Robust control of uncertain context-sensitive probabilistic Boolean networks

Uncertainty is an intrinsic phenomenon in control of gene regulatory networks (GRNs). The presence of uncertainty is related to impreciseness of GRN models due to: (1) Errors caused by imperfection of measurement devices and (2) Models' inability to fully capture a complex structure of the GRN. Consequently, there is a discrepancy between actual behaviour of the GRN and what is predicted by its mathematical model. This can result in false control signals, which can drive a cell to an undesirable state. To address the problem of control under uncertainties, a risk-sensitive control paradigm is proposed. Robustness is accomplished by minimisation of the mean exponential cost as opposed to, for instance, minimisation of the mean square cost by risk-neutral controllers. The authors derive an optimal risk-sensitive controller when a GRN is modelled by a context-sensitive probabilistic Boolean network (CSPBN). By using a relation between the relative entropy and free-energy, a relative stability of the cost achieved by the risk-sensitive controller is demonstrated when the distribution of the CSPBN attractors is perturbed, as opposed to the cost of the risk-neutral controller that exhibits increase. The use of the relation between the relative entropy and free-energy to analyse the influence of a particular attractor on the robustness of the controller is studied. The efficiency of the risk-sensitive controller is tested for the CSPBN obtained from the study of malignant melanoma.     

Optimal control with responses and control force constraints

Abstract

In order to assure safety and serviceability, structural responses are kept within certain limits by active control methods. Control forces as well as responses (displacement and velocity) are considered constraints in an optimal control system. The exact solution of the optimal control system under the aforementioned constraints is derived in the first part of the paper. An approximate solution is also suggested in the paper. An example dealing with a six‐story building is presented in the last part of the paper to demonstrate the application of the approximate approach.     

Production control policy for tandem workstations with constant service times and queue time constraints

Queue time constraints are commonly imposed to ensure product quality in contemporary production systems. We study the performance of two single stations with deterministic service times and a predetermined time window in between, where both stations suffer time-based pre-emptive breakdowns. To improve productivity, achieving higher capacity and lower rework rate are the two main objectives. While higher capacity requires a higher work-in-process (WIP)-level threshold, a lower rework rate requires a smaller one. To quantify the trade-off between the two objectives, an analytical model is derived. The model is then used to derive the WIP-level threshold control policy for a time-constrained system. We also show that system capacity diminishes with the decrease in WIP-level thresholds.     

Optimal pricing policy for a deteriorating product by dynamic tracking control

We study the optimal selling price of a deteriorating product under a deterministic situation in a finite time horizon where the time horizon is either known or unknown. Inventory holding cost is expressed as a quadratic function of the current inventory level. Given a known time horizon, we develop a model by considering the deterioration dynamics of the product, and show its equivalence to a generalised optimal control problem of a linear quadratic form, i.e. an optimal dynamic tracking problem with constraints on the control variable. An optimal pricing policy is derived based on the maximum value principle. The control policy takes a state feedback form; it exhibits a closed-loop relationship between the optimal selling price (control variable) and the optimal inventory level (state variable). Given an unknown time horizon, an optimal pricing policy is derived through a similar approach when the initial inventory level meets certain conditions. Numerical situations are conducted to illustrate the effectiveness of the derived price control policies. Some interesting managerial insights are discussed.     

Optimal production control policy in unreliable batch processing manufacturing systems with transportation delay

This paper considers the problem of production planning of unreliable batch processing manufacturing systems. The finished goods are produced in lots, and are then transported to a storage area in order to continuously meet a constant demand rate. The main objective of this work is to jointly determine the optimal lot sizing and optimal production control policy that minimise the total expected cost of inventory/backlog and transportation, over an infinite time horizon. The decision variables are the lot sizing and the production rate. The problem is formulated with a stochastic dynamic programming model and the impulse control theory is applied to establish the Hamilton–Jacobi–Bellman (HJB) equations. Based on a numerical resolution of the HJB equations, it is shown that the optimal control policy is governed by a base stock policy for production rate control and economic lot size for batch processing. A thorough analysis and practical issues are addressed with a simulation-based approach. Thus, a combined discrete–continuous simulation model is developed to determine the optimal parameters of the proposed policy when the failure and repair times follow general distributions. The results are illustrated with numerical examples and confirmed through sensitivity analysis.     

Reliability evaluation of capacitated-flow networks with budget constraints

Many real-world systems such as transportation systems and manufacturing systems can be regarded as flow networks whose arcs have independent, finite and multi-valued random capacities. Such a flow network is indeed a multistate system with multistate components and so its reliability for level (d,c), i.e., the probability that d units of flow can be transmitted from the source node to the sink node such that the total transmission cost is less than or equal to c, can be computed in terms of minimal path vectors to level (d,c) (named (d,c)-MPs here). The main objective of this paper is to present a simple algorithm to generate all (d,c)-MPs of such a system for each level (d,c) in terms of minimal pathsets. Two examples are given to illustrate how all (d,c)-MPs are generated by our algorithm and then the reliability of one example is computed.     

Optimal testing policy for a computer system with intermittent faults

A computer system with intermittent faults fails with probability p when it is used in hidden faults. Periodic tests are scheduled at times kT(k = 1, 2,…) to detect a hidden fault. The mean time, the expected number of tests and the expected cost until detection of a fault or system failure are derived, using the theory of Markov renewal processes. An optimal testing time T^* to minimize the expected cost is discussed. A finite T^* is given by a unique solution of an equation.     

Multi-element probabilistic collocation for sensitivity analysis in cellular signalling networks

The multi-element probabilistic collocation method (ME-PCM) as a tool for sensitivity analysis of differential equation models as applied to cellular signalling networks is formulated. This method utilises a simple, efficient sampling algorithm to quantify local sensitivities throughout the parameter space. The application of the ME-PCM to a previously published ordinary differential equation model of the apoptosis signalling network is presented. The authors verify agreement with the previously identified regions of sensitivity and then go on to analyse this region in greater detail with the ME-PCM. The authors demonstrate the generality of the ME-PCM by studying sensitivity of the system using a variety of biologically relevant markers in the system such as variation in one (or many) chemical species as a function of time, and total exposure of a single chemical species.     

Optimal replacement policy for a redundant system

Summary This paper considers the reliability of a system which consists ofn components connected in series, the i^thcomponent being supported by (m _i – 1) units in parallel and obtains the optimal policy of replacement by dynamic programming. In the case of a system consisting of identical pairs, the decision depends on the number of operable pairs. For this case sample calculations are presented.

---

Zusammenfassung Wir untersuchen die Zuverlässigkeit eines Seriensystems ausn Komponenten, bei dem für diei-te Komponentem _i – 1 Reserveelemente zur Verfügung stehen. Mit Hilfe der dynamischen Optimierung bestimmen wir die optimale Ersetzungspolitik. Besteht das System aus identischen Paaren, hängt die Entscheidung von der Zahl der intakten Paare ab. Für diesen Fall geben wir beispielhaft einige numerische Ergebnisse.

     

Tool replacement policy for probabilistic tool life and random wear process

This study presents a tool replacement model which determines the optimal initial tool setting, tool adjustment cycle and tool replacement time with regard to quality loss, adjustment cost, replacement cost and penalty cost for possible tool failure. A quadratic loss function is employed to characterize the costs representing part quality. The Bernstein probability density function (BDF) is applied to describe the usual characteristics of tool life and the tool wear process. Then, based on the considered costs, the best strategy for tool replacement can be determined. A possible application of the model is given that provides a method of off‐line tool management in a production environment. Copyright © 1999 John Wiley & Sons, Ltd.     

A unified framework for probabilistic sequential tolerance control

Sequential tolerance control (STC) is a methodology that uses available measurement information at the completion of one manufacturing operation to position the set point for subsequent operations. It has been shown that STC can lead to inferior solutions when the manufacturing process distributions are skewed. This paper presents an adaptive sphere-fitting method (ASF-STC) that adjusts for such skewness. ASF-STC requires as inputs both the direction of skewness and the probability distribution parameters for each operation. Heuristic methods for estimating each of these inputs are presented. Through computational testing, ASF-STC is shown to offer significant improvements over STC when such skewness exists.     

Mathematical programs with complementarity constraints in traffic and telecommunications networks

Given a suitably parametrized family of equilibrium models and a higher level criterion by which to measure an equilibrium state, mathematical programs with equilibrium constraints (MPECs) provide a framework for improving or optimizing the equilibrium state. An example is toll design in traffic networks, which attempts to reduce total travel time by choosing which arcs to toll and what toll levels to impose. Here, a Wardrop equilibrium describes the traffic response to each toll design. Communication networks also have a deep literature on equilibrium flows that suggest some MPECs. We focus on mathematical programs with complementarity constraints (MPCCs), a subclass of MPECs for which the lower level equilibrium system can be formulated as a complementarity problem and therefore, importantly, as a nonlinear program (NLP). Although MPECs and MPCCs are typically non-convex, which is a consequence of the upper level objective clashing with the users' objectives in the lower level equilibrium program, the last decade of research has paved the way for finding local solutions of MPCCs via standard NLP techniques.     

Optimal periodic preventive maintenance policy for leased equipment

For leased equipment the lessor incurs penalty costs for failures occurring over the lease period and for not rectifying such failures within a specified time limit. Through preventive maintenance actions the penalty costs can be reduced but this is achieved at the expense of increased maintenance costs. The paper looks at a periodic preventive maintenance policy which achieves a tradeoff between the penalty and maintenance costs.     

Optimal ordering and pricing policy for an inventory system with order cancellations

This paper investigates the problem of ordering and pricing over a finite time planning horizon for an inventory system with advance sales and spot sales. It is assumed that the planning horizon is divided into several sales cycles each of which is divided into an advance sales period and a spot sales period. During the advance sales period, all customers are required to make reservations for their orders and will receive them at the arrival time of the replenishment orders. In the case of the spot sales periods, all customers receive their orders at the time of the purchase. In actual practice, since customers with reservations may cancel their orders before receiving them, this paper considers this phenomenon and develops a continuous time inventory model to deal with the proposed problem. This paper maximizes the total profit over a finite time planning horizon by determining the optimal advance sales price, spot sales price, order size, and replenishment frequency. Analysis of results shows that a simple algorithm can be developed to arrive at an optimal decision.     

Optimal design of general stiffened composite circular cylinders for global buckling with strength constraints

A design strategy for optimal design of composite grid-stiffened cylinders subjected to global and local buckling constraints and strength constraints was developed using a discrete optimizer based on a genetic algorithm. An improved smeared stiffener theory was used for the global analysis. Local buckling of skin segments were assessed using a Rayleigh-Ritz method that accounts for material anisotropy. The local buckling of stiffener segments were also assessed. Constraints on the axial membrane strain in the skin and stiffener segments were imposed to include strength criteria in the grid-stiffened cylinder design. Design variables used in this study were the axial and transverse stiffener spacings, stiffener height and thickness, skin laminate stacking sequence and stiffening configuration, where stiffening configuration is a design variable that indicates the combination of axial, transverse and diagonal stiffener in the grid-stiffened cylinder. The design optimization process was adapted to identify the best suited stiffening configurations and stiffener spacings for grid-stiffened composite cylinder with the length and radius of the cylinder, the design in-plane loads and material properties as inputs. The effect of having axial membrane strain constraints in the skin and stiffener segments in the optimization process is also studied for selected stiffening configurations.     

Use of neural networks for fitting of FE probabilistic scaling model parameters

The probabilistic crack approach, based on the Monte Carlo method, was recently developed for finite element analysis of concrete cracking and related size effects. In this approach the heterogeneity of the material is taken into account by considering the material properties (tensile strength, Young modulus, etc.) to vary spatially following a normal distribution. N samples of the vector of random variables are generated from a specific probability density function, and the N samples corresponding to a simulation are functions of the mean value and of the standard deviation that define the Gauss density function. The problem is that these statistical moments are not known, a priori, for the characteristic volume of the finite elements used in the analysis. The paper proposes an inverse finite element analysis using neural networks for the determination of the statistical distribution parameters (e.g., for a normal distribution, the mean and the standard deviation) from a given response of the structure (for instance, an average load-displacement curve). From FE-analysis of 4-point bending beam tests, it is shown that the backanalysis technique developed in this paper is a powerful tool to determine the probabilistic distribution functions at the material level from structural tests for material volumes which are generally not accessible to direct testing. This revised version was published online in July 2006 with corrections to the Cover Date.     

Simplified probabilistic selectivity technique for earth fault detection in unearthed MV networks

A novel selectivity technique to identify the faulty feeder in unearthed Medium Voltage (MV) networks is introduced. The proposed technique is based on a simplified probabilistic method applied to transient features extracted from the residual currents only using the discrete wavelet transform (DWT). DWT will enhance the accuracy of localising fault events with different fault resistances and the probability will give the protection decision (detection/selectivity). The standard deviation of a window of the DWT detail coefficient is used to detect the fault and an adaptive threshold is used as input to the probability function to estimate the faulty feeder. A faulty unearthed 20 kV network is simulated by ATP/EMTP imitating practical fault cases such as arcing and tree leaning faults. The time-domain test cases provide evidence of the efficacy of the proposed technique.     

一种最优滑模控制算法及其仿真应用

针对某机电伺服跟踪系统存在扰动和系统参数摄动的情况,提出了一种带有积分补偿的最优滑模变结构控制算法,并给出了获得切换函数、积分增益和控制函数的方法。仿真结果表明,这种新的控制算法比经典控制方法的效果更好。它能够获得灵敏的跟踪效果,提高了系统响应的快速性,具有较好的动态性能和鲁棒性,并消除了抖振。此控制系统结构简单,易于实现。