Receding Horizon Control for a Class of Discrete-Event Systems With Real-Time Constraints

We consider discrete-event systems (DES) involving the control of tasks with real-time constraints. When future event time information is limited, we propose a receding horizon (RH) controller in which only some future information is available within a time window. Analyzing sample paths obtained under this scheme and comparing them to optimal sample paths (obtained when all event times are known), we derive a number of attractive properties of the RH controller, including: the fact that it still guarantees all real-time constraints; there are segments of its sample path over which all controls are still optimal; the error relative to the optimal task departure times is decreasing under certain conditions. Simulation results are included to verify the properties of the controller and show that its performance can be near-optimal even if the RH window size is relatively small     

Utilizing call admission control for pricing optimization of multiple service classes in wireless cellular networks

In this paper, we utilize admission control algorithms designed for revenue optimization with QoS guarantees to derive optimal pricing of multiple service classes in wireless cellular networks. A service provider typically adjusts pricing only periodically. Once a “global” optimal pricing is derived, it would stay static for a period of time, allowing users to be charged with the same rate while roaming. We utilize a hybrid partitioning-threshold admission control algorithm to analyze a pricing scheme that correlates service demand with pricing, and to periodically determine optimal pricing under which the system revenue is maximized while guaranteeing that QoS requirements of multiple service classes are satisfied.     

Optimal control of a delayed system subject to mixed control-state constraints with application to a within-host model of hepatitis virus B

This paper deals with the problem of optimal control of a delayed differential equation system subject to mixed control-state constraints with application to a within-host model of hepatitis B virus (HBV). Our method consists to transform the optimal control problem of a delayed differential equation with mixed control-state constraints to a nondelayed problem in order to derive necessary optimality conditions. This theory is applied to a deterministic HBV model with two constant delays. We derive conditions under which is optimal to eradicate HBV within the body of a host. Through numerical simulations, we found that the best control strategy to reduce the HBV infection is the combination of treatment in blocking new infection and treatment in inhibiting viral production.     

Optimal control of a production inventory system with deteriorating items

This paper is concerned with the problem of controlling the production rate of a production inventory system with deteriorating items. Under reasonable conditions, we derive the closed form optimal control of a cost minimization problem and a profit maximization problem. The derived analytical solutions yield good insight on how production planning tasks can be carried out. In addition, they also provide guidelines for further study and development of numerical methods for more complex systems involving more general exogenous functions.     

Comparisons of optimal release policies for software systems

In a software product management, it is an important problem to determine the optimal release timing which minimizes the total expected cost incurred in both testing and operation phases. In this paper, we compare the performance between two kinds of software release methods, which are referred to as the T-policy and the N-policy. Based on the existing software reliability growth models, we formulate the expected cost functions and derive analytically the optimal policies under these control methods. Also, we derive a criterion for which control methods should be adopted. Finally, in numerical examples, we calculate the optimal release policies for several cost parameters.     

Precautious-RM: a predictable non-preemptive scheduling algorithm for harmonic tasks

A major requirement of many real-time embedded systems is to have time-predictable interaction with the environment. More specifically, they need fixed or small sampling and I/O delays, and they cannot cope with large delay jitters. Non-preemptive execution is a known method to reduce the latter delay; however, the corresponding scheduling problem is NP-Hard for periodic tasks. In this paper, we present Precautious-RM as a predictable linear-time online non-preemptive scheduling algorithm for harmonic tasks which can also deal with the former delay, namely sampling delay. We derive conditions of optimality of Precautious-RM and show that satisfying those conditions, tight bounds for best- and worst-case response times of the tasks can be calculated in polynomial-time. More importantly, response time jitter of the tasks is analyzed and it is proven that under specific conditions, each task has either one or two values for response time, which leads to improving the predictability of the system interaction with the environment. Simulation results demonstrate efficiency of Precautious-RM in increasing accuracy of control applications.     

Initial condition of costate in linear optimal control using convex analysis

The Pontryagin Maximum Principle is one of the most important results in optimal control, and provides necessary conditions for optimality in the form of a mixed initial/terminal boundary condition on a pair of differential equations for the system state and its conjugate costate. Unfortunately, this mixed boundary value problem is usually difficult to solve, since the Pontryagin Maximum Principle does not give any information on the initial value of the costate. In this paper, we explore an optimal control problem with linear and convex structure and derive the associated dual optimization problem using convex duality, which is often much easier to solve than the original optimal control problem. We present that the solution to the dual optimization problem supplements the necessary conditions of the Pontryagin Maximum Principle, and elaborate the procedure of constructing the optimal control and its corresponding state trajectory in terms of the solution to the dual problem.     

Optimal control computation for nonlinear systems with state-dependent stopping criteria

In this paper, we consider a challenging optimal control problem in which the terminal time is determined by a stopping criterion. This stopping criterion is defined by a smooth surface in the state space; when the state trajectory hits this surface, the governing dynamic system stops. By restricting the controls to piecewise constant functions, we derive a finite-dimensional approximation of the optimal control problem. We then develop an efficient computational method, based on nonlinear programming, for solving the approximate problem. We conclude the paper with four numerical examples.     

Joint admission,production sequencing,and production rate control for a two-class make-to-order manufacturing system

Today's manufacturing industry faces a number of challenges related to the rapid delivery of products with a high degree of variety. Striking a balance between the effectiveness in capacity utilization and the rapidness in order-fulfillment is a substantial challenge for manufacturing companies. This work aims to provide a theoretical basis from which to address this practical question. To this end, we address the problem of coordinating the admission, production sequencing, and production rate controls for a two-class make-to-order manufacturing system. Formulating the problem as a Markov decision process model, we identify the structural properties of optimal control policies under both discounted and average profit criteria. We show that the cμ rule is optimal for production sequencing and the optimal admission and production rate control policies can be characterized by the state-dependent threshold levels, provided that the production times are not associated with customer class. We also show that the optimal production rates are monotone in the system state, as in the case of a single class queueing system, and that the lower priority class can be preferred to the higher priority class in order admission under a certain condition on the system parameters. Our numerical study demonstrates that a considerable economic benefit can be achieved if the production rate is dynamically controlled between the minimum and maximum rates rather than fixed to the mean rate of these values. Finally, we present a heuristic policy that is described by linear switching functions for the control of order admission and a selection rule for the control of production rate. We compare the performance of our heuristic to the optimal policy using a numerical experiment, revealing that the heuristic provides near optimal solutions to test example problems and is robust to the system parameters.     

Optimal controls of multidimensional modified Swift–Hohenberg equation

In this paper, we deal with the optimal control problem governed by multidimensional modified Swift–Hohenberg equation. After showing the relationship between the control problem and its approximation, we derive the optimality conditions for an optimal control of our original problem by using one of the approximate problems.     

Maximizing biogas production from the anaerobic digestion

This paper presents an optimal control law policy for maximizing biogas production of anaerobic digesters. In particular, using a simple model of the anaerobic digestion process, we derive a control law to maximize the biogas production over a period T using the dilution rate as the control variable. Depending on initial conditions and constraints on the actuator (the dilution rate D(·)), the search for a solution to the optimal control problem reveals very different levels of difficulty. In the present paper, we consider that there are no severe constraints on the actuator. In particular, the interval in which the input flow rate lives includes the value which allows the biogas to be maximized at equilibrium. For this case, we solve the optimal control problem using classical tools of differential equations analysis. Numerical simulations illustrate the robustness of the control law with respect to several parameters, notably with respect to initial conditions. We use these results to show that the heuristic control law proposed by Steyer et al., 1999 [20] is optimal in a certain sense. The optimal trajectories are then compared with those given by a purely numerical optimal control solver (i.e. the “BOCOP” toolkit) which is an open-source toolbox for solving optimal control problems. When the exact analytical solution to the optimal control problem cannot be found, we suggest that such numerical tool can be used to intuiter optimal solutions.     

First order conditions for generalized variational problems over time scales

In this paper, we derive the weak Pontryagin principle for generalized optimal control problems over time scales. Three types of problems are considered, namely (i) the problems involving the values of the state endpoints in the Lagrangian and the dynamics, (ii) the problems with an integral of the state in the Lagrangian and the dynamics, and (iii) the isoperimetric problems. As special cases, we obtain the first order optimality conditions for the corresponding calculus of variations problems, which have been of a recent interest in the literature. Our method is based on transforming the generalized optimal control or calculus of variations problem into a traditional optimal control problem on time scales, to which the known weak Pontryagin principle can be applied.     

Linear‐Quadratic Optimal Control Problem for Partially Observed Forward‐Backward Stochastic Differential Equations of Mean‐Field Type

This paper is concerned with the linear‐quadratic optimal control problem for partially observed forward‐backward stochastic differential equations (FBSDEs) of mean‐field type. Based on the classical spike variational method, backward separation approach as well as filtering technique, we first derive the necessary and sufficient conditions of the optimal control problem with the non‐convex domain. Nextly, by means of the decoupling technique, we obtain two Riccati equations, which are uniquely solvable under certain conditions. Also, the optimal cost functional is represented by the solutions of the Riccati equations for the special case.     

Controlling a Class of Nonlinear Systems on Rectangles

In this paper, we focus on a particular class of nonlinear affine control systems of the form xdot=f(x)+Bu, where the drift f is a multi-affine vector field (i.e., affine in each state component), the control distribution B is constant, and the control u is constrained to a convex set. For such a system, we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant. We then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time. If the control constraints are polyhedral, we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle. This work is motivated by the need to construct discrete abstractions for continuous and hybrid systems, in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs. We show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators     

Optimal timing control of discrete-time linear switched stochastic systems

Optimal switch-time control is the study that investigates how best to switch between different modes. In this paper, we investigate the optimal switch-time control problem for discrete-time linear switched stochastic systems. In particular, under the assumption that the sequence of active subsystems is pre-specified, we focus on the problem where the objective is to minimize a cost functional defined on the states and the switching times are the only control variables. For systems with one switching time, using calculus of variations, we firstly derive the difference formulae of the cost functional with respect to the switching time, which can be directly used to find the optimal switching instant. Then, a method is presented to deal with the problem with multiple switching times case. Finally, the viability of the proposed method is illustrated through two numerical examples.     

A multi-product EOQ model with pricing consideration—T. C.E. Cheng''s momdel revisited

We present two major revisions/corrections regarding a recent paper by T.C.E. Cheng [Computers ind. Engng 18, 529–534 (1990)]. Firstly, we note that a critical assumption of equal replenishment cycle length for all products is stated, but not incorporated into the mathematical formulation in Cheng. In this paper, we re-formulate the problem with equal replenishment cycle length incorporated and derive the corresponding Kuhn-Tucker optimality conditions. Next, under linear demand assumption, we show that the closed-form solutions provided by Cheng may result in non-optimal solutions. The reason is that Cheng failed to derive conditions under which closed-form solutions may be optimal. In this paper, by employing trigonometric methods {see e.g. Chap. 2 of Griffiths [Introduction of the Theory of Equations. Wiley, New York (1945).] or Appendix of Porteus [Mgmt Sci. 31, 998–1010 (1985)]}, we derive an optimal closed-form solution that is unqiue and obtain conditions under which the optimal closed-form solution is valid.     

一种无线传感器网络中多应用系统级分布式接入控制策略

本文从无线传感器网络中多应用场景下的能量管理问题出发提出了一个系统级分布式接入控制策略.其目的为在一定的网络初始配置能量的前提下最大化总用户收益期望值.在设计该策略时,本文将多应用接入控制问题抽象成为一个特殊的“动态随机背包问题”,并采用相关理论对该问题建立了模型.为了求解这一模型,本文还提出了一个估计应用Reward及Cost联合概率分布的方案.并采用相关算法对该模型进行了求解,并据此设计了最优系统级分布式接入控制策略.经实验表明,本文提出的分布式接入控制策略能在各种网络初始能量配置下获取显著的总用户收益期望值的提高.另外,经实验还发现可以采用一个最优阈值的＂阈值准则＂来获取次优的接入控制效果,从而大大简化最优接入控制策略的设计.     

Toward Maximizing the Quality of Results of Dependent Tasks Computed Unreliably

This paper studies the problem of maximizing the number of correct results of dependent tasks computed unreliably. We consider a distributed system composed of a reliable server that coordinates the computation of a massive number of unreliable workers. Any worker computes correctly with probability p < 1. Any incorrectly computed task corrupts all dependent tasks. The goal is to determine which tasks should be computed by the (reliable) server and which by the (unreliable) workers, and when, so as to maximize the expected number of correct results, under a constraint d on the computation time. This problem is motivated by distributed computing applications that persist partial results of computations for future use in other computations and that want to ensure that the persisted results are of high quality. We show that this optimization problem is NP-hard. Then we study optimal scheduling solutions for the mesh with the tightest deadline. We present combinatorial arguments that describe all optimal solutions for two ranges of values of worker reliability p, when p is close to zero and when p is close to one.     

Consensus of multi-agent systems based on sampled-data control

This article studies the consensus problem in directed networks, assuming that each agent is with double-integrator dynamics and only obtains the measurements of its positions relative to its neighbours at sampling instants. We propose a protocol based on sampled-data control and derive an equivalent characterisation of the solvability of the consensus problem under this protocol. In virtue of this equivalent characterisation, we further consider two cases: fixed topology and switching topology. For the first case, we present a set of sampling periods and feedback coefficients which ensure that the protocol can solve a consensus problem. For the second case, we derive sufficient conditions for the protocol to solve a consensus problem under arbitrary switching signals and under a class of switching signals, respectively. Finally, simulations are provided to illustrate the effectiveness of the theoretical results.     

Optimisation of strain selection in evolutionary continuous culture

In this work, we study a minimal time control problem for a perfectly mixed continuous culture with n ≥ 2 species and one limiting resource. The model that we consider includes a mutation factor for the microorganisms. Our aim is to provide optimal feedback control laws to optimise the selection of the species of interest. Thanks to Pontryagin's Principle, we derive optimality conditions on optimal controls and introduce a sub-optimal control law based on a most rapid approach to a singular arc that depends on the initial condition. Using adaptive dynamics theory, we also study a simplified version of this model which allows to introduce a near optimal strategy.