Discrete-time Theory Research Articles

Open Markov Type Population Models: From Discrete to Continuous Time

We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times.

Marked point processes in discrete time

We develop a general framework for stationary marked point processes in discrete time. We start with a careful analysis of the sample paths. Our initial representation is a sequence $$\{(t_j,k_j): j\in {\mathbb {Z}}\}$$ of times $$t_j\in {\mathbb {Z}}$$ and marks $$k_j\in {\mathbb {K}}$$ , with batch arrivals (i.e., $$t_j=t_{j+1}$$ ) allowed. We also define alternative interarrival time and sequence representations and show that the three different representations are topologically equivalent. Then, we develop discrete analogs of the familiar stationary stochastic constructs in continuous time: time-stationary and point-stationary random marked point processes, Palm distributions, inversion formulas and Campbell’s theorem with an application to the derivation of a periodic-stationary Little’s law. Along the way, we provide examples to illustrate interesting features of the discrete-time theory.

Self-tuning of a Neuro-Adaptive PID Controller for a SCARA Robot Based on Neural Network

In this paper a MIMO (Multiple-Input-Multiple-Output) adaptive neural PID (AN-PID) controller that can be applied to a nonlinear dynamics is proposed, and its use is shown in the control of a SCARA robot for two degrees of freedom. The AN-PID controller, including a neural network of the dynamic perceptron type, is designed. The proposed controller uses a RBF network to identify the model and back propagates the control error to the AN-PID controller, unlike other controllers, that use direct methods to back propagate such error. With these properties, an AN-PID controller corrects the tracking errors due to the uncertainties and variations in the robot arm dynamics. It is robust and with adaptive capacity in order to achieve a suitable control performance. Experimental results on the SCARA robot were obtained to illustrate the effectiveness of the proposed control strategy, including comparison with a classical PID. By using Lyapunov's discrete-time theory, it was demonstrated that the control error is semi-global uniformly ultimate bounded (SGUUB).

Segmented strings coupled to a B-field

In this paper we study segmented strings in AdS3 coupled to a background two-form whose field strength is proportional to the volume form. By changing the coupling, the theory interpolates between the Nambu-Goto string and the SL(2, ℝ) Wess-Zumino-Witten model. In terms of the kink momentum vectors, the action is independent of the coupling and the classical theory reduces to a single discrete-time Toda-type theory. The WZW model is a singular point in coupling space where the map into Toda variables degenerates.

Discrete-time control of bilateral teleoperation systems: a review

SUMMARYThe possibility of operating in remote environments using teleoperation systems has been considered widely in the control literature. This paper presents a review on the discrete-time teleoperation systems, including issues such as stability, passivity and time delays. Using discrete-time methods for a master-slave teleoperation system can simplify control implementation. Varieties of control schemes have been proposed for these systems and major concerns such as passivity, stability and transparency have been studied. Recently, unreliable communication networks affected by packet loss and variable transmission delays have been received much attention. Thus, it is worth considering discrete-time theories for bilateral teleoperation architectures, which are formulated on the same lines as the continuous-time systems. Despite the extensive amount of researches concerning continuous-time teleoperation systems, only a few papers have been published on the analysis and controller design for discrete bilateral forms. This paper takes into account the challenges for the discrete structure of bilateral teleoperation systems and notifies the recent contributions in this area. The effect of sampling time on the stability-transparency trade-off and the task performance is taken into consideration in this review. These studies can help to design guidelines to have better transparency and stable teleoperation systems.

Hydra effects in discrete-time models of stable communities

A species exhibits a hydra effect when, counter-intuitively, increased mortality of the species causes an increase in its abundance. Hydra effects have been studied in many continuous time (differential equation) multispecies models, but only rarely have hydra effects been observed in or studied with discrete time (difference equation) multispecies models. In addition most discrete time theory focuses on single-species models. Thus, it is unclear what unifying characteristics determine when hydra effects arise in discrete time models. Here, using discrete time multispecies models (where total abundance is the single variable describing each population), I show that a species exhibits a hydra effect in a stable system only when fixing that species’ density at its equilibrium density destabilizes the system. This general characteristic is referred to as subsystem instability. I apply this result to two-species models and identify specific mechanisms that cause hydra effects in stable communities, e.g., in host–-parasitoid models, host Allee effects and saturating parasitoid functional responses can cause parasitoid hydra effects. I discuss how the general characteristic can be used to identify mechanisms causing hydra effects in communities with three or more species. I also show that the condition for hydra effects at stable equilibria implies the system is reactive (i.e., density perturbations can grow before ultimately declining). This study extends previous work on conditions for hydra effects in single-species models by identifying necessary conditions for stable systems and sufficient conditions for cyclic systems. In total, these results show that hydra effects can arise in many more communities than previously appreciated and that hydra effects were present, but unrecognized, in previously studied discrete time models.

Time scale symplectic systems with analytic dependence on spectral parameter

This paper is devoted to the study of time scale symplectic systems with polynomial and analytic dependence on the complex spectral parameter . We derive fundamental properties of these systems (including the Lagrange identity) and discuss their connection with systems known in the literature, in particular with linear Hamiltonian systems. In analogy with the linear dependence on , we present a construction of the Weyl disks and determine the number of linearly independent square integrable solutions. These results extend the discrete time theory considered recently by the authors. To our knowledge, in the continuous time case this concept is new. We also establish the invariance of the limit circle case for a special quadratic dependence on and its extension to two (generally nonsymplectic) time scale systems, which yields new results also in the discrete case. The theory is illustrated by several examples.

ASYMPTOTIC EQUIVALENCE BETWEEN TWO LINEAR DYNAMIC SYSTEMS ON TIME SCALES

Abstract. In this paper we investigate asymptotic properties about as-ymptotic equilibrium and asymptotic equivalence for linear dynamic sys-tems on time scales by using the notion of u ∞ -similarity. Also, we givesome examples to illustrate our results. 1. IntroductionThe calculus on time scales was initiated by Aulbach and Hilger in orderto create a theory that can unify and extend discrete and continuous analysis[1, 2, 15]. The theory on time scales has been developed as a generalization ofboth continuous and discrete time theory and applied to many different fieldsof mathematics [1, 2, 3, 4].The notion of similarity is an effective tool to study the theory of stabilityfor differential systems and difference systems [5, 7, 10, 11, 12, 17, 18, 19].Markus [17] introduced the notion of kinematic similarity in the set of alln×n continuous matrices defined on [t 0 ,∞) and showed that the relationshipof kinematic similarity is an equivalence relation preserving the type numbersof the linear differential systems. Gohberg et al. [14] studied the problemto classify linear time-varying systems of difference equations under kinematicsimilarity. Conti [12] introduced the concept of t

Partial Derivatives and Optimization Problem on Time Scales

The optimization problem using time scales is studied. Time scale is a model of time. The language of time scales seems to be an ideal tool to unify the continuous-time and the discrete-time theories. In this work we present necessary conditions for a solution of an optimization problem on time scales. To obtain that result we use properties and results of the partial diamond-alpha derivatives for continuous-multivariable functions. These results are also presented here. Keywords—Lagrange multipliers, mathematical programming, optimization problem, time scales.

Derivative pricing methodology in continuous-time models

We show that the fundamental methodology (and practice) of evaluation of derivative securities in continuous-time models is consistent with discrete-time theory, in which a derivative price is based on the principle that adding this security to the market does not create a violation of the basic economic principle: no riskless profit with zero investment.

Oscillation theorems and Rayleigh principle for linear Hamiltonian and symplectic systems with general boundary conditions

The aim of this paper is to establish the oscillation theorems, Rayleigh principle, and coercivity results for linear Hamiltonian and symplectic systems with general boundary conditions, i.e., for the case of separated and jointly varying endpoints, and with no controllability (normality) and strong observability assumptions. Our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with Dirichlet boundary conditions, and the problem with jointly varying endpoints into a problem with separated endpoints. These more general results on time scales then provide new results for the continuous time linear Hamiltonian systems as well as for the discrete symplectic systems. This paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions. Furthermore, the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories.

Weyl‐Titchmarsh Theory for Time Scale Symplectic Systems on Half Line

We develop the Weyl‐Titchmarsh theory for time scale symplectic systems. We introduce the M(λ)‐function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl‐Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second‐order Sturm‐Liouville equations on time scales.

ON THE CONSISTENCY OF THE LUCAS PRICING FORMULA

In order to find the real market value of an asset in an exchange economy, one would typically apply the Lucas formula, developed in a discrete time framework. This theory has also been extended to continuous time models, in which case the same pricing formula has been universally applied.While the discrete time theory is rather transparent, there has been some confusion regarding the continuous time analogue. In particular, the continuous time pricing formula must contain a certain type of a square covariance term that does not readily follow from the discrete time formulation. As a result, this term has sometimes been missing in situations where it should have been included.In this paper, we reformulate the discrete time theory in such a way that this covariance term does not come as a mystery in the continuous time version.In most real life situations dividends are paid out in lump sums, not in rates. This leads to a discontinuous model, and adding a continuous time framework, it appears that our framework is a most natural one in finance.Finally, the Gordon growth formula is extended from its deterministic origin, to the present model of uncertainty, and it is indicated how this can be used to to possibly shed some light on the volatility puzzle.

Deriving Spin within a Discrete-Time Theory

We prove that the classical theory with a discrete time (chronon) is a particular case of a more general theory in which spinning particles are associated with generalized Lagrangians containing time-derivatives of any order (a theory that has been called "Non-Newtonian Mechanics"). As a consequence, we get, for instance, a classical kinematical derivation of Hamiltonian and spin vector for the mentioned chronon theory (e.g., in Caldirola et al.'s formulation).

On the Consistency of the Lucas Pricing Formula

Finally, the Gordon growth formula is extended from its deterministic origin, to the present model of uncertainty, and it is indicated how this can be used to to possibly shed some light on the volatility puzzle. In most real life situations dividends are paid out in lump sums, not in rates. This leads to a discontinuous model, and adding a continuous time framework, it appears that our framework is a most natural one in finance. In this paper, we reformulate the discrete time theory in such a way that this covariance term does not come as a mystery in the continuous time version. While the discrete time theory is rather transparent, there has been some confusion regarding the continuous time analogue. In particular, the continuous time pricing formula must contain a certain type of a square covariance term that does not readily follow from the discrete time formulation. As a result, this term has sometimes been missing in situations where it should have been included. In order to find the real market value of an asset in an exchange economy, one would typically apply the Lucas formula, developed in a discrete time framework. This theory has also been extended to continuous time models, in which case the same pricing formula has been universally applied. While the discrete time theory is rather transparent, there has been some confusion regarding the continuous time analogue. In particular, the continuous time pricing formula must contain a certain type of a square covariance term that does not readily follow from the discrete time formulation. As a result, this term has sometimes been missing in situations where it should have been included. In this paper, we reformulate the discrete time theory in such a way that this covariance term does not come as a mystery in the continuous time version. In most real life situations dividends are paid out in lump sums, not in rates. This leads to a discontinuous model, and adding a continuous time framework, it appears that our framework is a most natural one in finance. Finally, the Gordon growth formula is extended from its deterministic origin, to the present model of uncertainty, and it is indicated how this can be used to to possibly shed some light on the volatility puzzle.

Discrete-time timescale analysis via a new separation ratio and unified approach

In this paper, we first propose a modified definition for timescale separation ratio for the discrete-time system. Then, based on this new timescale separation ratio, a timescale analysis is presented to decompose the system into fast and slow subsystems. Next, using the difference (delta) operator, a timescale analysis is presented using a unified formulation (applicable to both continuous-time and discrete-time systems). Furthermore, it is shown that by using simple transformations, one can obtain the discrete-time results we developed earlier, and by similarly setting the sampling interval to zero, we obtain the continuous-time result developed previously by others. Thus, with this unified approach to a timescale system, a single technique is sufficient to develop the techniques for the continuous-time and discrete-time systems. Having a pair of routines for each task emphasizes the differences between discrete-time and continuous-time theory rather than the similarities. An F-8 fighter aircraft is given as an example to illustrate the methodology.

Discrete Optimal Control: Second Order Optimality Conditions

In this paper, we present a survey and refinement of our recent results in the discrete optimal control theory. For a general nonlinear discrete optimal control problem (P) , second order necessary and sufficient optimality conditions are derived via the nonnegativity ( I S 0) and positivity ( I >0) of the discrete quadratic functional I corresponding to its second variation. Thus, we fill the gap in the discrete-time theory by connecting the discrete control problems with the theory of conjugate intervals, Hamiltonian systems, and Riccati equations. Necessary conditions for I S 0 are formulated in terms of the positivity of certain partial discrete quadratic functionals, the nonexistence of conjugate intervals, the existence of conjoined bases of the associated linear Hamiltonian system, and the existence of solutions to Riccati matrix equations. Natural strengthening of each of these conditions yields a characterization of the positivity of I and hence, sufficiency criteria for the original problem (P) . Finally, open problems and perspectives are also discussed.

Efficient Design of Voice Carrying Fixed-Network Links in CDMA Mobile Communication Systems

Code Division Multiple Access (CDMA) technology is gaining momentum as the preferred wireless system for the next generation Personal Communication Systems (PCS). In CDMA systems, voice is encoded and packaged in variable length packets that are transported between the mobile station and the switching center. Although the packetization provides a great flexibility in resource allocation, it poses a Quality of Service (QoS) problem on voice. In this paper, we discuss link dimensioning for a typical CDMA system. We consider a T1/E1 link between a CDMA Basestation Transceiver System (BTS) and the Base Station Controller (BSC). Traffic from various voice sources is subject to a framing scheme, which presents a semi-periodic batch input at the T1/E1 interface cards. We analyze the resulting queuing system using discrete-time analysis and large deviations theory and verify our results by simulation. We provide results for the minimum link capacity needed to support a given number of CDMA voice sources. Our results show the potential statistical gain that can be achieved by voice packetization for all practical values of link capacities.

Repetitive Control Theory and Applications - A Survey

The aim of this paper is to present repetitive control, controllers for rejecting periodic disturbances or tracking periodic reference signals, This survey covers both continuous and discrete time theory, together with a number of applications

Discrete time process algebra

Abstract The axiom system ACP of [BeK84a] was extended with real time features in [BaB91]. Here we proceed to define a discrete time extension of ACP, along the lines of ATP [NiS94]. We present versions based on relative timing and on absolute timing. Both approaches are integrated using parametric timing. The time free ACP theory is embedded in the discrete time theory.