1ME451: Control SystemsME451: Control SystemsDr. Dr. JongeunJongeunChoiChoiDepartment of Mechanical EngineeringDepartment of Mechanical EngineeringMichigan State UniversityMichigan State UniversityLecture 7Lecture 7Linearization, time delaysLinearization, time delays2Course roadmapCourse roadmapLaplace transformLaplace transformTransfer functionTransfer functionModels for systemsModels for systems••electricalelectrical••mechanicalmechanical••electromechanicalelectromechanicalBlock diagramsBlock diagramsLinearizationLinearizationModelingModelingAnalysisAnalysisDesignDesignTime responseTime response••TransientTransient••Steady stateSteady stateFrequency responseFrequency response••Bode plotBode plotStabilityStability••RouthRouth--HurwitzHurwitz••NyquistNyquistDesign specsDesign specsRoot locusRoot locusFrequency domainFrequency domainPID & LeadPID & Lead--laglagDesign examplesDesign examples((MatlabMatlabsimulations &) laboratoriessimulations &) laboratories3What is a linear system?What is a linear system?A system having A system having Principle of SuperpositionPrinciple of SuperpositionA nonlinear system does not satisfy the principle of superposition.SystemSystem4Linear systemsLinear systemsEasier to understand and obtain solutionsEasier to understand and obtain solutionsLinear ordinary differential equations (Linear ordinary differential equations (ODEsODEs), ), Homogeneous solutionHomogeneous solutionand and particular solutionparticular solutionTransient solutionTransient solutionand and steady state solutionsteady state solutionSolution caused by initial valuesSolution caused by initial values, and , and forced solutionforced solutionAdd many simple solutions to get more complex Add many simple solutions to get more complex ones (use superposition!)ones (use superposition!)Easy to check the Easy to check the StabilityStabilityof stationary states of stationary states ((Laplace TransformLaplace Transform))5Why linearization?Why linearization?Real systems are inherently nonlinear. (Linear Real systems are inherently nonlinear. (Linear systems do not exist!) systems do not exist!) Ex.Ex.f(tf(t)=)=Kx(tKx(t), ), v(tv(t)=)=Ri(tRi(t))TF models are only for linear timeTF models are only for linear time--invariant (LTI) invariant (LTI) systems.systems.Many control analysis/design techniques are Many control analysis/design techniques are available for linear systems.available for linear systems.Nonlinear systems are difficult to deal with Nonlinear systems are difficult to deal with mathematically.mathematically.Often we Often we linearizelinearizenonlinear systems before nonlinear systems before analysis and design. How?analysis and design. How?6How to How to linearizelinearizeit?it?Nonlinearity Nonlinearity can be approximated by a can be approximated by a linear linear function function for small deviations for small deviations around an around an operating pointoperating pointUse a Taylor series expansionUse a Taylor series expansionLinear approximationNonlinear functionOperating pointOld coordinateNew coordinate7LinearizationLinearizationNonlinear system:Nonlinear system:Let Let uu00be a nominal input and let the resultant be a nominal input and let the resultant state be state be xx00Perturbation:Perturbation:Resultant perturb:Resultant perturb:Taylor series expansion:Taylor series expansion:8Linearization (cont.)Linearization (cont.)notice thatnotice that; hence; henceL. sys.L. sys.N. sysN. sys9Motion of the pendulumMotion of the pendulumLinearizeLinearizeit atit atFind Find uu00New coordinates:New coordinates:Linearization of a pendulum model Linearization of a pendulum model 10Linearization of a pendulum model (contLinearization of a pendulum model (cont’’))Taylor series expansionTaylor series expansionof of 11Time delay transfer functionTime delay transfer functionTF derivationTF derivationThe more time delay is, the more difficult to The more time delay is, the more difficult to control (Imagine that you are controlling the control (Imagine that you are controlling the temperature of your shower with a very long temperature of your shower with a very long hose. You will either get burned or frozen!)hose. You will either get burned or frozen!)(Memorize this!)12Summary and ExercisesSummary and ExercisesModeling ofModeling ofNonlinear systemsNonlinear systemsSystems with time delaySystems with time delayNextNextModeling of DC motorsModeling of DC motorsExercisesExercisesLinearizeLinearizethe pendulum model at the pendulum model at
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