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CALTECH CDS 101 - Lecture notesl

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CDS 101, Lecture 4.118 October 2004R. M. Murray, Caltech1CDS 101: Lecture 4.1Linear SystemsRichard M. Murray18 October 2004Goals:y Describe linear system models: properties, examples, and toolsy Characterize stability and performance of linear systems in terms of eigenvaluesy Compute linearization of a nonlinear systems around an equilibrium pointReading: y Åström and Murray, Analysis and Design of Feedback Systems, Ch 4y Packard, Poola and Horowitz, Dynamic Systems and Feedback, Sections 19, 20, 22 (available via course web page)18 Oct 04 R. M. Murray, Caltech CDS 2Lecture 3.1: Stability and PerformanceKey topics for this lecturey Stability of equilibrium pointsy Local versus global behaviory Performance specification via step and frequency response-2π02π-202x1x20 5 10 15 20 2500.511.50 5 10 15 20uyReview from Last WeekCDS 101, Lecture 4.118 October 2004R. M. Murray, Caltech218 Oct 04 R. M. Murray, Caltech CDS 3What is a Linear System?Linearity of functions: y Zero at the origin:y Addition:y Scaling: Linearity of systems: sums of solutions :nmf →\\(0) 0f=()()()fxy fx fy+= +() ()fxfxαα=()() ()fx yfxfyαβαβ+=+()fxAx=Canonical example:xAx=⇓xAx BuyCxDu=+=+⇓10 2012(0)() () ()xxxxtxtxtαβαβ=+→= +101(0)() ()xxxtxt=→=202(0)() ()xxxtxt=→=11(0) 0, ( ) ( )() ()xut u tyt y t==→=22(0) 0, ( ) ( )() ()xut u tyt y t==→=1212(0) 0, ( ) ( ) ( )() () ()xut u t u tyt y t y tαβαβ==+→= +Dynamical systemControl system18 Oct 04 R. M. Murray, Caltech CDS 4Linear SystemsInput/output linearity at x(0) = 0y Linear systems are linear in initial condition and input ⇒ need to use x(0) = 0 to add outputs togethery For different initial conditions, you need to be more careful (sounds like a good midterm question)Linear system ⇒ step response and frequency response scale with input amplitudey 2X input ⇒ 2X outputy Allows us to use ratios and percen-tages in step/freq response. These are independent of input amplitudey Limitation: input saturation ⇒ only holds up to certain input amplitudexAx BuyCxDu=+=+u10 5 10-101uy+ +0 5 10-202y1 + y2(0) 0x =0 5 10-101y1u20 5 10-1010 5 10-0.500.5y20 5 10-202u1 + u2CDS 101, Lecture 4.118 October 2004R. M. Murray, Caltech318 Oct 04 R. M. Murray, Caltech CDS 5bk3m1m2q1u(t)q2k2k1Why are Linear Systems Important?Many important toolsFrequency response, step response, etcy Traditional tools of control theoryy Developed in 1930’s at Bell Labs; intercontinental telecomClassical control design toolboxy Nyquist plots, gain/phase marginy Loop shapingOptimal control and estimatorsy Linear quadratic regulators y Kalman estimatorsRobust control designy H∞control designyµanalysis for structured uncertaintyMany important examplesElectronic circuitsy Especially true after feedbacky Frequency response is key performance specification (think telephones)Many mechanical systemsQuantum mechanics, Markov chains, …CDS101/110aCDS110bCDS110b/21218 Oct 04 R. M. Murray, Caltech CDS 6Solutions of Linear Systems: The Matrix ExponentialScalar linear system, with no inputMatrix version, with no inputMatrix exponentialy Analog to the scalar case; defined by series expansion: xAx BuyCxDu=+=+( ) ???yt=0(0)xaxxxycx===0()atxtex=0()atyt ce x=0(0)xAxxxyCx===0()Atxtex=0()Atyt Ce x=23112! 3!MeIMM M=+ + + +"P = expm(M)initial(A,B,C,D,x0);CDS 101, Lecture 4.118 October 2004R. M. Murray, Caltech418 Oct 04 R. M. Murray, Caltech CDS 7Stability of Linear SystemsStability is determined by the eigenvalues of the matrix Ay Simple case: diagonal systemy More generally: transform to “Jordan” formForm of eigenvalues determines system behaviorLinear systems are automatically globally stable or unstablexAx BuyCxDu=+=+0()Atxtex=Q: when is the systemasymptotically stable? 0lim ( ) 0txt→∞=100nxxλλ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦%⇒100()0nttextxeλλ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦%Stable if λi · 0Asy stable if λi< 0Unstable if λi> 01xTJTx−=100kJJJ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦%1010iiiJλλ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦%Asy stable if Re(λi) < 0Unstable if Re( λi )> 0Indeterminate if Re( λi )= 018 Oct 04 R. M. Murray, Caltech CDS 8Eigenstructure of Linear Systems-1 -0.5 0 0.5 1-1-0.8-0.6-0.4-0.200.20.40.60.81x1x2-1 -0.5 0 0.5 1-1-0.8-0.6-0.4-0.200.20.40.60.81x1x2-1 -0.5 0 0.5 1-1-0.8-0.6-0.4-0.200.20.40.60.81x1x2-1 -0.5 0 0.5 1-1-0.8-0.6-0.4-0.200.20.40.60.81x1x2Real e-valuesRe(λi) < 0Real e-valuesRe(λi) < 0Re(λj) > 0Complex e-valuesRe(λi) = 0Complex e-valuesRe(λi) < 0CDS 101, Lecture 4.118 October 2004R. M. Murray, Caltech518 Oct 04 R. M. Murray, Caltech CDS 9Step and Frequency ResponseEffect of eigenstructure on step responsey Complex eigenvalues with small real part lead to oscillatory response y Frequency of oscillations ≈ωiEffects of eigenstructure on frequency responsey Eigenvalues determine “break points” for frequency responsey Complex eigenvalues lead to peaks in response function near ωixAx BuyCxDu=+=+() 1()ut t=() sin( )ut A tω=Real AxisImag AxisPole-zero map-15 -10 -5 0 5-2-1012Time (sec.)AmplitudeStep Response0 5 10 15 20 25 3000.050.10.150.2Frequency (rad/sec)Phase (deg); Magnitude (dB)Bode Diagrams-100-50010-1100101-200-1000T≈2π/ωpωp≈ωp18 Oct 04 R. M. Murray, Caltech CDS 10Computing Frequency ResponsesTechnique #1: plot input and output, measure relative amplitude and phasey Use MATLAB or SIMULINK to generateresponse of system to sinusoidal outputy Gain = Ay/Auy Phase = 2π · ∆T/Ty Note: In general, gain and phase willdepend on the input amplitudeTechnique #2 (linear systems): use MATLAB bode commandy Assumes linear dynamics in statespace form:y Gain plotted on log-log scaleà dB = 20 log10(gain)y Phase plotted on linear-log scalexAxBuyCxDu=+=+0 5 10 15 20-1.5-1-0.500.511.5uyAuAy∆TTFrequency (rad/sec)Phase (deg)Magnitude (dB)-60-50-40-30-20-10010200.1 1 10-360-270-180-900bode(A,B,C,D)CDS 101, Lecture 4.118 October 2004R. M. Murray, Caltech618 Oct 04 R. M. Murray, Caltech CDS 11Example: Electrical CircuitDerivation based on Kirchoff’s laws for electrical circuits (Ph 2)y Sum of currents at nodes = 0:y Rewrite in terms of new states: vc1=v2, vc2=v3 – v121223112dv v v v vCdt R R−−=−31 3222()dv v v vCdt R−−=−1 2 3~vivo“Bridged Tee Circuit”R1R2C1C21111 2 1211 222222 2211 1 111 111ccicccvvCR R CRdvCR RvvdtVCR CR⎡⎤⎛⎞⎡⎤−+−⎛⎞⎢⎥⎜⎟+⎡⎤ ⎡⎤⎢⎥⎜⎟⎝⎠⎢⎥=+⎝⎠⎢⎥ ⎢⎥⎢⎥⎢⎥⎣⎦


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