Chapter 2System Modeling... I asked Fermi whether he was not impressed by the agreement betweenour calculated numbers and his measured numbers. He replied, “How manyarbitrary parameters did you use for your calculations?” I thought for amoment about our cut-off procedures and said, “Four.” He said, “I remembermy friend Johnny von Neumann used to say, with four parameters I can fitan elephant, and with five I can make him wiggle his trunk.”Freeman Dyson on describing the predictions of his model for meson-protonscattering to Enrico Fermo in 1953 [8].A model is a precise representation of a system’s dynamics used to an-swer questions via analysis and simulation. The model we choose dependson the questions that we wish to answer, and so there may be multiple mod-els for a single physical system, with different levels of fidelity depending onthe phenomena of interest. In this chapter we provide an introduction to theconcept of modeling, and provide some basic material on two specific meth-ods that are commonly used in feedback and control systems: differentialequations and difference equations.2.1 Modeling ConceptsA model is a mathematical representation of a physical, biological or in-formation system. Models allow us to reason about a system and makepredictions about who a system will behave. In this text, we will mainly beinterested in models describing the input/output behavior of systems andoften in so-called “state space” form.Roughly speaking, a dynamical system is one in which the effects of ac-tions do not occur immediately. For example, the velocity of a car does not3132 CHAPTER 2. SYSTEM MODELINGchange immediately when the gas pedal is pushed nor does the temperaturein a room rise instantaneously when an air conditioner is switched on. Sim-ilarly, a headache does not vanish right after an aspirin is taken, requiringtime to take effect. In business systems, increased funding for a developmentproject does not increase revenues in the short term, although it may do soin the long term (if it was a good investment). All of these are examples ofdynamical systems, in which the behavior of the system evolves with time.Dynamical systems can be viewed in two different ways: the internalview or the external view. The internal view attempts to describe the in-ternal workings of the system and originates from classical mechanics. Theprototype problem was describing the motion of the planets. For this prob-lem it was natural to give a complete characterization of the motion of allplanets. This involves careful analysis of the effects of gravitational pull andthe relative positions of the planets in a system.The other view on dynamics originated in electrical engineering. Theprototype problem was to describe electronic amplifiers. It was naturalto view an amplifier as a device that transforms input voltages to outputvoltages and disregard the internal detail of the amplifier. This resultedin the input/output view of systems. The two different views have beenamalgamated in control theory. Models based on the internal view are calledinternal descriptions, state models, or white box models. The external viewis associated with names such as external descriptions, input/output modelsor black box models. In this book we will mostly use the words state modelsand input/output models.In the remainder of this section we provide an overview of some of thekey concepts in modeling. The mathematical details introduced here areexplored more fully in the remainder of the chapter.The Heritage of MechanicsThe study of dynamics originated in the attempts to describe planetarymotion. The basis was detailed observations of the planets by Tycho Braheand the results of Kepler who found empirically that the orbits of the planetscould be well described by ellipses. Newton embarked on an ambitiousprogram to try to explain why the planets move in ellipses and he found thatthe motion could be explained by his law of gravitation and the formula thatforce equals mass times acceleration. In the process he also invented calculusand differential equations. Newton’s result was the first example of the ideaof reductionism, i.e. that seemingly complicated natural phenomena can beexplained by simple physical laws. This became the paradigm of natural2.1. MODELING CONCEPTS 33science for many centuries.One of the triumphs of Newton’s mechanics was the observation that themotion of the planets could be predicted based on the current positions andvelocities of all planets. It was not necessary to know the past motion. Thestate of a dynamical system is a collection of variables that characterize themotion of a system completely for the purpose of predicting future motion.For a system of planets the state is simply the positions and the velocitiesof the planets. We call the set of all possible states the state space.A common class of mathematical models for dynamical systems is or-dinary differential equations (ODEs). Mathematically, an ODE is writtenasdxdt= f(x). (2.1)Here x = (x1, x2, . . . , xn) ∈ Rnis a vector of real numbers that describes thecurrent state of the system and equation (2.1) describes the rate of changeof the state as a function of the state itself. Note that we do not bother towrite the vector x any differently than a scalar variable. It will generally beclear from context whether a variable is a vector or scalar quantity.An example of an ordinary differential equation is the van der Pol equa-tion,dx1dt= x1− x31− x2dx2dt= x1,(2.2)which is a model of an electronic oscillator. The state of the system isrepresented by two real numbers, x1and x2. The model (2.2) gives thevelocity of the state vector for each value of the state.The evolution of the states can be described using either a time plot or aphase plot, both of which are shown in Figure 2.1. The time plot, on the left,shows the values of the individual states as a function of time. The phaseplot, on the right, shows the vector field for the system, which gives thestate velocity (represented as an arrow) at every point in the state space.In addition, we have superimposed the traces of some of the states fromdifferent conditions. The phase plot gives a strong intuitive representationof the equation as a vector field or a flow. While systems of second order(two states) can be represented in this way, it is unfortunately difficult tovisualize equations of higher order using this approach.The ideas of dynamics and state have had a
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