Geometric existence theory for the control-affine nonlinear optimal regulator

AbstractFor infinite horizon nonlinear optimal control problems in which the control term enters linearly in the dynamics and quadratically in the cost, well-known conditions on the linearised problem guarantee existence of a smooth globally optimal feedback solution on a certain region of state space containing the equilibrium point. The method of proof is to demonstrate existence of a stable Lagrangian manifold M and then construct the solution from M in the region where M has a well-defined projection onto state space. We show that the same conditions also guarantee existence of a nonsmooth viscosity solution and globally optimal set-valued feedback on a much larger region. The method of proof is to extend the construction of a solution from M into the region where M no-longer has a well-defined projection onto state space

Analysis of Scheduling Policies for a M/G/I Queue with Rework

This thesis analyzes a multi-class M/G/1 priority queueing system in which distinct job types require one service cycle and, with non-zero probability, require a second service cycle. The main objective is to find a new heuristic scheduling policy that minimizes the long-run expected holding and preemption costs. Arrival rates, service rates, and the probability of undertaking second service are all class specific. A mean value analysis (MVA) approach was employed to derive the long- run mean time in queue for each job type under each policy, thereby providing the appropriate cost equations. Numerical experiments suggest that the preemptive resume scheduling policy yields the lowest cost most frequently

Robust Feedback Control of a Single Server Queueing System

This paper extends previous work of Ball et al. [BDKY] to control of a model of a simple queueing server. There are n queues of customers to be served by a single server who can service only one queue at a time. Each queue is subject to an unknown arrival rate, called a “disturbance” in accord with standard usage from H ∞ theory. An H ∞-type performance criterion is formulated. The resulting control problem has several novel features distinguishing it from the standard smooth case already studied in the control literature: the presence of constraining dynamics on the boundary of the state space to ensure the physical property that queue lengths remain nonnegative, and jump discontinuities in any nonconstant state-feedback law caused by the finiteness of the admissible control set (choice of queue to be served). We arrive at the solution to the appropriate Hamilton–Jacobi equation via an analogue of the stable invariant manifold for the associated Hamiltonian flow (as was done by van der Schaft for the smooth case) and relate this solution to the (lower) value of a restricted differential game, similar to that formulated by Soravia for problems without constraining dynamics. An additional example is included which shows that the projection dynamics used to maintain nonnegativity of the state variables must be handled carefully in more general models involving interactions among the different queues. Primary motivation comes from the application to traffic signal control. Other application areas, such as manufacturing systems and computer networks, are mentioned

Gradient projection anti-windup scheme

Thesis (Sc. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 205-217).It is a well-recognized fact that control saturation affects virtually all practical control systems. It leads to controller windup, which degrades/limits the system's closed-loop performance, and may cause catastrophic failures if it induces instability. Anti-windup compensation is one of two main approaches to mitigate the effects of windup, and is conceptually and practically attractive. For the idealized case of constrained linear time invariant (LTI) plants driven by LTI controllers, numerous anti-windup schemes exist. However, most practical control systems are inherently nonlinear, and anti-windup compensation for nonlinear systems remains largely an open problem. To this end, we propose the gradient projection anti-windup (GPAW) scheme, which is an extension of the conditional integration method to multi-input-multi-output (MIMO) nonlinear systems, using Rosen's gradient projection method for nonlinear programming. It achieves controller state-output consistency by projecting the controller state onto the unsaturated region induced by the control saturation constraints. The GPAW-compensated controller is a hybrid controller defined by the online solution to either a combinatorial optimization subproblem, a convex quadratic program, or a projection onto a convex polyhedral cone problem. We show that the GPAW-compensated system is obtained by modifying the uncompensated system with a passive operator. Qualitative weaknesses of some existing anti-windup results are established, which motivated a new paradigm to address the anti-windup problem. It is shown that for a constrained first order LTI plant driven by a first order LTI controller, GPAW compensation can only maintain/enlarge its region of attraction (ROA). In this new paradigm, we derived some ROA comparison and stability results for MIMO nonlinear as well as MIMO LTI systems. The thesis is not that the GPAW scheme solves a centuries-old open problem of immense practical importance, but rather, that it provides a potential path to a solution. We invite the reader to join us in this quest at the confluence of nonlinear systems, hybrid systems, projected dynamical systems, differential equations with discontinuous right-hand sides, combinatorial optimization, convex analysis and optimization, and passive systems.by Chun Sang Justin Teo.Sc.D