Analysis, Control and Optimization of Complex Dynamic Systems

This paper focuses on the problem of inventory control of production systems. The main contribution of the paper is that, for the first time, production systems are modeled as switched linear systems and the production problem is formulated as a switched \(H\infty \) control problem with a piecewise-affine control law. The switching variable for the production systems modeled in this paper is the stock level. When the stock level is positive, some of the produced parts are being stored. The stocked parts may deteriorate with time at a given rate. When the stock level is negative it leads to backorders, which means that orders for production of parts are coming in and there is no stocked parts to immediately meet the demand. A switched linear model is used and it is shown that the inventory control problem can be solved using switched control theory. More specifically, a state feedback controller that forces the stock level to be kept close to zero, even when there are fluctuations in the demand, will be designed in this paper using \(H\infty \), control theory. The synthesis of the gains of the state feedback controller that quadratically stabilizes the production dynamics and at the same time rejects the external demand fluctuation (treated as a disturbance) are determined by solving a given set of linear matrix inequalities (LMIs). A numerical example is provided to show the effectiveness of the developed method.

This work develops asymptotically optimal production planning strategies for a class of discrete-time manufacturing systems. To reflect uncertainty, finite-state Markov chains are used in the formulation. The state space of the underlying Markov chain is decomposed into a number of recurrent classes and a group of transient states. Using a hierarchical control approach, by aggregating the states in each recurrent class into a single state, a continuous-time limit control problem in which the resulting limit Markov chain has much smaller state space is derived. Using the optimal control of the limit problem, control policies for the original problem are constructed. Moreover, it is shown that the strategies so designed are nearly optimal.

This paper provides an analytical method for evaluating production rates in serial lines having finite buffers and unreliable machines with arbitrary unimodal distributions of up- and downtime. Provided that each buffer is capable of accommodating at least one downtime of all machines in the system, we show that the production rate (a) is relatively insensitive to the type of up- and downtime distributions and (b) can be approximated by a linear function of their coefficients of variation. The results obtained are verified using Weibull, gamma, and log-normal probability distributions of up- and downtime.

We propose a decision support framework for the Supply Chain management of a manufacturing enterprise. It utilizes structured inforiation sharing between a fluid approximation Master-Problem and facility specific Sub-Problems. We optimize weekly production schedules that minimize inventory and backlog costs subject to non-linear constraints on production imposed by weekly varying dynamic lead-times and inter-facility quality of service driven inventory hedging policies. Computational experience demonstrates that it is possible to achieve the same quality of service with significantly lower inventory and system times relative to static lead-time state of the art industry practice.

In this paper we consider the problem of providing statistical Quality of Service (QoS) guarantees defined in terms of packet loss when independent heterogeneous traffic streams access a network router of high capacity. By using a scaling technique we show how this problem becomes tractable when the server capacity is large and many traffic streams are present. In particular we show that we can define an effective bandwidth for the sources that allows us to map the model onto a multirate loss model. In particular we show several insights on the multiplexing problem as the capacity becomes large. We also provide numerical and simulation evidence to show how the largeness of networks can be used to advantage in providing very simple admission control schemes. The techniques are based on large deviations, local limit theorems, and the product-form associated with co-ordinate convex policies.

We consider in this paper the problem of combined flow control and routing in a noncooperative setting, where each user is faced with a multi-criteria optimization problem, formulated as the minimization of one criterion subject to constraints on others. We address here the basic questions of existence and uniqueness of equilibrium. We show that an equilibrium indeed exists, but it may not be unique due to the multi-criteria nature of the problem. We are able, however, to obtain uniqueness in some weaker sense under appropriate conditions; we show in particular that the link utilizations are uniquely determined at equilibrium and the normalized Nash equilibrium is unique.

The electricity pool is a fundamental coordination mechanism for scheduling short-term forward electricity markets. It is fundamental theoretically in the sense that its main goal is the best use of society's resources subject to basic reliability criteria. However, the pool scheduling and its associated marginal pricing procedure may not be able to unambiguously coordinate its self-interested participant generators. In other words, the centrally-determined operation and marginal price schedules may not lead to a competitive equilibrium. Several out-of-market mechanisms have been proposed to bridge the objectives of the pool with those of the market players. Among them are uplifts whose current use still does not induce true competitive equilibria. In this chapter, generalized uplifts are proposed as an alternative to traditional uplifts. An innovative mechanism to compute the generalized uplifts is developed. A simple illustrative example is presented.

We consider dynamic games in large population conditions where the agents evolve according to non-uniform dynamics and are weakly coupled via their dynamics and the individual costs. A state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an F-Nash equilibrium property. An attraction property of the mass behaviour is established. The methodology and the results contained in this paper reveal novel behavioral properties of the relationship of any given individual with respect to the mass of individuals in large-scale noncooperative systems of weakly coupled agents.