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Unformatted text preview: OptimizationBased Control Richard M. Murray Control and Dynamical Systems California Institute of Technology DRAFT v2.1a, January 3, 2010 c circlecopyrt California Institute of Technology All rights reserved. This manuscript is for review purposes only and may not be reproduced, in whole or in part, without written consent from the author. Chapter 1 Trajectory Generation and Tracking This chapter expands on Section 7.5 of Feedback Systems by ˚ Astr¨ om and Murray ( ˚ AM08), which introduces the use of feedforward compensation in control system design. We begin with a review of the two degree of freedom design approach and then focus on the problem of generating feasible trajectories for a (nonlinear) control system. We make use of the concept of differential flatness as a tool for generating feasible trajectories. Prerequisites. Readers should be familiar with modeling of input/output control systems using differential equations, linearization of a system around an equilibrium point and state space control of linear systems, including reachability and eigenvalue assignment. Although this material supplements concepts introduced in the context of output feedback and state estimation, no knowledge of observers is required. 1.1 Two Degree of Freedom Design A large class of control problems consist of planning and following a trajectory in the presence of noise and uncertainty. Examples include autonomous vehicles maneuvering in city streets, mobile robots performing tasks on factor floors (or other planets), manufacturing systems that regulate the flow of parts and materials through a plant or factory, and supply chain management systems that balance orders and inventories across an enterprise. All of these systems are highly nonlinear and demand accurate performance. To control such systems, we make use of the notion of two degree of freedom controller design. This is a standard technique in linear control theory that sepa rates a controller into a feedforward compensator and a feedback compensator. The feedforward compensator generates the nominal input required to track a given ref erence trajectory. The feedback compensator corrects for errors between the desired and actual trajectories. This is shown schematically in Figure 1.1. In a nonlinear setting, two degree of freedom controller design decouples the trajectory generation and asymptotic tracking problems. Given a desired output trajectory, we first construct a state space trajectory x d and a nominal input u d that satisfy the equations of motion. The error system can then be written as a time varying control system in terms of the error, e = x − x d . Under the assumption that that tracking error remains small, we can linearize this timevarying system about e = 0 and stabilize the e = 0 state. (Note: in ˚ AM08 the notation u ff was used for the desired [feedforward] input. We use u d here to match the desired state x d .) More formally, we assume that our process dynamics can be described by a 1.1. TWO DEGREE OF FREEDOM DESIGN1....
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This note was uploaded on 01/04/2012 for the course CDS 110b taught by Professor Murray,r during the Fall '08 term at Caltech.
 Fall '08
 Murray,R

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