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CALTECH CDS 101 - System Modeling

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CDS 101, Lecture 2.17 Oct 02R. M. Murray, Caltech1CDS 101: Lecture 2.1System ModelingRichard M. Murray7 October 2002Goals:y Describe what a model is and what types of questions it can be used to answery Introduce the concepts of state, dynamic, and inputs y Provide examples of common modeling techniques: finite state automata, difference equations, differential equations, Markov chainsy Describe common modeling tradeoffsReading: y K. J. Astrom, Control Systems Design, Sections 3.1-3.2, 3.6y Optional: Astrom, Section 3.37 Oct 02 R. M. Murray, Caltech CDS 230 Sep 02 R. M. Murray, Caltech CDS 0Lecture 1.1: Introduction to Feedback and ControlSenseComputeActuateControl =Sensing + Computation +ActuationFeedback Principlesy Robustness to Uncertaintyy Design of DynamicsMany examples of control and feedback in natural and engineered systems:BIOBIOESEESECSReview from last weekCDS 101, Lecture 2.17 Oct 02R. M. Murray, Caltech27 Oct 02 R. M. Murray, Caltech CDS 3ModelsModels are a mathematical representations of system dynamicsy Models allow the dynamics to be simulated and analyzed, without having to build the systemy Models are never exact, but they can be predictiveModels can be used in ways that the system can’ty Certain types of analysis (eg, parametric variations) can’t easily be done on the actual systemy In many cases, models can be run much more quickly than the original modelsThe model you use depends on the questions you want to answery A single system may have many modelsy Time and spatial scale must be chosen to suit the questions you want to answery Always formulate questions before building a modelExample: Weather Forecasting• Question 1: how much will it rain tomorrow?• Question 2: will it rain in the next 5-10 days?• Question 3: will we have a drought next summer?Different questions ⇒different models7 Oct 02 R. M. Murray, Caltech CDS 4Physical Concept of StateA key concept in modeling is the concept of stateThe state of a model of a dynamic system is a set of independent physical quantities, the specification of which (in the absence of excitation) completely determines the future evolution of the systemExample #1: car on a sloping roadà State: position and velocity of carà Angle of incline is not a state (not part of the model of the car)à Accelerator position is not a state (not intrinsic to the car)Example #2: predator prey (rabbits vs foxes)à State: number of rabbits and foxesà Amount of rabbit food is not a state (not intrinsic to the ecosystem as we have defined it)à Number of dead rabbits is not a state (not independent of number of live rabbits)Warning: objects in picture may not include any actual foxes.CDS 101, Lecture 2.17 Oct 02R. M. Murray, Caltech37 Oct 02 R. M. Murray, Caltech CDS 5DynamicsDynamics describes how the state evolvesThe dynamics of a model is an update rule for the system state that describes how the state evolves, as a function on the current state and any external inputsExample #1: car on a sloping roadà Dynamics: Newton’s law (F = ma)à Engine force modeled as external inputà Hill modeled as external inputExample #2: predator preyà Dynamics: empirically observed difference eqsà System of difference equationsengine hill() ()mx bx u t u t=−+ + [1] [] [] [][][1] [] [] [][]rfRk R k b R k aR k F kFk Fk d Fk aRkFk+=+ −+= − +7 Oct 02 R. M. Murray, Caltech CDS 6InputsInputs describe the external excitation of the dynamicsy Inputs are extrinsic to the system dynamics (externally specified)y Constant inputs are often considered to be parametersExample #1: car on a sloping roadExample #2: predator preyà Rabbit food can either be a parameter (if constant) or an external input (if nonconstant)engine hill() ()mx bx u t u t=−+ + Input #1 Input #2[1] [] [] [][][1] [] [] [][]fRkRk RkaRkFkFk Fk d Fk aRkFk+=+ −+= − +rb(u)CDS 101, Lecture 2.17 Oct 02R. M. Murray, Caltech47 Oct 02 R. M. Murray, Caltech CDS 7OutputsOutputs describe the directly measured variablesy Outputs are a function of the state and inputs ⇒ not independent variablesy Not all states are outputs; some states can’t be directly measuredExample #1: car on a sloping roadà Outputs: position and velocityà Measure velocity with speedometerà Measure position with odometer (or GPS)Example #2: motion of an airplane over USà States: position, altitude, linear + angular velà Dynamics: aerodynamics of flightà Inputs: thrust, rudder, elevator, flaps, windà Outputs (from radar): heading and speedà Roll, pitch and yaw are not directly measurable; part of state, but not part of outputshttp://www.flightview.com/7 Oct 02 R. M. Murray, Caltech CDS 8Modeling PropertiesChoice of state is not uniquey There may be many choices of variables that can act as the statey Predator prey example: look at number of rabbits plus the excess of foxes over rabbits:y Can also look at models for the same system with different numbers of statesà Ignore certain physical effects (and hence eliminate states)Choice of inputs and outputs depends on point of viewy Inputs: what factors are external to the model that you are buildingà Inputs in one model might be outputs of another model (eg, the output of a cruise controller provides the input to the vehicle model)y Outputs: what physical variables (often states) can you measureà Choice of outputs depends on what you can sense and what parts of the component model interact with other component modelsEFR=−[1] ([],[])[1] ([],[])reRkfRkFkFk f Rk Fk+=+=[1] ([],[])[1] ([],[])reRkfRkEkEk f Rk Ek+=+=CDS 101, Lecture 2.17 Oct 02R. M. Murray, Caltech57 Oct 02 R. M. Murray, Caltech CDS 9Model TypesModels are a mathematical representations of system dynamicsy Models allow the dynamics to be simulated and analyzed, without having to build the systemy Models are never exact, but they can be predictiveDifferent types of models are used for different purposesy Ordinary differential equations for rigid body mechanicsy Finite state machines for manufacturing, Internet, information flowy Partial differential differential equations for fluid flow, solid mechanics, etc7 Oct 02 R. M. Murray, Caltech CDS 10Model Type #1: Finite State MachinesFinite state machines model discrete transitions between finite number of statesy Represent each configuration of system as a statey Model transition between states using a graphy Inputs force transition between statesExample: Traffic light logicState:Inputs:Outputs:current pattern of lights that are on


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