Scheduling and discrete event control of flexible manufacturing systems based on Petri nets

A flexible manufacturing system (FMS) is a computerized production system that can simultaneously manufacture multiple types of products using various resources such as robots and multi-purpose machines. The central problems associated with design of flexible manufacturing systems are related to process planning, scheduling, coordination control, and monitoring. Many methods exist for scheduling and control of flexible manufacturing systems, although very few methods have addressed the complexity of whole FMS operations. This thesis presents a Petri net based method for deadlock-free scheduling and discrete event control of flexible manufacturing systems. A significant advantage of Petri net based methods is their powerful modeling capability. Petri nets can explicitly and concisely model the concurrent and asynchronous activities, multi-layer resource sharing, routing flexibility, limited buffers and precedence constraints in FMSs. Petri nets can also provide an explicit way for considering deadlock situations in FMSs, and thus facilitate significantly the design of a deadlock-free scheduling and control system. The contributions of this work are multifold. First, it develops a methodology for discrete event controller synthesis for flexible manufacturing systems in a timed Petri net framework. The resulting Petri nets have the desired qualitative properties of liveness, boundedness (safeness), and reversibility, which imply freedom from deadlock, no capacity overflow, and cyclic behavior, respectively. This precludes the costly mathematical analysis for these properties and reduces on-line computation overhead to avoid deadlocks. The performance and sensitivity of resulting Petri nets, thus corresponding control systems, are evaluated. Second, it introduces a hybrid heuristic search algorithm based on Petri nets for deadlock-free scheduling of flexible manufacturing systems. The issues such as deadlock, routing flexibility, multiple lot size, limited buffer size and material handling (loading/unloading) are explored. Third, it proposes a way to employ fuzzy dispatching rules in a Petri net framework for multi-criterion scheduling. Finally, it shows the effectiveness of the developed methods through several manufacturing system examples compared with benchmark dispatching rules, integer programming and Lagrangian relaxation approaches

Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs

We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton-Jacobi-Bellman equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite.Comment: 31 pages, 20 figure

Simple Approximations of Semialgebraic Sets and their Applications to Control

Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples

Sum-of-Squares approach to feedback control of laminar wake flows

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach to the solution of such optimisation problems, based on Sum-of-Squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re=100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, energy efficiency, and physical control mechanisms identified are analysed. Key elements, implications and future work are discussed

Socionic Multi-Agent Systems Based on Reflexive Petri Nets and Theories of Social Self-Organisation

This contribution summarises the core results of the transdisciplinary ASKO project, part of the German DFG's programme Sozionik, which combines sociologists' and computer scientists' skills in order to create improved theories and models of artificial societies. Our research group has (a) formulated a social theory, which is able to explain fundamental mechanisms of self-organisation in both natural and artificial societies, (b) modelled this in a mathematical way using a visual formalism, and (c) developed a novel multi-agent system architecture which is conceptually coherent, recursively structured (hence non-eclectic) and based on our social theory. The article presents an outline of both a sociological middle-range theory of social self-organisation in educational institutions, its formal, Petri net based model, including a simulation of one of its main mechanisms, and the multi-agent system architecture SONAR. It describes how the theory was created by a re-analysis of some grand social theories, by grounding it empirically, and finally how the theory was evaluated by modelling its concepts and statements.Multi-Agents Systems, Petri Nets, Self-Organisation, Social Theories

Belief propagation : an asymptotically optimal algorithm for the random assignment problem

The random assignment problem asks for the minimum-cost perfect matching in the complete $n\times n$ bipartite graph \Knn with i.i.d. edge weights, say uniform on $[0,1]$. In a remarkable work by Aldous (2001), the optimal cost was shown to converge to $\zeta(2)$ as $n\to\infty$, as conjectured by M\'ezard and Parisi (1987) through the so-called cavity method. The latter also suggested a non-rigorous decentralized strategy for finding the optimum, which turned out to be an instance of the Belief Propagation (BP) heuristic discussed by Pearl (1987). In this paper we use the objective method to analyze the performance of BP as the size of the underlying graph becomes large. Specifically, we establish that the dynamic of BP on \Knn converges in distribution as $n\to\infty$ to an appropriately defined dynamic on the Poisson Weighted Infinite Tree, and we then prove correlation decay for this limiting dynamic. As a consequence, we obtain that BP finds an asymptotically correct assignment in $O(n^2)$ time only. This contrasts with both the worst-case upper bound for convergence of BP derived by Bayati, Shah and Sharma (2005) and the best-known computational cost of $\Theta(n^3)$ achieved by Edmonds and Karp's algorithm (1972).Comment: Mathematics of Operations Research (2009