Frequency Response and Stability Analysis. Hans- Pe9er Halvorsen, M.Sc.

Transkript

1 Frequency Response and Stability Analysis Hans- Pe9er Halvorsen, M.Sc.

2 Tilstandsrom- modeller Dataverktøy SpesialElfelle MathScript LabVIEW Differensial - likninger Tidsplanet Laplace Blokk- diagrammer Transfer- funksjoner 2.orden 1.orden Analyse/Design Stabilitets- analyse Det komplekse plan 2. Frekvensrespons 1. Systemets poler Bodediagram Realisering/ Implementering Reguleringsteknikk Serie, Parallel, Feedback Det komplekse plan S- planet K = Forsterkning T=Tidskonstant Sprang- respons 1.orden med Edsforsinkelse Diskre@sering Reguleringssystem AsymptoEsk stabilt system Air Heater Tidsplanet Ustabilt system Marginalt stabilt system AsymptoEsk stabilt system Marginalt stabilt system Ustabilt system

3 Stabilitetssanalyse vha. Frekvensrespons Temaer: Frekvensrespons Hva er det? Frekvensrespons vha MathScript Sjekke om systemet er stabilt vha Bodediagram Sjekke systemets stabilitetsmarginer, dvs. hvor mye vi kan øke forsterkningen i systemet (reguleringssløyfa) før systemet blir ustabilt Reguleringssystemets båndbredde og følgeegenskaper

4 Frequency Response Simple Example Outside Temperature frequency 1 (year) T = 1 year frequency 2 (24 hours) - > Only the gain and phase are different T = 24 hours Assume the outdoor temperature is varying like a sine funceon during a year (frequency 1) or during 24 hours (frequency 2). Then the indoor temperature will be a sine as well, but with different gain. In addieon it will have a phase lag. Inside Temperature frequency 1 (year) frequency 2 (24 hours)

5 Frequency Response MathScript funceons

6 Bode Diagram The x- scale is logarithmic Gain ( Forsterkningen ) The y- scale is in [db] Phase lag ( Faseforkyvningen ) The y- scale is in [degrees] Vanligvis er enheten for frekvens Hertz [Hz], men i frekvensrespons/ Bodediagram brukes radianer ω [rad/s]. Sammenhengen mellom disse er:

7 Stability Analysis Loop Transfer (Sløyfetransferfunksjonen): Hr =... Hp =... Hm =... L = series(series(hr, Hp), Hm) Used in Frequency Response Stability Analysis (Bode Diagram) The Tracking Func@on (Følgeforholdet): L =... T = feedback(l, 1) The Sensivity Func@on (SensiEvitetsfunksjonen): T =... S = 1-T

8 Frequency Response and Stability Analysis Bode Diagram ωc and ω180 are called the crossover- frequencies ( kryssfrekvens ) ΔK is the gain margin (GM) of the system ( Forsterkningsmargin ). How much the loop gain can increase before the system becomes unstable ϕ is the phase margin (PM) of the system ( Fasemargin ). Hvor mye fasedreining systemet tåler før det blir ustabilt. We have the following: AsymptoEsk stabilt system Marginalt stabilt system Ustabilt system

9 Stability Analysis - Example Students: Find ωc, ω180, ΔK, φ from the plot (Bruk utdelt papir)

10 Stability Analysis Example - SoluEons

11 Gain Margin ΔK (GM) - Example Assume: Loop gain Loop gain ( Sløyfe- forsterkningen ) When becomes the system unstable? System becomes unstable when: Cont. next page - >

12 Example cont. stabilt system: Marginalt stabilt system: Ustabilt system:

13 Stability Analysis - Summary Tidsplanet AsymptoEsk stabilt system Frekvensrespons/Bodediagram Marginalt stabilt system Ustabilt system Det komplekse plan AsymptoEsk stabilt system Marginalt stabilt system Ustabilt system

14 Stability Analysis using Frequency Response within MathScript Hr =... Hp =... Hm =... L = series(series(hr, Hp), Hm) bode(l) margin(l) [gm, pm, w180, wc] = margin(h); gmdb = 20*log10(gm) bode margin

15 Stability Analysis MathScript Example Students: Set Kp=0.5 (in this example Kp will be the loop gain). Plot the Bode diagram. Find ωc, ω180, ΔK, φ both manually from the plot and compare with the results from MathScript (margin funceon).

16 Stability Analysis MathScript Example - SoluEons MathScript: clear, clc, clf % Define Transfer functions % The Process Transfer function Hp(s) num_p=[1]; den1=[1, 0]; den2=[1, 1]; den_p = conv(den1,den2); Hp = tf(num_p, den_p) % The Measurement Transfer function Hm(s) num_m=[1]; den_m=[3, 1]; Hm = tf(num_m, den_m) % The Controller Transfer function Hr(s) Kp = 0.5; Hr = tf(kp) % The Loop Transfer function L = series(series(hr, Hp), Hm) % Bode Diagram figure(1) bode(l) subplot(2,1,1) grid on subplot(2,1,2) grid on figure(2) margin(l) [gm, pm, w180, wc] = margin(l); wc w180 gmdb = 20*log10(gm) pm Cont. on next page - >

17 Stability Analysis Example - SoluEons Bode Diagram and Results: Tip: Adjust the scale on the axes to make it easier to read the different values GM wc PM w180

18 Stability Analysis Example cont. Results from previous page: Kp=0.5 ΔK Students: How much can we increase Kp before the system becomes unstable? Plot the Step Response and find the Poles for different Kp (asymtoecally stable, marginally stable, unstable)

19 Stability Analysis Example cont. - SoluEons Kp=0.5 - > ΔK=2.67 (from previous page) Kp (marginalt stabilt) = Kp x ΔK = 0.5 x 2.67 = 1.34 De9e gir følgende: Kp = > AsymptoEsk Stabilt System Kp = > Marginal Stabilt System MathScript Kode Kp = 2 (f.eks.) - > Ustabilt System Kp =... Kp > 2... t = 0:0.1:50; figure(4) step(t, t) p = poles(t) figure(5) pzmap(t) See plots on next page - >

20 Kp = 0.5 Kp = 1.34 Kp = 2

21 Golden rules of Stability Margins for a Control System Gain Margin: ( Forsterkningsmargin ) Phase Margin: ( Fasemargin ) Merk! Siden (alt)for dårlig stabilitet må unngås, mens (alt)for god stabilitet kan aksepteres, er de nedre grensene for GM og PM krieske, mens de øvre grensene er anbefalte, men ikke krieske.

22 Bandwidth 3 different Bandwidth definieons: Et reguleringssystems følgeegenskaper er gode for frekvenser under båndbredde- frekvensen, mens de er dårlige for frekvenser over båndbredde- frekvensen. Gode følge- egenskaper når: Følgende sees fra kurven

23 Bandwidth - Example Students: Find ωc, ωs, ωt from the plot (Bruk utdelt papir)

24 Bandwidth Example - SoluEons From the plot: ωc = 0.17 rad/s ωs = rad/s ωt = 0.3 rad/s

25 Bandwidth - MathScript Hr =... Hp =... Hm =... L = series(series(hr, Hp), Hm) T = feedback(l, 1) S = 1-T bodemag(l,t,s)

26 Bandwidth MathScript Example Students: Set Kp=0.5. Plot L(jω), T(jω) og S(jω). Find ωc, ωs og ωt.

27 Bandwidth Example MathScript - SoluEons clear, clc, clf % Define Transfer functions % The Process Transfer function Hp(s) num_p=[1]; den1=[1, 0]; den2=[1, 1]; den_p = conv(den1,den2); Hp = tf(num_p, den_p) % The Measurment Transfer function Hm(s) num_m=[1]; den_m=[3, 1]; Hm = tf(num_m, den_m) % The Controller Transfer function Hr(s) Kp = 0.5; Hr = tf(kp) % The Loop Transfer function L = series(series(hr, Hp), Hm) T = feedback (L, 1); S = 1-T; figure(3) bodemag(l, T, S) grid Use bodemag instead of bode if you just want to plot the gain funceon Same as previous example Cont. on next page - >

28 Bandwidth Example - SoluEons Tip: Adjust the scale on the axes to make it easier to read the different values From the plot:

29 Frequency Response MathScript funceons

30 Hans- PeYer Halvorsen, M.Sc. Telemark University College Faculty of Technology Department of Electrical Engineering, Technology and E- mail: Blog: hyp://home.hit.no/~hansha/ 30