Lecture 2 – Linear Systems

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iωts (e iωt ) d iωte iωe iωtdt Frequency domain analysisiωk tiωk tue y uH(iω)e k kkEE392m - Spring 2005GorinevskyControl Engineering2-17

Frequency domain description Frequency domain analysis1 1 iωtu u (ω )e dω y 2π2πuiωtPacket eofu (ω )sinusoidsH (iω ) (ω )e iωt dωH(iω)u 14243 y (ω )Packet eiωt ofy (ω )sinusoidsy Fourier transform – numerical analysisu(t) 0, for t 0 Laplace transform – complex analysis iωt u (ω ) u(t )e dω u(t )e st dω uˆ (iω ) s iωEE392m - Spring 2005GorinevskyControl Engineering2-18

Continuous systems in frequency domain x (ω ) iωtxtedt() 1x (t ) 2π iωt x(ω)edω [ , ] [ , ] Inverse Fourier transform ty (t ) Fourier transform h(t τ )u(τ )dt I/O impulse response model H ( s ) h(t )e st dt Transfer function y (ω ) H (iω )u (ω ) System frequency response0EE392m - Spring 2005GorinevskyControl Engineering2-19

Frequency domain descriptiony H (iω )eiωt Example:1H ( s) s 0.7 H is often measuredin dB– [dB] 20 log10 MEE392m - Spring 2005GorinevskyBode Diagram15Magnitude (dB)u eiωtϕ (ω ) arg H (iω )Phase (deg) Bode plots:M (ω ) H (iω )1050-50-45-90-135-180-210Control Engineering-110Frequency (rad/sec)0102-20

Model Approximation Model structure – physics, computational Determine parameters from data Step/impulse responses are close the input/outputmodels are close Example – fit step response Linearization of nonlinear modelEE392m - Spring 2005GorinevskyControl Engineering2-21

Black-box model from data Linear black-box model can be determined from the data,e.g., step response data, or frequency responseSTEP RESPONSEHEAT FLUX10.80.60.40.20020406080100TIME Example problem: fit an IIR model of a given order This is called model identification Considered in more detail in Lecture 8EE392m - Spring 2005GorinevskyControl Engineering2-22

Linear PDE models Include functions of spatial variables––––electromagnetic fieldsmass and heat transferfluid dynamicsstructural deformations Example: sideways heat equation T 2T k 2 x tT ( 0 ) u; Ty xEE392m - Spring 2005GorinevskyxTinside uToutside 0T (1) 0yheat fluxx 1Control Engineering2-23

Linear PDE System Example T 2T k 2 t xu T ( 0)T (1) 0 Heat transfer equation,– boundary temperature input u– heat flux output y Impulse response and step response Transfer function is not rational-2HEAT FLUX6 T xx 1PULSE RESPONSEx 10140.8200.6020406080100TIMESTEP RESPONSE1HEAT FLUXTEMPERATUREy 0.40.200.800.6150.50.40.20COORDINATE020EE392m - Spring 2005Gorinevsky4060TIME80100Control Engineering50110heatTIME flux2-24

Impulse response approximation Approximating impulseand step responses by alow order rationaltransfer function model Higher order model canprovide very accurateapproximation Methods:– trial and error– sampled time response fit,e.g., Matlab’s prony– identification, Lecture 8– formal model reductionapproaches - advancedEE392m - Spring 2005Gorinevsky0.08s 2 0.4 s 2.8H ( s ) 0.01 2s 0.34 s rol EngineeringTIME2-25

Validity of Model Approximation Why can we use an approximate model instead of the ‘real’model? Will the analysis hold? The input-output maps of two systems are ‘close’ if theconvolution kernels (impulse responses) are ‘close’ty (t ) h(t τ )u(τ )τ The closed-loop stability impact of the modeling error– Control robustness– Will be discussed in Lecture 9EE392m - Spring 2005GorinevskyControl Engineering2-26

Nonlinear map linearization Nonlinear - detailed model Linear - conceptual design model Differentiation, secant method Example:static map linearization fy f (u ) (u u0 ) uEE392m - Spring 2005GorinevskyControl Engineering2-27

Linearization Example: RTP RTP – Rapid ThermalProcessing Major semiconductormanufacturing processdT bu c1 (T 4 TF4 ) c2 (T TF )dtT – part temperatureu – IR heater powerTF – furnace temperature Stefan-Boltzmann law nonlinearity TF is assumed to be constantuinputheatersheated partfurnaceEE392m - Spring 2005GorinevskyControl EngineeringToutput2-28

RTP, cont’ddT f (T ) budtf (T ) c1 (T 4 TF4 ) c2 (T TF )f(T)Linearize around a steady state pointdTf L (T ) a (T T* ) d f L (T ) budtf (T* ) f (T* )d f (T* ) a f ′(T* ) 0b 1000,-200c1 1.1 10-10, -400c2 0.8,-600TF 300-800-1000-1200300EE392m - Spring 2005Gorinevsky400500600700800900TEMPERATUREControl Engineering1000110012002-291300

RTP, cont’dx& ax bu du kxx T T*Linear system with a polep ( a bk )p 11.7425T* 1000, a -1.7425, b 1000, k 0.01Simulate 0300Linearmodel,d 000.05EE392m - Spring 2005Gorinevsky0.10.150.20.25TIME0.3Control Engineering0.350.40.450.52-30

Nonlinear state space modellinearization Linearize the r.h.s. mapin a state-space modelx& f ( x, u ) f f( x x0 ) (u u0 )4243 u 14243 x 1qvq& Aq Bv Linearize around an equilibrium0 f ( x0 , u0 )j Secant methodf ( x0 s j , u0 ) f ( x0 , u0 )f x djs j [0 . d j . 0]{#j This is how Simulink computes linearizationEE392m - Spring 2005GorinevskyControl Engineering2-31

Example: F16 LongitudinalModelxdx f ( x, u )dt x1x2x3x4 αθx’State vector x- velocity V [ft/sec]- angle of attack α [rad]- pitch angle θ [rad]- pitch rate q [rad/sec]Control inputVy 57.3θ V α x q θ u - elevator deflection δe [deg].u δezδez’1V& ( Fx cos α Fz sin α )m1( Fx sin α Fz cos α ) qα& mVMyq& IyFx rC x ,t (α ) mg sin θ T&θ qFz rC z ,t (α , δ e ) mg cos θM y RC m ,t (α , δ e )For more detail see: Aircraft Control and Simulation by Stevens and LewisEE392m - Spring 2005GorinevskyControl Engineering2-32

Nonlinear Model of F16dx f ( x, u )dty g ( x) state evolutionobservationAircraft models are understood by groupsof peopleCould take many man-years worth ofeffortAerodynamics model is based onempirical dataf(x,u) available as a computationalfunction can be used without a deepunderstanding of the modelThe nonlinear model can be used forsimulation, or linearized for analysisEE392m - Spring 2005GorinevskyControl Engineering2-33

Linearized Longitudinal Model of F16 Assume trim condition V0 500 α 0.0393 x0 0 q0 0 θ0.0393 0 -velocity V [ft/sec]angle of attack α [rad]pitch rate q [rad/sec]pitch angle θ [rad Linearize the nonlinear function f(x,u) by a finite differencemethod (secant method). Step [1 0.001 0.01 0.001] 1.93 10 2 f 2.54 10 4 A x 0 12 2.95 10 32.2 0.48 1.0200.91 001 0.8201.08 8.820.17 3 f 2.15 10 B 0 u 0.18 These are the matrices we considered in the linear F16model exampleEE392m - Spring 2005GorinevskyControl Engineering2-34

Simulation-based validation Simulate with nonlinear model, compare with linear modelresults DoubletresponseEE392m - Spring 2005GorinevskyControl Engineering2-35

LTI models - summary ODE model State space linear model Linear system can be described by impulse response orstep response Linear system can be described by frequency response Fourier transform of the impulse response Linear model approximations can be obtained from morecomplex models– Approximation of a linear model response– Linearization of a nonlinear modelEE392m - Spring 2005GorinevskyControl Engineering2-36

Lecture 2 – Linear Systems This lecture: EE263 material recap some controls motivation Continuous time (physics) Linear state space model Transfer functions Black-box models; frequency domain analysis Linearization

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