Approximate IPA: Trading Unbiasedness for Simplicity

When Perturbation Analysis (PA) yields unbiased sensitivity estimators for expected-value performance functions in discrete event dynamic systems, it can be used for performance optimization of those functions. However, when PA is known to be unbiased, the complexity of its estimators often does not scale with the system's size. The purpose of this paper is to suggest an alternative approach to optimization which balances precision with computing efforts by trading off complicated, unbiased PA estimators for simple, biased approximate estimators. Furthermore, we provide guidelines for developing such estimators, that are largely based on the Stochastic Flow Modeling framework. We suggest that if the relative error (or bias) is not too large, then optimization algorithms such as stochastic approximation converge to a (local) minimum just like in the case where no approximation is used. We apply this approach to an example of balancing loss with buffer-cost in a finite-buffer queue, and prove a crucial upper bound on the relative error. This paper presents the initial study of the proposed approach, and we believe that if the idea gains traction then it may lead to a significant expansion of the scope of PA in optimization of discrete event systems.Comment: 8 pages, 8 figure

Congestion management in traffic-light intersections via Infinitesimal Perturbation Analysis

We present a flow-control technique in traffic-light intersections, aiming at regulating queue lengths to given reference setpoints. The technique is based on multivariable integrators with adaptive gains, computed at each control cycle by assessing the IPA gradients of the plant functions. Moreover, the IPA gradients are computable on-line despite the absence of detailed models of the traffic flows. The technique is applied to a two-intersection system where it exhibits robustness with respect to modeling uncertainties and computing errors, thereby permitting us to simplify the on-line computations perhaps at the expense of accuracy while achieving the desired tracking. We compare, by simulation, the performance of a centralized, joint two-intersection control with distributed control of each intersection separately, and show similar performance of the two control schemes for a range of parameters

Optimal control approaches for persistent monitoring problems.

Thesis (Ph. D.)--Boston UniversityPersistent monitoring tasks arise when agents must monitor a dynamically changing environment which cannot be fully covered by a stationary team of available agents. It differs from traditional coverage tasks due to the perpetual need to cover a changing environment, i.e., all areas of the mission space must be visited infinitely often. This dissertation presents an optimal control framework for persistent monitoring problems where the objective is to control the movement of multiple cooperating agents to minimize an uncertainty metric in a given mission space. In an one-dimensional mission space, it is shown that the optimal solution is for each agent to move at maximal speed from one switching point to the next, possibly waiting some time at each point before reversing its direction. Thus, the solution is reduced to a simpler parametric optimization problem: determining a sequence of switching locations and associated waiting times at these switching points for each agent. This amounts to a hybrid system which is analyzed using Infinitesimal Perturbation Analysis (IPA) , to obtain a complete on-line solution through a gradient-based algorithm. IPA is a method used to provide unbiased gradient estimates of performance metrics with respect to various controllable parameters in Discrete Event Systems (DES) as well as in Hybrid Systems (HS). It is also shown that the solution is robust with respect to the uncertainty model used, i.e., IPA provides an unbiased estimate of the gradient without any detailed knowledge of how uncertainty affects the mission space. In a two-dimensional mission space, such simple solutions can no longer be derived. An alternative is to optimally assign each agent a linear trajectory, motivated by the one dimensional analysis. It is proved, however, that elliptical trajectories outperform linear ones. With this motivation, the dissertation formulates a parametric optimization problem to determine such trajectories. It is again shown that the problem can be solved using IPA to obtain performance gradients on line and obtain a complete and scalable solution. Since the solutions obtained are generally locally optimal, a stochastic comparison algorithm is incorporated for deriving globally optimal elliptical trajectories. The dissertation also approaches the problem by representing an agent trajectory in terms of general function families characterized by a set of parameters to be optimized. The approach is applied to the family of Lissajous functions as well as a Fourier series representation of an agent trajectory. Numerical examples indicate that this scalable approach provides solutions that are near optimal relative to those obtained through a computationally intensive two point boundary value problem (TPBVP) solver. In the end, the problem is tackled using centralized and decentralized Receding Horizon Control (RHC) algorithms, which dynamically determine the control for agents by solving a sequence of optimization problems over a planning horizon and executing them over a shorter action horizon