Showing papers on "Discrete time and continuous time published in 2004"
TL;DR: In this paper, a discrete-time approximation for decoupled forward-backward stochastic dierential equations is proposed, and the L p norm of the error is shown to be of the order of the time step.
615 citations
TL;DR: An overview of developments in the scheduling of multiproduct/multipurpose batch and continuous processes is presented and various continuous-time models have been proposed in the literature and their strengths and limitations are examined.
614 citations
TL;DR: In this paper, an updated version of the semi-discretization method is presented for periodic systems with a single discrete time delay, where the delayed term is approximated as a weighted sum of two neighboring discrete delayed state values and the transition matrix over a single period is determined.
Abstract: An updated version of the semi-discretization method is presented for periodic systems with a single discrete time delay. The delayed term is approximated as a weighted sum of two neighbouring discrete delayed state values and the transition matrix over a single period is determined. Stability charts are constructed for the damped and delayed Mathieu equation for different time-period/time-delay ratios. The convergence of the method is investigated by examples. Stability charts are constructed for 1 and 2 degree of freedom milling models. The codes of the algorithm are also attached in the appendix. Copyright © 2004 John Wiley & Sons, Ltd.
566 citations
543 citations
TL;DR: A unified framework for design of stabilizing controllers for sampled-data differential inclusions via their approximate discrete-time models is presented and previous results in the literature are extended.
Abstract: A unified framework for design of stabilizing controllers for sampled-data differential inclusions via their approximate discrete-time models is presented. Both fixed and fast sampling are considered. In each case, sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete-time plant models also stabilizes the exact discrete-time plant model for sufficiently small integration and/or sampling periods. Previous results in the literature are extended to cover: 1) continuous-time plants modeled as differential inclusions; 2) general approximate discrete-time plant models; 3) dynamical discontinuous controllers modeled as difference inclusions; and 4) stability with respect to closed arbitrary (not necessarily compact) sets.
522 citations
20 Dec 2004
TL;DR: In this paper, the output-feedback stabilisation problem is solved for discrete-time systems with time-varying delay in the state, and a stability condition is first proposed, which is dependent on the minimum and maximum delay bounds.
Abstract: The output-feedback stabilisation problem is solved for discrete-time systems with time-varying delay in the state. A stability condition is first proposed, which is dependent on the minimum and maximum delay bounds. Based on this easily verifiable stability condition, the problems of stabilisation by static and dynamic output-feedback controllers are solved within the linear matrix inequality (LMI) framework. Since the obtained conditions for the existence of admissible controllers are not expressed as strict LMI conditions, the cone complementary linearisation procedure is exploited to solve the nonconvex feasibility problem. In addition, the obtained results, including stability analysis, static output-feedback stabilisation and dynamic output-feedback stabilisation are further extended to discrete time-delay systems with norm-bounded uncertain parameters. Numerical examples are also presented to illustrate the applicability of the developed results.
419 citations
Book•
01 Jan 2004
TL;DR: In this article, the authors give a mathematical treatment of the introduction to qualitative differential equations and discrete dynamical systems, including theoretical proofs, methods of calculation, and applications, and some new material has been included in both parts of the book.
Abstract: This book gives a mathematical treatment of the introduction to qualitative differential equations and discrete dynamical systems. The treatment includes theoretical proofs, methods of calculation, and applications. The two parts of the book, continuous time of differential equations and discrete time of dynamical systems, can be covered independently in one semester each or combined together into a year long course. The material on differential equations introduces the qualitative or geometric approach through a treatment of linear systems in any dimensions. There follows chapters where equilibria are the most important feature, where scalar (energy) functions is the principal tool, where periodic orbits appear, and finally chaotic systems of differential equations. The many different approaches are systematically introduced through examples and theorems. The material on discrete dynamical systems starts with maps of one variable and proceeds to systems in higher dimensions. The treatment starts with examples where the periodic points can be found explicitly and then introduces symbolic dynamics to analyze where they can be shown to exist but not given in explicit form. Chaotic systems are presented both mathematically and more computationally using Lyapunov exponents. With the one-dimensional maps as models, the multidimensional maps cover the same material in higher dimensions. This higher dimensional material is less computational and more conceptual and theoretical. The final chapter on fractals introduces various dimensions which is another computational tool for measuring the complexity of a system. It also treats iterated function systems which give examples of complicated sets. In the second edition of the book, much of the material has been rewritten to clarify the presentation. Also, some new material has been included in both parts of the book. This book can be used as a textbook for an advanced undergraduate course on ordinary differential equations and/or dynamical systems. Prerequisites are standard courses in calculus (single variable and multivariable), linear algebra, and introductory differential equations.
361 citations
TL;DR: In this paper, the robust H"2 and H"~ filtering problem for linear discrete-time systems with polytopic parameter uncertainty was studied and a matrix inequality condition was proposed to provide additional free parameters as compared to existing characterizations.
325 citations
TL;DR: This work investigates under what conditions, and in what sense, observer design for sampled-data nonlinear systems achieve convergence for the unknown exact discrete-time model.
214 citations
TL;DR: It is shown that the memory capacity of linear recurrent networks obeying discrete time dynamics scales with system size, and this temporal memory capacity for both distributed shift register and random orthogonal connectivity matrices is calculated.
Abstract: We study the ability of linear recurrent networks obeying discrete time dynamics to store long temporal sequences that are retrievable from the instantaneous state of the network. We calculate this temporal memory capacity for both distributed shift register and random orthogonal connectivity matrices. We show that the memory capacity of these networks scales with system size.
181 citations
20 Sep 2004
TL;DR: In this article, a language-theoretic approach to symbolic model checking of PCTL over discrete-time Markov chains is presented, where the probability with which a path formula is satisfied is represented by a regular expression, and a recursive evaluation of the regular expression yields an exact rational value when transition probabilities are rational and rational functions when some probabilities are left unspecified as parameters of the system.
Abstract: We present a language-theoretic approach to symbolic model checking of PCTL over discrete-time Markov chains. The probability with which a path formula is satisfied is represented by a regular expression. A recursive evaluation of the regular expression yields an exact rational value when transition probabilities are rational, and rational functions when some probabilities are left unspecified as parameters of the system. This allows for parametric model checking by evaluating the regular expression for different parameter values, for instance, to study the influence of a lossy channel in the overall reliability of a randomized protocol.
Book•
25 Oct 2004
TL;DR: In this paper, the authors present an overview of the application of Markov Decision Processes (MDPs) in the context of dynamic systems and their application in production planning.
Abstract: Prologue and Preliminaries.- Introduction, Overview, and Examples.- Mathematical Preliminaries.- Asymptotic Properties.- Asymptotic Expansions.- Occupation Measures.- Exponential Bounds.- Interim Summary and Extensions.- Applications.- Stability of Dynamic Systems.- Filtering.- Markov Decision Processes.- LQ Controls.- Mean-Variance Controls.- Production Planning.- Stochastic Approximation.
TL;DR: The communication limits over rapid phase-varying channels are investigated and the capacity of a discrete- time noncoherent additive white Gaussian noise (NCAWGN) channel under the average power constraint is considered.
Abstract: We investigate the communication limits over rapid phase-varying channels and consider the capacity of a discrete- time noncoherent additive white Gaussian noise (NCAWGN) channel under the average power constraint. We obtain necessary and sufficient conditions for the capacity-achieving input distribution and show that this distribution is discrete and possesses an infinite number of mass points. Using this characterization of the capacity-achieving distribution we compute a tight lower bound on the capacity of the channel based on examining suboptimal input distributions. In addition, we provide some easily computable lower and upper bounds on the channel capacity. Finally, we extend some of these results to the partially coherent channel, where it is assumed that a phase-locked loop (PLL) is used to track the carrier phase at the receiver, and that an ideal interleaver and de-interleaver are employed-rendering the Tikhonov distributed residual phase errors statistically independent from one symbol interval to another.
07 Jun 2004
TL;DR: A general methodology based on robust optimization to address the problem of optimally controlling a supply chain subject to stochastic demand in discrete time, which often outperforms dynamic programming based solutions for a wide range of parameters.
Abstract: We propose a general methodology based on robust optimization to address the problem of optimally controlling a supply chain subject to stochastic demand in discrete time. The attractive features of the proposed approach are: (a) It incorporates a wide variety of phenomena, including demands that are not identically distributed over time and capacity on the echelons and links; (b) it uses very little information on the demand distributions; (c) it leads to qualitatively similar optimal policies (basestock policies) as in dynamic programming; (d) it is numerically tractable for large scale supply chain problems even in networks, where dynamic programming methods face serious dimensionality problems; (e) in preliminary computational experiments, it often outperforms dynamic programming based solutions for a wide range of parameters.
01 Jan 2004
TL;DR: This paper exploits set invariance and parametric programming to devise an efficient robust time optimal control scheme and shows how to compute /spl Omega//spl tilde//sub /spl infin// and derive conditions for finite time computation.
Abstract: Piecewise affine (PWA) systems are useful models for describing non-linear and hybrid systems. One of the key problems in designing controllers for these systems is the inherent computational complexity of controller synthesis and analysis. These problems are amplified in the presence of state and input constraints and additive but bounded disturbances. In this paper, we exploit set invariance and parametric programming to devise an efficient robust time optimal control scheme. Specifically, the state is driven into the maximal robust invariant set /spl Omega//sub /spl infin// in minimum time. We show how to compute /spl Omega//spl tilde//sub /spl infin// and derive conditions for finite time computation.
TL;DR: In this paper, the convergence of continuous-time BAM neural networks is studied and sufficient conditions are obtained for the networks to converge exponentially toward the equilibrium associated with the constant input sources.
Abstract: First, convergence of continuous-time Bidirectional Associative Memory (BAM) neural networks are studied. By using Lyapunov functionals and some analysis technique, the delay-independent sufficient conditions are obtained for the networks to converge exponentially toward the equilibrium associated with the constant input sources. Second, discrete-time analogues of the continuous-time BAM networks are formulated and studied. It is shown that the convergence characteristics of the continuous-time systems are preserved by the discrete-time analogues without any restriction imposed on the uniform discretionary step size. An illustrative example is given to demonstrate the effectiveness of the obtained results.
TL;DR: This work decomposes the overall problem spatially into three domains: the crude-oil unloading and blending, the production unit operations and the product blending and delivery, and proposes a comprehensive mathematical programming model for the efficient scheduling of oil-refinery operations.
TL;DR: The polytopic uncertainty type is considered in this work and a new alternative to design static or dynamic output feedback controllers is developed for the certain and uncertain systems.
Abstract: In this paper we deal with the class of uncertain discrete-time linear systems. The polytopic uncertainty type is considered in this work. For the certain and uncertain systems a new alternative to design static or dynamic output feedback controllers is developed. The design of the corresponding controller is formulated as an LMI problem that includes some slack variables. For the feasibility of the LMI problem, the controller is obtained by simple matrix calculation. The proposed design is performed into two steps. The first step is devoted to a classical state feedback controller design whereas the second one is the solution of the LMI problem.
TL;DR: In this article, an adaptive grid scheme is used for finding the global solutions of discrete time Hamilton-Jacobi-Bellman equations and an adapting iteration for the discretization of the state space is developed.
15 Nov 2004
TL;DR: A potential extension of formal verification methodology in order to deal with time-domain properties of analog and mixed-signal circuits whose dynamic behavior is described by differential algebraic equations is demonstrated.
Abstract: In this paper we demonstrate a potential extension of formal verification methodology in order to deal with time-domain properties of analog and mixed-signal circuits whose dynamic behavior is described by differential algebraic equations To model and analyze such circuits under all possible input signals and all values of parameters, we build upon two techniques developed in the context of hybrid (discrete-continuous) control systems First, we extend our algorithm for approximating sets of reachable sets for dense-time continuous systems to deal with differential algebraic equations (DAEs) and apply it to a biquad low-pass filter To analyze more complex circuits, we resort to bounded horizon verification We use optimal control techniques to check whether a Δ-Σ modulator, modeled as a discrete-time hybrid automaton, admits an input sequence of bounded length that drives it to saturation
TL;DR: A new model predictive control algorithm for nonlinear systems is presented, in which the plant under control, the state and control constraints, and the performance index to be minimized are described in continuous time, while the manipulated variables are allowed to change at fixed and uniformly distributed sampling times.
Abstract: A new model predictive control (MPC) algorithm for nonlinear systems is presented. The plant under control, the state and control constraints, and the performance index to be minimized are described in continuous time, while the manipulated variables are allowed to change at fixed and uniformly distributed sampling times. In so doing, the optimization is performed with respect to sequences, as in discrete-time nonlinear MPC, but the continuous-time evolution of the system is considered as in continuous-time nonlinear MPC.
TL;DR: It is shown that the robust filtering problem for linear uncertain discrete time-delay systems with Markovian jump parameters can be solved in terms of the solutions to a set of coupled matrix Riccati-like inequalities.
Abstract: In this letter, we study the robust filtering problem for linear uncertain discrete time-delay systems with Markovian jump parameters. The system under consideration is subjected to time-varying norm-bounded parameter uncertainties, time-delay in the state, and Markovian jump parameters in all system matrices. A filter is designed to guarantee that the dynamics of the estimation error is robustly stochastically stable in the mean square, irrespective of the admissible uncertainties as well as the time-delay. It is shown that the problem addressed can be solved in terms of the solutions to a set of coupled matrix Riccati-like inequalities.
TL;DR: Based on linear matrix inequalities, delay-dependent solutions are obtained by using a descriptor model transformation of the system and by applying a new bounding technique for cross terms.
TL;DR: In this paper, the central limit theorem and almost sure invariance principle for the underlying discrete time system are inherited by the suspension flow, and the results of Denker and Philipp (1984) for Axiom A flows are recovered.
Abstract: In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function. In this paper, we give conditions under which statistical laws, such as the central limit theorem and almost sure invariance principle, for the underlying discrete time system are inherited by the suspension flow. As a consequence, we give a simpler proof of the results of Ratner (1973) and recover the results of Denker and Philipp (1984) for Axiom A flows. Morcover, we obtain several new results for nonuniformly and partially hyperbolic flows, including frame flows on negatively curved manifolds satisfying a pinching condition.
01 Jan 2004
TL;DR: Observability conditions are found to distinguish the system mode in the presence of bounded system and measurement noises, which allow one to construct an estimator that relies on the combination of the identification of the discrete state with the estimation of the state variables by minimizing a receding-horizon quadratic cost function.
Abstract: Receding-horizon state estimation is addressed for a class of discrete-time systems that may switch among different modes taken from a finite set. The dynamics and measurement equations for each mode are assumed to be linear and perfectly known, but the current mode of the system is unknown, and the state variables are not perfectly measurable and are affected by disturbances. The system mode is regarded as an unknown discrete state to be estimated together with the state vector. Observability conditions have been found to distinguish the system mode in the presence of bounded system and measurement noises. These results allow one to construct a receding-horizon estimator that relies on the combination of the identification of the discrete state with the estimation of the state variables by minimizing a receding-horizon quadratic cost function. The convergence properties of such an estimator are studied, and simulation results are reported to show the effectiveness of the proposed approach.
07 Jun 2004
TL;DR: It is shown that passivity can be maintained in the face of varying delay and packet loss but that it depends fundamentally on the mechanism used to handle missing packets and a novel buffering and interpolation scheme is introduced which not only preserves passivity but has been shown to improve tracking performance and transparency in a single-degree-of-freedom teleoperator system.
Abstract: In this paper, we investigate issues in the discrete-time implementation of passivity based control of bilateral teleoperators. The usual scattering formalism which, in continuous time, guarantees passivity for any constant delay, is extended in several important ways to the discrete domain, in particular to the case where communication between the master and slave robots occurs over a packet-switched network. We first show that passivity can be maintained in the face of varying delay and packet loss but that it depends fundamentally on the mechanism used to handle missing packets. Passivity alone is not sufficient to guarantee good performance. Therefore, we also introduce a novel buffering and interpolation scheme which not only preserves passivity but has been shown through simulation and experiments to improve tracking performance and transparency in a single-degree-of-freedom teleoperator system.
Posted Content•
TL;DR: In this article, the authors reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem and apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and estimate the convergence rate and sample complexity.
Abstract: We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on high-dimensional state spaces. We reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem. Within this setup we apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and to estimate the convergence rate and sample complexity. Our results strengthen and extend earlier results.
TL;DR: The algorithm combines multi-parametric quadratic programming with reachability analysis to obtain the optimal piecewise affine (PWA) feedback law and reduces the time necessary to compute the PWA solution for the CLQR when compared to other approaches.
TL;DR: In this article, a discrete-time semi-Markov model is defined and a computation procedure for solving the corresponding Markov renewal equation, necessary for all reliability measurements, and the reliability and its related measures are applied to a three-state system.
Abstract: In this paper, we define a discrete-time semi-Markov model and propose a computation procedure for solving the corresponding Markov renewal equation, necessary for all our reliability measurements. Then, we compute the reliability and its related measures, and we apply the results to a three-state system.
01 Jan 2004
TL;DR: In this article, a neural network-based output feedback controller with magnitude constraints is designed to deliver a desired tracking performance for a class of MIMO discrete-time strict feedback nonlinear systems.
Abstract: A novel neural network (NN) -based output feedback controller with magnitude constraints is designed to deliver a desired tracking performance for a class of multi-input-multi-output (MIMO) discrete-time strict feedback nonlinear systems. Reinforcement learning in discrete time is proposed for the output feedback controller, which uses three NN: 1) a NN observer to estimate the system states with the input-output data; 2) a critic NN to approximate certain strategic utility function; and 3) an action NN to minimize both the strategic utility function and the unknown dynamics estimation errors. The magnitude constraints are manifested as saturation nonlinearities in the output feedback controller design. Using the Lyapunov approach, the uniformly ultimate boundedness (UUB) of the state estimation errors, the tracking errors and weight estimates is shown.