success story, rather than a failure) is LeMessurier’s Citicorp Building 2 , in which an engineering design change was made, but tested in a model in which the wind came only from a limited set of directions. 2 See “The Fifty-Nine Story Crisis”, The New Yorker , May 29, 1995, pp 45-53.
Chapter 1 Course Overview 6.01— Spring 2011— April 25, 2011 14 Another important dimension in modeling is whether the model is deterministic or not. We might, for example, model the effect of the robot executing a command to set its velocity as mak- ing an instantaneous change to the commanded velocity. But, of course, there is likely to be some delay and some error. We have to decide whether to ignore that delay and error in our model and treat it as if it were ideal; or, for example, to make a probabilistic model of the possible results of that command. Generally speaking, the more different the possible outcomes of a particular action or command, and the more diffuse the probability distribution over those outcomes, the more important it is to explicitly include uncertainty in the model. Once we have a model, we can use it in a variety of different ways, which we explore below. Analytical models By far the most prevalent use of models is in analysis: Given a circuit design, what will the voltage on this terminal be? Given a computer program, will it compute the desired answer? How long will it take? Given a control system, implemented as a circuit or as a program, will the system stop at the right distance from the light bulb? Analytical tools are important. It can be hard to verify the correctness of a system by trying it in all possible initial conditions with all possible inputs; sometimes it is easier to develop a mathematical model and then prove a theorem about the model. For some systems, such as pure software computations, or the addition circuitry in a calculator, it is possible to analyze correctness or speed with just a model of the system in question. For other systems, such as fuel injectors, it is impossible to analyze the correctness of the controller without also modeling the environment (or “plant”) to which it is connected, and then analyzing the behavior of the coupled system of the controller and environment. To demonstrate some of these tradeoffs, we can do a very simple analysis of a robot moving to- ward a lamp. Imagine that we arrange it so that the robot’s velocity at time t , V [ t ] , is proportional to the difference between the actual light level, X [ t ] , and a desired light level (that is, the light level we expect to encounter when the robot is the desired distance from the lamp), X desired ; that is, we can model our control system with the difference equation V [ t ] = k ( X desired − X [ t ]) where k is the constant of proportionality, or gain , of the controller.