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CALTECH CDS 101 - System Modeling

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Chapter 2System Modeling2.1 IntroductionIn this chapter we introduce the notion of a dynamical system and describehow to model system systems. Roughly speaking, a dynamical system isone in which the effects of actions do not occur immediately. For example,the velocity of a car does not change immediately when the gas pedal ispushed nor does the temperature in a room rise instantaneously when anair conditioner is switched on. Similarly, a headache does not vanish rightafter an aspirin is taken, requiring time to take effect. In business systems,increased funding for a development project does not increase revenues inthe short term, although it may do so in the long term (if it was a goodinvestment). All of these are examples of dynamical systems, in which thebehavior of the system evolves with time.Modeling is the method by which we deseribe a dynamical system in aprecise mathematical form, for the purpose of analysis and simulation. Amodel of a system is a representation of the system dynamics and it is usedto answer questions about that system. The model we choose depends onthe questions that we wish to answer, and so there may be multiple modelsfor a single physical system, with different levels of fidelity depending on thephenomena 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 different equations.2122 CHAPTER 2. SYSTEM MODELING2.2 Two Views on DynamicsDynamical systems can be viewed from two different ways: the internalview or the external view. The internal view which attempts to describe theinternal workings of the system originates from classical mechanics. Theprototype problem was the problem to describe the motion of the planets.For this problem it was natural to give a complete characterization of themotion of all planets. This involves careful analysis of the effects of gravi-tational pull and the 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.The Heritage of MechanicsDynamics originated in the attempts to describe planetary motion. Thebasis was detailed observations of the planets by Tycho Brahe and the resultsof Kepler who found empirically that the orbits could be well describedby ellipses. Newton embarked on an ambitious program to try to explainwhy the planets move in ellipses and he found that the motion could beexplained by his law of gravitation and the formula that force equals masstimes acceleration. In the process he also invented calculus and differentialequations. Newtons results was the first example of the idea of reductionism,i.e. that seemingly complicated natural phenomena can be explained bysimple physical laws. This became the paradigm of natural science for manycenturies.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. A mathematical model simply gives the rate of change of the2.2. TWO VIEWS ON DYNAMICS 23x ’ = M x − y − x3y ’ = x M = 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52xyFigure 2.1: Illustration of a state model. A state model gives the rate ofchange of the state as a function of the state. The velocity of the state aredenoted by arrows.state as a function of the state itself, i.e. a differential equation.dxdt= f(x) (2.1)This is illustrated in Figure 2.1 for a system with two state variables. Theparticular system represented in the figure is the van der Pol equation:dx1dt= x1− x31− x2dx2dt= x1,which is a model of an electronic oscillator. The model (2.1) gives thevelocity of the state vector for each value of the state. These are representedby the arrows in the figure. The figure gives a strong intuitive representationof the equation as a vector field or a flow. Systems of second order can berepresented in this way. It is unfortunately difficult to visualize equationsof higher order in this way.The ideas of dynamics and state have had a profound influence on phi-losophy where it inspired the idea of predestination. If the state of a naturalsystem is known at some time, its future development is complete deter-mined. The vital development of dynamics has continued in the 20th cen-tury. One of the interesting outcomes is chaos theory. It was discovered that24 CHAPTER 2. SYSTEM MODELINGSystemInputOutputFigure 2.2: Illustration of the input-output view of a dynamical system.there are simple dynamical systems that are extremely sensitive to initialconditions, small perturbations may lead to drastic changes in the behaviorof the system. The behavior of the system could also be extremely compli-cated. The emergence of chaos also resolved the problem of determinism,even if the solution is uniquely determined by the initial conditions it is inpractice impossible to make predictions because of the sensitivity of initialconditions.The differential equation (2.1) is called an autonomous system becausethere are no external influences. Such a model is natural to use for celestialmechanics, because it is difficult to influence the motion of the planets. Thesituation in control is quite different because the external influences are quiteimportant. One way to capture this is to replace equation (2.1) bydxdt= f(x, u) (2.2)where u represents the effect of external influences. The model (2.2) is calleda controlled differential equation. The model implies that


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