Variational Dualities in the Linear Regulator and Estimation Problems With and Without Time Delay

A unified approach to the variational dualities in the stateregulator of control theory and linear estimation with and withouttime delay is developed by the complementary variational technique.New variational duals for the state regulator with time-delayand estimation problems with and without time delay are presented.     

时滞广义时变系统的容许性分析与镇定

研究了时滞广义时变系统的容许性与镇定性问题.首先,基于广义Lyapunov不等式、线性矩阵不等式和受限等价方法,建立时滞广义时变系统的Lyapunov不等式,将时滞广义时变系统的容许性问题转化为求解时滞广义时变系统的Lyapunov不等式问题,得到了系统容许的充分条件.然后,根据充分条件进一步研究了时滞广义时变系统的镇定问题,给出了状态反馈镇定器的设计方法.最后,通过数值算例验证了所得结论的有效性.     

Stability for the damped wave equation with neutral delay

Of concern is a wave equation with a distributed neutral delay. We prove that, despite the destructive nature of delays in general, solutions may go back to the equilibrium state in an exponential manner as time goes to infinity. Reasonable conditions on the distributed neutral delay are established. This type of problems appear in the study of wave propagation in viscoelastic media and in acoustic wave propagation. It is not well studied so far.     

Multiple-interval pseudospectral approximation for nonlinear optimal control problems with time-varying delays

A multiple-interval pseudospectral scheme is developed for solving nonlinear optimal control problems with time-varying delays, which employs collocation at the shifted flipped Jacobi-Gauss–Radau points. The new pseudospectral scheme has the following distinctive features/abilities: (i) it can directly and flexibly solve nonlinear optimal control problems with time-varying delays without the tedious quasilinearization procedure and the uniform mesh restriction on time domain decomposition, and (ii) it provides a smart approach to compute the values of state delay efficiently and stably, and a unified framework for solving standard and delay optimal control problems. Numerical results on benchmark delay optimal control problems including challenging practical engineering problems demonstrate that the proposed pseudospectral scheme is highly accurate, efficient and flexible.     

Developments in delay time analysis for modelling plant maintenance

Delay time (DT) analysis is a pragmatic mathematical concept readily embraced by engineers which has been developed as a means to model maintenance decision problems. Attention is focused upon the maintenance engineering decisions of what to do, as opposed to the logistical decisions of how to do it. This paper reviews the cumulative knowledge and experience of delay time modelling. The decision environment within which delay time (DT) models are intended as decision aids is briefly reviewed, and the initial development of simple DT models for a repairable component and a complex plant presented. Variations on the basic model are outlined and discussed including perfect and non-perfect inspection, steady state and non-steady state conditions, and homogeneous and non-homogeneous Poisson arrival rate of defects. Attention is given to the parameter estimation process, and both subjective and objective estimation techniques are outlined. Case sketches present practical experience in using the DT concept to model actual plant, to assess the benefits obtained, and to validate modelling and parameter assessment.     

Optimality of monotonic policies for two-action Markovian decision processes,with applications to control of queues with delayed information

We consider a discrete-time Markov decision process with a partially ordered state space and two feasible control actions in each state. Our goal is to find general conditions, which are satisfied in a broad class of applications to control of queues, under which an optimal control policy is monotonic. An advantage of our approach is that it easily extends to problems with both information and action delays, which are common in applications to high-speed communication networks, among others. The transition probabilities are stochastically monotone and the one-stage reward submodular. We further assume that transitions from different states are coupled, in the sense that the state after a transition is distributed as a deterministic function of the current state and two random variables, one of which is controllable and the other uncontrollable. Finally, we make a monotonicity assumption about the sample-path effect of a pairwise switch of the actions in consecutive stages. Using induction on the horizon length, we demonstrate that optimal policies for the finite- and infinite-horizon discounted problems are monotonic. We apply these results to a single queueing facility with control of arrivals and/or services, under very general conditions. In this case, our results imply that an optimal control policy has threshold form. Finally, we show how monotonicity of an optimal policy extends in a natural way to problems with information and/or action delay, including delays of more than one time unit. Specifically, we show that, if a problem without delay satisfies our sufficient conditions for monotonicity of an optimal policy, then the same problem with information and/or action delay also has monotonic (e.g., threshold) optimal policies.     

Robust stability and H∞ control for nonlinear discrete‐time switched systems with interval time‐varying delay

This paper considers the problems of the robust stability analysis and H_∞ controller synthesis for uncertain discrete‐time switched systems with interval time‐varying delay and nonlinear disturbances. Based on the system transformation and by introducing a switched Lyapunov‐Krasovskii functional, the novel sufficient conditions, which guarantee that the uncertain discrete‐time switched system is robust asymptotically stable are obtained in terms of linear matrix inequalities. Then, the robust H_∞ control synthesis via switched state feedback is studied for a class of discrete‐time switched systems with uncertainties and nonlinear disturbances. We designed a switched state feedback controller to stabilize asymptotically discrete‐time switched systems with interval time‐varying delay and H_∞ disturbance attenuation level based on matrix inequality conditions. Examples are provided to illustrate the advantage and effectiveness of the proposed method.     

Finite‐time stability and boundedness of switched nonlinear time‐delay systems under state‐dependent switching

This article investigates the finite‐time stability, stabilization, and boundedness problems for switched nonlinear systems with time‐delay. Unlike the existing average dwell‐time technique based on time‐dependent switching strategy, largest region function strategy, that is, state‐dependent switching control strategy is adopted to design the switching signal, which does not require the switching instants to be given in advance. Some sufficient conditions which guarantee finite‐time stable, stabilization, and boundedness of switched nonlinear systems with time‐delay are presented in terms of linear matrix inequalities. Detail proofs are given using multiple Lyapunov‐like functions. A numerical example is given to illustrate the effectiveness of the proposed methods. © 2014 Wiley Periodicals, Inc. Complexity 21: 267–275, 2015     

Stochastic control problems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay. We state conditions for certain classes of such systems under which the stochastic control problems become finite-dimensional. These conditions are illustrated with three applications. First, we solve some linear quadratic problems with delay. Then we find the optimal consumption rate in a financial market with delay. Finally, we solve explicitly a deterministic fluid problem with delay which arises from admission control in ATM communication networks.     

Efficient estimates of the solutions of perturbed control problems

In this paper, we obtain estimates of the solutions for a sequence of strongly convex extremal problems. As applications of our abstract results, we consider optimal control problems with various types of perturbations. We estimate the solutions of problems with perturbations in the state equation and in the control constraining set. A singularly perturbed problem and a problem with perturbed time delay parameter are studied.     

Stability switch and Hopf bifurcation for a diffusive prey-predator system with delay

The increasing time delay usually destabilizes any dynamical system. In this paper we give an example that in some special cases the opposite effect can be experienced if the time delay is sufficiently great. We investigate the effect of both the parameter in the time delay kernel and diffusion coefficient on the stability of the positive steady state for a diffusive prey-predator system with delay. We obtain the condition of the occurrence of the stability switches of the positive steady state.     

On Stochastic Stabilization of Discrete‐Time Markovian Jump Systems with Delay in State

Abstract

In this paper, we investigate the stochastic stabilization problem for a class of linear discrete time‐delay systems with Markovian jump parameters. The jump parameters considered here is modeled by a discrete‐time Markov chain. Our attention is focused on the design of linear state feedback memoryless controller such that stochastic stability of the resulting closed‐loop system is guaranteed when the system under consideration is either with or without parameter uncertainties. Sufficient conditions are proposed to solve the above problems, which are in terms of a set of solutions of coupled matrix inequalities.     

New methods for proving the existence and stability of periodic solutions in singularly perturbed delay systems

We carry out a detailed analysis of the existence, asymptotics, and stability problems for periodic solutions that bifurcate from the zero equilibrium state in systems with large delay. The account is based on a specific meaningful example given by a certain scalar nonlinear second-order differential-difference equation that is a mathematical model of a single-circuit RCL oscillator with delay in a feedback loop.     

Control of linear processes with distributed lags using dynamic programming from first principles

A simple dynamic programming argument is presented for the quadratic-cost controller synthesis problem for discrete-time linear processes with delay. Distributed delays are allowed in both state and control. The solution obtained has a discrete-time Riccati difference structure closely analogous to the Riccati differential structure associated with delay problems in continuous time. Extensions are provided for the cases of varying lag-limits, performance criterion dependent on past variables, and the time-invariant regulator problem. A feedback solution is also obtained for a continuous-time problem with distributed delays in the control, by passage to limit from the discrete results.This work was supported by the Operations Research Center, University of California, Berkeley, California, under NSF Grant No. GP-30961X2. The author would like to thank Professor S. E. Dreyfus for guidance and helpful suggestions.     

State constrained optimal control problems with time delays

In a recent, related, paper, necessary conditions in the form of a Maximum Principle were derived for optimal control problems with time delays in both state and control variables. Different versions of the necessary conditions covered fixed end-time problems and, under additional hypotheses, free end-time problems. These conditions improved on previous conditions in the following respects. They provided the first fully non-smooth Pontryagin Maximum Principle for problems involving delays in both state and control variables, only special cases of which were previously available. They provide a strong version of the Weierstrass condition for general problems with possibly non-commensurate control delays, whereas the earlier literature does so only under structural assumptions about the dynamic constraint. They also provided a new ‘two-sided’ generalized transversality condition, associated with the optimal end-time. This paper provides an extension of the Pontryagin Maximum Principle of the earlier paper for time delay systems, to allow for the presence of a unilateral state constraint. The new results fully recover the necessary conditions of the earlier paper when the state constraint is absent, and therefore retain all their advantages but in a setting of greater generality.     

一类具有时滞的捕食者食饵模型的稳定性和Hopf分支

针对种群的发展趋势不仅依赖于当前的状态,而且还依赖于过去某一时间段的状态.研究了一类带时滞的捕食者-食饵模型.首先应用线性化方法分析该模型正平衡点的局部渐近稳定性,然后以时滞为分支参数讨论Hopf分支的存在性,最后对模型进行数值模拟证实了以上相关结果.     

Real-time calculation of the current state estimates for a class of delay systems

The linear optimal observation problem is examined for one type of nonstationary delay system with an uncertainty in the initial state. A fast implementation of the dual method is proposed for calculating estimates of the initial state. This implementation is based on the quasi-reduction of the fundamental matrix of solutions to the mathematical model of delay systems. It is shown that an iteration step of the dual method only requires that auxiliary systems of ordinary differential equations be integrated on small time intervals. An algorithm is described for the real-time calculation of current state estimates. The results are illustrated by the optimal observation problem for a third-order stationary delay system.     

Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.     

State estimation of memristor‐based recurrent neural networks with time‐varying delays based on passivity theory

This article deals with the state estimation problem of memristor‐based recurrent neural networks (MRNNs) with time‐varying delay based on passivity theory. The main purpose is to estimate the neuron states, through available output measurements such that for all admissible time delay, the dynamics of the estimation error is passive from the control input to the output error. Based on the Lyapunov–Krasovskii functional (LKF) involving proper triple integral terms, convex combination technique, and reciprocal convex technique, a delay‐dependent state estimation of MRNNs with time‐varying delay is established in terms of linear matrix inequalities (LMIs). The information about the neuron activation functions and lower bound of the time‐varying delays is fully used in the LKF. Then, the desired estimator gain matrix is accomplished by solving LMIs. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed theoretical results. © 2013 Wiley Periodicals, Inc. Complexity 19: 32–43, 2014     

A Bliss-Type Multiplier Rule for Constrained Variational Problems with Time Delay

This paper gives a Bliss-type multiplier rule for general constrained variational problems with time delay. Our formulation allows the time delay terms and their derivatives to be present in both the objective functional and the equality and inequality constraint equations. Our major results include a Lagrange Multiplier Rule involving Euler–Lagrange equations for delay differential equations and transversality conditions. Of special note is that these results will be used in later work by the authors to obtain new analytical techniques to solve general constrained problems involving delay differential equations for the Calculus of Variations and for Optimal Control Theory. They will also lead to new general, accurate, and efficient numerical methods to solve these important problems where no such methods exist at this time. Thus, for the first time, we will be able to solve this important class of problems obtaining analytic solutions for simpler problems and accurate numerical results for more complex problems when such a solution exists.