Lecture 2 – Linear Systems - Stanford University

1 EE392m - Spring 2005 GorinevskyControl Engineering2-1 Lecture 2 Linear SystemsThis Lecture : EE263 material recap + some controls motivation Continuous time (physics) Linear state space model Transfer functions Black-box models; frequency domain analysis LinearizationEE392m - Spring 2005 GorinevskyControl Engineering2-2 Modeling and Analysis This Lecture considers Linear models. More detail on modeling in Lecture 7 Simulation: computing state evolution and output signal Stability: does the solution diverge after some time? Approximate Linear modelsSystem Model[internal states]Input signal(control)Output signal(observation)state evolutionEE392m - Spring 2005 GorinevskyControl Engineering2-3 Linear Models Model is a mathematical representations of a system Models allow simulating the system Models can be used for conceptual analysis Models are never exact Linear models Have simple structure Can be analyzed using powerful mathematical tools Can be matched against real data using known procedures Many simple physics models are Linear They are just models, not the real Systems EE392m - Spring 2005 GorinevskyControl Engineering2-4 State space model Generic state space model is described by ODEs , physics-based system model state vector x observation vector y control vectoru),()

2 ,,(txgytuxfdtdx== state evolution observationExample: F16 Longitudinal Model . 180081 8201095210152910021105421705802328281093 112342 + == + =+ + =&&&&zx e Veu = Control and Simulationby Stevens and Lewisx1- velocity V [ft/sec] x2- angle of attack [rad] x3- pitch angle [rad] x4- pitch rate q [rad/sec] e- elevator deflection [deg]EE392m - Spring 2005 GorinevskyControl Engineering2-5 Linear state space model Linear Time Invariant (LTI) state space model: Can be integrated analytically or numerically (simulation) Can be well analyzed: stability, responseCxyBuAxdtdx=+= state evolution observationsExample: F16 Longitudinal Model []xyuxdtdx = + = - Spring 2005 GorinevskyControl Engineering2-6 Integrating Linear autonomous system Matrix exponential Can be computed in Matlab as expm(A) The definition corresponds to integrating the ODE by Euler methodAxdtdx=()ttttAIAt +=/0lim)exp(())()()()(TxtAIttAxTxtTx += + +Example: >>A = [ ; 0 ;0 0 0 1; 0 ];>> expm(A)ans = ())()exp()()(TxtAnTxtAItnTxn += +EE392m - Spring 2005 GorinevskyControl Engineering2-7 Integrating Linear autonomous system Initial condition response0)0(,xxAxdtdx==0)exp()(xAttx=Exa mple: %Take A, B, C from the F16 example>> k = >> G = A + k*B*C;>> x0 = [ ; ;0; ];>> for j = 1:length(t); x(:,j)=expm(G*t(j))*x0; end.)

3 MEEE392m - Spring 2005 GorinevskyControl Engineering2-8 Eigenvalues and Stability Consider eigenvalues of the matrix A Suppose Ahas all different and nonzero eigenvalues, then The system solution is exponentially stable if If Ahas eigenvalues with multiplicity more than 1, things are a bit more complicated: Jordan blocks, polynomials in t Still the condition of exponentially stability is 0)det();eig(}{= =AIAj 1}diag{ =VVAj 0Re<j 1}diag{)exp( =VeVAttj Example: % take A from % the F16 example>> eig(A)ans = + - <j EE392m - Spring 2005 GorinevskyControl Engineering2-9 Input-output models Black-box models describe system Pas an operatorPxuinput signalyoutput signalinternal state (hidden) Historically (50 years ago) Black-box models EE State-space models ME, AAEE392m - Spring 2005 GorinevskyControl Engineering2-10 Linear System (input-output) Linearity Linear Time-Invariant Systems - LTI )()(11 yuP)()(TyTuP )()()()(2121 + + byaybuauP)()(22 yuP PutytEE392m - Spring 2005 GorinevskyControl Engineering2-11 Convolution representation Convolution integral impulse response Step response .

4 U = 1 for t > 0 =tduthty )()()()()()()(thtyttu= = = =ttdhdthtg00)()()( )()(tgdtdth=uhy*=signal processing notationEE392m - Spring 2005 GorinevskyControl Engineering2-12 impulse response for State Space Model impulse response for the state x System impulse response {1)0()(tuBtAxtx + BtuBtx= )(BAttx)exp()(=CxyBuAxdtdx=+= state evolution observationExample: >> A = [ 0; 0;0 0];>> B = [ ; ; 0];>> C = [0, 0, 1]; )exp()(=EE392m - Spring 2005 GorinevskyControl Engineering2-13 Formal transfer function Rational transfer function = IIR ( infinite impulse response ) model Broad class of input-output Linear models Differentiation operator Formal transfer function rational function of s For a causal system m nsdtd ubdtudbdtudbyadtydadtydannnnnmmmmm111211 1121+ + +++=+++KKusDsNusHy = =)()()(11)(++++=mmmasasasNK11)(++++=nnnb sbsbsDKEE392m - Spring 2005 GorinevskyControl Engineering2-14 Poles, impulse response ExpandThen Quasi-polynomial impulse response see a textbook Example:usDsNy =)()(0)(=sN0)(=sD- zeros- polesKMKM pspssD)()()(11 =KupssNupssNyKMKKM ++ =)()()()(111K()()tpMKMKtpMMKKK ectcectcth1,11,1,111,1111)(+ + ++++++=KKKudtyd=22usy =2100)(xtvth+=Transfer function: impulse response .}

5 EE392m - Spring 2005 GorinevskyControl Engineering2-15 Transfer Function for State Space Characteristic polynomial Poles are the same as eigenvalues of the state-space matrix A For stability we need Re pk= Re k< 0()()BAsICsHuBAsIy11)( = =Poles eigenvalues()0det= AsI()0det)(= =AsIsNCxyBuAxsx=+= Formal transfer function for a state space model sdtd EE392m - Spring 2005 GorinevskyControl Engineering2-16 Laplace transform Laplace integral transform: Laplace transform of the convolution integral yields Transfer function: function of complex variable s analytical in a right half-plane Res a for a stable system a 0 for an IIR model =0)()(dtethsHst)( )()( susHsy=)( sxsdtdx dtetxsxtxst = 0)()( )(Re sImags)(sH)()()(sDsNsH=EE392m - Spring 2005 GorinevskyControl Engineering2-17 Frequency decomposition Sinusoids are eigenfunctions of an LTI system.

6 LTIP lant Frequency domain analysisusHy)(= = tikktikkkeiHuyeu )(tie tieiH )(()tititieiedtdes = EE392m - Spring 2005 GorinevskyControl Engineering2-18 Frequency domain description = = deuiHydeuutiyti43421)(~)(~)(21)(~21)(~ uetiuPacket of sinusoids)(~ yetiPacket of sinusoids)( iHy Frequency domain analysis)( )()()(~ iudetudetuuisstti= === Fourier transform numerical analysis Laplace transform complex analysis u(t) = 0, for t < 0 EE392m - Spring 2005 GorinevskyControl Engineering2-19 Continuous Systems in frequency domain Fourier transform Inverse Fourier transform I/O impulse response model Transfer function System frequency response )(~)()(~)()()()()()(~21)()()(~0 uiHydtethsHdtuthtydextxdtetxxstttiti== === ],[],[ EE392m - Spring 2005 GorinevskyControl Engineering2-20 Frequency domain description Bode plots.

7 TitieiHyeu )(== )( =ssH Bode DiagramFrequency (rad/sec) Phase (deg) Magnitude (dB) -505101510-210-1100-180-135-90-450 |H|is often measured in dB [dB] = 20 log10M)(arg)()()( iHiHM==EE392m - Spring 2005 GorinevskyControl Engineering2-21 Model Approximation Model structure physics, computational Determine parameters from data Step/ impulse responses are close the input/output models are close Example fit step response Linearization of nonlinear model EE392m - Spring 2005 GorinevskyControl Engineering2-22 Black-box model from data Linear black-box model can be determined from the data, , step response data, or frequency response Example problem: fit an IIR model of a given order This is called model identification Considered in more detail in Lecture 8 FLUXSTEP RESPONSEEE392m - Spring 2005 GorinevskyControl Engineering2-23 Linear PDE models Include functions of spatial variables electromagnetic fields mass and heat transfer fluid dynamics structural deformations Example: sideways heat equation1220)1(;)0(= === = xxTyTuTxTktTyheat fluxxToutside=0 Tinside=uEE392m - Spring 2005 GorinevskyControl Engineering2-24 Linear PDE System Example Heat transfer equation, boundary temperature input u heat flux output y impulse response and step response Transfer function is not rational0)1()0(22== = TTuxTktT1= =xxTy0204060801000246x 10-2 TIMEHEAT FLUXPULSE FLUXSTEP RESPONSE00.

8 I NATE heat fluxEE392m - Spring 2005 GorinevskyControl Engineering2-25 impulse response approximation Approximating impulse and step responses by a low order rational transfer function model Higher order model can provide very accurate approximation Methods: trial and error sampled time response fit, , Matlab sprony identification, Lecture 8 formal model reduction approaches - )(22+++ = MEEE392m - Spring 2005 GorinevskyControl Engineering2-26 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 the convolution kernels ( impulse responses) are close The closed-loop stability impact of the modeling error Control robustness Will be discussed in Lecture 9 =tuthty )()()(EE392m - Spring 2005 GorinevskyControl Engineering2-27 Nonlinear map linearization Nonlinear - detailed model Linear - conceptual design model Differentiation, secant method Example: static map linearization)()(0uuufufy =EE392m - Spring 2005 GorinevskyControl Engineering2-28 Linearization Example.

9 RTP RTP Rapid Thermal Processing Major semiconductor manufacturing process uinput heatersheated partT output furnace)()(2441 FFTTcTTcbudtdT =T part temperatureu IR heater powerTF furnace temperature Stefan-Boltzmann law nonlinearity TFis assumed to be constant EE392m - Spring 2005 GorinevskyControl Engineering2-29 RTP, cont dLinearize around a steady state point b = 1000, c1= 10-10, c2= , TF = 300buTfdtdT+=)(3004005006007008009001000 110012001300-1200-1000-800-600-400-2000 TEMPERATUREf(T))()()(2441 FFTTcTTcTf = + =)()()(**TfTfTfadTTaTfL+ =)()(*)(*Tfd=buTfdtdTL+=)(EE392m - Spring 2005 GorinevskyControl Engineering2-30 RTP, cont dSimulate performance:kxudbuaxx =++=&)(bkap+ = Linear system with a poleT*= 1000, a = , b = 1000, k = 00. METL inear model, d= 0 Non- Linear model*TTx = =pp1EE392m - Spring 2005 GorinevskyControl Engineering2-31 Nonlinear state space model linearization Linearize the map in a state-space modelBvAqquuufxxxfuxfxvq+= + =&4342143421&)()(),(00{] [),(),(#0000jjjjjjdsduxfusxfxf= += ),(000uxf= Linearize around an equilibrium Secant method This is how Simulink computes linearization EE392m - Spring 2005 GorinevskyControl Engineering2-32 State vector xx1 - velocity V [ft/sec] x2 - angle of attack [rad] x3 - pitch angle [rad] x4 - pitch rate q [rad/sec] Control inputu- elevator deflection e [deg].}

10 Z x e Vzxeu = ++ =+= &&&&)cossin(1)sincos(1),(cos),(sin)(,,,e tmyetzztxxRCMmgrCFTmgrCF =+=+ = = qVx),(uxfdtdx=Example: F16 Longitudinal Model For more detail see:Aircraft Control and Simulationby Stevens and LewisEE392m - Spring 2005 GorinevskyControl Engineering2-33 Nonlinear Model of F16 Aircraft models are understood by groups of people Could take many man-years worth of effort Aerodynamics model is based on empirical data f(x,u) available as a computational function can be used without a deep understanding of the model The nonlinear model can be used for simulation, or linearized for analysis )(),(xgyuxfdtdx== state evolution observationEE392m - Spring 2005 GorinevskyControl Engineering2-34 Linearized Longitudinal Model of F16 Assume trim condition Linearize the nonlinear function f(x,u) by a finite difference method (secant method).