Involving Time Delays Research Articles

Global stability in chemostat models involving time delays and wall growth

A chemostat model with distributed time delays both in material recycling and biotic species growth has been considered. A concept of wall growth is introduced and its effect on the global stability of the positive equilibrium is discussed. Several easily verifiable independent sets of global stability conditions are presented.

Stability of Dead Zone Bidirectional Associative Memory Neural Networks Involving Time Delays

Stability of Dead Zone Bidirectional Associative Memory Neural Networks Involving Time Delays

LMI-based robust H∞ control of uncertain linear jump systems with time-delays

This paper examines the robust mean square stabilization and robust H ∞ control problem for a class of linear systems. The systems under consideration involve time delays, parameter uncertainties, and jumping parameters which may be seen as, for example, failures that occur in the systems at certain probabilities. The uncertain parameters are assumed to be time-varying and satisfy boundness conditions. A systematic approach via linear matrix inequalities (LMI) is proposed to solve the problem of concern. Sufficient conditions for robust mean square stabilization and robust H ∞ control are given in the form of LMI and the control law is computed using standard LMI techniques.

Certain classes of properly relaxed delayed controls

Optimal control problems involving time delays in the control variables can be formulated by treating the control functions as independent variables which satisfy certain compatibility conditions in terms of the delays. By generalizing these conditions in the corresponding space of relaxed controls, Warga proposed a ‘weak’ model of relaxation and proved existence of weakly relaxed minimizers. However, several examples were found, involving commensurate delays, for which weakly relaxed controls cannot be approximated with original controls, so that this extension fails to be ‘proper’. The case of commensurate delays was solved by the introduction of a ‘strong’ model, but the question of how to properly relax noncommensurately delayed controls has remained unsolved, and a natural candidate has been precisely that of weakly relaxed controls. In this paper we show that, for the noncommensurate case, certain classes of weakly relaxed controls belong to the (weak star) closure of the space of ordinary delayed controls. The proof is based on previous results obtained for a different, larger space of controls for which the notion of weak relaxation can also be applied, and on a recent result related to periodic measurable functions. Also, we extend those previous results obtained for the larger space of weakly relaxed controls, which we call the ‘R-weak’ procedure, and exhibit, for the first time, certain cases of delays for which both the space of R-weakly relaxed controls and the closure of the space of ordinary controls coincide. In other words, we prove that the R-weak model is, for certain delays, a proper relaxation procedure.

Stability and Hopf Bifurcation of Four-Wheel-Steering Vehicles Involving Driver's Delay

A mathematical model is presented for four-wheel-steeringvehicles, with the time delay in driver's response and the nonlinearityin lateral tyre forces taken into account. It is proved that thevehicle-driver system has a trivial steady state motion, as well aseight non-trivial steady state motions due to the nonlinearity of tyreforces. The asymptotic stability and Hopf bifurcation of the trivialsteady state are analyzed for two control strategies ofrear-wheel-steering. It is shown through the numerical simulations thatthe four-wheel-steering technique based on the bilinear control strategyworks better when the driver's response involves time delay.

A new approximation method for time-delay systems

We develop a rational approximant B/sub n//sup /spl alpha// of e/sup -ds/ and make a detailed analysis on the approximant B/sub n//sup /spl alpha//G of e/sup -ds/G, where G/spl isin/RL/sup 2/. We prove L/sup p=2,/spl infin// error convergence and provide corresponding error bounds explicitly. Comparative study with other approximants is performed in order to make clear the properties of B/sub n//sup /spl alpha//. Making full use of its recursive structure, B/sub n//sup /spl alpha// is shown to be a suitable approximant for a class of H/sup /spl infin// problems involving time delay.

Echo-volumar imaging (EVI) of the brain at 3.0 T: first normal volunteer and functional imaging results.

Our goal was to present the first echo-volumar brain images obtained at 3.0 T, together with the first functional imaging results using echo volumar imaging. The results presented were obtained on volunteers using an in-house designed and constructed 3.0 T echo-planar/volumar imager. The results demonstrate the feasibility of obtaining snapshot volumar images comprising up to 64 x 64 x 8 voxels corresponding to a spatial resolution of 3.0 x 3.0 x 2.5 mm3. Results are also presented showing local cortical changes in signal in response to an external visual stimulus for both the left and the right brain hemispheres. The snapshot acquisition of a whole-volume data set has a number of advantages when considering motional effects or functional image changes that may involve time delays or phase effects within different cortical regions. Instantaneous acquisition of the whole data set means accurate phase information may be straightforwardly obtained.

Computation of optimal reduced-order models with time delay

This paper considers the problem of optimally approximating high-order transfer functions by low-order ones with a time delay. The performance index used for optimization is the integral of squared error (ISE) between the unit-step responses of the reduced-order and original transfer functions. To facilitate using gradient-based methods to search optimal model parameters, an effective numerical method is adapted for evaluating the ISE and its gradient with respect to parameters. The method is based on using Parseval's identify and the variable transformation ▪, such that the accurate evaluation of ISE involving time delay is performed via solving a first-order differential equation. In determining the optimal denominator polynomial for the reduced-order transfer function, the Routh stability parameters rather than the polynomial coefficients are searched. Such a substitution avoids the effort of stability check at each iteration of parameter search.

Minimizing approximate original solutions for commensurate delayed controls

The theory of relaxation of optimal control problems involving time delays in the control functions has received an increasing attention over the last few years. Different models of relaxation have been proposed and, for a wide range of problems, properness of some procedures has been established. However, not until recently, a constructive methodology for approximating relaxed minimizers with ordinary commensurate delayed controls had not been achieved. In this paper, we analyze a recently proposed methodology which explicitly derives the construction of approximate original solutions. We compare it with previous techniques and explain why it turns out to be the only one which, for the general case, is useful from an algorithmic point of view. We also include a program, based on this technique, for the construction of minimizing approximate original solutions for constant controls involving one delay.

Proper relaxation of optimal control problems

In a recent paper, we proposed different relaxation procedures for optimal control problems involving time delays in the control functions. We introduced a notion of proper relaxation, applicable to problems without side constraints, where we required the minimum cost of the relaxed problem to coincide with the infimum cost of the original one. In this paper, a new and appropriate notion of relaxation for problems with side constraints is introduced. As examples of proper extensions, in the new sense, we describe in detail the standard procedure for delay-free problems and the procedure for one-delay systems which we recently proposed.

Relaxation procedures for time delay systems

Despite the attention that optimal control problems involving time delays have received over the last two decades, little attention has been paid to the important question of how such problems should be relaxed in order to assure existence of minimizers. Recently, Warga has proposed a relaxation procedure for fully nonlinear problems with delays in the state and control variables and showed that the resulting relaxed problem has a solution. Nevertheless, the effect of this relaxation may reduce the infimum cost as we show through an example which belongs to a class of problems with “commensurate” delays; i.e., the quotient of any two delays is rational. For problems of this nature we provide a new relaxation procedure for which the extension is “proper”; i.e., the infimum costs coincide. We also present an abstract relaxation procedure applicable to problems with possibly noncommensurate delays.

On Control Laws for Vehicle Suspensions Accounting for Input Correlations

SUMMARY Input correlations involving time delays are common in active vehicle suspension system problems. One approach to control law derivation fur such systems is to restrict attention to slate feedback laws in the interests of practicality and it is then of interest to determine the law which is, in some sense, the best. Under assumptions which are common in this area. relating to input, system and cost Function forms, a new derivation of the expression for the cost, accounting for time delays, is given. The use of the expression in numerical procedures for determining effective control gains is discussed and an example for a half car planar vehicle model is described. By comparing results with existing ones which are truly optimal, an estimate is made of the loss of performance which results from the restriction of the control law form in this case. Some generalisation of the results is attempted and they are placed in a contemporary context at the conclusion.

Controller design for integrating and runaway processes involving time delay

AbstractThis paper presents an efficient method for the design of controllers for integrating and runaway processes. The method is based on model matching in the frequency domain. The presence of open‐loop instability as well as pure time delay in the process models make the design task challenging for these classes of processes.The goal is to achieve low‐order, easily implementable cascade controllers in a unity‐output‐feedback configuration. It is shown that the central problem is in the selection of appropriate reference models. Several key constraints are developed which relate a given process model to a class of reference models for achieving total stability. Typical design examples are presented to clearly illustrate the various mathematical techniques.

Design of set point regulators for processes involving time delay

AbstractAn efficient frequency domain methodology for the design of set point regulators for high‐order process models involving relatively large pure time delay is presented. The unique advantage of the methodology is that it enables the designer to rapidly visualize the structure of the required controller as soon as the time and frequency domain specifications are stipulated for the closed‐loop system. Optimizing the parameters of an appropriate structure improves the robustness of the controller. The method does not require order reduction of the original model, and time delay is handled directly. Also, only output feedback is utilized.

Optimal control computation for parabolic systems with boundary conditions involving time delays

In this paper, we consider a class of optimal control problems involving a second-order, linear parabolic partial differential equation with Neumann boundary conditions. The time-delayed arguments are assumed to appear in the boundary conditions. A necessary and sufficient condition for optimality is derived, and an iterative method for solving this optimal control problem is proposed. The convergence property of this iterative method is also investigated.

Solvent-peak-suppressed NMR: Correction of baseline distortions and use of strong-pulse excitation

Pulse excitation sequences which involve time delays lead to spectra suffering from so-called frequency-dependent phase shifts. The phenomenon is analyzed in a model case, and the above description is shown to be somewhat improper. Rather, the spectrum should be described as a sum of Lorentzian lines, each of which has its own, frequency-independent, phase shift. The analysis leads to procedures for correcting the baseline distortions of solvent-suppressed NMR based on the Redfield 2-1-4 sequence, or on analogous strong-pulse sequences which are described. Goals and methods of solvent-signal suppression are discussed in this context.

Filtering for linear systems involving time delays in the noise process

The filtered estimate of a linear system with delays in the noise satisfies a stochastic functional differential equation. The optimal filter is characterized by two gains which are the unique solution of two coupled Riccati-type differential equations.

Simple stability criteria for single and composite linear systems with time delays

Easy ways to test the stability of systems involving time delays have been sought. In this paper, several stability conditions with an extremely simple form are provided. First, some criteria for single linear systems with time delays are presented. Then these results are extended to the composite linear systems with time delays which exist in each subsystem and in the interconnections between subsystems. Among these criteria, those which are expressed by scalar inequalities also permit us to assess the transient behaviour of systems.

Modeling of time delay and its application to estimation of nonstationary delays

A method to model a time delay by a finite impulse response filter is presented. It is useful in simulation work that involves time delays and transforms the time delay estimation problem into one of parameter estimation. The benefits of this approach are the elimination of spectral estimation, a choice of many parameter estimation algorithms, and the capability to track time-varying delays. Two examples of estimating nonstationary time delays are also given.

Time-optimal control of parabolic systems with boundary conditions involving time delays

The present paper is concerned with the control of certain parabolic systems whose boundary conditions involve time delays. The optimal controls are characterized in terms of an adjoint system and shown to be unique and bang-bang. These results extend to certain cases of nonlinear control and to fixed-time, minimum-norm control problems.