systems. Secondly, a new discontinuous switching controller is presented in the presence of uncertainties and disturbances, it is rigorously proved that the corresponding closed-loop system can be stabilized to the origin equilibrium point in a finite time. Finally, the simulation results show the effectiveness of the proposed control design approach.

On improving wave energy conversion, part II: Development of latching control technologies

In the first part of the investigation, a new latching control technology is proposed, and it has been shown that the new latching control technique is capable of greatly improving wave energy conversion in regular waves. In this part of the research, a new analysis technique is developed for studying the latching control technology. A ‘time-out’ method is developed and employed for ‘re-packing’ the dynamic system, hence the analysis of the latching control technology can be changed from a complete nonlinear dynamics into a simple linear dynamic system, and it is further proven that the ‘re-packed’ dynamic system can be transformed back to frequency domain for further analysis. In the research, we could prove how the phase optimal condition can be attained. Further on, the new latching control technology will be used in irregular waves. Unlike many other latching control technologies, the new latching control does not need the detailed future information. In the development of the technology, we will show how we can obtain the latching duration for irregular waves for improving wave power extraction. As a result, we could remove one barrier in implementing latching control strategy while the wave energy conversion can still be much improved.

Quantification of Highly Coupled Dynamic Fault Tree Using IRVPM and SBDD

Dynamic fault tree (DFT) is a commonly used method to model systems having sequence-dependent and function-dependent failure behaviors. The failure structure function of a DFT can be expressed by logic OR of all minimal cut sequences, that is, minimal cut sequence set (MCSS). The occurrence probability to the top event of a DFT can be calculated using inclusion–exclusion (IE) principle based on enumerating the MCSS. However, the IE-based approach would have exponential evaluation complexity. Then, a sequential binary decision diagram (SBDD)-based method is proposed and successfully applied to analyze simple dynamic systems. This method is more efficient than IE-based method in asymptotic analysis. But this method cannot handle complex systems modeled by different highly coupled dynamic gates. In this paper, we put forward using Independent Random Variable Probabilistic Model-based plus SBDD-based methods to quantify an MCSS to obtain the failure probability of a complex DFT. The results obtained by the proposed method are exactly matched with those obtained by the existing methods. In addition, this method enhances the analyzing ability of the original SBDD and retains the advantage of high computational efficiency. The application and advantage of our proposed method is demonstrated by a case study. Copyright © 2014 John Wiley & Sons, Ltd.

Autonomous learning by simple dynamical systems with delayed feedback.

A general scheme for the construction of dynamical systems able to learn generation of the desired kinds of dynamics through adjustment of their internal structure is proposed. The scheme involves intrinsic time-delayed feedback to steer the dynamics towards the target performance. As an example, a system of coupled phase oscillators, which can, by changing the weights of connections between its elements, evolve to a dynamical state with the prescribed (low or high) synchronization level, is considered and investigated.

Catchments as simple dynamical systems: A case study on methods and data requirements for parameter identification

AbstractIn many rainfall‐runoff models, at least some calibration of model parameters has to take place. Especially for ungauged or poorly gauged basins this can be problematic, because there is little or no data available for calibration. A possible solution to overcome the problems caused by data scarcity is to set up a measurement campaign for a limited time period. In this study, we determine the minimum amount of data required to determine robust parameter values for a simple model with two parameters. The model is constructed such that the parameters can be determined not only with automatic calibration, but also by recession analysis and a priori from Boussinesq theory. The model has been applied to a research catchment in Switzerland. For automatic calibration and recession analysis, one season (5 months) is found to be sufficient to give robust parameters for simulation of high flows over the full observation period. For automatic calibration, this should be the season with the highest precipitation, for recession analysis the season with least evapotranspiration. The Boussinesq equation is able to give good parameter estimates for modeling high flows, but detailed in situ knowledge of the catchment is required. Automatic calibration outperforms recession analysis and Boussinesq theory by far when it comes to parameter estimation with a focus on prediction of low flows. It was shown that a single set of parameters cannot simultaneously describe high and low flows with a reasonable accuracy, suggesting that more than two parameters are needed to characterize subsurface properties.

Reply to "comment on 'how to obtain extreme multistability in coupled dynamical systems' ".

In this Reply we answer the two major issues raised by the Comment. First, we point out that the idea of constructing extreme multistability in simple dynamical systems is not new and has been demonstrated previously by other authors. Furthermore, we emphasize the importance of the concept of a conserved quantity and its consequences for the dynamics, which applies to all the examples in the Comment. Second, we show that the design of controllers to achieve extreme multistability in coupled systems is as general as described in Phys. Rev. E 85, 035202(R) (2012) by providing two examples which do not lead to a master-slave dynamics.

Galaxies as simple dynamical systems: observational data disfavor dark matter and stochastic star formation

According to modern theory, galactic evolution is driven by the dynamics of dark matter and stochastic star formation, but galaxies are observed to be simple systems. The existence of dark matter particles is a key hypothesis in present-day cosmology and galactic dynamics. Given the large body of high-quality work within the standard model of cosmology (SMoC), the validity of this hypothesis is challenged significantly by two independent arguments: (i) The dual dwarf galaxy theorem must be true in any realistic cosmological model. But the data now available appear to falsify this postulate when the dark-matter-based model is compared with the observational data. A consistency test for this conclusion comes from the significantly anisotropic distributions of satellite galaxies (baryonic mass <108M⊙) that orbit in the same direction around their hosting galaxies in disk-like structures, which cannot be derived from dark matter models. (ii) The action of dynamical friction due to expansive and massive dark matter halos must be evident in the galaxy population. The evidence for dynamical friction is poor or even absent. Indendently of this, the long history of failures of the SMoC have reduced the likelihood that it describes the observed Universe to less than 10−4%. The implication for fundamental physics is that exotic dark matter particles do not exist and that consequently effective gravitational physics on the scales of galaxies and beyond ought to be non-Newtonian and (or) non-Einsteinian. An analysis of the kinematic data in galaxies shows them to be described elegantly in the weak-gravitational regime by scale-invariant dynamics, as discovered by Milgrom. The full classical regime of gravitation is effectively described by Milgromian dynamics. This leads to a natural emergence of the simple laws that galaxies are indeed observed to obey. Such success has not been forthcoming in dark-matter-based models. Observations of stellar populations in galaxies suggest that the galaxy-wide initial mass function varies with star formation rate and that stochastic descriptions of star formation are inconsistent with the data. This requires a reinterpretation of the stellar mass assembly in galaxies and thus of the accretion rates onto galaxies. A consequence of this understanding of galactic astrophysics is that most dwarf satellite galaxies are formed as tidal dwarf galaxies in galaxy–galaxy encounters, that they follow the mass–metallicity relation, that galactic mergers are rare, that galaxies immersed in external potentials are physically larger than isolated galaxies, and that star-forming galaxies form a main sequence. Eight predictions are offered that will allow the concepts raised here to be tested. A very conservative, cold- and warm-dark-matter-free cosmological model may be emerging from these considerations.

Generating pseudo-random numbers by combining two systems with complex behaviors

Cellular automata (CA) due to its complex behavior has several applications such as generating random numbers and cryptography. Because of the intrinsic self-organized property, a pure CA cannot produce a long sequence of random numbers. Non-uniform, controllable/programmable CA or a combination of several automata can be used to increase the length of a produced sequence of random numbers. In this paper, a non-uniform CA as a random number generator has been combined with Langton's ants in order to generate random numbers. Langton's ant is a simple discrete dynamical system, with a surprisingly complex behavior. The combination of some Langton's ants gives them a chaotic behavior and combination of this behavior with complex behavior of cellular automata causes a great efficiency in generating random sequences. Experimental results show that, results given by the combination of ants and CA does not depend on initial value and shows a random behavior, resulting in cycles with very long period lengths and a limited number of cells such that a cycle length of 23n is obtained by n cells. Moreover, some tests such as entropy, avalanche, diehard, NIST and several basic statistical tests have been performed and all of them have been successfully passed.

About implementation of physics approach to dynamic systems simulation

Advantages of a physics approach to dynamic systems simulation are discussed. A great need for the approach at the modern state of system engineering is pointed out. In particular, such an approach may be useful for simulation of control and self-organizing systems. A situation-event formalism of an interacting hybrid process specification is briefly stated and some ways of its use for physic simulation model implementation are shown. Facilities of the considered methods are illustrated by examples of some simple dynamic system models implementations. — Bibl. 9 items.

Approximation of a linear dynamic process model using the frequency approach and a non-quadratic measure of the model error

Abstract The paper presents a novel approach to approximation of a linear transfer function model, based on dynamic properties represented by a frequency response, e.g., determined as a result of discrete-time identification. The approximation is derived for minimization of a non-quadratic performance index. This index can be determined as an exponent or absolute norm of an error. Two algorithms for determination of the approximation coefficients are considered, a batch processing one and a recursive scheme, based on the well-known on-line identification algorithm. The proposed approach is not sensitive to local outliers present in the original frequency response. Application of the approach and its features are presented on examples of two simple dynamic systems.

Generalised dynamical systems soil deformation model

When sheared past failure, particles in a soil element move into the steady-state structure at random shear strains, behaving as a simple friction-based dynamical system that closely models test data for a wide range of soils, stress paths and drainage conditions. This paper reviews this dynamical systems approach and generalises it as follows: for any stress path, particles move at random shear strains into the final structure corresponding to the applied stresses; that is, the same friction-based dynamical system-based model of shear to failure applies to any stress path, not just to those that lead to failure. The model being general, it can be specialised for the particular case of one-dimensional consolidation – that is, shear along the K0 line. Such a specialisation reveals the commonly observed linear relationship between e and [Formula: see text] for one-dimensional consolidation and hence supports the generalisation.

Dynamics near a minimal-mass soliton for a Korteweg–de Vries equation

We study soliton solutions to a generalized Korteweg–de Vries (KdV) equation with a saturated nonlinearity, following the line of inquiry of Marzuola, Raynor and Simpson for the nonlinear Schrödinger equation (NLS). KdV with such a nonlinearity is known to possess a minimal-mass soliton. We consider a small perturbation of a minimal-mass soliton and numerically shadow a system of ordinary differential equation (ODEs), which models the behaviour of the perturbation for short times. This connects nicely to analytic works of Comech, Cuccagna and Pelinovsky as well as of Grimshaw and Pelinovsky. These ODEs form a simple dynamical system with a single unstable hyperbolic fixed point with two possible dynamical outcomes. A particular feature of the dynamics is that they are non-oscillatory. This distinguishes the KdV problem from the analogous NLS one.

Vertical distribution of elements in regolith over mineral deposits and implications for mapping geochemical weak anomalies in covered areas

Several case studies are presented to demonstrate that significant migration of elements in regolith over mineral deposits can reach the earth surface through thick layers of superimposed regolith. Data obtained from drill-holes in Mo-W and Mo-Ag mineral deposits in Eastern Inner Mongolia, China using portable X-ray fluorescence (pXRF, Niton XL3t 950) and data from other areas reported in the literature have been modeled using power-law decay functions that describe the regolith decay trends with increasing distance from the underlying altered rocks or saprocks. The results demonstrate that the element concentration in the surface media can be very low due to decay and mask effects, even if a thin layer of overburden exists. In order to characterize the decay behaviour of geochemical concentration of an element in a vertical regolith profile caused by complex mechanisms, a new non-linear differential equation was proposed which assumes the decay rate of concentration is negatively proportional to the concentration itself, with a functional coefficient dependent on vertical distance from the underlying surface of mineralised rocks. Applying Taylor series expansion to the coefficient function, the differential equation can be approximated by four simple dynamic systems, each with explicit solutions including Gaussian functions, exponential functions, power-law functions and exponential functions with inverse distance. These functions can be utilized as either separate or combined models to fit observed data by means of simple linear regression or multivariate regression. The combined mode is useful for evaluating the comprehensive effect of element concentration distribution due to various mechanisms.

Dynamical sectors of a relativistic two particle model

We reconsider a model of two relativistic particles interacting via a multiplicative potential, as an example of a simple dynamical system with sectors, or branches, with different dynamics and degrees of freedom.The presence or absence of sectors depends on the values of rest masses. Some aspects of the canonical quantization are described. The model could be interpreted as a bigravity model in one dimension.

ON SMOOTH SOLUTIONS OF NON LINEAR DYNAMICAL SYSTEMS, f<sub>n+1</sub> = u(f<sub>n</sub>), PART I

We consider the dynamical system, fn+1 = u(fn), (1) (where usually n, is time) defined by a continuous map u. Our target is to find a flow of the system for each initial state f0, i.e., we seek continuous solutions of (1), with the same smoothness degree as u. We start with the introduction of continued forms which are a generalization of continued fractions. With the use of continued forms and a modulator function (i.e., weight function) m, we construct a sequence of smooth functions, which come arbitrarily close to a smooth flow of (1). The limit of this sequence is a functional transform, Km[u], of u, with respect to m. The functional transform is a solution of (1), in the sense that, Km[u] (y + c), is a flow of (1) for each translation constant c. Here we present the first part of our work where we consider a subclass of dissipative dynamical systems in the sence that they have wandering sets of positive measure. In particular we consider strictly increasing real univariate maps, u: D⟶D, D = (a+∞), where, a≤0, or, a=-∞, with the property, u(x)-x≥ε>0, which implies that u, has no real fixed points. We briefly give some mathematical and physical applications and we discuss some open problems. We demonstrate the method on the simple non-linear dynamical system, fn+1 = (fn)2+1.

Tide-surge adjoint modeling: A new technique to understand forecast uncertainty

[1] For a simple dynamical system, such as a pendulum, it is easy to deduce where and when applied forcing might produce a particular response. However, for a complex nonlinear dynamical system such as the ocean or atmosphere, this is not as obvious. Knowing when or where the system is most sensitive, to observational uncertainty or otherwise, is key to understanding the physical processes, improving and providing reliable forecasts. We describe the application of adjoint modeling to determine the sensitivity of sea level at a UK coastal location, Sheerness, to perturbations in wind stress preceding an extreme North Sea storm surge event on 9 November 2007. Sea level at Sheerness is one of the most important factors used to decide whether to close the Thames Flood Barrier, which protects London. Adjoint modeling has been used by meteorologists since the 1990s, but is a relatively new technique for ocean modeling. It may be used to determine system sensitivity beyond the scope of ensemble modeling and in a computationally efficient way. Using estimates of wind stress error from Met Office forecasts, we find that for this event total sea level at Sheerness is most sensitive in the 3 h preceding the time of its unperturbed maximum level and over a radius of approximately 300 km. We also find that the pattern of sensitivity follows a simple sequence when considered in the reverse-time direction.

On correlation decay in low-dimensional systems

While the decay of correlations in dynamical systems has been discussed in the physics and mathematics literature for several decades, there exists virtually no nontrivial example where the actual decay rates can be computed explicitly. We construct a class of simple dynamical systems for which all correlation properties, in particular, the entire spectrum of the Perron-Frobenius operator, are accessible by analytical means. As an application we discuss an example of an interval map with a small number of branches where the decay of correlations can be made arbitrarily fast. The example can be viewed as the simplest toy model which gives rise to a crossover from transient to asymptotic behaviour with predictable crossover time scale. The model class introduced here points towards relations between correlation decay and Lyapunov spectra.