S. Boyd EE102Lecture 6Qualitative properties of signals & Laplacetransforms• qualitative behavior from pole locations• damping & quality factor• dominant poles• stability of autonomous LCCODEs• initial value theorem, final value theorem6–1Inverse Laplace transform of rational Fsuppose F (s) = b(s)/a(s) is rational and strictly proper with L−1(F ) = feach term in partial fraction expansion of F gives a term in f:• for single pole at s = λ,L−1µ1s − λ¶= eλt• for pole at s = λ of multiplicity k,L−1µ1(s − λ)k¶=1(k − 1)!tk−1eλt• the poles of F determine the types of terms that appear in f• the zeros (or residues) of F determine the coefficients multiplying eachterm, or the amplitude & phase of oscillatory termsQualitative properties of signals & Laplace transforms 6–2Qualitative properties of terms• real, positive poles correspond to growing exponential terms• real, negative poles correspond to decaying exponential terms• a pole at s = 0 corresponds to a constant term• complex pole pairs with positive real part correspond to exponentiallygrowing sinusoidal terms• complex pole pairs with negative real part correspond to exponentiallydecaying sinusoidal terms• pure imaginary pole pairs correspond to sinusoidal terms• repeated poles yield same types of terms, multiplied by powers of tQualitative properties of signals & Laplace transforms 6–3Quantitative properties of termspole λ = σ + jω (and λ, if ω 6= 0) gives time-domain term aeσtcos(ωt + φ)PSfrag replacements<=osc. freq.osc. freq.decay ratedecay rategrowth rate• the real part of a pole gives the growth rate (if positive) or decay rate(if negative) of the corresponding term in f• the imaginary part gives the oscillation frequencyQualitative properties of signals & Laplace transforms 6–4f(t) = eσtcos(ωt)rows: ω = 30, 15, 0; columns: σ = −1.5, −0.75, 0, 0.75, 1.50 1−2020 1−2020 1−2020 1−2020 1−2020 1−2020 1−2020 1−2020 1−2020 1−2020 1−2020 1−2020 1−2020 1−2020 1−202Qualitative properties of signals & Laplace transforms 6–5Complex poles: damping ratio and Qpole at s = λ = σ + jω (hence also at λ) with σ < 0F (s) =rs − λ+rs − λ, f(t) = aeσtcos(ωt + φ)two measures of decay rate per cycle of oscillation:• damping ratioζ =−σ√σ2+ ω2• quality factorQ =12rσ2+ ω2σ2=12ζQualitative properties of signals & Laplace transforms 6–6damping ratio (or Q) are related to angle of pole in complex plane:PSfrag replacements<=ω|σ|sin−1ζ• underdamped: ζ < 1 (Q > 1/2)• critically damped: ζ = 1 (Q = 1/2)Qualitative properties of signals & Laplace transforms 6–7example: underdamped parallel RLC circuit of page 4-31PSfrag replacementsRL Cviσ =−12RC, ω =r1LC−14R2C2givesQ =RpL/C, ζ =pL/C2RQualitative properties of signals & Laplace transforms 6–8interpretation: Q is a measure of number of cycles to decay• time to decay to 1% amplitude is about 4.6/|σ|• period of oscillation: 2π/ω• number of cycles to decay to 1% amplitudeN1%≈4.6/|σ|2π/ω= 1.46ω2|σ|rule of thumb (accurate for Q > 2 or so):N1%≈ 1.46Qother rule of thumb: N4%≈ QQualitative properties of signals & Laplace transforms 6–9example0 5 10 15 20 25 30−1010 5 10 15 20 25 30−101PSfrag replacementsnumber of cyclesQ = 20Q = 10Qualitative properties of signals & Laplace transforms 6–10Dominant polessuppose the poles of F are p1, . . . , pnthe asymptotic growth (or decay if < 0) rate of f is determined by themaximum real part:α = max{<p1, . . . , <pn}• pole (or poles) which achieve this max real part are called dominant• as t → ∞, these terms become larger and larger compared to the otherterms, no matter what the residuesPSfrag replacements<=<s = αQualitative properties of signals & Laplace transforms 6–11example:F (s) =100s + 2+1s + 1, f(t) = 100e−2t+ e−t• asymptotic decay rate determined by dominant pole at s = −1• asymptotically, f decays like e−t• even though residue for nondominant pole is 100 times larger, termassociated with dominant pole is larger for t > 4.6Qualitative properties of signals & Laplace transforms 6–12Stability of autonomous LCCODEany(n)+ an−1y(n−1)+ ··· + a0y = 0is stable if all solutions converge to zero, regardless of initial conditiontake Laplace transform:an³snY (s) − sn−1y(0) − sn−2y0(0) − ··· − y(n−1)(0)´+ an−1³sn−1Y (s) − sn−2y(0) − ··· − y(n−2)(0)´+ ··· + a0Y (s) = 0Y (s) =bn−1sn−1+ bn−2sn−2+ ··· + b1s + b0ansn+ an−1sn−1+ ··· + a1s + a0=b(s)a(s)where b depends on initial conditionsLCCODE is stable only when all poles of Y have negative real part, i.e.,roots of a are in left half planeQualitative properties of signals & Laplace transforms 6–13Initial value theorema general property of Laplace transforms (not just for rational F ):lims→∞sF (s) = f(0+)(can take s real in the limit)makes connection between f(t) for small t, and F (s) for large sreason: for large (real) s, se−stis bunched up near t = 0, sosF (s) =Z∞0se−stf(t) dt ≈ f(0+)Z∞0se−stdt = f(0+)Qualitative properties of signals & Laplace transforms 6–14examples• f(t) = eat, so F (s) = 1/(s − a)lims→∞sF (s) = lims→∞ss − a= 1 = f(0)• f is unit step at t = 0, so F (s) = 1/slims→∞sF (s) = 1 = f(0+)Qualitative properties of signals & Laplace transforms 6–15Final value theoremmakes connection between f(t) for large t and F (s) for small slimt→∞f(t) = sF (s)|s=0if the limit existsreason: from relation between Laplace transforms and derivatives,sF (s) −f(0) = L(f0) =Z∞0f0(t)e−stdtsF (s)|s=0− f(0) =Z∞0f0(t)dt = limt→∞f(t) − f(0)sF (s)|s=0= limt→∞f(t)Qualitative properties of signals & Laplace transforms 6–16examples• f(t) = 1 − e−t, so F (s) =1s−1s + 1, andlimt→∞f(t) = 1 = sF (s)|s=0• F (s) =ss2+ ω2, so f(t) = cos ωt and limt→∞f(t) does not exist;the final value theorem does not apply hereQualitative properties of signals & Laplace transforms