CDS 101/110: Lecture 6.2 Transfer Functions
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1 CDS 11/11: Lecture 6.2 Transfer Functions November 2, 216 Goals: Continued study of Transfer functions Review Laplace Transform Block Diagram Algebra Bode Plot Intro Reading: Åström and Murray, Feedback Systems-2e, Chapter 9
2 Transfer Functions Exponential response of a linear state space system yy tt = CCee AAAA xx ssss AA 1 BB + CC ssss AA 1 BB + DD ee ssss transient steady state Transfer function Steady state response is proportional to exponential input => look at input/output ratio y(s)/u(s) GG ss = CC ssss AA 1 BB + DD is the transfer function between input and output Frequency response (system response to pure sinusoidal input) uu tt = MMMMMMMMMMMM = MM 2ii (eeiiiiii e iωωt ) 2 yy ssss tt = MM 2ii GG iiii ee iiiiii GG iiii ee iiiiii = MM GG iiii sin(ωωωω + arg GG(iiii)) gain phase
3 Laplace Transform Review Definition: given a suitable function yy tt the Laplace Transform of yy is: L yy tt Defined on entire complex plane Properties: = yy ττ ee ssss dddd Linearity: L αα ff tt + ββ gg(tt) = αα L ff tt + ββ L{gg tt }, for αα, ββββr Derivative: L dd xx tt = ssl xx tt xx() dddd More generally, L Proof: dd dddd (xx tt ee ssss ) ddtt = xx ddnn xx tt = ddtt nn ssnn L xx tt nn kk=1 ss kk 1 xx nn kk () xx tt ee ssss dddd = dddd ee ssss dddd ss xx tt ee ssss dddd = L dddd ssl xx tt dddd
4 Laplace Transform Review tt 1 Integral: L xx tt dddd = L xx tt Convolution: L h tt gg tt = HH ss GG(ss) Proof: L h tt gg tt = h tt gg tt ee ssss dddd ss If h(t) is causal (h(t) = for t<), then h tt gg tt = tt h tt ττ gg ττ dddd L h tt gg tt = Exponential: = tt h tt ττ gg(ττ)ee ssss ddττ dddd = h μμ gg(ττ) ee ss(μμ+ττ) ddμμ ddττ = 1 L ee aaaa = ee ss+aa tt dddd = L ee AAtt = ssss AA 1 = aaaaaa(ssss AA) det(ssss AA) ss+aa ee ss+aa h μμ ee ssss dddd = 1 ss+aa ττ h tt ττ gg(ττ)ee ssss ddddddττ gg ττ ee ssss dddd = HH ss GG ss
5 Laplace Transform Review Transfer Function: xx = AAAA + BBBB L ssss ss xx = AAAA ss + BBBB SS yy = CCCC + DDDD L YY ss = CCCC ss + DDDD(ss) XX ss = ssss AA 1 BB UU ss + ssss AA 1 xx() YY ss = CC ssss AA 1 BB + DD UU ss + CC ssss AA 1 xx When xx =, YY(ss) UU(ss) = GG ss = CC ssss AA 1 BB + DD Impulse Response: y t = h tt ττ uu ττ dddd YY ss = L yy tt = ee ssss yy tt dddd = ee ssss h tt ττ uu ττ dddd dddd = tt ee ss tt ττ ee ssss h tt ττ uu ττ dddddddd = ee ssss uu ττ dddd ee ssss h tt dddd = HH ss UU(ss) Therefore, GG ss = L h tt =Laplace Transform of system Impulse Response
6 Laplace Transform Review Constant Coefficient O.D.E.: Laplace Transform (assuming zero initial conditions) L dd nn ddtt nn yy tt + aa dd nn 1 1 ddtt nn 1 yy tt + + aa dd nn 1 nnyy tt = bb 1 ddtt nn 1 uu tt + + bb nnuu tt ss nn + aa 1 ss nn aa nn 1 ss + aa nn YY ss = bb 1 ss nn bb nn UU ss (*) GG ss = YY(ss) UU(ss) = bb 1 ss nn bb nn ss nn + aa 1 ss nn aa nn 1 ss + aa nn = nn(ss) dd(ss) Roots of dd ss are called the poles of transfer function GG ss If pp is a system pole, then yy = ee pppp is a solution to (*) with uu tt = Poles are strictly defined by matrix AA.. Roots of nn ss are called the zeros of G(s) If ss is a pole of GG(ss), then GG ss ee ssss is an output if dd ss. Out put is zero at ss if nn ss =.
7 Poles and Zeros Roots of d(s) are called poles of G(s) Roots of n(s) are called zeros of G(s) Poles of G(s) determine the stability of the (closed loop) system Denominator of transfer function = characteristic polynomial of state space system Provides easy method for computing stability of systems Right half plane (RHP) poles (Re > ) correspond to unstable systems Zeros of G(s) related to frequency ranges with limited transmission A pure imaginary zero at s = iω blocks any output at that frequency (G(iω) = ) Zeros provide limits on performance, especially RHP zeros MATLAB: pole(g), zero (G), pzmap(g) pzmap(g) 7 Imag Axis Real Axis
8 Series Interconnections Q: what happens when we connect two systems together in series? Magnitude Magnitude Phase (deg) Phase (deg) Frequency (rad/sec) Frequency (rad/sec) A: Transfer functions multiply Gains multiply, phases add Generally: transfer functions well formulated for frequency domain interconnections - Convolution multiplication MATLAB/python: G = series(g1, G2) 8
9 Feedback Interconnection -1 a -1 a State space derivation Transfer function derivation Frequency response 9 Frequency response MATLAB: G = feedback(sys, a) works for either state space or transfer functions
10 Block Diagram Algebra Basic idea: treat transfer functions as multiplication, write down equations d r + e C(s) + u P(s) y -1 Manipulate equations to compute desired signals Note: linearity gives superposition of terms Algebra works because we are working in frequency domain Time domain (ODE) representations are not as easy to work with Formally, all of this works because of Laplace transforms
11 Type Diagram Transfer function Series Block Diagram Algebra Parallel Feedback 12 These are the basic manipulations needed; some others are possible Formally, could work all of this out using the original ODEs ( nothing really new)
12 Example: Engine Control of a GM Astro 13
13 Example: Engine Control of a GM Astro 14
14 Example: Engine Control of a GM Astro 15
15 Example: Engine Control of a GM Astro 16
16 Example: Engine Control of a GM Astro 17
17 Example: Engine Control of a GM Astro 18
18 Example: Engine Control of a GM Astro 19
19 Sketching the Bode Plot for a Transfer Function (1/2) Evaluate transfer function on imaginary axis Plot gain (M) on log/log scale Plot phase (φ) on log/linear scale Piecewise linear approximations available 2
20 Sketching the Bode Plot for a Transfer Function (2/2) Complex poles Ratios of products 21
21 Example: Coupled Masses u q 2 q 1 m m k k k b Frequency Response Poles (Hq1f and Hq2f ) -.2 ±.7743j -.2 ±.4468j Zeros (Hq2f ) -.2 ±.6321j Interpretation Zeros in Hq2f give low response at ω.6321 Phase [deg] Gain (log scale) MATLAB: bode(h) Frequency [rad/sec]
22 MATLAB/Python manipulation of transfer functions Creating transfer functions [num, den] = ss2tf(a, B, C, D) sys = tf(num, den) num, den = [1 a b] s2 + as + b Interconnecting blocks sys= series(sys1, sys2), parallel, feedback Computing poles and zeros pole(sys), zero(sys) pzmap(sys) I/O response initial(sys) step(sys) lsim(sys) bode(sys) Amplitude Real Axis
23 Control Analysis and Design Using Transfer Functions disturbance d ref + Control + System r + e C(s) + u P(s) y -1-1 Transfer functions provide a method for block diagram algebra Easy to compute transfer functions between various inputs and outputs - Her(s) is the transfer function between the reference and the error - Hed(s) is the transfer function between the disturbance and the error Transfer functions provide a method for performance specification 24 Since transfer functions provide frequency response directly, it is convenient to work in the frequency domain - H er (s) should be small in the frequency range to 1 Hz (good tracking) Key idea: perform all analysis and design for linear systems in frequency domain Convert specifications on time response to specifications on frequency response Shape the frequency response by design of controller transfer function (Ch 1-13)
24 Summary: Frequency Response & Transfer Functions 11 Magnitude Phase (deg) Frequency (rad/sec) d F + e C(s) 25 + u P(s) y -c -1 -k
25 Example: Engine Control of a GM Astro 26
26 Example: Engine Control of a GM Astro 27
27 Example: Engine Control of a GM Astro 28
28 Example: Engine Control of a GM Astro 29
29 Example: Engine Control of a GM Astro 3
30 Example: Engine Control of a GM Astro 31
31 Example: Engine Control of a GM Astro 32
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