Continuous-time Approximation Research Articles

Detrending moving average algorithm: Frequency response and scaling performances.

The Detrending Moving Average (DMA) algorithm has been widely used in its several variants for characterizing long-range correlations of random signals and sets (one-dimensional sequences or high-dimensional arrays) over either time or space. In this paper, mainly based on analytical arguments, the scaling performances of the centered DMA, including higher-order ones, are investigated by means of a continuous time approximation and a frequency response approach. Our results are also confirmed by numerical tests. The study is carried out for higher-order DMA operating with moving average polynomials of different degree. In particular, detrending power degree, frequency response, asymptotic scaling, upper limit of the detectable scaling exponent, and finite scale range behavior will be discussed.

Approximation of linear distributed parameter systems by delay systems

The present work addresses continuous-time approximation of distributed parameter systems governed by linear one-dimensional partial differential equations. While approximation is usually realized by lumped systems, that is finite dimensional systems, we propose to approximate the plant by a time-delay system. Within the graph topology, we prove that, if the plant admits a coprime factorization in the algebra of BIBO-stable systems, any linear distributed parameter plant can be approximated by a time-delay system, governed by coupled differential–difference equations. Considerations on stabilization and state-space realization are carried out. A numerical method for constructive approximation is also proposed and illustrated.

The Post-Disaster Debris Clearance Problem Under Incomplete Information

Debris management is one of the most time consuming and complicated activities among post-disaster operations. Debris clearance is aimed at pushing the debris to the sides of the roads so that relief distribution and search-and-rescue operations can be maintained in a timely manner. Given the limited resources, uncertainty, and urgency during disaster response, efficient and effective planning of debris clearance to achieve connectivity between relief demand and supply is important. In this paper, we define the stochastic debris clearance problem (SDCP), which captures post-disaster situations where the limited information on the debris amounts along the roads is updated as clearance activities proceed. The main decision in SDCP is to determine a sequence of roads to clear in each period such that benefit accrued by satisfying relief demand is maximized. To solve SDCP to optimality, we develop a partially observable Markov decision process model. We then propose a heuristic based on a continuous-time approximation, and we further reduce the computational burden by applying a limited look ahead on the search tree and heuristic pruning. The performance of these approaches is tested on randomly generated instances that reflect various geographical and information settings, and instances based on a real-world earthquake scenario. The results of these experiments underline the importance of applying a stochastic approach and indicate significant improvements over heuristics that mimic the current practice for debris clearance.

Grid Power-Delay Tradeoff for Energy Harvesting Wireless Communication Systems with Finite Renewable Energy Storage

In this paper, we study the grid power-delay tradeoff in a point-to-point energy harvesting wireless communication system with finite energy storage capacity serving delay-sensitive applications. This communication system is powered by both grid and renewable power sources. First, we consider the average grid power consumption minimization subject to the data queue stability and renewable energy availability constraints. By exploring the optimality property and using the theory of random walks, we transform the grid power minimization problem to an asymptotically equivalent problem. Using the Lyapunov drift approach, we obtain an online dynamic power control policy to solve the asymptotically equivalent problem. Then, we introduce a novel analysis framework to study the grid power-delay tradeoff relationship of the online power control in the small delay regime. Specifically, using continuous-time approximation, dynamic programming and sample-path approach, we obtain bounds on the average delay and grid power consumption, which are asymptotically tight in the small delay regime. Based upon the derived closed-form expressions, we quantify the impacts of energy storage capacity and some other system parameters on the grid power-delay tradeoff.

Multi-objective optimal design of active vibration absorber with delayed feedback

In this paper, a multi-objective optimal design of delayed feedback control of an actively tuned vibration absorber for a stochastically excited linear structure is investigated. The simple cell mapping (SCM) method is used to obtain solutions of the multi-objective optimization problem (MOP). The continuous time approximation (CTA) method is applied to analyze the delayed system. Stability is imposed as a constraint for MOP. Three conflicting objective functions including the peak frequency response, vibration energy of primary structure and control effort are considered. The Pareto set and Pareto front for the optimal feedback control design are presented for two examples. Numerical results have found that the Pareto optimal solutions provide effective delayed feedback control design.

Implicit numerical methods for highly nonlinear neutral stochastic differential equations with time-dependent delay

This paper represents the continuation of the analysis from papers Milošević (2011) [10] and Milošević (2013) [11]. The main aim of this paper is to establish certain results for the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay. For that purpose, the split-step backward Euler method, which represents an extension of the backward Euler method, is introduced for this class of equations. Conditions under which the split-step backward Euler method, and thus the backward Euler method, is well defined are revealed. Moreover, the convergence in probability of the backward Euler method is proved under certain nonlinear growth conditions including the one-sided Lipschitz condition. This result is proved using the technique which is based on the application of the continuous-time approximation. For this reason, the discrete forward–backward Euler method is involved since it allows its continuous version to be well defined from the aspect of measurability. The convergence in probability is established for the continuous forward–backward Euler solution, which is essential for proving the same result for both discrete forward–backward and backward Euler methods. Additionally, it is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable, without the linear growth condition on the drift coefficient of the equation. As usual, the whole consideration is affected by the presence and properties of the delay function.

A 4-Stated DICE: Quantitatively Addressing Uncertainty Effects in Climate Change

We introduce a version of the DICE-2007 model designed for uncertainty analysis. DICE is a wide-spread deterministic integrated assessment model of climate change. Climate change, long-term economic development, and their interactions are highly uncertain. The quantitative analysis of optimal mitigation policy under uncertainty requires a recursive dynamic programming implementation of integrated assessment models. Such implementations are subject to the curse of dimensionality. Every increase in the dimension of the state space is paid for by a combination of (exponentially) increasing processor time, lower quality of the value or policy function approximations, and reductions of the uncertainty domain. The paper promotes a state-reduced, recursive dynamic programming implementation of the DICE-2007 model. We achieve the reduction by simplifying the carbon cycle and the temperature delay equations. We compare our model’s performance and that of the DICE model to the scientific AOGCM models emulated by MAGICC 6.0 and find that our simplified model performs equally well as the original DICE model. Our implementation solves the infinite planning horizon problem in an arbitrary time step. The paper is the first to carefully analyze the quality of the value function approximation using two different types of basis functions and systematically varying the dimension of the basis. We present the closed form, continuous time approximation to the exogenous (discretely and inductively defined) processes in DICE, and we present a numerically more efficient re-normalized Bellman equation that, in addition, can disentangle risk attitude from the propensity to smooth consumption over time.

A 4-Stated Dice: Quantitatively Addressing Uncertainty Effects in Climate Change

We introduce a version of the DICE-2007 model designed for uncertainty analysis. DICE is a wide-spread deterministic integrated assessment model of climate change. Climate change, long-term economic development, and their interactions are highly uncertain. The quantitative analysis of optimal mitigation policy under uncertainty requires a recursive dynamic programming implementation of integrated assessment models. Such implementations are subject to the curse of dimensionality. Every increase in the dimension of the state space is paid for by a combination of (exponentially) increasing processor time, lower quality of the value or policy function approximations, and reductions of the uncertainty domain. The paper promotes a state reduced, recursive dynamic programming implementation of the DICE-2007 model. We achieve the reduction by simplifying the carbon cycle and the temperature delay equations. We compare our model's performance and that of the DICE model to the scientific AOGCM models emulated by MAGICC 6.0 and find that our simplified model performs equally well as the original DICE model. Our implementation solves the infinite planning horizon problem in an arbitrary time step. The paper is the first to carefully analyze the quality of the value function approximation using two different types of basis functions and systematically varying the dimension of the basis. We present the closed form, continuous time approximation to the exogenous (discretely and inductively defined) processes in DICE, and we present a numerically more efficient re-normalized Bellman equation that, in addition, can disentangle risk attitude from the propensity to smooth consumption over time.

Binary Recurrent Statistical Experiments with Persistent Linear Regression

We construct a stochastic approximation of binary statistical experiments with persistent regression by the autoregression process with normal perturbations. The continuous-time diffusion approximation of the Ornstein–Uhlenbeck type is obtained. The stationary distributions of the approximation processes are calculated.

Numerical stability analysis of linear stochastic delay differential equations using Chebyshev spectral continuous time approximation

In this paper, a numerical approach is developed for the stability analysis of linear stochastic delay differential equations (SDDEs) in the parameter space based on the Chebyshev Spectral Continuous Time Approximation (CSCTA) technique. The CSCTA method is used to approximate an infinite-dimensional linear SDDE as a set of linear stochastic differential equations (SDEs). The mean and mean-square stability concepts are employed for the stochastic stability analysis of the resulting SDE. For this purpose, a set of linear deterministic differential equations for both the first and second moments are obtained using the Ito differential rule. Two examples are provided: a first order SDDE with multiplicative stochastic excitation and a second order SDDE with both additive and multiplicative stochastic excitation. In both examples the stability charts obtained from the proposed approach match those obtained using the stochastic semi-discretization method as described by Elbeyli et al. (Commun Nonlinear Sci Numer Simul 10(1):85–94, 2005). In the first example the stability results obtained from both numerical approaches are found to be less conservative than the Lyapunov-based stability region obtained by Samiei et al. (Int J Dyn Control 1(1):64–80, 2013).

Characteristics of LNA Operation in Direct Delta–Sigma Receivers

This brief analyzes the dual role and operation of the low noise amplifier (LNA) in the recently introduced direct delta-sigma receiver (DDSR). First, the LNA functions as a transconductor in an integrator stage, and in this role, we explore the effects of LNA output impedance on quantization noise shaping by the system. In the second role of a voltage preamplifier, we show how the closed-loop DDSR structure impacts LNA voltage gain and system noise. LNA and system properties are thus intertwined and lead to the need for careful codesign. The reliability of the utilized continuous-time DDSR approximation is verified by simulating a sample receiver model.

Optimal feedback control strategies for periodic delayed systems

In this study, three strategies based on infinite-dimensional Floquet theory, Chebyshev spectral collocation, and the Lyapunov–Floquet transformation (LFT) are proposed for optimal feedback control of linear time periodic delay differential equations using periodic control gains. First, a periodic-gain discrete-delayed feedback control is implemented where optimization of the control gains is included to obtain the minimum spectral radius of the closed-loop response. Second, a large set of ODEs is obtained using the Chebyshev spectral continuous time approximation, after which optimal (time-varying LQR) control is used to obtain a periodic-gain distributed-delayed feedback control. The third strategy involves the use of both CSCTA and the reduced LFT, along with either pole-placement or time-invariant LQR used on a linear time invariant auxiliary system, to obtain a periodic-gain non-delayed feedback control that asymptotically stabilizes the original system. The delayed Mathieu equation is used as an illustrative example for all three control strategies.

Escape dynamics: A continuous-time approximation

We extend a continuous-time approximation approach to the analysis of escape dynamics in economic models with constant gain adaptive learning. This approach is based on the application of the results of continuous-time version of large deviations theory to the linear diffusion approximation of the original discrete-time dynamics under learning. We characterize escape dynamics by analytically deriving the most probable escape point and mean escape time. The approximation is tested on the Phelps problem of a government controlling inflation while adaptively learning a misspecified Phillips curve, studied previously by Sargent (1999) and Cho et al. (2002) (henceforth, CWS), among others. We compare our results with simulations extended to very low values of the constant gain and show that, for the lowest gains, our approach approximates simulations relatively well. We express reservations regarding the applicability of any approach based on large deviations theory to characterizing escape dynamics for economically plausible values of constant gain in the model of CWS when escapes are not rare. We show that for these values of the gain it is possible to derive first passage times for learning dynamics reduced to one dimension without resort to large deviations theory. This procedure delivers mean escape time results that fit the simulations closely. We explain inapplicability of large deviations theory by insufficient averaging near the point of self-confirming equilibrium for relatively large gains which makes escapes relatively frequent, suggest the changes which might help approaches based on the theory to work better in this gain interval, and describe a simple heuristic method for determining the range of constant gain values for which large deviations theory could be applicable.

Escape Dynamics: A Continuous-Time Approximation

We extend a continuous-time approach to the analysis of escape dynamics in economic models with adaptive learning with constant gain. This approach is based on applying results of continuous time version of large deviations theory to the diffusion approximation of the original discrete-time dynamics under learning. We characterize escape dynamics by analytically deriving the most probable escape point and mean escape time. The continuous-time approach is tested on the Phelps problem of a government controlling inflation while adaptively learning the approximate Phillips curve, studied previously by Sargent (1999) and Cho, Williams and Sargent (2002) (henceforth, CWS). We compare the results with simulations and the results obtained by CWS. We express reservations regarding applicability of escape dynamics theory to characterization of mean escape time for economically plausible values of constant gain in the model of CWS.We show that for these values of the gain simple considerations and formulae generate much better mean escape time results than the large deviations theory. We explain it by insufficient averaging near the point of self confirming equilibrium for relatively large gains and suggest two changes which might help the approaches based on large deviation theory to work better in this gain interval.

A deformation/blurring-based spatio-temporal model

A key objective in spatio-temporal modeling consists of providing an appropriate representation of complexity in interactive spatio-temporal dynamics inherent to real phenomena. Propagated effect of dynamical spatial deformation provides a meaningful way to describe certain forms of heterogeneous behaviour; in particular, in relation to processes evolving in unstable media, or to account for the possible effect of covariates, to mention some significant interpretations. In this paper, the formulation of a discrete time and continuous space spatio-temporal interaction model with autoregressive dynamics, incorporating the effect of continuous deformation of the spatial support over time, is studied. Among other fields, this approach provides a suitable representation for a variety of geophysical and environmental applications. In particular, a vast family of heterogeneous models is generated from models which display homogeneity in the absence of deformation. Structural characteristics and variability properties, as well as self-consistency conditions for a limiting continuous-time approximation, are analyzed.

Sharp approximations of ruin probabilities in the discrete time models

According to Solvency II directive, each insurance company could determine solvency capital requirements using its own, tailor made, internal model. This highlights the urgency of having fast numerical tools based on practically-oriented mathematical models. From the Solvency II perspective discrete time framework seems to be the most relevant one. In this paper, we propose a number of fast and accurate approximations of ruin probabilities involving some integral operator and examine them along strictly theoretical as well as numerical lines. For a few claim distributions the approximations are shown to be exact. In general, we prove that they converge with an exponential rate to the exact ruin probabilities without any restrictive assumptions on the claim distribution. A fast algorithm to approximate ruin probabilities by a numerical fixed point of the involved integral operator is given. As an application, ruin probabilities for, e.g. normally and Weibull – distributed claims are computed. Comparisons with discrete time counterparts of some continuous time approximation methods are also carried out. Numerical studies show that our approximations are precise both for small and large values of the initial surplus u. In contrast, the empirical De Vylder-type ones strongly depend on the claim distributions and are less precise for small and medium values of u.

Delay, state, and parameter estimation in chaotic and hyperchaotic delayed systems with uncertainty and time-varying delay

A novel approach is proposed for simultaneous estimation of states, delay and parameters of nonlinear chaotic and hyperchaotic delayed systems with constant delay as well as simultaneous estimation of states and parameters for such delayed systems with time-varying delay. The approach exploits continuous time approximation and stochastic optimal filtering. Also, an innovative technique is proposed to approximately compute the Lyapunov exponents of a nonlinear delayed system in order to determine the parameter values for which the system becomes chaotic or hyperchaotic. The model used in this approach contains two different source of considerable uncertainty. The approach is successfully implemented for state, parameter and delay estimation on various forms of time delayed Lorenz system and delayed Hopfield neural network including chaotic and hyperchaotic cases with constant and time-varying delays. In case of delayed Hopfield neural network, the performance of the approach is shown to be superior compared with two other existing approaches.

A note on the stability of linear dynamical systems with time delay

This paper presents a stability study of linear time-invariant and periodic systems with time delay. The methods of semi-discretization, continuous time approximation and Lyapunov stability theory are used to study the stability of two benchmark systems. It is found that for linear time-invariant systems, the Lyapunov method is usually conservative leading to a much smaller domain of stability in a parameter space than the true solution, with the exception of the complete Lyapunov functional due to Gu, which gives highly accurate predictions with little conservatism. For periodic systems, it is difficult to find appropriate Lyapunov–Krasovskii functionals. Numerical methods such as semi-discretization and continuous time approximation are more appealing, and can compute geometrically complex stability boundaries in the parameter space with high accuracy.

Recurrent statistical experiments with persistent linear regression

We construct a stochastic approximation of binary statistical experiments with persistent regression by the autoregression process with normal perturbations. The continuous-time diffusion approximation of the Ornstein–Uhlenbeck type is obtained. The stationary distributions of the approximation processes are calculated.

Stochastic parameter estimation in nonlinear time-delayed vibratory systems with distributed delay

The stochastic estimation of parameters and states in linear and nonlinear time-delayed vibratory systems with distributed delay is explored. The approach consists of first employing a continuous time approximation to approximate the delayed integro-differential system with a large set of ordinary differential equations having stochastic excitations. Then the problem of state and parameter estimation in the resulting stochastic ordinary differential system is represented as an optimal filtering problem using a state augmentation technique. By adapting the extended Kalman–Bucy filter to the augmented filtering problem, the unknown parameters of the time-delayed system are estimated from noise-corrupted, possibly incomplete measurements of the states. Similarly, the upper bound of the distributed delay can also be estimated by the proposed technique. As an illustrative example to a practical problem in vibrations, the parameter, delay upper bound, and state estimation from noise-corrupted measurements in a distributed force model widely used for modeling machine tool vibrations in the turning operation is investigated.