Systeme-Eingenschaften im Zeit und Frequenzbereich

Präsentation zum Thema: "Systeme-Eingenschaften im Zeit und Frequenzbereich"—  Präsentation transkript:

1 Systeme-Eingenschaften im Zeit und Frequenzbereich
M d Q Tavares , TB425

2 Applied Mathematics (SiSy)
SiSy Overview Signals step / impulse / rect / sinc continuous / discret periodic / aperiodic deterministic / random representation in time / frequency domains power and energy in freq domain Systems linearisation feedback and stability discretisation Transforms FR ; FT ; DFT/FFT : Laplace : Z-Transformation : DCT : Applications Biomedical / Operational Research/ Sensorics & Messtechnik Automation / Location & Telecommunication Systems Data Compression & Cryptography / Image Processing / Audio Synthesis & Analysis Statistical Signal Processing LTI DGl; BSB; ZVD; h(t); g(t); G(ω); G(s) u(t) U(ω) u[n] U(z) y(t) Y(ω) y[n] Y(z) LTD DzGl; BSB; ZVD; g[n]; G(ω); G(z) ω t f S-Plane Control (RT) Telecomm (NTM) SigProc (DSV, ASV) Z-Plane n Applied Mathematics (SiSy) x y Mathematics

3 Inhalt Relationship LTI views & System Properties
System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

4 LTI Systeme : Modellierung & Darstellung
Physikalisches Prozess Modell-bildung Differential- gleichung Messung (Schrittantwort) Schrittantwort (analytisch) Impulsantwort Frequenzgang (Bodediagramm) Block schaltbild Zustands variable Genauigkeit? Messverfahren Kontinuierliche Systeme Antwort beliebige Eingang (Faltung) Stationäre Antwort zum Schwingungseingang Numerische Simulation Physikalisches Modell nicht genug

5 State Space (ZVD) System free-response & Eingenvalues of the state matrix State Space (ZVD) representation of LTI system nth-order Free-response, with u(t) =0 to satisfy differential equation, x(t) must have exponential form λi = eigenvalues of state matrix A vi = eigenvectors of state matrix A Solving for λ (non-trivial solution) nth-order polynom characteristic equation

6 State Space (ZVD) Free-response & Eingenvalues
Solving for v (eigenvectors) satisfy this equation with 1 normalised component Given superposition (linearity) principle, if n distinct eigenvalues Apply initial conditions to solve For the αi factors (*) Compare to known free-response of normalised system 2.order with parameters : k, d and ω0 ; or σ and ωe . (*) For larger n, symbolic solution gets complicated, but there are computationally efficient numerical techniques.

7 State Space (ZVD) Forced response: response to a step input
Complete solution (sum) With homogeneous solution And input or stimuli Simplifying for t > 0 The particular solution has form similar to input, and for t>0 derivative equals zero, so: Therefore complete solution Obs.: use initial conditions to solve for αi

8 State Space (ZVD) System free-response & Eingenvalues of the state matrix State Space (ZVD) representation of LTI system nth-order System response as homogeneous plus particular solution with λi = eigenvalues of state matrix A vi = eigenvectors of state matrix A αi = constants determined by n initial conditions

9 Inhalt Relationship LTI views & System Properties
System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

10 State Space (ZVD) ZVD with Laplace Operator & Transfer Function G(s)
representation of LTI system nth-order & Laplace Transform (with initial conditions equal zero) Manipulating to isolate the transfer function : G(s) = Y(s) / U(s) G(s) Übung 2 : Aufgabe 3

11 State Space (ZVD) ZVD with Laplace Operator & Transfer Function (Übertragungsfunktion) Examples: Generic 2nd order system (from BSB exercise with direct form I & II) - Sallen-Key Butterworth TPF

12 ZVD & System-Antwort u(t) y(t) b2 1/s -a1 b1 -a0 b0
Example 1: Generic 2nd order system direct form II 1/s -a1 -a0 b1 b2 b0 u(t) y(t)

13 ZVD & System-Antwort Example 1: Generic 2nd order system
Calculating the eigenvalues: and the respective eigenvectors: Gives the complete system response:

14 ZVD & Übertragungsfunktion
Example 1: Generic 2nd order system Übung 2 - Auf 3 Similar but easier with b2 =0

15 ZVD & Übertragungsfunktion
Example 2: Sallen-Key Butterworth TPF For Butterworth pattern:

16 Inhalt Relationship LTI views & System Properties
System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

17 LTI Systeme : Modellierung & Darstellung
Physikalisches Prozess Modell-bildung Differential- gleichung Messung (Schrittantwort) Schrittantwort (analytisch) Impulsantwort Frequenzgang (Bodediagramm) Block schaltbild Zustands variable Genauigkeit? Messverfahren Kontinuierliche Systeme Antwort beliebige Eingang (Faltung) Stationäre Antwort zum Schwingungseingang Numerische Simulation Physikalisches Modell nicht genug

18 Relationship among LTI views
Skript: Kapitel 6 S. 73

19 Relationship among LTI views
Known relationships (visualisation : Matlab ltiview) Differential Equation Frequence Response G(jω) Differential Equation Transfer Function G(s) Frequency Response Impulse Response g(t) Impulse Response System Output y(t) Transfer Function System Output Y(s) Step Response h(t) Impulse Response g(t) LTI u(t) U(s) y(t) Y(s) Fourier Transformation + isolate {Y(jω) / U(jω)} s=jω Laplace Transformation Inv. Fourier Transformation Convolution with input fct Multiplication with input fct Derivative in time domain

20 Inhalt Relationship LTI views & System Properties
System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

21 Laplace Transformation : PN-Map
Relationship: Transfer Function & Frequency Response (Übertragungsfunktion & Frequenzgang) S-Plane (S-Ebene) Im{s} X Re{s} X Sweep ω from 0 to +∞ , and check for minimum & maximum points

22 Laplace Transformation : PN-Map & Freq.Response
Examples: determine the filter type (LP, HP, BP, BS) given the PN-Map Re{s} Im{s} Re{s} Im{s} x x o Re{s} Im{s} Re{s} Im{s} x x o o x o x Sweep ω from 0 to +∞ , and check for minimum & maximum points

23 Laplace Transformation : PN-Map & Freq.Response
Examples: determine the filter type (LP, HP, BP, BS) given the PN-Map Re{s} Im{s} Re{s} Im{s} Tiefpass Hochpass x x o Re{s} Im{s} Re{s} Im{s} x x Bandpass o Bandsperre o x o x Sweep ω from 0 to +∞ , and check for minimum & maximum points

24 Laplace Transformation : PN-Map & Freq.Response
Examples: determine the filter type (LP, HP, BP, BS) given the PN-Map Re{s} Im{s} Re{s} Im{s} Tiefpass Hochpass x x o Re{s} Im{s} Re{s} Im{s} x x Bandpass o Bandsperre o x o x Sweep ω from 0 to +∞ , and check for minimum & maximum points

25 Inhalt Relationship LTI views & System Properties
System Free-Response & Eigenvalues of State Matrix SS (ZVD) with Laplace Notation (=> Transfer Function) Overview known relationships PN-Map and Frequency Response PN-Map and System Response in time domain

26 Pol-Nullstellen-Darstellung
Beispiel 1

27 Pol-Nullstellen-Darstellung
Beispiel 2

28 Pol-Nullstellen-Darstellung
Beispiel 2

29 Pol-Nullstellen-Darstellung
Beispiel 3

30 Pol-Nullstellen-Darstellung
Beispiel 3

31 Pol-Nullstellen-Darstellung
Erkenntnisse

32 Pol-Nullstellen-Darstellung
Erkenntnisse

33 Pol-Nullstellen-Darstellung
Erkenntnisse

34 Pol-Nullstellen-Darstellung
Erkenntnisse

35 Pol-Nullstellen-Darstellung
Erkenntnisse

36 Pol-Nullstellen-Darstellung
Beispiel 4

37 Pol-Nullstellen-Darstellung
Beispiel 4

38 Pol-Nullstellen-Darstellung
Beispiel 4

39 Pol-Nullstellen-Darstellung
Beispiel 4

40 Pol-Nullstellen-Darstellung
Beispiel 4